diff --git a/Integration Test Suite/SanityCheck.m b/Integration Test Suite/SanityCheck.m
index 1b99801..99171eb 100644
--- a/Integration Test Suite/SanityCheck.m	
+++ b/Integration Test Suite/SanityCheck.m	
@@ -1,16 +1,6 @@
 (* ::Package:: *)
 
-(* ::Title::Closed:: *)
-(*Integrands of the form (c x)^m (a+b x)^n*)
-
-
-(* ::Section::Closed:: *)
-(*Integrands of the form (b x)^n*)
-
-
-(* ::Subsection::Closed:: *)
-(*Integrands of the form b*)
-
+(* Section 1 *)
 
 {0, x, 1, 0}
 {1, x, 1, x}
@@ -22,11 +12,6 @@
 {3*a, x, 1, 3*a*x}
 {Pi/Sqrt[16 - E^2], x, 1, (Pi*x)/Sqrt[16 - E^2]}
 
-
-(* ::Subsection::Closed:: *)
-(*Integrands of the form x^n*)
-
-
 {x^100, x, 1, x^101/101}
 {x^3, x, 1, x^4/4}
 {x^2, x, 1, x^3/3}
@@ -38,14 +23,151 @@
 {1/x^4, x, 1, -(1/(3*x^3))}
 {1/x^100, x, 1, -1/(99*x^99)}
 
-
-(* ::Subsection::Closed:: *)
-(*Integrands of the form (b x)^(n/2)*)
-
-
 {x^(5/2), x, 1, 2*x^(7/2)/7}
 {x^(3/2), x, 1, 2*x^(5/2)/5}
 {x^(1/2), x, 1, 2*x^(3/2)/3}
 {1/x^(1/2), x, 1, 2*Sqrt[x]}
 {1/x^(3/2), x, 1, -2/Sqrt[x]}
 {1/x^(5/2), x, 1, -2/(3*x^(3/2))}
+
+(* Section 2 *)
+
+{F^(c*(a + b*x))*(d + e*x)^m, x, 1, (F^(c*(a - (b*d)/e))*(d + e*x)^m*Gamma[1 + m, -((b*c*(d + e*x)*Log[F])/e)])/((-((b*c*(d + e*x)*Log[F])/e))^m*(b*c*Log[F]))}
+
+{F^(c*(a + b*x))*(d + e*x)^4, x, 5, (24*e^4*F^(c*(a + b*x)))/(b^5*c^5*Log[F]^5) - (24*e^3*F^(c*(a + b*x))*(d + e*x))/(b^4*c^4*Log[F]^4) + (12*e^2*F^(c*(a + b*x))*(d + e*x)^2)/(b^3*c^3*Log[F]^3) - (4*e*F^(c*(a + b*x))*(d + e*x)^3)/(b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^4)/(b*c*Log[F])}
+{F^(c*(a + b*x))*(d + e*x)^3, x, 4, -((6*e^3*F^(c*(a + b*x)))/(b^4*c^4*Log[F]^4)) + (6*e^2*F^(c*(a + b*x))*(d + e*x))/(b^3*c^3*Log[F]^3) - (3*e*F^(c*(a + b*x))*(d + e*x)^2)/(b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^3)/(b*c*Log[F])}
+{F^(c*(a + b*x))*(d + e*x)^2, x, 3, (2*e^2*F^(c*(a + b*x)))/(b^3*c^3*Log[F]^3) - (2*e*F^(c*(a + b*x))*(d + e*x))/(b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^2)/(b*c*Log[F])}
+{F^(c*(a + b*x))*(d + e*x)^1, x, 2, -((e*F^(c*(a + b*x)))/(b^2*c^2*Log[F]^2)) + (F^(c*(a + b*x))*(d + e*x))/(b*c*Log[F])}
+{F^(c*(a + b*x))*(d + e*x)^0, x, 1, F^(c*(a + b*x))/(b*c*Log[F])}
+{F^(c*(a + b*x))/(d + e*x)^1, x, 1, (F^(c*(a - (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e])/e}
+{F^(c*(a + b*x))/(d + e*x)^2, x, 2, -(F^(c*(a + b*x))/(e*(d + e*x))) + (b*c*F^(c*(a - (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F])/e^2}
+{F^(c*(a + b*x))/(d + e*x)^3, x, 3, -(F^(c*(a + b*x))/(2*e*(d + e*x)^2)) - (b*c*F^(c*(a + b*x))*Log[F])/(2*e^2*(d + e*x)) + (b^2*c^2*F^(c*(a - (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F]^2)/(2*e^3)}
+{F^(c*(a + b*x))/(d + e*x)^4, x, 4, -(F^(c*(a + b*x))/(3*e*(d + e*x)^3)) - (b*c*F^(c*(a + b*x))*Log[F])/(6*e^2*(d + e*x)^2) - (b^2*c^2*F^(c*(a + b*x))*Log[F]^2)/(6*e^3*(d + e*x)) + (b^3*c^3*F^(c*(a - (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F]^3)/(6*e^4)}
+{F^(c*(a + b*x))/(d + e*x)^5, x, 5, -(F^(c*(a + b*x))/(4*e*(d + e*x)^4)) - (b*c*F^(c*(a + b*x))*Log[F])/(12*e^2*(d + e*x)^3) - (b^2*c^2*F^(c*(a + b*x))*Log[F]^2)/(24*e^3*(d + e*x)^2) - (b^3*c^3*F^(c*(a + b*x))*Log[F]^3)/(24*e^4*(d + e*x)) + (b^4*c^4*F^(c*(a - (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F]^4)/(24*e^5)}
+
+(*
+(* Section 3 *)
+{x^3*Log[c*x], x, 1, -x^4/16 + (x^4*Log[c*x])/4}
+{x^2*Log[c*x], x, 1, -x^3/9 + (x^3*Log[c*x])/3}
+{x^1*Log[c*x], x, 1, -x^2/4 + (x^2*Log[c*x])/2}
+{x^0*Log[c*x], x, 1, -x + x*Log[c*x]}
+{Log[c*x]/x^1, x, 1, Log[c*x]^2/2}
+{Log[c*x]/x^2, x, 1, -x^(-1) - Log[c*x]/x}
+{Log[c*x]/x^3, x, 1, -1/(4*x^2) - Log[c*x]/(2*x^2)}
+
+
+{x^3*Log[c*x]^2, x, 2, x^4/32 - (x^4*Log[c*x])/8 + (x^4*Log[c*x]^2)/4}
+{x^2*Log[c*x]^2, x, 2, (2*x^3)/27 - (2*x^3*Log[c*x])/9 + (x^3*Log[c*x]^2)/3}
+{x^1*Log[c*x]^2, x, 2, x^2/4 - (x^2*Log[c*x])/2 + (x^2*Log[c*x]^2)/2}
+{x^0*Log[c*x]^2, x, 2, 2*x - 2*x*Log[c*x] + x*Log[c*x]^2}
+{Log[c*x]^2/x^1, x, 2, Log[c*x]^3/3}
+{Log[c*x]^2/x^2, x, 2, -2/x - (2*Log[c*x])/x - Log[c*x]^2/x}
+{Log[c*x]^2/x^3, x, 2, -1/(4*x^2) - Log[c*x]/(2*x^2) - Log[c*x]^2/(2*x^2)}
+
+(* Section 4 *)
+{Sin[a + b*x]^1, x, 1, -(Cos[a + b*x]/b)}
+{Sin[a + b*x]^2, x, 2, x/2 - (Cos[a + b*x]*Sin[a + b*x])/(2*b)}
+{Sin[a + b*x]^3, x, 2, -(Cos[a + b*x]/b) + Cos[a + b*x]^3/(3*b)}
+{Sin[a + b*x]^4, x, 3, (3*x)/8 - (3*Cos[a + b*x]*Sin[a + b*x])/(8*b) - (Cos[a + b*x]*Sin[a + b*x]^3)/(4*b)}
+{Sin[a + b*x]^5, x, 2, -(Cos[a + b*x]/b) + (2*Cos[a + b*x]^3)/(3*b) - Cos[a + b*x]^5/(5*b)}
+{Sin[a + b*x]^6, x, 4, (5*x)/16 - (5*Cos[a + b*x]*Sin[a + b*x])/(16*b) - (5*Cos[a + b*x]*Sin[a + b*x]^3)/(24*b) - (Cos[a + b*x]*Sin[a + b*x]^5)/(6*b)}
+{Sin[a + b*x]^7, x, 2, -(Cos[a + b*x]/b) + Cos[a + b*x]^3/b - (3*Cos[a + b*x]^5)/(5*b) + Cos[a + b*x]^7/(7*b)}
+{Sin[a + b*x]^8, x, 5, (35*x)/128 - (35*Cos[a + b*x]*Sin[a + b*x])/(128*b) - (35*Cos[a + b*x]*Sin[a + b*x]^3)/(192*b) - (7*Cos[a + b*x]*Sin[a + b*x]^5)/(48*b) - (Cos[a + b*x]*Sin[a + b*x]^7)/(8*b)}
+
+{Sin[b*x]^(7/2), x, 3, -((10*EllipticF[Pi/4 - (b*x)/2, 2])/(21*b)) - (10*Cos[b*x]*Sqrt[Sin[b*x]])/(21*b) - (2*Cos[b*x]*Sin[b*x]^(5/2))/(7*b)}
+{Sin[b*x]^(5/2), x, 2, -((6*EllipticE[Pi/4 - (b*x)/2, 2])/(5*b)) - (2*Cos[b*x]*Sin[b*x]^(3/2))/(5*b)}
+{Sin[b*x]^(3/2), x, 2, -((2*EllipticF[Pi/4 - (b*x)/2, 2])/(3*b)) - (2*Cos[b*x]*Sqrt[Sin[b*x]])/(3*b)}
+{Sin[b*x]^(1/2), x, 1, -((2*EllipticE[Pi/4 - (b*x)/2, 2])/b)}
+{1/Sin[b*x]^(1/2), x, 1, -((2*EllipticF[Pi/4 - (b*x)/2, 2])/b)}
+{1/Sin[b*x]^(3/2), x, 2, (2*EllipticE[Pi/4 - (b*x)/2, 2])/b - (2*Cos[b*x])/(b*Sqrt[Sin[b*x]])}
+{1/Sin[b*x]^(5/2), x, 2, -((2*EllipticF[Pi/4 - (b*x)/2, 2])/(3*b)) - (2*Cos[b*x])/(3*b*Sin[b*x]^(3/2))}
+{1/Sin[b*x]^(7/2), x, 3, (6*EllipticE[Pi/4 - (b*x)/2, 2])/(5*b) - (2*Cos[b*x])/(5*b*Sin[b*x]^(5/2)) - (6*Cos[b*x])/(5*b*Sqrt[Sin[b*x]])}
+
+(* Section 5 *)
+{x^6*(a + b*ArcCsc[c*x]), x, 7, (5*b*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(112*c^5) + (5*b*Sqrt[1 - 1/(c^2*x^2)]*x^4)/(168*c^3) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^6)/(42*c) + (x^7*(a + b*ArcCsc[c*x]))/7 + (5*b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(112*c^7)}
+{x^5*(a + b*ArcCsc[c*x]), x, 4, (4*b*Sqrt[1 - 1/(c^2*x^2)]*x)/(45*c^5) + (2*b*Sqrt[1 - 1/(c^2*x^2)]*x^3)/(45*c^3) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^5)/(30*c) + (x^6*(a + b*ArcCsc[c*x]))/6}
+{x^4*(a + b*ArcCsc[c*x]), x, 6, (3*b*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(40*c^3) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^4)/(20*c) + (x^5*(a + b*ArcCsc[c*x]))/5 + (3*b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(40*c^5)}
+{x^3*(a + b*ArcCsc[c*x]), x, 3, (b*Sqrt[1 - 1/(c^2*x^2)]*x)/(6*c^3) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^3)/(12*c) + (x^4*(a + b*ArcCsc[c*x]))/4}
+{x^2*(a + b*ArcCsc[c*x]), x, 5, (b*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(6*c) + (x^3*(a + b*ArcCsc[c*x]))/3 + (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(6*c^3)}
+{x*(a + b*ArcCsc[c*x]), x, 2, (b*Sqrt[1 - 1/(c^2*x^2)]*x)/(2*c) + (x^2*(a + b*ArcCsc[c*x]))/2}
+{a + b*ArcCsc[c*x], x, 5, a*x + b*x*ArcCsc[c*x] + (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/c}
+{(a + b*ArcCsc[c*x])/x, x, 6, ((I/2)*(a + b*ArcCsc[c*x])^2)/b - (a + b*ArcCsc[c*x])*Log[1 - E^((2*I)*ArcCsc[c*x])] + (I/2)*b*PolyLog[2, E^((2*I)*ArcCsc[c*x])]}
+{(a + b*ArcCsc[c*x])/x^2, x, 2, -(b*c*Sqrt[1 - 1/(c^2*x^2)]) - (a + b*ArcCsc[c*x])/x}
+{(a + b*ArcCsc[c*x])/x^3, x, 4, -(b*c*Sqrt[1 - 1/(c^2*x^2)])/(4*x) + (b*c^2*ArcCsc[c*x])/4 - (a + b*ArcCsc[c*x])/(2*x^2)}
+{(a + b*ArcCsc[c*x])/x^4, x, 4, -(b*c^3*Sqrt[1 - 1/(c^2*x^2)])/3 + (b*c^3*(1 - 1/(c^2*x^2))^(3/2))/9 - (a + b*ArcCsc[c*x])/(3*x^3)}
+{(a + b*ArcCsc[c*x])/x^5, x, 5, -(b*c*Sqrt[1 - 1/(c^2*x^2)])/(16*x^3) - (3*b*c^3*Sqrt[1 - 1/(c^2*x^2)])/(32*x) + (3*b*c^4*ArcCsc[c*x])/32 - (a + b*ArcCsc[c*x])/(4*x^4)}
+{(a + b*ArcCsc[c*x])/x^6, x, 4, -(b*c^5*Sqrt[1 - 1/(c^2*x^2)])/5 + (2*b*c^5*(1 - 1/(c^2*x^2))^(3/2))/15 - (b*c^5*(1 - 1/(c^2*x^2))^(5/2))/25 - (a + b*ArcCsc[c*x])/(5*x^5)}
+{(a + b*ArcCsc[c*x])/x^7, x, 6, -(b*c*Sqrt[1 - 1/(c^2*x^2)])/(36*x^5) - (5*b*c^3*Sqrt[1 - 1/(c^2*x^2)])/(144*x^3) - (5*b*c^5*Sqrt[1 - 1/(c^2*x^2)])/(96*x) + (5*b*c^6*ArcCsc[c*x])/96 - (a + b*ArcCsc[c*x])/(6*x^6)}
+
+
+{x^3*(a + b*ArcCsc[c*x])^2, x, 5, (b^2*x^2)/(12*c^2) + (b*Sqrt[1 - 1/(c^2*x^2)]*x*(a + b*ArcCsc[c*x]))/(3*c^3) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^3*(a + b*ArcCsc[c*x]))/(6*c) + (x^4*(a + b*ArcCsc[c*x])^2)/4 + (b^2*Log[x])/(3*c^4)}
+{x^2*(a + b*ArcCsc[c*x])^2, x, 8, (b^2*x)/(3*c^2) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^2*(a + b*ArcCsc[c*x]))/(3*c) + (x^3*(a + b*ArcCsc[c*x])^2)/3 + (2*b*(a + b*ArcCsc[c*x])*ArcTanh[E^(I*ArcCsc[c*x])])/(3*c^3) - ((I/3)*b^2*PolyLog[2, -E^(I*ArcCsc[c*x])])/c^3 + ((I/3)*b^2*PolyLog[2, E^(I*ArcCsc[c*x])])/c^3}
+{x*(a + b*ArcCsc[c*x])^2, x, 4, (b*Sqrt[1 - 1/(c^2*x^2)]*x*(a + b*ArcCsc[c*x]))/c + (x^2*(a + b*ArcCsc[c*x])^2)/2 + (b^2*Log[x])/c^2}
+{(a + b*ArcCsc[c*x])^2, x, 7, x*(a + b*ArcCsc[c*x])^2 + (4*b*(a + b*ArcCsc[c*x])*ArcTanh[E^(I*ArcCsc[c*x])])/c - ((2*I)*b^2*PolyLog[2, -E^(I*ArcCsc[c*x])])/c + ((2*I)*b^2*PolyLog[2, E^(I*ArcCsc[c*x])])/c}
+{(a + b*ArcCsc[c*x])^2/x, x, 6, ((I/3)*(a + b*ArcCsc[c*x])^3)/b - (a + b*ArcCsc[c*x])^2*Log[1 - E^((2*I)*ArcCsc[c*x])] + I*b*(a + b*ArcCsc[c*x])*PolyLog[2, E^((2*I)*ArcCsc[c*x])] - (b^2*PolyLog[3, E^((2*I)*ArcCsc[c*x])])/2}
+{(a + b*ArcCsc[c*x])^2/x^2, x, 4, (2*b^2)/x - 2*b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]) - (a + b*ArcCsc[c*x])^2/x}
+{(a + b*ArcCsc[c*x])^2/x^3, x, 4, b^2/(4*x^2) + (a*b*c^2*ArcCsc[c*x])/2 + (b^2*c^2*ArcCsc[c*x]^2)/4 - (b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]))/(2*x) - (a + b*ArcCsc[c*x])^2/(2*x^2)}
+{(a + b*ArcCsc[c*x])^2/x^4, x, 5, (2*b^2)/(27*x^3) + (4*b^2*c^2)/(9*x) - (4*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]))/9 - (2*b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]))/(9*x^2) - (a + b*ArcCsc[c*x])^2/(3*x^3)}
+{(a + b*ArcCsc[c*x])^2/x^5, x, 5, b^2/(32*x^4) + (3*b^2*c^2)/(32*x^2) + (3*a*b*c^4*ArcCsc[c*x])/16 + (3*b^2*c^4*ArcCsc[c*x]^2)/32 - (b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]))/(8*x^3) - (3*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]))/(16*x) - (a + b*ArcCsc[c*x])^2/(4*x^4)}
+
+(* Section 6 *)
+{(c + d*x)^4*Sinh[a + b*x], x, 5, (24*d^4*Cosh[a + b*x])/b^5 + (12*d^2*(c + d*x)^2*Cosh[a + b*x])/b^3 + ((c + d*x)^4*Cosh[a + b*x])/b - (24*d^3*(c + d*x)*Sinh[a + b*x])/b^4 - (4*d*(c + d*x)^3*Sinh[a + b*x])/b^2}
+{(c + d*x)^3*Sinh[a + b*x], x, 4, (6*d^2*(c + d*x)*Cosh[a + b*x])/b^3 + ((c + d*x)^3*Cosh[a + b*x])/b - (6*d^3*Sinh[a + b*x])/b^4 - (3*d*(c + d*x)^2*Sinh[a + b*x])/b^2}
+{(c + d*x)^2*Sinh[a + b*x], x, 3, (2*d^2*Cosh[a + b*x])/b^3 + ((c + d*x)^2*Cosh[a + b*x])/b - (2*d*(c + d*x)*Sinh[a + b*x])/b^2}
+{(c + d*x)*Sinh[a + b*x], x, 2, ((c + d*x)*Cosh[a + b*x])/b - (d*Sinh[a + b*x])/b^2}
+{Sinh[a + b*x]/(c + d*x), x, 3, (CoshIntegral[(b*c)/d + b*x]*Sinh[a - (b*c)/d])/d + (Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/d}
+{Sinh[a + b*x]/(c + d*x)^2, x, 4, (b*Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x])/d^2 - Sinh[a + b*x]/(d*(c + d*x)) + (b*Sinh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/d^2}
+{Sinh[a + b*x]/(c + d*x)^3, x, 5, -(b*Cosh[a + b*x])/(2*d^2*(c + d*x)) + (b^2*CoshIntegral[(b*c)/d + b*x]*Sinh[a - (b*c)/d])/(2*d^3) - Sinh[a + b*x]/(2*d*(c + d*x)^2) + (b^2*Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/(2*d^3)}
+
+
+{(c + d*x)^4*Sinh[a + b*x]^2, x, 6, (-3*d^4*x)/(4*b^4) - (d*(c + d*x)^3)/(2*b^2) - (c + d*x)^5/(10*d) + (3*d^4*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^5) + (3*d^2*(c + d*x)^2*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^3) + ((c + d*x)^4*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) - (3*d^3*(c + d*x)*Sinh[a + b*x]^2)/(2*b^4) - (d*(c + d*x)^3*Sinh[a + b*x]^2)/b^2}
+{(c + d*x)^3*Sinh[a + b*x]^2, x, 4, (-3*c*d^2*x)/(4*b^2) - (3*d^3*x^2)/(8*b^2) - (c + d*x)^4/(8*d) + (3*d^2*(c + d*x)*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^3) + ((c + d*x)^3*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) - (3*d^3*Sinh[a + b*x]^2)/(8*b^4) - (3*d*(c + d*x)^2*Sinh[a + b*x]^2)/(4*b^2)}
+{(c + d*x)^2*Sinh[a + b*x]^2, x, 4, -(d^2*x)/(4*b^2) - (c + d*x)^3/(6*d) + (d^2*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^3) + ((c + d*x)^2*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) - (d*(c + d*x)*Sinh[a + b*x]^2)/(2*b^2)}
+{(c + d*x)*Sinh[a + b*x]^2, x, 2, -(c*x)/2 - (d*x^2)/4 + ((c + d*x)*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) - (d*Sinh[a + b*x]^2)/(4*b^2)}
+{Sinh[a + b*x]^2/(c + d*x), x, 5, (Cosh[2*a - (2*b*c)/d]*CoshIntegral[(2*b*c)/d + 2*b*x])/(2*d) - Log[c + d*x]/(2*d) + (Sinh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*x])/(2*d)}
+{Sinh[a + b*x]^2/(c + d*x)^2, x, 5, (b*CoshIntegral[(2*b*c)/d + 2*b*x]*Sinh[2*a - (2*b*c)/d])/d^2 - Sinh[a + b*x]^2/(d*(c + d*x)) + (b*Cosh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*x])/d^2}
+{Sinh[a + b*x]^2/(c + d*x)^3, x, 7, (b^2*Cosh[2*a - (2*b*c)/d]*CoshIntegral[(2*b*c)/d + 2*b*x])/d^3 - (b*Cosh[a + b*x]*Sinh[a + b*x])/(d^2*(c + d*x)) - Sinh[a + b*x]^2/(2*d*(c + d*x)^2) + (b^2*Sinh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*x])/d^3}
+{Sinh[a + b*x]^2/(c + d*x)^4, x, 7, -b^2/(3*d^3*(c + d*x)) + (2*b^3*CoshIntegral[(2*b*c)/d + 2*b*x]*Sinh[2*a - (2*b*c)/d])/(3*d^4) - (b*Cosh[a + b*x]*Sinh[a + b*x])/(3*d^2*(c + d*x)^2) - Sinh[a + b*x]^2/(3*d*(c + d*x)^3) - (2*b^2*Sinh[a + b*x]^2)/(3*d^3*(c + d*x)) + (2*b^3*Cosh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*x])/(3*d^4)}
+
+(* Section 7 *)
+{x^4*ArcSinh[a*x], x, 4, -(Sqrt[1 + a^2*x^2]/(5*a^5)) + (2*(1 + a^2*x^2)^(3/2))/(15*a^5) - (1 + a^2*x^2)^(5/2)/(25*a^5) + (1/5)*x^5*ArcSinh[a*x]}
+{x^3*ArcSinh[a*x], x, 4, (3*x*Sqrt[1 + a^2*x^2])/(32*a^3) - (x^3*Sqrt[1 + a^2*x^2])/(16*a) - (3*ArcSinh[a*x])/(32*a^4) + (1/4)*x^4*ArcSinh[a*x]}
+{x^2*ArcSinh[a*x], x, 4, Sqrt[1 + a^2*x^2]/(3*a^3) - (1 + a^2*x^2)^(3/2)/(9*a^3) + (1/3)*x^3*ArcSinh[a*x]}
+{x^1*ArcSinh[a*x], x, 3, -((x*Sqrt[1 + a^2*x^2])/(4*a)) + ArcSinh[a*x]/(4*a^2) + (1/2)*x^2*ArcSinh[a*x]}
+{x^0*ArcSinh[a*x], x, 2, -(Sqrt[1 + a^2*x^2]/a) + x*ArcSinh[a*x]}
+{ArcSinh[a*x]/x^1, x, 5, (-(1/2))*ArcSinh[a*x]^2 + ArcSinh[a*x]*Log[1 - E^(2*ArcSinh[a*x])] + (1/2)*PolyLog[2, E^(2*ArcSinh[a*x])]}
+{ArcSinh[a*x]/x^2, x, 4, -(ArcSinh[a*x]/x) - a*ArcTanh[Sqrt[1 + a^2*x^2]]}
+{ArcSinh[a*x]/x^3, x, 2, -((a*Sqrt[1 + a^2*x^2])/(2*x)) - ArcSinh[a*x]/(2*x^2)}
+{ArcSinh[a*x]/x^4, x, 5, -((a*Sqrt[1 + a^2*x^2])/(6*x^2)) - ArcSinh[a*x]/(3*x^3) + (1/6)*a^3*ArcTanh[Sqrt[1 + a^2*x^2]]}
+{ArcSinh[a*x]/x^5, x, 3, -((a*Sqrt[1 + a^2*x^2])/(12*x^3)) + (a^3*Sqrt[1 + a^2*x^2])/(6*x) - ArcSinh[a*x]/(4*x^4)}
+{ArcSinh[a*x]/x^6, x, 6, -((a*Sqrt[1 + a^2*x^2])/(20*x^4)) + (3*a^3*Sqrt[1 + a^2*x^2])/(40*x^2) - ArcSinh[a*x]/(5*x^5) - (3/40)*a^5*ArcTanh[Sqrt[1 + a^2*x^2]]}
+
+
+{x^4*ArcSinh[a*x]^2, x, 7, (16*x)/(75*a^4) - (8*x^3)/(225*a^2) + (2*x^5)/125 - (16*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(75*a^5) + (8*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(75*a^3) - (2*x^4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(25*a) + (1/5)*x^5*ArcSinh[a*x]^2}
+{x^3*ArcSinh[a*x]^2, x, 6, (-3*x^2)/(32*a^2) + x^4/32 + (3*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(16*a^3) - (x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(8*a) - (3*ArcSinh[a*x]^2)/(32*a^4) + (x^4*ArcSinh[a*x]^2)/4}
+{x^2*ArcSinh[a*x]^2, x, 5, -((4*x)/(9*a^2)) + (2*x^3)/27 + (4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(9*a^3) - (2*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(9*a) + (1/3)*x^3*ArcSinh[a*x]^2}
+{x*ArcSinh[a*x]^2, x, 4, x^2/4 - (x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*a) + ArcSinh[a*x]^2/(4*a^2) + (x^2*ArcSinh[a*x]^2)/2}
+{ArcSinh[a*x]^2, x, 3, 2*x - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/a + x*ArcSinh[a*x]^2}
+{ArcSinh[a*x]^2/x, x, 6, -ArcSinh[a*x]^3/3 + ArcSinh[a*x]^2*Log[1 - E^(2*ArcSinh[a*x])] + ArcSinh[a*x]*PolyLog[2, E^(2*ArcSinh[a*x])] - PolyLog[3, E^(2*ArcSinh[a*x])]/2}
+{ArcSinh[a*x]^2/x^2, x, 7, -(ArcSinh[a*x]^2/x) - 4*a*ArcSinh[a*x]*ArcTanh[E^ArcSinh[a*x]] - 2*a*PolyLog[2, -E^ArcSinh[a*x]] + 2*a*PolyLog[2, E^ArcSinh[a*x]]}
+{ArcSinh[a*x]^2/x^3, x, 3, -((a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/x) - ArcSinh[a*x]^2/(2*x^2) + a^2*Log[x]}
+{ArcSinh[a*x]^2/x^4, x, 9, -(a^2/(3*x)) - (a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(3*x^2) - ArcSinh[a*x]^2/(3*x^3) + (2/3)*a^3*ArcSinh[a*x]*ArcTanh[E^ArcSinh[a*x]] + (1/3)*a^3*PolyLog[2, -E^ArcSinh[a*x]] - (1/3)*a^3*PolyLog[2, E^ArcSinh[a*x]]}
+{ArcSinh[a*x]^2/x^5, x, 5, -a^2/(12*x^2) - (a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(6*x^3) + (a^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(3*x) - ArcSinh[a*x]^2/(4*x^4) - (a^4*Log[x])/3}
+
+(* Section 8 *)
+{x^5*Erf[b*x], x, 5, (5*x)/(E^(b^2*x^2)*(8*b^5*Sqrt[Pi])) + (5*x^3)/(E^(b^2*x^2)*(12*b^3*Sqrt[Pi])) + x^5/(E^(b^2*x^2)*(6*b*Sqrt[Pi])) - (5*Erf[b*x])/(16*b^6) + (1/6)*x^6*Erf[b*x]}
+{x^3*Erf[b*x], x, 4, (3*x)/(E^(b^2*x^2)*(8*b^3*Sqrt[Pi])) + x^3/(E^(b^2*x^2)*(4*b*Sqrt[Pi])) - (3*Erf[b*x])/(16*b^4) + (1/4)*x^4*Erf[b*x]}
+{x^1*Erf[b*x], x, 3, x/(E^(b^2*x^2)*(2*b*Sqrt[Pi])) - Erf[b*x]/(4*b^2) + (1/2)*x^2*Erf[b*x]}
+{Erf[b*x]/x^1, x, 1, (2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, (-b^2)*x^2])/Sqrt[Pi]}
+{Erf[b*x]/x^3, x, 3, -(b/(E^(b^2*x^2)*(Sqrt[Pi]*x))) - b^2*Erf[b*x] - Erf[b*x]/(2*x^2)}
+{Erf[b*x]/x^5, x, 4, -(b/(E^(b^2*x^2)*(6*Sqrt[Pi]*x^3))) + b^3/(E^(b^2*x^2)*(3*Sqrt[Pi]*x)) + (1/3)*b^4*Erf[b*x] - Erf[b*x]/(4*x^4)}
+{Erf[b*x]/x^7, x, 5, -(b/(E^(b^2*x^2)*(15*Sqrt[Pi]*x^5))) + (2*b^3)/(E^(b^2*x^2)*(45*Sqrt[Pi]*x^3)) - (4*b^5)/(E^(b^2*x^2)*(45*Sqrt[Pi]*x)) - (4/45)*b^6*Erf[b*x] - Erf[b*x]/(6*x^6)}
+
+{x^6*Erf[b*x], x, 5, 6/(E^(b^2*x^2)*(7*b^7*Sqrt[Pi])) + (6*x^2)/(E^(b^2*x^2)*(7*b^5*Sqrt[Pi])) + (3*x^4)/(E^(b^2*x^2)*(7*b^3*Sqrt[Pi])) + x^6/(E^(b^2*x^2)*(7*b*Sqrt[Pi])) + (1/7)*x^7*Erf[b*x]}
+{x^4*Erf[b*x], x, 4, 2/(E^(b^2*x^2)*(5*b^5*Sqrt[Pi])) + (2*x^2)/(E^(b^2*x^2)*(5*b^3*Sqrt[Pi])) + x^4/(E^(b^2*x^2)*(5*b*Sqrt[Pi])) + (1/5)*x^5*Erf[b*x]}
+{x^2*Erf[b*x], x, 3, 1/(E^(b^2*x^2)*(3*b^3*Sqrt[Pi])) + x^2/(E^(b^2*x^2)*(3*b*Sqrt[Pi])) + (1/3)*x^3*Erf[b*x]}
+{x^0*Erf[b*x], x, 1, 1/(E^(b^2*x^2)*(b*Sqrt[Pi])) + x*Erf[b*x]}
+{Erf[b*x]/x^2, x, 2, -(Erf[b*x]/x) + (b*ExpIntegralEi[(-b^2)*x^2])/Sqrt[Pi]}
+{Erf[b*x]/x^4, x, 3, -(b/(E^(b^2*x^2)*(3*Sqrt[Pi]*x^2))) - Erf[b*x]/(3*x^3) - (b^3*ExpIntegralEi[(-b^2)*x^2])/(3*Sqrt[Pi])}
+{Erf[b*x]/x^6, x, 4, -(b/(E^(b^2*x^2)*(10*Sqrt[Pi]*x^4))) + b^3/(E^(b^2*x^2)*(10*Sqrt[Pi]*x^2)) - Erf[b*x]/(5*x^5) + (b^5*ExpIntegralEi[(-b^2)*x^2])/(10*Sqrt[Pi])}
+*)
diff --git a/IntegrationRules/2 Exponentials/2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m b/IntegrationRules/2 Exponentials/2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m
new file mode 100755
index 0000000..0d57eed
--- /dev/null
+++ b/IntegrationRules/2 Exponentials/2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m	
@@ -0,0 +1,19 @@
+
+(* ::Subsection::Closed:: *)
+(* 2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p *)
+$UseGamma = False;
+Int[(c_. + d_.*x_)^m_.*(b_.*F_^(g_.*(e_. + f_.*x_)))^n_., x_Symbol] := (c + d*x)^m*(b*F^(g*(e + f*x)))^n/(f*g*n*Log[F]) - d*m/(f*g*n*Log[F])* Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x)))^n, x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] && Not[TrueQ[$UseGamma]]
+Int[(c_. + d_.*x_)^m_*(b_.*F_^(g_.*(e_. + f_.*x_)))^n_., x_Symbol] := (c + d*x)^(m + 1)*(b*F^(g*(e + f*x)))^n/(d*(m + 1)) - f*g*n*Log[F]/(d*(m + 1))* Int[(c + d*x)^(m + 1)*(b*F^(g*(e + f*x)))^n, x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] && Not[TrueQ[$UseGamma]]
+Int[F_^(g_.*(e_. + f_.*x_))/(c_. + d_.*x_), x_Symbol] := F^(g*(e - c*f/d))/d*ExpIntegralEi[f*g*(c + d*x)*Log[F]/d] /; FreeQ[{F, c, d, e, f, g}, x] && Not[TrueQ[$UseGamma]]
+Int[(c_. + d_.*x_)^m_.*F_^(g_.*(e_. + f_.*x_)), x_Symbol] := (-d)^m*F^(g*(e - c*f/d))/(f^(m + 1)*g^(m + 1)*Log[F]^(m + 1))* Gamma[m + 1, -f*g*Log[F]/d*(c + d*x)] /; FreeQ[{F, c, d, e, f, g}, x] && IntegerQ[m]
+Int[F_^(g_.*(e_. + f_.*x_))/Sqrt[c_. + d_.*x_], x_Symbol] := 2/d*Subst[Int[F^(g*(e - c*f/d) + f*g*x^2/d), x], x, Sqrt[c + d*x]] /; FreeQ[{F, c, d, e, f, g}, x] && Not[TrueQ[$UseGamma]]
+Int[(c_. + d_.*x_)^m_*F_^(g_.*(e_. + f_.*x_)), x_Symbol] := -F^(g*(e - c*f/d))*(c + d*x)^ FracPart[ m]/(d*(-f*g*Log[F]/d)^(IntPart[m] + 1)*(-f*g* Log[F]*(c + d*x)/d)^FracPart[m])* Gamma[m + 1, (-f*g*Log[F]/d)*(c + d*x)] /; FreeQ[{F, c, d, e, f, g, m}, x] && Not[IntegerQ[m]]
+Int[(c_. + d_.*x_)^m_.*(b_.*F_^(g_.*(e_. + f_.*x_)))^n_, x_Symbol] := (b*F^(g*(e + f*x)))^n/F^(g*n*(e + f*x))* Int[(c + d*x)^m*F^(g*n*(e + f*x)), x] /; FreeQ[{F, b, c, d, e, f, g, m, n}, x]
+Int[(c_. + d_.*x_)^m_.*(a_ + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.)^p_., x_Symbol] := Int[ExpandIntegrand[(c + d*x)^m, (a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
+Int[(c_. + d_.*x_)^m_./(a_ + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.), x_Symbol] := (c + d*x)^(m + 1)/(a*d*(m + 1)) - b/a*Int[(c + d*x)^ m*(F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n), x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
+(* Int[(c_.+d_.*x_)^m_./(a_+b_.*(F_^(g_.*(e_.+f_.*x_)))^n_.),x_Symbol]  := -(c+d*x)^m/(a*f*g*n*Log[F])*Log[1+a/(b*(F^(g*(e+f*x)))^n)] + d*m/(a*f*g*n*Log[F])*Int[(c+d*x)^(m-1)*Log[1+a/(b*(F^(g*(e+f*x)))^n) ],x] /; FreeQ[{F,a,b,c,d,e,f,g,n},x] && IGtQ[m,0] *)
+Int[(c_. + d_.*x_)^m_.*(a_ + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.)^p_, x_Symbol] := 1/a*Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x] - b/a* Int[(c + d*x)^m*(F^(g*(e + f*x)))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && ILtQ[p, 0] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*(a_ + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.)^p_, x_Symbol] := With[{u = IntHide[(a + b*(F^(g*(e + f*x)))^n)^p, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[(c + d*x)^(m - 1)*u, x]] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0] && LtQ[p, -1]
+Int[u_^m_.*(a_. + b_.*(F_^(g_.*v_))^n_.)^p_., x_Symbol] := Int[NormalizePowerOfLinear[u, x]^ m*(a + b*(F^(g*ExpandToSum[v, x]))^n)^p, x] /; FreeQ[{F, a, b, g, n, p}, x] && LinearQ[v, x] && PowerOfLinearQ[u, x] && Not[LinearMatchQ[v, x] && PowerOfLinearMatchQ[u, x]] && IntegerQ[m]
+Int[u_^m_.*(a_. + b_.*(F_^(g_.*v_))^n_.)^p_., x_Symbol] := Module[{uu = NormalizePowerOfLinear[u, x], z}, z = If[PowerQ[uu] && FreeQ[uu[[2]], x], uu[[1]]^(m*uu[[2]]), uu^m]; uu^m/z*Int[z*(a + b*(F^(g*ExpandToSum[v, x]))^n)^p, x]] /; FreeQ[{F, a, b, g, m, n, p}, x] && LinearQ[v, x] && PowerOfLinearQ[u, x] && Not[LinearMatchQ[v, x] && PowerOfLinearMatchQ[u, x]] && Not[IntegerQ[m]]
+Int[(c_. + d_.*x_)^m_.*(a_ + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.)^p_., x_Symbol] := Unintegrable[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
diff --git a/IntegrationRules/2 Exponentials/2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p.m b/IntegrationRules/2 Exponentials/2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p.m
new file mode 100755
index 0000000..e53b624
--- /dev/null
+++ b/IntegrationRules/2 Exponentials/2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p.m	
@@ -0,0 +1,7 @@
+
+(* ::Subsection::Closed:: *)
+(* 2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p *)
+Int[(c_. + d_.*x_)^ m_.*(F_^(g_.*(e_. + f_.*x_)))^ n_./(a_ + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.), x_Symbol] := (c + d*x)^m/(b*f*g*n*Log[F])*Log[1 + b*(F^(g*(e + f*x)))^n/a] - d*m/(b*f*g*n*Log[F])* Int[(c + d*x)^(m - 1)*Log[1 + b*(F^(g*(e + f*x)))^n/a], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*(F_^(g_.*(e_. + f_.*x_)))^ n_.*(a_. + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.)^p_., x_Symbol] := (c + d*x)^ m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1)/(b*f*g*n*(p + 1)* Log[F]) - d*m/(b*f*g*n*(p + 1)*Log[F])* Int[(c + d*x)^(m - 1)*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, -1]
+Int[(c_. + d_.*x_)^m_.*(F_^(g_.*(e_. + f_.*x_)))^ n_.*(a_. + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.)^p_., x_Symbol] := Unintegrable[(c + d*x)^m*(F^(g*(e + f*x)))^ n*(a + b*(F^(g*(e + f*x)))^n)^p, x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x]
+Int[(c_. + d_.*x_)^m_.*(k_.*G_^(j_.*(h_. + i_.*x_)))^ q_.*(a_. + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.)^p_., x_Symbol] := (k*G^(j*(h + i*x)))^q/(F^(g*(e + f*x)))^n* Int[(c + d*x)^m*(F^(g*(e + f*x)))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x] /; FreeQ[{F, a, b, c, d, e, f, g, h, i, j, k, m, n, p, q}, x] && EqQ[f*g*n*Log[F] - i*j*q*Log[G], 0] && NeQ[(k*G^(j*(h + i*x)))^q - (F^(g*(e + f*x)))^n, 0]
diff --git a/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m b/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m
new file mode 100755
index 0000000..3344c1f
--- /dev/null
+++ b/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m	
@@ -0,0 +1,115 @@
+
+(* ::Subsection::Closed:: *)
+(* 2.3 Miscellaneous exponentials *)
+Int[(F_^(c_.*(a_. + b_.*x_)))^n_., x_Symbol] := (F^(c*(a + b*x)))^n/(b*c*n*Log[F]) /; FreeQ[{F, a, b, c, n}, x]
+Int[u_*F_^(c_.*v_), x_Symbol] := Int[ExpandIntegrand[u*F^(c*ExpandToSum[v, x]), x], x] /; FreeQ[{F, c}, x] && PolynomialQ[u, x] && LinearQ[v, x] && TrueQ[$UseGamma]
+Int[u_*F_^(c_.*v_), x_Symbol] := Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[u, x] && LinearQ[v, x] && Not[TrueQ[$UseGamma]]
+Int[u_^m_.*F_^(c_.*v_)*w_, x_Symbol] := With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0], e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, g*u^(m + 1)*F^(c*v)/(b*c*e*Log[F]) /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x]
+Int[w_*u_^m_.*F_^(c_.*v_), x_Symbol] := Int[ExpandIntegrand[ w*NormalizePowerOfLinear[u, x]^m*F^(c*ExpandToSum[v, x]), x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x] && IntegerQ[m] && TrueQ[$UseGamma]
+Int[w_*u_^m_.*F_^(c_.*v_), x_Symbol] := Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePowerOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x] && IntegerQ[m] && Not[TrueQ[$UseGamma]]
+Int[w_*u_^m_.*F_^(c_.*v_), x_Symbol] := Module[{uu = NormalizePowerOfLinear[u, x], z}, z = If[PowerQ[uu] && FreeQ[uu[[2]], x], uu[[1]]^(m*uu[[2]]), uu^m]; uu^m/z*Int[ExpandIntegrand[w*z*F^(c*ExpandToSum[v, x]), x], x]] /; FreeQ[{F, c, m}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x] && Not[IntegerQ[m]]
+Int[F_^(c_.*(a_. + b_.*x_))* Log[d_.*x_]^n_.*(e_ + h_.*(f_. + g_.*x_)*Log[d_.*x_]), x_Symbol] := e*x*F^(c*(a + b*x))*Log[d*x]^(n + 1)/(n + 1) /; FreeQ[{F, a, b, c, d, e, f, g, h, n}, x] && EqQ[e - f*h*(n + 1), 0] && EqQ[g*h*(n + 1) - b*c*e*Log[F], 0] && NeQ[n, -1]
+Int[x_^m_.*F_^(c_.*(a_. + b_.*x_))* Log[d_.*x_]^n_.*(e_ + h_.*(f_. + g_.*x_)*Log[d_.*x_]), x_Symbol] := e*x^(m + 1)*F^(c*(a + b*x))*Log[d*x]^(n + 1)/(n + 1) /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*(m + 1) - f*h*(n + 1), 0] && EqQ[g*h*(n + 1) - b*c*e*Log[F], 0] && NeQ[n, -1]
+Int[F_^(a_. + b_.*(c_. + d_.*x_)), x_Symbol] := F^(a + b*(c + d*x))/(b*d*Log[F]) /; FreeQ[{F, a, b, c, d}, x]
+Int[F_^(a_. + b_.*(c_. + d_.*x_)^2), x_Symbol] := F^a*Sqrt[Pi]* Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2]) /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]
+Int[F_^(a_. + b_.*(c_. + d_.*x_)^2), x_Symbol] := F^a*Sqrt[Pi]* Erf[(c + d*x)*Rt[-b*Log[F], 2]]/(2*d*Rt[-b*Log[F], 2]) /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]
+Int[F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol] := (c + d*x)*F^(a + b*(c + d*x)^n)/d - b*n*Log[F]*Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n] && ILtQ[n, 0]
+Int[F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol] := With[{k = Denominator[n]}, k/d* Subst[Int[x^(k - 1)*F^(a + b*x^(k*n)), x], x, (c + d*x)^(1/k)]] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n] && Not[IntegerQ[n]]
+Int[F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol] := -F^a*(c + d*x)* Gamma[1/n, -b*(c + d*x)^n*Log[F]]/(d* n*(-b*(c + d*x)^n*Log[F])^(1/n)) /; FreeQ[{F, a, b, c, d, n}, x] && Not[IntegerQ[2/n]]
+Int[(e_. + f_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol] := (e + f*x)^n*F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F]) /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] && EqQ[d*e - c*f, 0]
+Int[F_^(a_. + b_.*(c_. + d_.*x_)^n_)/(e_. + f_.*x_), x_Symbol] := F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]]/(f*n) /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]
+Int[(c_. + d_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol] := 1/(d*(m + 1))*Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1)]
+Int[(c_. + d_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol] := (c + d*x)^(m - n + 1)*F^(a + b*(c + d*x)^n)/(b*d*n*Log[F]) - (m - n + 1)/(b*n*Log[F])* Int[(c + d*x)^(m - n)*F^(a + b*(c + d*x)^n), x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*(m + 1)/n] && LtQ[0, (m + 1)/n, 5] && IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])
+Int[(c_. + d_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol] := (c + d*x)^(m - n + 1)*F^(a + b*(c + d*x)^n)/(b*d*n*Log[F]) - (m - n + 1)/(b*n*Log[F])* Int[(c + d*x)^Simplify[m - n]*F^(a + b*(c + d*x)^n), x] /; FreeQ[{F, a, b, c, d, m, n}, x] && IntegerQ[2*Simplify[(m + 1)/n]] && LtQ[0, Simplify[(m + 1)/n], 5] && Not[RationalQ[m]] && SumSimplerQ[m, -n]
+Int[(c_. + d_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol] := (c + d*x)^(m + 1)*F^(a + b*(c + d*x)^n)/(d*(m + 1)) - b*n*Log[F]/(m + 1)* Int[(c + d*x)^(m + n)*F^(a + b*(c + d*x)^n), x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*(m + 1)/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[ n] && (GtQ[n, 0] && LtQ[m, -1] || GtQ[-n, 0] && LeQ[-n, m + 1])
+Int[(c_. + d_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol] := (c + d*x)^(m + 1)*F^(a + b*(c + d*x)^n)/(d*(m + 1)) - b*n*Log[F]/(m + 1)* Int[(c + d*x)^Simplify[m + n]*F^(a + b*(c + d*x)^n), x] /; FreeQ[{F, a, b, c, d, m, n}, x] && IntegerQ[2*Simplify[(m + 1)/n]] && LtQ[-4, Simplify[(m + 1)/n], 5] && Not[RationalQ[m]] && SumSimplerQ[m, n]
+Int[(c_. + d_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol] := With[{k = Denominator[n]}, k/d* Subst[Int[x^(k*(m + 1) - 1)*F^(a + b*x^(k*n)), x], x, (c + d*x)^(1/k)]] /; FreeQ[{F, a, b, c, d, m, n}, x] && IntegerQ[2*(m + 1)/n] && LtQ[0, (m + 1)/n, 5] && Not[IntegerQ[n]]
+Int[(e_. + f_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol] := (e + f*x)^m/(c + d*x)^m*Int[(c + d*x)^m*F^(a + b*(c + d*x)^n), x] /; FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0] && IntegerQ[2*Simplify[(m + 1)/n]] && Not[IntegerQ[m]] && NeQ[f, d] && NeQ[c*e, 0]
+Int[(e_. + f_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol] := With[{p = Simplify[(m + 1)/n]}, -F^a*(f/d)^m/(d*n*(-b*Log[F])^p)* Simplify[FunctionExpand[Gamma[p, -b*(c + d*x)^n*Log[F]]]] /; IGtQ[p, 0]] /; FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0] && Not[TrueQ[$UseGamma]]
+Int[(e_. + f_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol] := (*-F^a*(e+f*x)^(m+1)/(f*n)*ExpIntegralE[1-(m+1)/n,-b*(c+d*x)^n*Log[F] ] *) -F^a*(e + f*x)^(m + 1)/(f*n*(-b*(c + d*x)^n*Log[F])^((m + 1)/n))* Gamma[(m + 1)/n, -b*(c + d*x)^n*Log[F]] /; FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]
+Int[(e_. + f_.*x_)^m_*F_^(a_. + b_.*(c_. + d_.*x_)^2), x_Symbol] := f*(e + f*x)^(m - 1)*F^(a + b*(c + d*x)^2)/(2*b*d^2*Log[F]) + (d*e - c*f)/d*Int[(e + f*x)^(m - 1)*F^(a + b*(c + d*x)^2), x] - (m - 1)*f^2/(2*b*d^2*Log[F])* Int[(e + f*x)^(m - 2)*F^(a + b*(c + d*x)^2), x] /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && FractionQ[m] && GtQ[m, 1]
+Int[(e_. + f_.*x_)^m_*F_^(a_. + b_.*(c_. + d_.*x_)^2), x_Symbol] := f*(e + f*x)^(m + 1)*F^(a + b*(c + d*x)^2)/((m + 1)*f^2) + 2*b*d*(d*e - c*f)*Log[F]/(f^2*(m + 1))* Int[(e + f*x)^(m + 1)*F^(a + b*(c + d*x)^2), x] - 2*b*d^2*Log[F]/(f^2*(m + 1))* Int[(e + f*x)^(m + 2)*F^(a + b*(c + d*x)^2), x] /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && LtQ[m, -1]
+Int[(e_. + f_.*x_)^m_*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol] := (e + f*x)^(m + 1)*F^(a + b*(c + d*x)^n)/(f*(m + 1)) - b*d*n*Log[F]/(f*(m + 1))* Int[(e + f*x)^(m + 1)*(c + d*x)^(n - 1)*F^(a + b*(c + d*x)^n), x] /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && IGtQ[n, 2] && LtQ[m, -1]
+Int[F_^(a_. + b_./(c_. + d_.*x_))/(e_. + f_.*x_), x_Symbol] := d/f*Int[F^(a + b/(c + d*x))/(c + d*x), x] - (d*e - c*f)/f*Int[F^(a + b/(c + d*x))/((c + d*x)*(e + f*x)), x] /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0]
+Int[(e_. + f_.*x_)^m_*F_^(a_. + b_./(c_. + d_.*x_)), x_Symbol] := (e + f*x)^(m + 1)*F^(a + b/(c + d*x))/(f*(m + 1)) + b*d*Log[F]/(f*(m + 1))* Int[(e + f*x)^(m + 1)*F^(a + b/(c + d*x))/(c + d*x)^2, x] /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && ILtQ[m, -1]
+Int[F_^(a_. + b_.*(c_. + d_.*x_)^n_)/(e_. + f_.*x_), x_Symbol] := Unintegrable[F^(a + b*(c + d*x)^n)/(e + f*x), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && NeQ[d*e - c*f, 0]
+Int[u_^m_.*F_^v_, x_Symbol] := Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] && LinearQ[u, x] && BinomialQ[v, x] && Not[LinearMatchQ[u, x] && BinomialMatchQ[v, x]]
+Int[u_*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol] := Int[ExpandLinearProduct[F^(a + b*(c + d*x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]
+Int[u_.*F_^(a_. + b_.*v_), x_Symbol] := Int[u*F^(a + b*NormalizePowerOfLinear[v, x]), x] /; FreeQ[{F, a, b}, x] && PolynomialQ[u, x] && PowerOfLinearQ[v, x] && Not[PowerOfLinearMatchQ[v, x]]
+(* Int[u_.*F_^(a_.+b_.*v_^n_),x_Symbol] := Int[u*F^(a+b*ExpandToSum[v,x]^n),x] /; FreeQ[{F,a,b,n},x] && PolynomialQ[u,x] && LinearQ[v,x] &&  Not[LinearMatchQ[v,x]] *)
+(* Int[u_.*F_^u_,x_Symbol] := Int[u*F^ExpandToSum[u,x],x] /; FreeQ[F,x] && PolynomialQ[u,x] && BinomialQ[u,x] &&  Not[BinomialMatchQ[u,x]] *)
+Int[F_^(a_. + b_./(c_. + d_.*x_))/((e_. + f_.*x_)*(g_. + h_.*x_)), x_Symbol] := -d/(f*(d*g - c*h))* Subst[Int[F^(a - b*h/(d*g - c*h) + d*b*x/(d*g - c*h))/x, x], x, (g + h*x)/(c + d*x)] /; FreeQ[{F, a, b, c, d, e, f}, x] && EqQ[d*e - c*f, 0]
+Int[(g_. + h_.*x_)^m_.*F_^(e_. + f_.*(a_. + b_.*x_)/(c_. + d_.*x_)), x_Symbol] := F^(e + f*b/d)*Int[(g + h*x)^m, x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m}, x] && EqQ[b*c - a*d, 0]
+Int[(g_. + h_.*x_)^m_.*F_^(e_. + f_.*(a_. + b_.*x_)/(c_. + d_.*x_)), x_Symbol] := Int[(g + h*x)^m*F^((d*e + b*f)/d - f*(b*c - a*d)/(d*(c + d*x))), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m}, x] && NeQ[b*c - a*d, 0] && EqQ[d*g - c*h, 0]
+Int[F_^(e_. + f_.*(a_. + b_.*x_)/(c_. + d_.*x_))/(g_. + h_.*x_), x_Symbol] := d/h*Int[F^(e + f*(a + b*x)/(c + d*x))/(c + d*x), x] - (d*g - c*h)/h* Int[F^(e + f*(a + b*x)/(c + d*x))/((c + d*x)*(g + h*x)), x] /; FreeQ[{F, a, b, c, d, e, f, g, h}, x] && NeQ[b*c - a*d, 0] && NeQ[d*g - c*h, 0]
+Int[(g_. + h_.*x_)^m_*F_^(e_. + f_.*(a_. + b_.*x_)/(c_. + d_.*x_)), x_Symbol] := (g + h*x)^(m + 1)*F^(e + f*(a + b*x)/(c + d*x))/(h*(m + 1)) - f*(b*c - a*d)*Log[F]/(h*(m + 1))* Int[(g + h*x)^(m + 1)*F^(e + f*(a + b*x)/(c + d*x))/(c + d*x)^2, x] /; FreeQ[{F, a, b, c, d, e, f, g, h}, x] && NeQ[b*c - a*d, 0] && NeQ[d*g - c*h, 0] && ILtQ[m, -1]
+Int[F_^(e_. + f_.*(a_. + b_.*x_)/(c_. + d_.*x_))/((g_. + h_.*x_)*(i_. + j_.*x_)), x_Symbol] := -d/(h*(d*i - c*j))* Subst[Int[ F^(e + f*(b*i - a*j)/(d*i - c*j) - (b*c - a*d)*f*x/(d*i - c*j))/ x, x], x, (i + j*x)/(c + d*x)] /; FreeQ[{F, a, b, c, d, e, f, g, h}, x] && EqQ[d*g - c*h, 0]
+Int[F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol] := F^(a - b^2/(4*c))*Int[F^((b + 2*c*x)^2/(4*c)), x] /; FreeQ[{F, a, b, c}, x]
+Int[F_^v_, x_Symbol] := Int[F^ExpandToSum[v, x], x] /; FreeQ[F, x] && QuadraticQ[v, x] && Not[QuadraticMatchQ[v, x]]
+Int[(d_. + e_.*x_)*F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol] := e*F^(a + b*x + c*x^2)/(2*c*Log[F]) /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]
+Int[(d_. + e_.*x_)^m_*F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol] := e*(d + e*x)^(m - 1)*F^(a + b*x + c*x^2)/(2*c*Log[F]) - (m - 1)*e^2/(2*c*Log[F])* Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0] && GtQ[m, 1]
+Int[F_^(a_. + b_.*x_ + c_.*x_^2)/(d_. + e_.*x_), x_Symbol] := 1/(2*e)*F^(a - b^2/(4*c))* ExpIntegralEi[(b + 2*c*x)^2*Log[F]/(4*c)] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]
+Int[(d_. + e_.*x_)^m_*F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol] := (d + e*x)^(m + 1)*F^(a + b*x + c*x^2)/(e*(m + 1)) - 2*c*Log[F]/(e^2*(m + 1))* Int[(d + e*x)^(m + 2)*F^(a + b*x + c*x^2), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0] && LtQ[m, -1]
+Int[(d_. + e_.*x_)*F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol] := e*F^(a + b*x + c*x^2)/(2*c*Log[F]) - (b*e - 2*c*d)/(2*c)*Int[F^(a + b*x + c*x^2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0]
+Int[(d_. + e_.*x_)^m_*F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol] := e*(d + e*x)^(m - 1)*F^(a + b*x + c*x^2)/(2*c*Log[F]) - (b*e - 2*c*d)/(2*c)* Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2), x] - (m - 1)*e^2/(2*c*Log[F])* Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]
+Int[(d_. + e_.*x_)^m_*F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol] := (d + e*x)^(m + 1)*F^(a + b*x + c*x^2)/(e*(m + 1)) - (b*e - 2*c*d)*Log[F]/(e^2*(m + 1))* Int[(d + e*x)^(m + 1)*F^(a + b*x + c*x^2), x] - 2*c*Log[F]/(e^2*(m + 1))* Int[(d + e*x)^(m + 2)*F^(a + b*x + c*x^2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && LtQ[m, -1]
+Int[(d_. + e_.*x_)^m_.*F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol] := Unintegrable[(d + e*x)^m*F^(a + b*x + c*x^2), x] /; FreeQ[{F, a, b, c, d, e, m}, x]
+Int[u_^m_.*F_^v_, x_Symbol] := Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] && LinearQ[u, x] && QuadraticQ[v, x] && Not[LinearMatchQ[u, x] && QuadraticMatchQ[v, x]]
+Int[x_^m_.*F_^(e_.*(c_. + d_.*x_))*(a_. + b_.*F_^v_)^p_, x_Symbol] := With[{u = IntHide[F^(e*(c + d*x))*(a + b*F^v)^p, x]}, Dist[x^m, u, x] - m*Int[x^(m - 1)*u, x]] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[v, 2*e*(c + d*x)] && GtQ[m, 0] && ILtQ[p, 0]
+Int[(F_^(e_.*(c_. + d_.*x_)))^ n_.*(a_ + b_.*(F_^(e_.*(c_. + d_.*x_)))^n_.)^p_., x_Symbol] := 1/(d*e*n*Log[F])* Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n] /; FreeQ[{F, a, b, c, d, e, n, p}, x]
+Int[(G_^(h_. (f_. + g_.*x_)))^ m_.*(a_ + b_.*(F_^(e_.*(c_. + d_.*x_)))^n_.)^p_., x_Symbol] := (G^(h*(f + g*x)))^m/(F^(e*(c + d*x)))^n* Int[(F^(e*(c + d*x)))^n*(a + b*(F^(e*(c + d*x)))^n)^p, x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[d*e*n*Log[F], g*h*m*Log[G]]
+Int[G_^(h_. (f_. + g_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_., x_Symbol] := With[{m = FullSimplify[g*h*Log[G]/(d*e*Log[F])]}, Denominator[m]*G^(f*h - c*g*h/d)/(d*e*Log[F])* Subst[Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^(e*(c + d*x)/Denominator[m])] /; LeQ[m, -1] || GeQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]
+Int[G_^(h_. (f_. + g_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_., x_Symbol] := With[{m = FullSimplify[d*e*Log[F]/(g*h*Log[G])]}, Denominator[m]/(g*h*Log[G])* Subst[Int[ x^(Denominator[m] - 1)*(a + b*F^(c*e - d*e*f/g)*x^Numerator[m])^p, x], x, G^(h*(f + g*x)/Denominator[m])] /; LtQ[m, -1] || GtQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]
+Int[G_^(h_. (f_. + g_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_., x_Symbol] := Int[Expand[G^(h*(f + g*x))*(a + b*F^(e*(c + d*x)))^p, x], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h}, x] && IGtQ[p, 0]
+Int[G_^(h_. (f_. + g_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_, x_Symbol] := a^p*G^(h*(f + g*x))/(g*h*Log[G])* Hypergeometric2F1[-p, g*h*Log[G]/(d*e*Log[F]), g*h*Log[G]/(d*e*Log[F]) + 1, Simplify[-b/a*F^(e*(c + d*x))]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] || GtQ[a, 0])
+Int[G_^(h_. (f_. + g_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_, x_Symbol] := (a + b*F^(e*(c + d*x)))^p/(1 + (b/a)*F^(e*(c + d*x)))^p* Int[G^(h*(f + g*x))*(1 + b/a*F^(e*(c + d*x)))^p, x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && Not[ILtQ[p, 0] || GtQ[a, 0]]
+Int[G_^(h_. u_)*(a_ + b_.*F_^(e_.*v_))^p_, x_Symbol] := Int[G^(h*ExpandToSum[u, x])*(a + b*F^(e*ExpandToSum[v, x]))^p, x] /; FreeQ[{F, G, a, b, e, h, p}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
+(* Int[(c_.+d_.*x_)^m_.*F_^(g_.*(e_.+f_.*x_))/(a_+b_.*F_^(h_.*(e_.+f_. *x_))),x_Symbol] := 1/b*Int[(c+d*x)^m*F^((g-h)*(e+f*x)),x] - a/b*Int[(c+d*x)^m*F^((g-h)*(e+f*x))/(a+b*F^(h*(e+f*x))),x] /; FreeQ[{F,a,b,c,d,e,f,g,h,m},x] && LeQ[0,g/h-1,g/h] *)
+(* Int[(c_.+d_.*x_)^m_.*F_^(g_.*(e_.+f_.*x_))/(a_+b_.*F_^(h_.*(e_.+f_. *x_))),x_Symbol] := 1/a*Int[(c+d*x)^m*F^(g*(e+f*x)),x] - b/a*Int[(c+d*x)^m*F^((g+h)*(e+f*x))/(a+b*F^(h*(e+f*x))),x] /; FreeQ[{F,a,b,c,d,e,f,g,h,m},x] && LeQ[g/h,g/h+1,0] *)
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*F_^u_)^p_.*(c_. + d_.*F_^v_)^q_., x_Symbol] := With[{w = ExpandIntegrand[(e + f*x)^m, (a + b*F^u)^p*(c + d*F^v)^q, x]}, Int[w, x] /; SumQ[w]] /; FreeQ[{F, a, b, c, d, e, f, m}, x] && IntegersQ[p, q] && LinearQ[{u, v}, x] && RationalQ[Simplify[u/v]]
+Int[G_^(h_. (f_. + g_.*x_))* H_^(t_. (r_. + s_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_., x_Symbol] := With[{m = FullSimplify[(g*h*Log[G] + s*t*Log[H])/(d*e*Log[F])]}, Denominator[m]*G^(f*h - c*g*h/d)*H^(r*t - c*s*t/d)/(d*e*Log[F])* Subst[Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^(e*(c + d*x)/Denominator[m])] /; RationalQ[m]] /; FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t, p}, x]
+Int[G_^(h_. (f_. + g_.*x_))* H_^(t_. (r_. + s_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_., x_Symbol] := G^((f - c*g/d)*h)* Int[H^(t*(r + s*x))*(b + a*F^(-e*(c + d*x)))^p, x] /; FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t}, x] && EqQ[d*e*p*Log[F] + g*h*Log[G], 0] && IntegerQ[p]
+Int[G_^(h_. (f_. + g_.*x_))* H_^(t_. (r_. + s_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_., x_Symbol] := Int[Expand[ G^(h*(f + g*x))*H^(t*(r + s*x))*(a + b*F^(e*(c + d*x)))^p, x], x] /; FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t}, x] && IGtQ[p, 0]
+Int[G_^(h_. (f_. + g_.*x_))* H_^(t_. (r_. + s_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_, x_Symbol] := a^p*G^(h*(f + g*x))*H^(t*(r + s*x))/(g*h*Log[G] + s*t*Log[H])* Hypergeometric2F1[-p, (g*h*Log[G] + s*t*Log[H])/(d*e* Log[F]), (g*h*Log[G] + s*t*Log[H])/(d*e*Log[F]) + 1, Simplify[-b/a*F^(e*(c + d*x))]] /; FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t}, x] && ILtQ[p, 0]
+Int[G_^(h_. (f_. + g_.*x_))* H_^(t_. (r_. + s_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_, x_Symbol] := G^(h*(f + g*x))* H^(t*(r + s*x))*(a + b*F^(e*(c + d*x)))^ p/((g*h*Log[G] + s*t*Log[H])*((a + b*F^(e*(c + d*x)))/a)^p)* Hypergeometric2F1[-p, (g*h*Log[G] + s*t*Log[H])/(d*e* Log[F]), (g*h*Log[G] + s*t*Log[H])/(d*e*Log[F]) + 1, Simplify[-b/a*F^(e*(c + d*x))]] /; FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t, p}, x] && Not[IntegerQ[p]]
+Int[G_^(h_. u_)*H_^(t_. w_)*(a_ + b_.*F_^(e_.*v_))^p_, x_Symbol] := Int[G^(h*ExpandToSum[u, x])* H^(t*ExpandToSum[w, x])*(a + b*F^(e*ExpandToSum[v, x]))^p, x] /; FreeQ[{F, G, H, a, b, e, h, t, p}, x] && LinearQ[{u, v, w}, x] && Not[LinearMatchQ[{u, v, w}, x]]
+Int[F_^(e_.*(c_. + d_.*x_))*(a_.*x_^n_. + b_.*F_^(e_.*(c_. + d_.*x_)))^p_., x_Symbol] := (a*x^n + b*F^(e*(c + d*x)))^(p + 1)/(b*d*e*(p + 1)*Log[F]) - a*n/(b*d*e*Log[F])* Int[x^(n - 1)*(a*x^n + b*F^(e*(c + d*x)))^p, x] /; FreeQ[{F, a, b, c, d, e, n, p}, x] && NeQ[p, -1]
+Int[x_^m_.* F_^(e_.*(c_. + d_.*x_))*(a_.*x_^n_. + b_.*F_^(e_.*(c_. + d_.*x_)))^ p_., x_Symbol] := x^m*(a*x^n + b*F^(e*(c + d*x)))^(p + 1)/(b*d*e*(p + 1)*Log[F]) - a*n/(b*d*e*Log[F])* Int[x^(m + n - 1)*(a*x^n + b*F^(e*(c + d*x)))^p, x] - m/(b*d*e*(p + 1)*Log[F])* Int[x^(m - 1)*(a*x^n + b*F^(e*(c + d*x)))^(p + 1), x] /; FreeQ[{F, a, b, c, d, e, m, n, p}, x] && NeQ[p, -1]
+Int[(f_. + g_.*x_)^m_./(a_. + b_.*F_^u_ + c_.*F_^v_), x_Symbol] := With[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[(f + g*x)^m/(b - q + 2*c*F^u), x] - 2*c/q*Int[(f + g*x)^m/(b + q + 2*c*F^u), x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
+Int[(f_. + g_.*x_)^m_.*F_^u_/(a_. + b_.*F_^u_ + c_.*F_^v_), x_Symbol] := With[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[(f + g*x)^m*F^u/(b - q + 2*c*F^u), x] - 2*c/q*Int[(f + g*x)^m*F^u/(b + q + 2*c*F^u), x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
+Int[(f_. + g_.*x_)^m_.*(h_ + i_.*F_^u_)/(a_. + b_.*F_^u_ + c_.*F_^v_), x_Symbol] := With[{q = Rt[b^2 - 4*a*c, 2]}, (Simplify[(2*c*h - b*i)/q] + i)* Int[(f + g*x)^m/(b - q + 2*c*F^u), x] - (Simplify[(2*c*h - b*i)/q] - i)* Int[(f + g*x)^m/(b + q + 2*c*F^u), x]] /; FreeQ[{F, a, b, c, f, g, h, i}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
+Int[x_^m_./(a_.*F_^(c_. + d_.*x_) + b_.*F_^v_), x_Symbol] := With[{u = IntHide[1/(a*F^(c + d*x) + b*F^v), x]}, x^m*u - m*Int[x^(m - 1)*u, x]] /; FreeQ[{F, a, b, c, d}, x] && EqQ[v, -(c + d*x)] && GtQ[m, 0]
+Int[u_/(a_ + b_.*F_^v_ + c_.*F_^w_), x_Symbol] := Int[u*F^v/(c + a*F^v + b*F^(2*v)), x] /; FreeQ[{F, a, b, c}, x] && EqQ[w, -v] && LinearQ[v, x] && If[RationalQ[Coefficient[v, x, 1]], GtQ[Coefficient[v, x, 1], 0], LtQ[LeafCount[v], LeafCount[w]]]
+Int[F_^(g_.*(d_. + e_.*x_)^n_.)/(a_. + b_.*x_ + c_.*x_^2), x_Symbol] := Int[ExpandIntegrand[F^(g*(d + e*x)^n), 1/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x]
+Int[F_^(g_.*(d_. + e_.*x_)^n_.)/(a_ + c_.*x_^2), x_Symbol] := Int[ExpandIntegrand[F^(g*(d + e*x)^n), 1/(a + c*x^2), x], x] /; FreeQ[{F, a, c, d, e, g, n}, x]
+Int[u_^m_.*F_^(g_.*(d_. + e_.*x_)^n_.)/(a_. + b_.*x_ + c_*x_^2), x_Symbol] := Int[ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x] && PolynomialQ[u, x] && IntegerQ[m]
+Int[u_^m_.*F_^(g_.*(d_. + e_.*x_)^n_.)/(a_ + c_*x_^2), x_Symbol] := Int[ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + c*x^2), x], x] /; FreeQ[{F, a, c, d, e, g, n}, x] && PolynomialQ[u, x] && IntegerQ[m]
+Int[F_^((a_. + b_.*x_^4)/x_^2), x_Symbol] := Sqrt[Pi]*Exp[2*Sqrt[-a*Log[F]]*Sqrt[-b*Log[F]]]* Erf[(Sqrt[-a*Log[F]] + Sqrt[-b*Log[F]]*x^2)/x]/ (4*Sqrt[-b*Log[F]]) - Sqrt[Pi]*Exp[-2*Sqrt[-a*Log[F]]*Sqrt[-b*Log[F]]]* Erf[(Sqrt[-a*Log[F]] - Sqrt[-b*Log[F]]*x^2)/x]/ (4*Sqrt[-b*Log[F]]) /; FreeQ[{F, a, b}, x]
+Int[x_^m_.*(E^x_ + x_^m_.)^n_, x_Symbol] := -(E^x + x^m)^(n + 1)/(n + 1) + Int[(E^x + x^m)^(n + 1), x] + m*Int[x^(m - 1)*(E^x + x^m)^n, x] /; RationalQ[m, n] && GtQ[m, 0] && LtQ[n, 0] && NeQ[n, -1]
+Int[u_.*F_^(a_.*(v_. + b_.*Log[z_])), x_Symbol] := Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a, b}, x]
+Int[F_^(f_.*(a_. + b_.*Log[c_.*(d_. + e_.*x_)^n_.]^2)), x_Symbol] := (d + e*x)/(e*n*(c*(d + e*x)^n)^(1/n))* Subst[Int[E^(a*f*Log[F] + x/n + b*f*Log[F]*x^2), x], x, Log[c*(d + e*x)^n]] /; FreeQ[{F, a, b, c, d, e, f, n}, x]
+Int[(g_. + h_.*x_)^m_.* F_^(f_.*(a_. + b_.*Log[c_.*(d_. + e_.*x_)^n_.]^2)), x_Symbol] := (g + h*x)^(m + 1)/(h*n*(c*(d + e*x)^n)^((m + 1)/n))* Subst[Int[E^(a*f*Log[F] + ((m + 1)*x)/n + b*f*Log[F]*x^2), x], x, Log[c*(d + e*x)^n]] /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*g - d*h, 0]
+Int[(g_. + h_.*x_)^m_.* F_^(f_.*(a_. + b_.*Log[c_.*(d_. + e_.*x_)^n_.]^2)), x_Symbol] := 1/e^(m + 1)* Subst[Int[ ExpandIntegrand[F^(f*(a + b*Log[c*x^n]^2)), (e*g - d*h + h*x)^m, x], x], x, d + e*x] /; FreeQ[{F, a, b, c, d, e, f, g, h, n}, x] && IGtQ[m, 0]
+Int[(g_. + h_.*x_)^m_.* F_^(f_.*(a_. + b_.*Log[c_.*(d_. + e_.*x_)^n_.]^2)), x_Symbol] := Unintegrable[(g + h*x)^m*F^(f*(a + b*Log[c*(d + e*x)^n]^2)), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x]
+Int[F_^(f_.*(a_. + b_.*Log[c_.*(d_. + e_.*x_)^n_.])^2), x_Symbol] := c^(2*a*b*f*Log[F])* Int[(d + e*x)^(2*a*b*f*n*Log[F])* F^(a^2*f + b^2*f*Log[c*(d + e*x)^n]^2), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && IntegerQ[2*a*b*f*Log[F]]
+Int[F_^(f_.*(a_. + b_.*Log[c_.*(d_. + e_.*x_)^n_.])^2), x_Symbol] := (c*(d + e*x)^n)^(2*a*b*f*Log[F])/(d + e*x)^(2*a*b*f*n*Log[F])* Int[(d + e*x)^(2*a*b*f*n*Log[F])* F^(a^2*f + b^2*f*Log[c*(d + e*x)^n]^2), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && Not[IntegerQ[2*a*b*f*Log[F]]]
+Int[(g_. + h_.*x_)^m_.* F_^(f_.*(a_. + b_.*Log[c_.*(d_. + e_.*x_)^n_.])^2), x_Symbol] := h^m*c^(2*a*b*f*Log[F])/e^m* Int[(d + e*x)^(m + 2*a*b*f*n*Log[F])* F^(a^2*f + b^2*f*Log[c*(d + e*x)^n]^2), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*g - d*h, 0] && IntegerQ[2*a*b*f*Log[F]] && (IntegerQ[m] || EqQ[h, e])
+Int[(g_. + h_.*x_)^m_.* F_^(f_.*(a_. + b_.*Log[c_.*(d_. + e_.*x_)^n_.])^2), x_Symbol] := (g + h*x)^ m*(c*(d + e*x)^n)^(2*a*b*f*Log[F])/(d + e*x)^(m + 2*a*b*f*n*Log[F])* Int[(d + e*x)^(m + 2*a*b*f*n*Log[F])* F^(a^2*f + b^2*f*Log[c*(d + e*x)^n]^2), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*g - d*h, 0]
+Int[(g_. + h_.*x_)^m_.* F_^(f_.*(a_. + b_.*Log[c_.*(d_. + e_.*x_)^n_.])^2), x_Symbol] := 1/e^(m + 1)* Subst[Int[ ExpandIntegrand[F^(f*(a + b*Log[c*x^n])^2), (e*g - d*h + h*x)^m, x], x], x, d + e*x] /; FreeQ[{F, a, b, c, d, e, f, g, h, n}, x] && IGtQ[m, 0]
+Int[(g_. + h_.*x_)^m_.* F_^(f_.*(a_. + b_.*Log[c_.*(d_. + e_.*x_)^n_.])^2), x_Symbol] := Unintegrable[(g + h*x)^m*F^(f*(a + b*Log[c*(d + e*x)^n])^2), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x]
+Int[Log[a_ + b_.*(F_^(e_.*(c_. + d_.*x_)))^n_.], x_Symbol] := 1/(d*e*n*Log[F])* Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
+Int[Log[a_ + b_.*(F_^(e_.*(c_. + d_.*x_)))^n_.], x_Symbol] := x*Log[a + b*(F^(e*(c + d*x)))^n] - b*d*e*n*Log[F]* Int[x*(F^(e*(c + d*x)))^n/(a + b*(F^(e*(c + d*x)))^n), x] /; FreeQ[{F, a, b, c, d, e, n}, x] && Not[GtQ[a, 0]]
+(* Int[u_.*(a_.*F_^v_)^n_,x_Symbol] := a^n*Int[u*F^(n*v),x] /; FreeQ[{F,a},x] && IntegerQ[n] *)
+Int[u_.*(a_.*F_^v_)^n_, x_Symbol] := (a*F^v)^n/F^(n*v)*Int[u*F^(n*v), x] /; FreeQ[{F, a, n}, x] && Not[IntegerQ[n]]
+Int[u_, x_Symbol] := With[{v = FunctionOfExponential[u, x]}, v/D[v, x]* Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v]] /; FunctionOfExponentialQ[u, x] && Not[ MatchQ[u, w_*(a_.*v_^n_)^m_ /; FreeQ[{a, m, n}, x] && IntegerQ[m*n]]] && Not[ MatchQ[u, E^(c_.*(a_. + b_.*x))*F_[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]]
+Int[u_.*(a_.*F_^v_ + b_.*F_^w_)^n_, x_Symbol] := Int[u*F^(n*v)*(a + b*F^ExpandToSum[w - v, x])^n, x] /; FreeQ[{F, a, b, n}, x] && ILtQ[n, 0] && LinearQ[{v, w}, x]
+Int[u_.*(a_.*F_^v_ + b_.*G_^w_)^n_, x_Symbol] := Int[u*F^(n*v)*(a + b*E^ExpandToSum[Log[G]*w - Log[F]*v, x])^n, x] /; FreeQ[{F, G, a, b, n}, x] && ILtQ[n, 0] && LinearQ[{v, w}, x]
+Int[u_.*(a_.*F_^v_ + b_.*F_^w_)^n_, x_Symbol] := (a*F^v + b*F^w)^n/(F^(n*v)*(a + b*F^ExpandToSum[w - v, x])^n)* Int[u*F^(n*v)*(a + b*F^ExpandToSum[w - v, x])^n, x] /; FreeQ[{F, a, b, n}, x] && Not[IntegerQ[n]] && LinearQ[{v, w}, x]
+Int[u_.*(a_.*F_^v_ + b_.*G_^w_)^n_, x_Symbol] := (a*F^v + b*G^w)^ n/(F^(n*v)*(a + b*E^ExpandToSum[Log[G]*w - Log[F]*v, x])^n)* Int[u*F^(n*v)*(a + b*E^ExpandToSum[Log[G]*w - Log[F]*v, x])^n, x] /; FreeQ[{F, G, a, b, n}, x] && Not[IntegerQ[n]] && LinearQ[{v, w}, x]
+Int[u_.*F_^v_*G_^w_, x_Symbol] := With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x], x] /; BinomialQ[z, x] || PolynomialQ[z, x] && LeQ[Exponent[z, x], 2]] /; FreeQ[{F, G}, x]
+Int[F_^u_*(v_ + w_)*y_., x_Symbol] := With[{z = v*y/(Log[F]*D[u, x])}, F^u*z /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
+Int[F_^u_*v_^n_.*w_, x_Symbol] := With[{z = Log[F]*v*D[u, x] + (n + 1)*D[v, x]}, Coefficient[w, x, Exponent[w, x]]/ Coefficient[z, x, Exponent[z, x]]*F^u*v^(n + 1) /; EqQ[Exponent[w, x], Exponent[z, x]] && EqQ[w*Coefficient[z, x, Exponent[z, x]], z*Coefficient[w, x, Exponent[w, x]]]] /; FreeQ[{F, n}, x] && PolynomialQ[u, x] && PolynomialQ[v, x] && PolynomialQ[w, x]
+Int[(a_. + b_.*F_^(c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]))^ n_./(A_. + B_.*x_ + C_.*x_^2), x_Symbol] := 2*e*g/(C*(e*f - d*g))* Subst[Int[(a + b*F^(c*x))^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]] /; FreeQ[{a, b, c, d, e, f, g, A, B, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[B*e*g - C*(e*f + d*g), 0] && IGtQ[n, 0]
+Int[(a_. + b_.*F_^(c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]))^ n_./(A_ + C_.*x_^2), x_Symbol] := 2*e*g/(C*(e*f - d*g))* Subst[Int[(a + b*F^(c*x))^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]] /; FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && IGtQ[n, 0]
+Int[(a_. + b_.*F_^(c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]))^ n_/(A_. + B_.*x_ + C_.*x_^2), x_Symbol] := Unintegrable[(a + b*F^(c*Sqrt[d + e*x]/Sqrt[f + g*x]))^ n/(A + B*x + C*x^2), x] /; FreeQ[{a, b, c, d, e, f, g, A, B, C, F, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[B*e*g - C*(e*f + d*g), 0] && Not[IGtQ[n, 0]]
+Int[(a_. + b_.*F_^(c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]))^ n_/(A_ + C_.*x_^2), x_Symbol] := Unintegrable[(a + b*F^(c*Sqrt[d + e*x]/Sqrt[f + g*x]))^ n/(A + C*x^2), x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && Not[IGtQ[n, 0]]
diff --git a/IntegrationRules/3 Logarithms/3.1.1 (a+b log(c x^n))^p.m b/IntegrationRules/3 Logarithms/3.1.1 (a+b log(c x^n))^p.m
new file mode 100755
index 0000000..405ae8b
--- /dev/null
+++ b/IntegrationRules/3 Logarithms/3.1.1 (a+b log(c x^n))^p.m	
@@ -0,0 +1,9 @@
+
+(* ::Subsection::Closed:: *)
+(* 3.1.1 (a+b log(c x^n))^p *)
+Int[Log[c_.*x_^n_.], x_Symbol] := x*Log[c*x^n] - n*x /; FreeQ[{c, n}, x]
+Int[(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := x*(a + b*Log[c*x^n])^p - b*n*p*Int[(a + b*Log[c*x^n])^(p - 1), x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
+Int[(a_. + b_.*Log[c_.*x_^n_.])^p_, x_Symbol] := x*(a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)) - 1/(b*n*(p + 1))*Int[(a + b*Log[c*x^n])^(p + 1), x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] && IntegerQ[2*p]
+Int[1/Log[c_.*x_], x_Symbol] := LogIntegral[c*x]/c /; FreeQ[c, x]
+Int[(a_. + b_.*Log[c_.*x_^n_.])^p_, x_Symbol] := 1/(n*c^(1/n))*Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]
+Int[(a_. + b_.*Log[c_.*x_^n_.])^p_, x_Symbol] := x/(n*(c*x^n)^(1/n))* Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]] /; FreeQ[{a, b, c, n, p}, x]
diff --git a/IntegrationRules/3 Logarithms/3.1.2 (d x)^m (a+b log(c x^n))^p.m b/IntegrationRules/3 Logarithms/3.1.2 (d x)^m (a+b log(c x^n))^p.m
new file mode 100755
index 0000000..76e8cb0
--- /dev/null
+++ b/IntegrationRules/3 Logarithms/3.1.2 (d x)^m (a+b log(c x^n))^p.m	
@@ -0,0 +1,15 @@
+
+(* ::Subsection::Closed:: *)
+(* 3.1.2 (d x)^m (a+b log(c x^n))^p *)
+Int[(a_. + b_.*Log[c_.*x_^n_.])/x_, x_Symbol] := (a + b*Log[c*x^n])^2/(2*b*n) /; FreeQ[{a, b, c, n}, x]
+Int[(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol] := 1/(b*n)*Subst[Int[x^p, x], x, a + b*Log[c*x^n]] /; FreeQ[{a, b, c, n, p}, x]
+Int[(d_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := b*(d*x)^(m + 1)*Log[c*x^n]/(d*(m + 1)) /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && EqQ[a*(m + 1) - b*n, 0]
+Int[(d_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := (d*x)^(m + 1)*(a + b*Log[c*x^n])/(d*(m + 1)) - b*n*(d*x)^(m + 1)/(d*(m + 1)^2) /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
+Int[(d_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := (d*x)^(m + 1)*(a + b*Log[c*x^n])^p/(d*(m + 1)) - b*n*p/(m + 1)*Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]
+Int[(d_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.])^p_, x_Symbol] := (d*x)^(m + 1)*(a + b*Log[c*x^n])^(p + 1)/(b*d*n*(p + 1)) - (m + 1)/(b*n*(p + 1))*Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]
+Int[x_^m_./Log[c_.*x_^n_], x_Symbol] := 1/n*Subst[Int[1/Log[c*x], x], x, x^n] /; FreeQ[{c, m, n}, x] && EqQ[m, n - 1]
+Int[(d_*x_)^m_./Log[c_.*x_^n_], x_Symbol] := (d*x)^m/x^m*Int[x^m/Log[c*x^n], x] /; FreeQ[{c, d, m, n}, x] && EqQ[m, n - 1]
+Int[x_^m_.*(a_. + b_.*Log[c_.*x_])^p_, x_Symbol] := 1/c^(m + 1)*Subst[Int[E^((m + 1)*x)*(a + b*x)^p, x], x, Log[c*x]] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]
+Int[(d_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.])^p_, x_Symbol] := (d*x)^(m + 1)/(d*n*(c*x^n)^((m + 1)/n))* Subst[Int[E^((m + 1)/n*x)*(a + b*x)^p, x], x, Log[c*x^n]] /; FreeQ[{a, b, c, d, m, n, p}, x]
+Int[(d_.*x_^q_)^m_*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := (d*x^q)^m/x^(m*q)*Int[x^(m*q)*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p, q}, x]
+Int[(d1_.*x_^q1_)^m1_*(d2_.*x_^q2_)^m2_*(a_. + b_.*Log[c_.*x_^n_.])^ p_., x_Symbol] := (d1*x^q1)^m1*(d2*x^q2)^m2/x^(m1*q1 + m2*q2)* Int[x^(m1*q1 + m2*q2)*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d1, d2, m1, m2, n, p, q1, q2}, x]
diff --git a/IntegrationRules/3 Logarithms/3.1.3 (d+e x^r)^q (a+b log(c x^n))^p.m b/IntegrationRules/3 Logarithms/3.1.3 (d+e x^r)^q (a+b log(c x^n))^p.m
new file mode 100755
index 0000000..33e962d
--- /dev/null
+++ b/IntegrationRules/3 Logarithms/3.1.3 (d+e x^r)^q (a+b log(c x^n))^p.m	
@@ -0,0 +1,25 @@
+
+(* ::Subsection::Closed:: *)
+(* 3.1.3 (d+e x^r)^q (a+b log(c x^n))^p *)
+Int[(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := With[{u = IntHide[(d + e*x^r)^q, x]}, u*(a + b*Log[c*x^n]) - b*n*Int[SimplifyIntegrand[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0]
+Int[(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := With[{u = IntHide[(d + e*x^r)^q, x]}, Dist[(a + b*Log[c*x^n]), u] - b*n*Int[SimplifyIntegrand[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0]
+Int[(d_ + e_.*x_^r_.)^q_*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := x*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])/d - b*n/d*Int[(d + e*x^r)^(q + 1), x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
+(* Int[(a_.+b_.*Log[c_.*x_^n_.])^p_./(d_+e_.*x_^r_.),x_Symbol] := 1/e*Int[(a+b*Log[c*x^n])^p/x^r,x] - d/e*Int[(a+b*Log[c*x^n])^p/(x^r*(d+e*x^r)),x] /; FreeQ[{a,b,c,d,e,n,r},x] && IGtQ[p,0] && IGtQ[r,0] *)
+Int[Log[c_.*x_]/(d_ + e_.*x_), x_Symbol] := -1/e*PolyLog[2, 1 - c*x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
+Int[(a_. + b_.*Log[c_.*x_])/(d_ + e_.*x_), x_Symbol] := (a + b*Log[-c*d/e])*Log[d + e*x]/e + b*Int[Log[-e*x/d]/(d + e*x), x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[-c*d/e, 0]
+Int[(a_. + b_.*Log[c_.*x_^n_.])^p_./(d_ + e_.*x_), x_Symbol] := Log[1 + e*x/d]*(a + b*Log[c*x^n])^p/e - b*n*p/e*Int[Log[1 + e*x/d]*(a + b*Log[c*x^n])^(p - 1)/x, x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*Log[c_.*x_^n_.])^p_./(d_ + e_.*x_)^2, x_Symbol] := x*(a + b*Log[c*x^n])^p/(d*(d + e*x)) - b*n*p/d*Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[p, 0]
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := (d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p/(e*(q + 1)) - b*n*p/(e*(q + 1))* Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || IntegersQ[2*p, 2*q] && Not[IGtQ[q, 0]] || EqQ[p, 2] && NeQ[q, 1])
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_, x_Symbol] := x*(d + e*x)^q*(a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)) + d*q/(b*n*(p + 1))* Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^(p + 1), x] - (q + 1)/(b*n*(p + 1))* Int[(d + e*x)^q*(a + b*Log[c*x^n])^(p + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && LtQ[p, -1] && GtQ[q, 0]
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := x*(d + e*x^2)^q*(a + b*Log[c*x^n])/(2*q + 1) - b*n/(2*q + 1)*Int[(d + e*x^2)^q, x] + 2*d*q/(2*q + 1)*Int[(d + e*x^2)^(q - 1)*(a + b*Log[c*x^n]), x] /; FreeQ[{a, b, c, d, e, n}, x] && GtQ[q, 0]
+Int[(a_. + b_.*Log[c_.*x_^n_.])/(d_ + e_.*x_^2)^(3/2), x_Symbol] := x*(a + b*Log[c*x^n])/(d*Sqrt[d + e*x^2]) - b*n/d*Int[1/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, n}, x]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := -x*(d + e*x^2)^(q + 1)*(a + b*Log[c*x^n])/(2*d*(q + 1)) + b*n/(2*d*(q + 1))*Int[(d + e*x^2)^(q + 1), x] + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*Log[c*x^n]), x] /; FreeQ[{a, b, c, d, e, n}, x] && LtQ[q, -1]
+Int[(a_. + b_.*Log[c_.*x_^n_.])/(d_ + e_.*x_^2), x_Symbol] := With[{u = IntHide[1/(d + e*x^2), x]}, u*(a + b*Log[c*x^n]) - b*n*Int[u/x, x]] /; FreeQ[{a, b, c, d, e, n}, x]
+Int[(a_. + b_.*Log[c_.*x_^n_.])/Sqrt[d_ + e_.*x_^2], x_Symbol] := ArcSinh[Rt[e, 2]*x/Sqrt[d]]*(a + b*Log[c*x^n])/Rt[e, 2] - b*n/Rt[e, 2]*Int[ArcSinh[Rt[e, 2]*x/Sqrt[d]]/x, x] /; FreeQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && PosQ[e]
+Int[(a_. + b_.*Log[c_.*x_^n_.])/Sqrt[d_ + e_.*x_^2], x_Symbol] := ArcSin[Rt[-e, 2]*x/Sqrt[d]]*(a + b*Log[c*x^n])/Rt[-e, 2] - b*n/Rt[-e, 2]*Int[ArcSin[Rt[-e, 2]*x/Sqrt[d]]/x, x] /; FreeQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && NegQ[e]
+Int[(a_. + b_.*Log[c_.*x_^n_.])/Sqrt[d_ + e_.*x_^2], x_Symbol] := Sqrt[1 + e/d*x^2]/Sqrt[d + e*x^2]* Int[(a + b*Log[c*x^n])/Sqrt[1 + e/d*x^2], x] /; FreeQ[{a, b, c, d, e, n}, x] && Not[GtQ[d, 0]]
+Int[(a_. + b_.*Log[c_.*x_^n_.])/(Sqrt[d1_ + e1_.*x_]* Sqrt[d2_ + e2_.*x_]), x_Symbol] := Sqrt[1 + e1*e2/(d1*d2)*x^2]/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])* Int[(a + b*Log[c*x^n])/Sqrt[1 + e1*e2/(d1*d2)*x^2], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0]
+Int[(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := With[{u = IntHide[(d + e*x^r)^q, x]}, Dist[(a + b*Log[c*x^n]), u, x] - b*n*Int[SimplifyIntegrand[u/x, x], x] /; EqQ[r, 1] && IntegerQ[q - 1/2] || EqQ[r, 2] && EqQ[q, -1] || InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && IntegerQ[2*q] && IntegerQ[r]
+Int[(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || IGtQ[p, 0] && IntegerQ[r])
+Int[(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := Unintegrable[(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
+Int[u_^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := Int[ExpandToSum[u, x]^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p, q}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
diff --git a/IntegrationRules/3 Logarithms/3.1.4 (f x)^m (d+e x^r)^q (a+b log(c x^n))^p.m b/IntegrationRules/3 Logarithms/3.1.4 (f x)^m (d+e x^r)^q (a+b log(c x^n))^p.m
new file mode 100755
index 0000000..c456c66
--- /dev/null
+++ b/IntegrationRules/3 Logarithms/3.1.4 (f x)^m (d+e x^r)^q (a+b log(c x^n))^p.m	
@@ -0,0 +1,35 @@
+
+(* ::Subsection::Closed:: *)
+(* 3.1.4 (f x)^m (d+e x^r)^q (a+b log(c x^n))^p *)
+Int[x_^m_.*(d_ + e_./x_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := Int[(e + d*x)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[m, q] && IntegerQ[q]
+Int[x_^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, u*(a + b*Log[c*x^n]) - b*n*Int[SimplifyIntegrand[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IGtQ[m, 0]
+Int[x_^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Dist[(a + b*Log[c*x^n]), u] - b*n*Int[SimplifyIntegrand[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && Not[EqQ[q, 1] && EqQ[m, -1]]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_.)^q_*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := (f*x)^(m + 1)*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])/(d* f*(m + 1)) - b*n/(d*(m + 1))*Int[(f*x)^m*(d + e*x^r)^(q + 1), x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m + r*(q + 1) + 1, 0] && NeQ[m, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_)^q_.*(a_. + b_.*Log[c_.*x_^n_])^p_., x_Symbol] := f^m/n*Subst[Int[(d + e*x)^q*(a + b*Log[c*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && EqQ[r, n]
+Int[(f_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.])^p_./(d_ + e_.*x_^r_), x_Symbol] := f^m*Log[1 + e*x^r/d]*(a + b*Log[c*x^n])^p/(e*r) - b*f^m*n*p/(e*r)* Int[Log[1 + e*x^r/d]*(a + b*Log[c*x^n])^(p - 1)/x, x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := f^m*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^p/(e*r*(q + 1)) - b*f^m*n*p/(e*r*(q + 1))* Int[(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^(p - 1)/x, x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]
+Int[(f_*x_)^m_.*(d_ + e_.*x_^r_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := (f*x)^m/x^m*Int[x^m*(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && Not[(IntegerQ[m] || GtQ[f, 0])]
+(* Int[x_^m_.*(a_.+b_.*Log[c_.*x_^n_.])^p_./(d_+e_.*x_^r_.),x_Symbol]  := 1/e*Int[x^(m-r)*(a+b*Log[c*x^n])^p,x] - d/e*Int[(x^(m-r)*(a+b*Log[c*x^n])^p)/(d+e*x^r),x] /; FreeQ[{a,b,c,d,e,m,n,r},x] && IGtQ[p,0] && IGtQ[r,0] && IGeQ[m-r,0] *)
+Int[(a_. + b_.*Log[c_.*x_^n_])/(x_*(d_ + e_.*x_^r_.)), x_Symbol] := 1/n*Subst[Int[(a + b*Log[c*x])/(x*(d + e*x^(r/n))), x], x, x^n] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[r/n]
+(* Int[(a_.+b_.*Log[c_.*x_^n_.])^p_./(x_*(d_+e_.*x_)),x_Symbol] := 1/d*Int[(a+b*Log[c*x^n])^p/x,x] -  e/d*Int[(a+b*Log[c*x^n])^p/(d+e*x),x] /; FreeQ[{a,b,c,d,e,n},x] && IGtQ[p,0] *)
+(* Int[(a_.+b_.*Log[c_.*x_^n_.])^p_./(x_*(d_+e_.*x_^r_.)),x_Symbol] := (r*Log[x]-Log[1+(e*x^r)/d])*(a+b*Log[c*x^n])^p/(d*r) - b*n*p/d*Int[Log[x]*(a+b*Log[c*x^n])^(p-1)/x,x] + b*n*p/(d*r)*Int[Log[1+(e*x^r)/d]*(a+b*Log[c*x^n])^(p-1)/x,x] /; FreeQ[{a,b,c,d,e,n,r},x] && IGtQ[p,0] *)
+Int[(a_. + b_.*Log[c_.*x_^n_.])^p_./(x_*(d_ + e_.*x_^r_.)), x_Symbol] := -Log[1 + d/(e*x^r)]*(a + b*Log[c*x^n])^p/(d*r) + b*n*p/(d*r)* Int[Log[1 + d/(e*x^r)]*(a + b*Log[c*x^n])^(p - 1)/x, x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
+Int[x_^m_.*(a_. + b_.*Log[c_.*x_^n_.])^p_./(d_ + e_.*x_^r_.), x_Symbol] := 1/d*Int[x^m*(a + b*Log[c*x^n])^p, x] - e/d*Int[(x^(m + r)*(a + b*Log[c*x^n])^p)/(d + e*x^r), x] /; FreeQ[{a, b, c, d, e, m, n, r}, x] && IGtQ[p, 0] && IGtQ[r, 0] && ILtQ[m, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_)^q_*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := -(f*x)^(m + 1)*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^ p/(d*f*(q + 1)) + b*n*p/(d*(q + 1))* Int[(f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
+Int[x_^m_.*(d_ + e_.*x_)^q_*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := With[{u = IntHide[x^m*(d + e*x)^q, x]}, Dist[(a + b*Log[c*x^n]), u, x] - b*n*Int[SimplifyIntegrand[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_)^q_*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := -(f*x)^(m + 1)*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^ p/(d*f*(q + 1)) + (m + q + 2)/(d*(q + 1))* Int[(f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p, x] + b*n*p/(d*(q + 1))* Int[(f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1] && GtQ[m, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := (f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])/(e*(q + 1)) - f/(e*(q + 1))* Int[(f*x)^(m - 1)*(d + e*x)^(q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := -(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*Log[c*x^n])/(2*d* f*(q + 1)) + 1/(2*d*(q + 1))* Int[(f*x)^ m*(d + e*x^2)^(q + 1)*(a*(m + 2*q + 3) + b*n + b*(m + 2*q + 3)*Log[c*x^n]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && ILtQ[m, 0]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := d^IntPart[q]*(d + e*x^2)^FracPart[q]/(1 + e/d*x^2)^FracPart[q]* Int[x^m*(1 + e/d*x^2)^q*(a + b*Log[c*x^n]), x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[m/2] && IntegerQ[q - 1/2] && Not[LtQ[m + 2*q, -2] || GtQ[d, 0]]
+Int[x_^m_.*(d1_ + e1_.*x_)^q_*(d2_ + e2_.*x_)^ q_*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := (d1 + e1*x)^q*(d2 + e2*x)^q/(1 + e1*e2/(d1*d2)*x^2)^q* Int[x^m*(1 + e1*e2/(d1*d2)*x^2)^q*(a + b*Log[c*x^n]), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[m] && IntegerQ[q - 1/2]
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol] := d*Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p/x, x] + e*Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]
+Int[(d_ + e_.*x_)^q_*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol] := 1/d*Int[(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p/x, x] - e/d*Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
+Int[(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.])/x_, x_Symbol] := With[{u = IntHide[(d + e*x^r)^q/x, x]}, u*(a + b*Log[c*x^n]) - b*n*Int[Dist[1/x, u, x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[q - 1/2]
+Int[(d_ + e_.*x_^r_.)^q_*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol] := 1/d*Int[(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^p/x, x] - e/d*Int[x^(r - 1)*(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0] && ILtQ[q, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[(a + b*Log[c*x^n]), u, x] - b*n*Int[SimplifyIntegrand[u/x, x], x] /; (EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2] || InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && (IntegerQ[m] && IntegerQ[r] || IGtQ[q, 0])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := With[{u = ExpandIntegrand[(a + b*Log[c*x^n]), (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || IntegerQ[m] && IntegerQ[r])
+Int[x_^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x^(r/n))^ q*(a + b*Log[c*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, e, m, n, p, q, r}, x] && IntegerQ[q] && IntegerQ[r/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[p, 0])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^ p_., x_Symbol] := With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[ q] && (GtQ[q, 0] || IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^ p_., x_Symbol] := Unintegrable[(f*x)^m*(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x]
+Int[(f_.*x_)^m_.*u_^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := Int[(f*x)^m*ExpandToSum[u, x]^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, f, m, n, p, q}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_)^q_*(a_. + b_.*Log[c_.*x_^n_.])^ p_., x_Symbol] := (f + g*x)^(m + 1)*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^ p/((q + 1)*(e*f - d*g)) - b*n*p/((q + 1)*(e*f - d*g))* Int[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1)/x, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && NeQ[e*f - d*g, 0] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
diff --git a/IntegrationRules/3 Logarithms/3.1.5 u (a+b log(c x^n))^p.m b/IntegrationRules/3 Logarithms/3.1.5 u (a+b log(c x^n))^p.m
new file mode 100755
index 0000000..bd2779b
--- /dev/null
+++ b/IntegrationRules/3 Logarithms/3.1.5 u (a+b log(c x^n))^p.m	
@@ -0,0 +1,63 @@
+
+(* ::Subsection::Closed:: *)
+(* 3.1.5 u (a+b log(c x^n))^p *)
+Int[(A_. + B_.*Log[c_.*(d_. + e_.*x_)^n_.])/ Sqrt[a_ + b_.*Log[c_.*(d_. + e_.*x_)^n_.]], x_Symbol] := B*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]]/(b*e) + (2*A*b - B*(2*a + b*n))/(2*b)* Int[1/Sqrt[a + b*Log[c*(d + e*x)^n]], x] /; FreeQ[{a, b, c, d, e, A, B, n}, x]
+Int[x_^m_.*(d_ + e_./x_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := Int[(e + d*x)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[m, q] && IntegerQ[q]
+Int[x_^m_.*(d_ + e_.*x_^r_.)^q_.*Log[c_.*x_^n_.], x_Symbol] := With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Dist[Log[c*x^n], u, x] - n*Int[SimplifyIntegrand[u/x, x], x]] /; FreeQ[{c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && Not[EqQ[q, 1] && EqQ[m, -1]]
+Int[x_^m_.*(d_ + e_.*x_^r_.)^q_.*(a_ + b_.*Log[c_.*x_^n_.]), x_Symbol] := With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, u*(a + b*Log[c*x^n]) - b*n*Int[SimplifyIntegrand[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && Not[EqQ[q, 1] && EqQ[m, -1]]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_.)^q_*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := (f*x)^(m + 1)*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])/(d* f*(m + 1)) - b*n/(d*(m + 1))*Int[(f*x)^m*(d + e*x^r)^(q + 1), x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m + r*(q + 1) + 1, 0] && NeQ[m, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_)^q_.*(a_. + b_.*Log[c_.*x_^n_])^p_., x_Symbol] := f^m/n*Subst[Int[(d + e*x)^q*(a + b*Log[c*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && EqQ[r, n]
+Int[(f_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.])^p_./(d_ + e_.*x_^r_), x_Symbol] := f^m*Log[1 + e*x^r/d]*(a + b*Log[c*x^n])^p/(e*r) - b*f^m*n*p/(e*r)* Int[Log[1 + e*x^r/d]*(a + b*Log[c*x^n])^(p - 1)/x, x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := f^m*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^p/(e*r*(q + 1)) - b*f^m*n*p/(e*r*(q + 1))* Int[(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^(p - 1)/x, x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]
+Int[(f_*x_)^m_.*(d_ + e_.*x_^r_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := (f*x)^m/x^m*Int[x^m*(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && Not[(IntegerQ[m] || GtQ[f, 0])]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := (f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])/(e*(q + 1)) - f/(e*(q + 1))* Int[(f*x)^(m - 1)*(d + e*x)^(q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := -(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*Log[c*x^n])/(2*d* f*(q + 1)) + 1/(2*d*(q + 1))* Int[(f*x)^ m*(d + e*x^2)^(q + 1)*(a*(m + 2*q + 3) + b*n + b*(m + 2*q + 3)*Log[c*x^n]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && ILtQ[m, 0]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := d^IntPart[q]*(d + e*x^2)^FracPart[q]/(1 + e/d*x^2)^FracPart[q]* Int[x^m*(1 + e/d*x^2)^q*(a + b*Log[c*x^n]), x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[m/2] && IntegerQ[q - 1/2] && Not[LtQ[m + 2*q, -2] || GtQ[d, 0]]
+Int[x_^m_.*(d1_ + e1_.*x_)^q_*(d2_ + e2_.*x_)^ q_*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := (d1 + e1*x)^q*(d2 + e2*x)^q/(1 + e1*e2/(d1*d2)*x^2)^q* Int[x^m*(1 + e1*e2/(d1*d2)*x^2)^q*(a + b*Log[c*x^n]), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[m] && IntegerQ[q - 1/2]
+Int[(a_. + b_.*Log[c_.*x_^n_])/(x_*(d_ + e_.*x_^r_.)), x_Symbol] := 1/n*Subst[Int[(a + b*Log[c*x])/(x*(d + e*x^(r/n))), x], x, x^n] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[r/n]
+Int[(a_. + b_.*Log[c_.*x_^n_.])^p_./(x_*(d_ + e_.*x_)), x_Symbol] := 1/d*Int[(a + b*Log[c*x^n])^p/x, x] - e/d*Int[(a + b*Log[c*x^n])^p/(d + e*x), x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
+(* Int[(a_.+b_.*Log[c_.*x_^n_.])^p_./(x_*(d_+e_.*x_^r_.)),x_Symbol] := (r*Log[x]-Log[1+(e*x^r)/d])*(a+b*Log[c*x^n])^p/(d*r) - b*n*p/d*Int[Log[x]*(a+b*Log[c*x^n])^(p-1)/x,x] + b*n*p/(d*r)*Int[Log[1+(e*x^r)/d]*(a+b*Log[c*x^n])^(p-1)/x,x] /; FreeQ[{a,b,c,d,e,n,r},x] && IGtQ[p,0] *)
+Int[(a_. + b_.*Log[c_.*x_^n_.])^p_./(x_*(d_ + e_.*x_^r_.)), x_Symbol] := -Log[1 + d/(e*x^r)]*(a + b*Log[c*x^n])^p/(d*r) + b*n*p/(d*r)* Int[Log[1 + d/(e*x^r)]*(a + b*Log[c*x^n])^(p - 1)/x, x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol] := d*Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p/x, x] + e*Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]
+Int[(d_ + e_.*x_)^q_*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol] := 1/d*Int[(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p/x, x] - e/d*Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
+Int[(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.])/x_, x_Symbol] := With[{u = IntHide[(d + e*x^r)^q/x, x]}, u*(a + b*Log[c*x^n]) - b*n*Int[Dist[1/x, u, x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[q - 1/2]
+Int[(d_ + e_.*x_^r_.)^q_*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol] := 1/d*Int[(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^p/x, x] - e/d*Int[x^(r - 1)*(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0] && ILtQ[q, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[(a + b*Log[c*x^n]), u, x] - b*n*Int[SimplifyIntegrand[u/x, x], x] /; (EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2] || InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && (IntegerQ[m] && IntegerQ[r] || IGtQ[q, 0])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := With[{u = ExpandIntegrand[(a + b*Log[c*x^n]), (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || IntegerQ[m] && IntegerQ[r])
+Int[x_^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x^(r/n))^ q*(a + b*Log[c*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, e, m, n, p, q, r}, x] && IntegerQ[q] && IntegerQ[r/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[p, 0])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^ p_., x_Symbol] := With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[ q] && (GtQ[q, 0] || IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^ p_., x_Symbol] := Unintegrable[(f*x)^m*(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x]
+Int[(f_.*x_)^m_.*u_^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := Int[(f*x)^m*ExpandToSum[u, x]^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, f, m, n, p, q}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[Polyx_*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := Int[ExpandIntegrand[Polyx*(a + b*Log[c*x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]
+Int[RFx_*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
+Int[RFx_*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := With[{u = ExpandIntegrand[RFx*(a + b*Log[c*x^n])^p, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
+Int[AFx_*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := Unintegrable[AFx*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p}, x] && AlgebraicFunctionQ[AFx, x, True]
+Int[(a_. + b_.*Log[c_.*x_^n_.])^p_.*(d_ + e_.*Log[c_.*x_^n_.])^q_., x_Symbol] := Int[ExpandIntegrand[(a + b*Log[c*x^n])^p*(d + e*Log[c*x^n])^q, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[p] && IntegerQ[q]
+Int[(a_. + b_.*Log[c_.*x_^n_.])^p_.*(d_. + e_.*Log[f_.*x_^r_.]), x_Symbol] := With[{u = IntHide[(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - e*r*Int[SimplifyIntegrand[u/x, x], x]] /; FreeQ[{a, b, c, d, e, f, n, p, r}, x]
+Int[(a_. + b_.*Log[c_.*x_^n_.])^p_.*(d_. + e_.*Log[f_.*x_^r_.])^q_., x_Symbol] := x*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^q - e*q*r*Int[(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^(q - 1), x] - b*n*p*Int[(a + b*Log[c*x^n])^(p - 1)*(d + e*Log[f*x^r])^q, x] /; FreeQ[{a, b, c, d, e, f, n, r}, x] && IGtQ[p, 0] && IGtQ[q, 0]
+Int[(a_. + b_.*Log[c_.*x_^n_.])^p_.*(d_. + e_.*Log[f_.*x_^r_.])^q_., x_Symbol] := Unintegrable[(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^q, x] /; FreeQ[{a, b, c, d, e, f, n, p, q, r}, x]
+Int[(a_. + b_.*Log[v_])^p_.*(c_. + d_.*Log[v_])^q_., x_Symbol] := 1/Coeff[v, x, 1]* Subst[Int[(a + b*Log[x])^p*(c + d*Log[x])^q, x], x, v] /; FreeQ[{a, b, c, d, p, q}, x] && LinearQ[v, x] && NeQ[Coeff[v, x, 0], 0]
+Int[(a_. + b_.*Log[c_.*x_^n_.])^p_.*(d_. + e_.*Log[c_.*x_^n_.])^q_./ x_, x_Symbol] := 1/n*Subst[Int[(a + b*x)^p*(d + e*x)^q, x], x, Log[c*x^n]] /; FreeQ[{a, b, c, d, e, n, p, q}, x]
+Int[(g_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.])^ p_.*(d_. + e_.*Log[f_.*x_^r_.]), x_Symbol] := With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[(d + e*Log[f*x^r]), u, x] - e*r*Int[SimplifyIntegrand[u/x, x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] && Not[EqQ[p, 1] && EqQ[a, 0] && NeQ[d, 0]]
+Int[(g_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.])^ p_.*(d_. + e_.*Log[f_.*x_^r_.])^q_., x_Symbol] := (g*x)^(m + 1)*(a + b*Log[c*x^n])^ p*(d + e*Log[f*x^r])^q/(g*(m + 1)) - e*q*r/(m + 1)* Int[(g*x)^m*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^(q - 1), x] - b*n*p/(m + 1)* Int[(g*x)^m*(a + b*Log[c*x^n])^(p - 1)*(d + e*Log[f*x^r])^q, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, r}, x] && IGtQ[p, 0] && IGtQ[q, 0] && NeQ[m, -1]
+Int[(g_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.])^ p_.*(d_. + e_.*Log[f_.*x_^r_.])^q_., x_Symbol] := Unintegrable[(g*x)^m*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^q, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r}, x]
+Int[u_^m_.*(a_. + b_.*Log[v_])^p_.*(c_. + d_.*Log[v_])^q_., x_Symbol] := With[{e = Coeff[u, x, 0], f = Coeff[u, x, 1], g = Coeff[v, x, 0], h = Coeff[v, x, 1]}, 1/h* Subst[Int[(f*x/h)^m*(a + b*Log[x])^p*(c + d*Log[x])^q, x], x, v] /; EqQ[f*g - e*h, 0] && NeQ[g, 0]] /; FreeQ[{a, b, c, d, m, p, q}, x] && LinearQ[{u, v}, x]
+Int[Log[d_.*(e_ + f_.*x_^m_.)^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := With[{u = IntHide[Log[d*(e + f*x^m)^r], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - b*n*p*Int[Dist[(a + b*Log[c*x^n])^(p - 1)/x, u, x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && RationalQ[ m] && (EqQ[p, 1] || FractionQ[m] && IntegerQ[1/m] || EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 1])
+Int[Log[d_.*(e_ + f_.*x_^m_.)^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := With[{u = IntHide[(a + b*Log[c*x^n])^p, x]}, Dist[Log[d*(e + f*x^m)^r], u, x] - f*m*r*Int[Dist[x^(m - 1)/(e + f*x^m), u, x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && IntegerQ[m]
+Int[Log[d_.*(e_ + f_.*x_^m_.)^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := Unintegrable[Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, f, r, m, n, p}, x]
+Int[Log[d_.*u_^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := Int[Log[d*ExpandToSum[u, x]^r]*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, r, n, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[Log[d_.*(e_ + f_.*x_^m_.)]*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol] := -PolyLog[2, -d*f*x^m]*(a + b*Log[c*x^n])^p/m + b*n*p/m* Int[PolyLog[2, -d*f*x^m]*(a + b*Log[c*x^n])^(p - 1)/x, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]
+Int[Log[d_.*(e_ + f_.*x_^m_.)^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol] := Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)) - f*m*r/(b*n*(p + 1))* Int[x^(m - 1)*(a + b*Log[c*x^n])^(p + 1)/(e + f*x^m), x] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && NeQ[d*e, 1]
+Int[(g_.*x_)^q_.* Log[d_.*(e_ + f_.*x_^m_.)^r_.]*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[(a + b*Log[c*x^n]), u, x] - b*n*Int[Dist[1/x, u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || RationalQ[m] && RationalQ[q]) && NeQ[q, -1]
+Int[(g_.*x_)^q_.* Log[d_.*(e_ + f_.*x_^m_.)]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - b*n*p*Int[Dist[(a + b*Log[c*x^n])^(p - 1)/x, u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ[p, 1] || FractionQ[m] && IntegerQ[(q + 1)/m] || IGtQ[q, 0] && IntegerQ[(q + 1)/m] && EqQ[d*e, 1])
+Int[(g_.*x_)^q_.* Log[d_.*(e_ + f_.*x_^m_.)^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := With[{u = IntHide[(g*x)^q*(a + b*Log[c*x^n])^p, x]}, Dist[Log[d*(e + f*x^m)^r], u, x] - f*m*r*Int[Dist[x^(m - 1)/(e + f*x^m), u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q]
+Int[(g_.*x_)^q_.* Log[d_.*(e_ + f_.*x_^m_.)^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := Unintegrable[(g*x)^q*Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, p, q}, x]
+Int[(g_.*x_)^q_.*Log[d_.*u_^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := Int[(g*x)^q*Log[d*ExpandToSum[u, x]^r]*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, g, r, n, p, q}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[PolyLog[k_, e_.*x_^q_.]*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := -b*n*x*PolyLog[k, e*x^q] + x*PolyLog[k, e*x^q]*(a + b*Log[c*x^n]) + b*n*q*Int[PolyLog[k - 1, e*x^q], x] - q*Int[PolyLog[k - 1, e*x^q]*(a + b*Log[c*x^n]), x] /; FreeQ[{a, b, c, e, n, q}, x] && IGtQ[k, 0]
+Int[PolyLog[k_, e_.*x_^q_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := Unintegrable[PolyLog[k, e*x^q]*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, e, n, p, q}, x]
+Int[PolyLog[k_, e_.*x_^q_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol] := PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^p/q - b*n*p/q*Int[PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(p - 1)/x, x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
+Int[PolyLog[k_, e_.*x_^q_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol] := PolyLog[k, e*x^q]*(a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)) - q/(b*n*(p + 1))* Int[PolyLog[k - 1, e*x^q]*(a + b*Log[c*x^n])^(p + 1)/x, x] /; FreeQ[{a, b, c, e, k, n, q}, x] && LtQ[p, -1]
+Int[(d_.*x_)^m_.*PolyLog[k_, e_.*x_^q_.]*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := -b*n*(d*x)^(m + 1)*PolyLog[k, e*x^q]/(d*(m + 1)^2) + (d*x)^(m + 1)* PolyLog[k, e*x^q]*(a + b*Log[c*x^n])/(d*(m + 1)) + b*n*q/(m + 1)^2*Int[(d*x)^m*PolyLog[k - 1, e*x^q], x] - q/(m + 1)* Int[(d*x)^m*PolyLog[k - 1, e*x^q]*(a + b*Log[c*x^n]), x] /; FreeQ[{a, b, c, d, e, m, n, q}, x] && IGtQ[k, 0]
+Int[(d_.*x_)^m_.* PolyLog[k_, e_.*x_^q_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol] := Unintegrable[(d*x)^m*PolyLog[k, e*x^q]*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x]
+Int[Px_.*F_[d_.*(e_. + f_.*x_)]^m_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := With[{u = IntHide[Px*F[d*(e + f*x)]^m, x]}, Dist[(a + b*Log[c*x^n]), u, x] - b*n*Int[Dist[1/x, u, x], x]] /; FreeQ[{a, b, c, d, e, f, n}, x] && PolynomialQ[Px, x] && IGtQ[m, 0] && MemberQ[{ArcSin, ArcCos, ArcSinh, ArcCosh}, F]
+Int[Px_.*F_[d_.*(e_. + f_.*x_)]*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol] := With[{u = IntHide[Px*F[d*(e + f*x)], x]}, Dist[(a + b*Log[c*x^n]), u, x] - b*n*Int[Dist[1/x, u, x], x]] /; FreeQ[{a, b, c, d, e, f, n}, x] && PolynomialQ[Px, x] && MemberQ[{ArcTan, ArcCot, ArcTanh, ArcCoth}, F]
diff --git a/IntegrationRules/3 Logarithms/3.2.1 (f+g x)^m (A+B log(e ((a+b x) over (c+d x))^n))^p.m b/IntegrationRules/3 Logarithms/3.2.1 (f+g x)^m (A+B log(e ((a+b x) over (c+d x))^n))^p.m
new file mode 100755
index 0000000..b307d85
--- /dev/null
+++ b/IntegrationRules/3 Logarithms/3.2.1 (f+g x)^m (A+B log(e ((a+b x) over (c+d x))^n))^p.m	
@@ -0,0 +1,28 @@
+
+(* ::Subsection::Closed:: *)
+(* 3.2.1 (f+g x)^m (A+B log(e ((a+b x) over (c+d x))^n))^p *)
+Int[(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])^p_., x_Symbol] := (a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^p/b - B*n*p*(b*c - a*d)/b* Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^(p - 1)/(c + d*x), x] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0]
+Int[(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])^p_., x_Symbol] := (a + b*x)*(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^p/b - B*n*p*(b*c - a*d)/b* Int[(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^(p - 1)/(c + d*x), x] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && IGtQ[p, 0]
+(* Int[(A_.+B_.*Log[e_.*((a_.+b_.*x_)^n1_.*(c_.+d_.*x_)^n2_)^n_.])^p_. ,x_Symbol] := (a+b*x)*(A+B*Log[e*((a+b*x)^n1/(c+d*x)^n1)^n])^p/b - B*n*p*(b*c-a*d)/b*Int[(A+B*Log[e*((a+b*x)^n1/(c+d*x)^n1)^n])^(p-1)/( c+d*x),x] /; FreeQ[{a,b,c,d,e,A,B,n},x] && EqQ[n1+n2,0] && GtQ[n1,0] && (EqQ[n1,1]  || EqQ[n,1]) && NeQ[b*c-a*d,0] && IGtQ[p,0] *)
+Int[(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])^p_, x_Symbol] := Unintegrable[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^p, x] /; FreeQ[{a, b, c, d, e, A, B, n, p}, x]
+Int[(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])^p_, x_Symbol] := Unintegrable[(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, A, B, n, p}, x] && EqQ[n + mn, 0]
+Int[(A_. + B_.*Log[e_.*(u_/v_)^n_.])^p_., x_Symbol] := Int[(A + B*Log[e*(ExpandToSum[u, x]/ExpandToSum[v, x])^n])^p, x] /; FreeQ[{e, A, B, n, p}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
+Int[(A_. + B_.*Log[e_.*u_^n_.*v_^mn_])^p_., x_Symbol] := Int[(A + B*Log[e*ExpandToSum[u, x]^n/ExpandToSum[v, x]^n])^p, x] /; FreeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
+Int[(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])/(f_. + g_.*x_), x_Symbol] := -Log[-(b*c - a*d)/(d*(a + b*x))]*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/g + B*n*(b*c - a*d)/g* Int[Log[-(b*c - a*d)/(d*(a + b*x))]/((a + b*x)*(c + d*x)), x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0]
+Int[(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])/(f_. + g_.*x_), x_Symbol] := -Log[-(b*c - a*d)/(d*(a + b*x))]*(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/g + B*n*(b*c - a*d)/g* Int[Log[-(b*c - a*d)/(d*(a + b*x))]/((a + b*x)*(c + d*x)), x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0]
+Int[(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])/(f_. + g_.*x_), x_Symbol] := -Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/g + B*n*(b*c - a*d)/g* Int[Log[(b*c - a*d)/(b*(c + d*x))]/((a + b*x)*(c + d*x)), x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[d*f - c*g, 0]
+Int[(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])/(f_. + g_.*x_), x_Symbol] := -Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/g + B*n*(b*c - a*d)/g* Int[Log[(b*c - a*d)/(b*(c + d*x))]/((a + b*x)*(c + d*x)), x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && EqQ[d*f - c*g, 0]
+Int[(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])/(f_. + g_.*x_), x_Symbol] := Log[f + g*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/g - b*B*n/g*Int[Log[f + g*x]/(a + b*x), x] + B*d*n/g*Int[Log[f + g*x]/(c + d*x), x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && NeQ[b*c - a*d, 0]
+Int[(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])/(f_. + g_.*x_), x_Symbol] := Log[f + g*x]*(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/g - b*B*n/g*Int[Log[f + g*x]/(a + b*x), x] + B*d*n/g*Int[Log[f + g*x]/(c + d*x), x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0]
+Int[(f_. + g_.*x_)^ m_.*(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.]), x_Symbol] := (f + g*x)^(m + 1)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(g*(m + 1)) - B*n*(b*c - a*d)/(g*(m + 1))* Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x] /; FreeQ[{a, b, c, d, e, f, g, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, -2]
+Int[(f_. + g_.*x_)^ m_.*(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_]), x_Symbol] := (f + g*x)^(m + 1)*(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/(g*(m + 1)) - B*n*(b*c - a*d)/(g*(m + 1))* Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x] /; FreeQ[{a, b, c, d, e, f, g, A, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && Not[EqQ[m, -2] && IntegerQ[n]]
+Int[(f_. + g_.*x_)^ m_.*(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])^p_., x_Symbol] := (b*c - a*d)^(m + 1)*(g/b)^m* Subst[Int[x^m*(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x], x, (a + b*x)/(c + d*x)] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
+Int[(f_. + g_.*x_)^ m_.*(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])^ p_., x_Symbol] := (b*c - a*d)^(m + 1)*(g/b)^m* Subst[Int[x^m*(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x], x, (a + b*x)/(c + d*x)] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
+Int[(f_. + g_.*x_)^ m_.*(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])^p_., x_Symbol] := (b*c - a*d)^(m + 1)*(g/d)^m* Subst[Int[(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x], x, (a + b*x)/(c + d*x)] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
+Int[(f_. + g_.*x_)^ m_.*(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])^ p_., x_Symbol] := (b*c - a*d)^(m + 1)*(g/d)^m* Subst[Int[(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x], x, (a + b*x)/(c + d*x)] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
+Int[(f_. + g_.*x_)^ m_.*(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])^p_., x_Symbol] := (b*c - a*d)* Subst[Int[(b*f - a*g - (d*f - c*g)*x)^ m*(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x], x, (a + b*x)/(c + d*x)] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
+Int[(f_. + g_.*x_)^ m_.*(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])^ p_., x_Symbol] := (b*c - a*d)* Subst[Int[(b*f - a*g - (d*f - c*g)*x)^ m*(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x], x, (a + b*x)/(c + d*x)] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
+Int[(f_. + g_.*x_)^ m_.*(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])^p_., x_Symbol] := Unintegrable[(f + g*x)^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, A, B, m, n, p}, x]
+Int[(f_. + g_.*x_)^ m_.*(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])^ p_., x_Symbol] := Unintegrable[(f + g*x)^m*(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, A, B, m, n, p}, x] && EqQ[n + mn, 0] && IntegerQ[n]
+Int[w_^m_.*(A_. + B_.*Log[e_.*(u_/v_)^n_.])^p_., x_Symbol] := Int[ExpandToSum[w, x]^ m*(A + B*Log[e*(ExpandToSum[u, x]/ExpandToSum[v, x])^n])^p, x] /; FreeQ[{e, A, B, m, n, p}, x] && LinearQ[{u, v, w}, x] && Not[LinearMatchQ[{u, v, w}, x]]
+Int[w_^m_.*(A_. + B_.*Log[e_.*u_^n_.*v_^mn_])^p_., x_Symbol] := Int[ExpandToSum[w, x]^ m*(A + B*Log[e*ExpandToSum[u, x]^n/ExpandToSum[v, x]^n])^p, x] /; FreeQ[{e, A, B, m, n, p}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && LinearQ[{u, v, w}, x] && Not[LinearMatchQ[{u, v, w}, x]]
diff --git a/IntegrationRules/3 Logarithms/3.2.2 (f+g x)^m (h+i x)^q (A+B log(e ((a+b x) over (c+d x))^n))^p.m b/IntegrationRules/3 Logarithms/3.2.2 (f+g x)^m (h+i x)^q (A+B log(e ((a+b x) over (c+d x))^n))^p.m
new file mode 100755
index 0000000..69bf785
--- /dev/null
+++ b/IntegrationRules/3 Logarithms/3.2.2 (f+g x)^m (h+i x)^q (A+B log(e ((a+b x) over (c+d x))^n))^p.m	
@@ -0,0 +1,24 @@
+
+(* ::Subsection::Closed:: *)
+(* 3.2.2 (f+g x)^m (h+i x)^q (A+B log(e ((a+b x) over (c+d x))^n))^p *)
+Int[(f_. + g_.*x_)^ m_.*(h_. + i_.*x_)*(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.]), x_Symbol] := (f + g*x)^(m + 1)*(h + i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(g*(m + 2)) + i*(b*c - a*d)/(b*d*(m + 2))* Int[(f + g*x)^m*(A - B*n + B*Log[e*((a + b*x)/(c + d*x))^n]), x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IGtQ[m, -2]
+Int[(f_. + g_.*x_)^ m_.*(h_. + i_.*x_)*(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_]), x_Symbol] := (f + g*x)^(m + 1)*(h + i*x)*(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/(g*(m + 2)) + i*(b*c - a*d)/(b*d*(m + 2))* Int[(f + g*x)^m*(A - B*n + B*Log[e*(a + b*x)^n/(c + d*x)^n]), x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, m, n}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IGtQ[m, -2]
+Int[(f_. + g_.*x_)^m_.*(h_. + i_.*x_)^ q_.*(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])^p_., x_Symbol] := (b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q* Subst[Int[x^m*(A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2), x], x, (a + b*x)/(c + d*x)] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
+Int[(f_. + g_.*x_)^m_.*(h_. + i_.*x_)^ q_.*(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])^ p_., x_Symbol] := (b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q* Subst[Int[x^m*(A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2), x], x, (a + b*x)/(c + d*x)] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
+Int[(f_. + g_.*x_)^m_.*(h_. + i_.*x_)^ q_.*(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])^p_., x_Symbol] := d^2*(g*(a + b*x)/b)^ m/(i^2*(b*c - a*d)*(i*(c + d*x)/d)^m*((a + b*x)/(c + d*x))^m)* Subst[Int[x^m*(A + B*Log[e*x^n])^p, x], x, (a + b*x)/(c + d*x)] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && EqQ[m + q + 2, 0]
+Int[(f_. + g_.*x_)^m_.*(h_. + i_.*x_)^ q_.*(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])^ p_., x_Symbol] := d^2*(g*(a + b*x)/b)^ m/(i^2*(b*c - a*d)*(i*(c + d*x)/d)^m*((a + b*x)/(c + d*x))^m)* Subst[Int[x^m*(A + B*Log[e*x^n])^p, x], x, (a + b*x)/(c + d*x)] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, m, n, p, q}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && EqQ[m + q + 2, 0]
+(* Int[(f_.+g_.*x_)^m_.*(h_.+i_.*x_)^q_.*(A_.+B_.*Log[e_.*(a_.+b_.*x_) ^n_.*(c_.+d_.*x_)^mn_])^p_.,x_Symbol] := b*d*(f+g*x)^(m+1)/(g*i*(b*c-a*d)*(h+i*x)^(m+1)*((a+b*x)/(c+d*x))^(m+ 1))* Subst[Int[x^m*(A+B*Log[e*x^n])^p,x],x,(a+b*x)/(c+d*x)] /; FreeQ[{a,b,c,d,e,f,g,h,i,A,B,m,n,p,q},x] && EqQ[n+mn,0] &&  NeQ[b*c-a*d,0] && EqQ[b*f-a*g,0] && EqQ[d*h-c*i,0] && EqQ[m+q+2,0] *)
+Int[(f_. + g_.*x_)^m_.*(h_. + i_.*x_)^ q_.*(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])^p_., x_Symbol] := (b*c - a*d)^(q + 1)*(i/d)^q* Subst[Int[(b*f - a*g - (d*f - c*g)*x)^ m*(A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2), x], x, (a + b*x)/(c + d*x)] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n}, x] && NeQ[b*c - a*d, 0] && IntegersQ[m, q] && IGtQ[p, 0] && EqQ[d*h - c*i, 0]
+Int[(f_. + g_.*x_)^m_.*(h_. + i_.*x_)^ q_.*(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])^ p_., x_Symbol] := (b*c - a*d)^(q + 1)*(i/d)^q* Subst[Int[(b*f - a*g - (d*f - c*g)*x)^ m*(A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2), x], x, (a + b*x)/(c + d*x)] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, q] && IGtQ[p, 0] && EqQ[d*h - c*i, 0]
+Int[(f_. + g_.*x_)^m_.*(h_. + i_.*x_)^ q_.*(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])^p_., x_Symbol] := (b*c - a*d)* Subst[Int[(b*f - a*g - (d*f - c*g)*x)^ m*(b*h - a*i - (d*h - c*i)*x)^ q*(A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2), x], x, (a + b*x)/(c + d*x)] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n}, x] && NeQ[b*c - a*d, 0] && IntegersQ[m, q] && IGtQ[p, 0]
+Int[(f_. + g_.*x_)^m_.*(h_. + i_.*x_)^ q_.*(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])^ p_., x_Symbol] := (b*c - a*d)* Subst[Int[(b*f - a*g - (d*f - c*g)*x)^ m*(b*h - a*i - (d*h - c*i)*x)^ q*(A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2), x], x, (a + b*x)/(c + d*x)] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, q] && IGtQ[p, 0]
+Int[(f_. + g_.*x_)^m_.*(h_. + i_.*x_)^ q_.*(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])^p_., x_Symbol] := Unintegrable[(f + g*x)^m*(h + i*x)^ q*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, m, n, p, q}, x]
+Int[(f_. + g_.*x_)^m_.*(h_. + i_.*x_)^ q_.*(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])^ p_., x_Symbol] := Unintegrable[(f + g*x)^m*(h + i*x)^ q*(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, m, n, p, q}, x] && EqQ[n + mn, 0] && IntegerQ[n]
+Int[w_^m_.*y_^q_.*(A_. + B_.*Log[e_.*(u_/v_)^n_.])^p_., x_Symbol] := Int[ExpandToSum[w, x]^m* ExpandToSum[y, x]^ q*(A + B*Log[e*(ExpandToSum[u, x]/ExpandToSum[v, x])^n])^p, x] /; FreeQ[{e, A, B, m, n, p, q}, x] && LinearQ[{u, v, w, y}, x] && Not[LinearMatchQ[{u, v, w, y}, x]]
+Int[w_^m_.*y_^q_.*(A_. + B_.*Log[e_.*u_^n_.*v_^mn_])^p_., x_Symbol] := Int[ExpandToSum[w, x]^m* ExpandToSum[y, x]^ q*(A + B*Log[e*ExpandToSum[u, x]^n/ExpandToSum[v, x]^n])^p, x] /; FreeQ[{e, A, B, m, n, p, q}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && LinearQ[{u, v, w, y}, x] && Not[LinearMatchQ[{u, v, w, y}, x]]
+Int[w_.*(A_. + B_.*Log[e_.*u_^n_.*v_^mn_])^p_., x_Symbol] := Subst[Int[w*(A + B*Log[e*(u/v)^n])^p, x], e*(u/v)^n, e*u^n/v^n] /; FreeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] && Not[IntegerQ[n]]
+(* Int[w_.*(A_.+B_.*Log[e_.*(f_.*u_^q_.*v_^mq_)^n_.])^p_.,x_Symbol] := Subst[Int[w*(A+B*Log[e*f^n*(u/v)^(n*q)])^p,x],e*f^n*(u/v)^(n*q),e*( f*(u^q/v^q))^n] /; FreeQ[{e,f,A,B,n,p,q},x] && EqQ[q+mq,0] && LinearQ[{u,v},x] &&  Not[IntegerQ[n]] *)
+Int[(f_. + g_.*x_ + h_.*x_^2)^ m_.*(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])^p_., x_Symbol] := h^m/(b^m*d^m)* Int[(a + b*x)^m*(c + d*x)^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^ p, x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, n, p}, x] && EqQ[b*d*f - a*c*h, 0] && EqQ[b*d*g - h*(b*c + a*d), 0] && IntegerQ[m]
+Int[(f_. + g_.*x_ + h_.*x_^2)^ m_.*(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])^ p_., x_Symbol] := h^m/(b^m*d^m)* Int[(a + b*x)^m*(c + d*x)^m*(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^ p, x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, n, p}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && EqQ[b*d*f - a*c*h, 0] && EqQ[b*d*g - h*(b*c + a*d), 0] && IntegerQ[m]
+Int[P2x_^m_.*(A_. + B_.*Log[e_.*((a_. + b_.*x_)/(c_. + d_.*x_))^n_.])^ p_., x_Symbol] := With[{f = Coeff[P2x, x, 0], g = Coeff[P2x, x, 1], h = Coeff[P2x, x, 2]}, (b*c - a*d)* Subst[ Int[(b^2*f - a*b*g + a^2*h - (2*b*d*f - b*c*g - a*d*g + 2*a*c*h)* x + (d^2*f - c*d*g + c^2*h)*x^2)^m*(A + B*Log[e*x^n])^p/ (b - d*x)^(2*(m + 1)), x], x, (a + b*x)/(c + d*x)]] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && PolyQ[P2x, x, 2] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
+Int[P2x_^ m_.*(A_. + B_.*Log[e_.*(a_. + b_.*x_)^n_.*(c_. + d_.*x_)^mn_])^ p_., x_Symbol] := With[{f = Coeff[P2x, x, 0], g = Coeff[P2x, x, 1], h = Coeff[P2x, x, 2]}, (b*c - a*d)* Subst[ Int[(b^2*f - a*b*g + a^2*h - (2*b*d*f - b*c*g - a*d*g + 2*a*c*h)* x + (d^2*f - c*d*g + c^2*h)*x^2)^m*(A + B*Log[e*x^n])^p/ (b - d*x)^(2*(m + 1)), x], x, (a + b*x)/(c + d*x)]] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && PolyQ[P2x, x, 2] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
diff --git a/IntegrationRules/3 Logarithms/3.2.3 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m b/IntegrationRules/3 Logarithms/3.2.3 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m
new file mode 100755
index 0000000..1713019
--- /dev/null
+++ b/IntegrationRules/3 Logarithms/3.2.3 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m	
@@ -0,0 +1,26 @@
+
+(* ::Subsection::Closed:: *)
+(* 3.2.3 u log(e (f (a+b x)^p (c+d x)^q)^r)^s *)
+Int[u_.*Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol] := Int[u*Log[e*(b^p*f/d^p*(c + d*x)^(p + q))^r]^s, x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p]
+Int[Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol] := (a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/b - r*s*(p + q)* Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1), x] + q*r*s*(b*c - a*d)/b* Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(c + d*x), x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && NeQ[p + q, 0] && IGtQ[s, 0] && LtQ[s, 4]
+Int[Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]/(g_. + h_.*x_), x_Symbol] := Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/h - b*p*r/h*Int[Log[g + h*x]/(a + b*x), x] - d*q*r/h*Int[Log[g + h*x]/(c + d*x), x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r}, x] && NeQ[b*c - a*d, 0]
+Int[(g_. + h_.*x_)^m_.* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.], x_Symbol] := (g + h*x)^(m + 1)* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(h*(m + 1)) - b*p*r/(h*(m + 1))*Int[(g + h*x)^(m + 1)/(a + b*x), x] - d*q*r/(h*(m + 1))*Int[(g + h*x)^(m + 1)/(c + d*x), x] /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1]
+Int[Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^2/(g_. + h_.*x_), x_Symbol] := Int[(Log[(a + b*x)^(p*r)] + Log[(c + d*x)^(q*r)])^2/(g + h*x), x] + (Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - Log[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)])* (2*Int[Log[(c + d*x)^(q*r)]/(g + h*x), x] + Int[(Log[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)] + Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(g + h*x), x]) /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[b*g - a*h, 0]
+(* Int[Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^2/(g_.+h_. *x_),x_Symbol] := Int[(Log[(a+b*x)^(p*r)]+Log[(c+d*x)^(q*r)])^2/(g+h*x),x] + (Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]-Log[(a+b*x)^(p*r)]-Log[(c+d*x)^(q* r)])* Int[(Log[(a+b*x)^(p*r)]+Log[(c+d*x)^(q*r)]+Log[e*(f*(a+b*x)^p*(c+ d*x)^q)^r])/(g+h*x),x] /; FreeQ[{a,b,c,d,e,f,g,h,p,q,r},x] && NeQ[b*c-a*d,0] && EqQ[b*g-a*h,0] *)
+Int[Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^2/(g_. + h_.*x_), x_Symbol] := Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/h - 2*b*p*r/h* Int[Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(a + b*x), x] - 2*d*q*r/h* Int[Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(c + d*x), x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r}, x] && NeQ[b*c - a*d, 0]
+Int[(g_. + h_.*x_)^m_.* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_, x_Symbol] := (g + h*x)^(m + 1)* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(h*(m + 1)) - b*p*r*s/(h*(m + 1))* Int[(g + h*x)^(m + 1)* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(a + b*x), x] - d*q*r*s/(h*(m + 1))* Int[(g + h*x)^(m + 1)* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(c + d*x), x] /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && NeQ[m, -1]
+Int[(s_. + t_.*Log[i_.*(g_. + h_.*x_)^n_.])^m_.* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]/(j_. + k_.*x_), x_Symbol] := (s + t*Log[i*(g + h*x)^n])^(m + 1)* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(k*n*t*(m + 1)) - b*p*r/(k*n*t*(m + 1))* Int[(s + t*Log[i*(g + h*x)^n])^(m + 1)/(a + b*x), x] - d*q*r/(k*n*t*(m + 1))* Int[(s + t*Log[i*(g + h*x)^n])^(m + 1)/(c + d*x), x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, m, n, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[h*j - g*k, 0] && IGtQ[m, 0]
+Int[(s_. + t_.*Log[i_.*(g_. + h_.*x_)^n_.])* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]/(j_. + k_.*x_), x_Symbol] := (Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - Log[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)])* Int[(s + t*Log[i*(g + h*x)^n])/(j + k*x), x] + Int[(Log[(a + b*x)^(p*r)]*(s + t*Log[i*(g + h*x)^n]))/(j + k*x), x] + Int[(Log[(c + d*x)^(q*r)]*(s + t*Log[i*(g + h*x)^n]))/(j + k*x), x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, n, p, q, r}, x] && NeQ[b*c - a*d, 0]
+Int[(s_. + t_.*Log[i_.*(g_. + h_.*x_)^n_.])^m_.* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^ u_./(j_. + k_.*x_), x_Symbol] := Unintegrable[(s + t*Log[i*(g + h*x)^n])^m* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^u/(j + k*x), x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, m, n, p, q, r, u}, x] && NeQ[b*c - a*d, 0]
+Int[u_*Log[v_]* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol] := With[{g = Simplify[(v - 1)*(c + d*x)/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, -h*PolyLog[2, 1 - v]* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(b*c - a*d) + h*p*r*s* Int[PolyLog[2, 1 - v]* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/((a + b*x)*(c + d*x)), x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]
+Int[v_*Log[i_.*(j_.*(g_. + h_.*x_)^t_.)^u_.]* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol] := With[{k = Simplify[v*(a + b*x)*(c + d*x)]}, k*Log[i*(j*(g + h*x)^t)^u]* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(p* r*(s + 1)*(b*c - a*d)) - k*h*t*u/(p*r*(s + 1)*(b*c - a*d))* Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(g + h*x), x] /; FreeQ[k, x]] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, p, q, r, s, t, u}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[s, -1]
+Int[u_*PolyLog[n_, v_]* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol] := With[{g = Simplify[v*(c + d*x)/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, h*PolyLog[n + 1, v]* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(b*c - a*d) - h*p*r*s* Int[PolyLog[n + 1, v]* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/((a + b*x)*(c + d*x)), x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b, c, d, e, f, n, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]
+Int[(a_. + b_.*Log[c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]])^ n_./(A_. + B_.*x_ + C_.*x_^2), x_Symbol] := 2*e*g/(C*(e*f - d*g))* Subst[Int[(a + b*Log[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]] /; FreeQ[{a, b, c, d, e, f, g, A, B, C, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[B*e*g - C*(e*f + d*g), 0]
+Int[(a_. + b_.*Log[c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]])^ n_./(A_. + C_.*x_^2), x_Symbol] := g/(C*f)* Subst[Int[(a + b*Log[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]] /; FreeQ[{a, b, c, d, e, f, g, A, C, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0]
+Int[RFx_.*Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.], x_Symbol] := p*r*Int[RFx*Log[a + b*x], x] + q*r*Int[RFx*Log[c + d*x], x] - (p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])*Int[RFx, x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] && Not[ MatchQ[RFx, u_.*(a + b*x)^m_.*(c + d*x)^n_. /; IntegersQ[m, n]]]
+(* Int[RFx_*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.],x_ Symbol] := With[{u=IntHide[RFx,x]}, u*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r] - b*p*r*Int[u/(a+b*x),x] -  d*q*r*Int[u/(c+d*x),x] /; NonsumQ[u]] /; FreeQ[{a,b,c,d,e,f,p,q,r},x] && RationalFunctionQ[RFx,x] &&  NeQ[b*c-a*d,0] *)
+Int[RFx_*Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol] := With[{u = ExpandIntegrand[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x] && IGtQ[s, 0]
+Int[RFx_*Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol] := Unintegrable[RFx*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x]
+Int[u_.*Log[e_.*(f_.*v_^p_.*w_^q_.)^r_.]^s_., x_Symbol] := Int[u*Log[e*(f*ExpandToSum[v, x]^p*ExpandToSum[w, x]^q)^r]^s, x] /; FreeQ[{e, f, p, q, r, s}, x] && LinearQ[{v, w}, x] && Not[LinearMatchQ[{v, w}, x]] && AlgebraicFunctionQ[u, x]
+Int[u_.*Log[e_.*(f_.*(g_ + v_./w_))^r_.]^s_., x_Symbol] := Int[u*Log[e*(f*ExpandToSum[v + g*w, x]/ExpandToSum[w, x])^r]^s, x] /; FreeQ[{e, f, g, r, s}, x] && LinearQ[w, x] && (FreeQ[v, x] || LinearQ[v, x]) && AlgebraicFunctionQ[u, x]
+(* Int[Log[g_.*(h_.*(a_.+b_.*x_)^p_.)^q_.]*Log[i_.*(j_.*(c_.+d_.*x_)^ r_.)^s_.]/(e_+f_.*x_),x_Symbol] := 1/f*Subst[Int[Log[g*(h*Simp[-(b*e-a*f)/f+b*x/f,x]^p)^q]*Log[i*(j* Simp[-(d*e-c*f)/f+d*x/f,x]^r)^s]/x,x],x,e+f*x] /; FreeQ[{a,b,c,d,e,f,g,h,i,j,p,q,r,s},x] *)
diff --git a/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x)^n))^p.m b/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x)^n))^p.m
new file mode 100755
index 0000000..221d127
--- /dev/null
+++ b/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x)^n))^p.m	
@@ -0,0 +1,65 @@
+
+(* ::Subsection::Closed:: *)
+(* 3.3 u (a+b log(c (d+e x)^n))^p *)
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol] := 1/e*Subst[Int[(a + b*Log[c*x^n])^p, x], x, d + e*x] /; FreeQ[{a, b, c, d, e, n, p}, x]
+Int[(f_ + g_. x_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol] := 1/e*Subst[Int[(f*x/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x] && EqQ[e*f - d*g, 0]
+Int[Log[c_.*(d_ + e_.*x_^n_.)]/x_, x_Symbol] := -PolyLog[2, -c*e*x^n]/n /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)])/x_, x_Symbol] := (a + b*Log[c*d])*Log[x] + b*Int[Log[1 + e*x/d]/x, x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[c*d, 0]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)])/(f_. + g_. x_), x_Symbol] := 1/g*Subst[Int[(a + b*Log[1 + c*e*x/g])/x, x], x, f + g*x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g + c*(e*f - d*g), 0]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])/(f_. + g_. x_), x_Symbol] := Log[e*(f + g*x)/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n])/g - b*e*n/g*Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]
+Int[(f_. + g_.*x_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.]), x_Symbol] := (f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])/(g*(q + 1)) - b*e*n/(g*(q + 1))*Int[(f + g*x)^(q + 1)/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_/(f_. + g_. x_), x_Symbol] := Log[e*(f + g*x)/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n])^p/g - b*e*n*p/g* Int[Log[(e*(f + g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[p, 1]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_/(f_. + g_.*x_)^2, x_Symbol] := (d + e*x)*(a + b*Log[c*(d + e*x)^n])^p/((e*f - d*g)*(f + g*x)) - b*e*n*p/(e*f - d*g)* Int[(a + b*Log[c*(d + e*x)^n])^(p - 1)/(f + g*x), x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0]
+Int[(f_. + g_.*x_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_, x_Symbol] := (f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p/(g*(q + 1)) - b*e*n*p/(g*(q + 1))* Int[(f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && (Not[IGtQ[q, 0]] || EqQ[p, 2] && NeQ[q, 1])
+Int[(f_. + g_.*x_)^q_./(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.]), x_Symbol] := Int[ExpandIntegrand[(f + g*x)^q/(a + b*Log[c*(d + e*x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && IGtQ[q, 0]
+Int[(f_. + g_.*x_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_, x_Symbol] := (d + e*x)*(f + g*x)^ q*(a + b*Log[c*(d + e*x)^n])^(p + 1)/(b*e*n*(p + 1)) + q*(e*f - d*g)/(b*e*n*(p + 1))* Int[(f + g*x)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1), x] - (q + 1)/(b*n*(p + 1))* Int[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && LtQ[p, -1] && GtQ[q, 0]
+Int[(f_. + g_.*x_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_, x_Symbol] := Int[ExpandIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[q, 0]
+Int[Log[c_./(d_ + e_.*x_)]/(f_ + g_.*x_^2), x_Symbol] := -e/g*Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
+Int[(a_. + b_.*Log[c_./(d_ + e_.*x_)])/(f_ + g_.*x_^2), x_Symbol] := (a + b*Log[c/(2*d)])*Int[1/(f + g*x^2), x] + b*Int[Log[2*d/(d + e*x)]/(f + g*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e^2*f + d^2*g, 0] && GtQ[c/(2*d), 0]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])/Sqrt[f_ + g_.*x_^2], x_Symbol] := With[{u = IntHide[1/Sqrt[f + g*x^2], x]}, u*(a + b*Log[c*(d + e*x)^n]) - b*e*n*Int[SimplifyIntegrand[u/(d + e*x), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && GtQ[f, 0]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])/(Sqrt[f1_ + g1_.*x_]* Sqrt[f2_ + g2_.*x_]), x_Symbol] := With[{u = IntHide[1/Sqrt[f1*f2 + g1*g2*x^2], x]}, u*(a + b*Log[c*(d + e*x)^n]) - b*e*n*Int[SimplifyIntegrand[u/(d + e*x), x], x]] /; FreeQ[{a, b, c, d, e, f1, g1, f2, g2, n}, x] && EqQ[f2*g1 + f1*g2, 0] && GtQ[f1, 0] && GtQ[f2, 0]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])/Sqrt[f_ + g_.*x_^2], x_Symbol] := Sqrt[1 + g/f*x^2]/Sqrt[f + g*x^2]* Int[(a + b*Log[c*(d + e*x)^n])/Sqrt[1 + g/f*x^2], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && Not[GtQ[f, 0]]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])/(Sqrt[f1_ + g1_.*x_]* Sqrt[f2_ + g2_.*x_]), x_Symbol] := Sqrt[1 + g1*g2/(f1*f2)*x^2]/(Sqrt[f1 + g1*x]*Sqrt[f2 + g2*x])* Int[(a + b*Log[c*(d + e*x)^n])/Sqrt[1 + g1*g2/(f1*f2)*x^2], x] /; FreeQ[{a, b, c, d, e, f1, g1, f2, g2, n}, x] && EqQ[f2*g1 + f1*g2, 0]
+Int[(f_. + g_.*x_^r_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol] := With[{k = Denominator[r]}, k*Subst[ Int[x^(k - 1)*(f + g*x^(k*r))^q*(a + b*Log[c*(d + e*x^k)^n])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x] && FractionQ[r] && IGtQ[p, 0]
+Int[(f_ + g_.*x_^r_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || IntegerQ[r] && NeQ[r, 1])
+Int[x_^m_.*Log[c_.*(d_ + e_.*x_)]/(f_ + g_. x_), x_Symbol] := Int[ExpandIntegrand[Log[c*(d + e*x)], x^m/(f + g*x), x], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[e*f - d*g, 0] && EqQ[c*d, 1] && IntegerQ[m]
+Int[(f_. + g_. x_)^q_.*(h_. + i_. x_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol] := 1/e*Subst[ Int[(g*x/e)^q*((e*h - d*i)/e + i*x/e)^r*(a + b*Log[c*x^n])^p, x], x, d + e*x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
+Int[x_^m_.*(f_ + g_./x_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^ p_., x_Symbol] := Int[(g + f*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q}, x] && EqQ[m, q] && IntegerQ[q]
+Int[x_^m_.*(f_. + g_.*x_^r_.)^ q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol] := (f + g*x^r)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p/(g*r*(q + 1)) - b*e*n*p/(g*r*(q + 1))* Int[(f + g*x^r)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f, g, m, n, q, r}, x] && EqQ[m, r - 1] && NeQ[q, -1] && IGtQ[p, 0]
+Int[x_^m_.*(f_ + g_.*x_^r_.)^ q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.]), x_Symbol] := With[{u = IntHide[x^m*(f + g*x^r)^q, x]}, Dist[(a + b*Log[c*(d + e*x)^n]), u, x] - b*e*n*Int[SimplifyIntegrand[u/(d + e*x), x], x] /; InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q, r}, x] && IntegerQ[m] && IntegerQ[q] && IntegerQ[r]
+Int[x_^m_.*(f_. + g_.*x_^r_)^ q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol] := With[{k = Denominator[r]}, k*Subst[ Int[x^(k*(m + 1) - 1)*(f + g*x^(k*r))^ q*(a + b*Log[c*(d + e*x^k)^n])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x] && FractionQ[r] && IGtQ[p, 0] && IntegerQ[m]
+Int[(h_.*x_)^m_.*(f_ + g_.*x_^r_.)^ q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^ p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
+Int[Polyx_*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol] := Int[ExpandIntegrand[Polyx*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && PolynomialQ[Polyx, x]
+Int[RFx_*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol] := With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[RFx, x] && IntegerQ[p]
+Int[RFx_*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol] := With[{u = ExpandIntegrand[RFx*(a + b*Log[c*(d + e*x)^n])^p, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[RFx, x] && IntegerQ[p]
+Int[AFx_*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol] := Unintegrable[AFx*(a + b*Log[c*(d + e*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x] && AlgebraicFunctionQ[AFx, x, True]
+Int[u_^q_.*(a_. + b_.*Log[c_.*v_^n_.])^p_., x_Symbol] := Int[ExpandToSum[u, x]^q*(a + b*Log[c*ExpandToSum[v, x]^n])^p, x] /; FreeQ[{a, b, c, n, p, q}, x] && BinomialQ[u, x] && LinearQ[v, x] && Not[BinomialMatchQ[u, x] && LinearMatchQ[v, x]]
+Int[Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.]), x_Symbol] := -x*(m - Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]) + b*e*m*n*Int[x/(d + e*x), x] - b*e*n*Int[(x*Log[f*x^m])/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
+Int[Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_, x_Symbol] := With[{u = IntHide[(a + b*Log[c*(d + e*x)^n])^p, x]}, Dist[Log[f*x^m], u, x] - m*Int[Dist[1/x, u, x], x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 1]
+Int[Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol] := Unintegrable[Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
+Int[Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])/x_, x_Symbol] := Log[f*x^m]^2*(a + b*Log[c*(d + e*x)^n])/(2*m) - b*e*n/(2*m)*Int[Log[f*x^m]^2/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
+Int[(g_.*x_)^q_.* Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.]), x_Symbol] := -1/(g*(q + 1))*(m*(g*x)^(q + 1)/(q + 1) - (g*x)^(q + 1)* Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]) + b*e*m*n/(g*(q + 1)^2)*Int[(g*x)^(q + 1)/(d + e*x), x] - b*e*n/(g*(q + 1))*Int[(g*x)^(q + 1)*Log[f*x^m]/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && NeQ[q, -1]
+Int[Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_./x_, x_Symbol] := Log[f*x^m]^2*(a + b*Log[c*(d + e*x)^n])^p/(2*m) - b*e*n*p/(2*m)* Int[Log[f*x^m]^2*(a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
+Int[(g_.*x_)^q_.* Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_, x_Symbol] := With[{u = IntHide[(g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x]}, Dist[Log[f*x^m], u, x] - m*Int[Dist[1/x, u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 1] && IGtQ[q, 0]
+(* Int[(g_.*x_)^q_.*Log[f_.*x_^m_.]*(a_.+b_.*Log[c_.*(d_+e_.*x_)^n_.]) ^p_,x_Symbol] := With[{u=IntHide[(a+b*Log[c*(d+e*x)^n])^p,x]}, Dist[(g*x)^q*Log[f*x^m],u,x] - g*m*Int[Dist[(g*x)^(q-1),u,x],x] -  g*q*Int[Dist[(g*x)^(q-1)*Log[f*x^m],u,x],x]] /; FreeQ[{a,b,c,d,e,f,g,m,n,q},x] && IGtQ[p,1] *)
+Int[(g_.*x_)^q_.* Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol] := Unintegrable[(g*x)^q*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q}, x]
+Int[Log[f_.*(g_. + h_.*x_)^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol] := 1/e*Subst[Int[Log[f*(g*x/d)^m]*(a + b*Log[c*x^n])^p, x], x, d + e*x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[e*f - d*g, 0]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])*(f_. + g_.*Log[c_.*(d_ + e_.*x_)^n_.]), x_Symbol] := x*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]) - e*n* Int[(x*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f, g, n}, x]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^ p_.*(f_. + g_.*Log[h_.*(i_. + j_.*x_)^m_.]), x_Symbol] := x*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m]) - g*j*m*Int[x*(a + b*Log[c*(d + e*x)^n])^p/(i + j*x), x] - b*e*n*p* Int[x*(a + b*Log[c*(d + e*x)^n])^(p - 1)*(f + g*Log[h*(i + j*x)^m])/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^ p_.*(f_. + g_.*Log[h_.*(i_. + j_.*x_)^m_.])^q_., x_Symbol] := Unintegrable[(a + b*Log[c*(d + e*x)^n])^ p*(f + g*Log[h*(i + j*x)^m])^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n, p}, x]
+Int[(k_. + l_.*x_)^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^ p_.*(f_. + g_.*Log[h_.*(i_. + j_.*x_)^m_.]), x_Symbol] := 1/e*Subst[ Int[(k*x/d)^r*(a + b*Log[c*x^n])^ p*(f + g*Log[h*((e*i - d*j)/e + j*x/e)^m]), x], x, d + e*x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 0]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])*(f_. + g_.*Log[c_.*(d_ + e_.*x_)^n_.])/x_, x_Symbol] := Log[x]*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]) - e*n* Int[(Log[x]*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f, g, n}, x]
+Int[x_^m_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])*(f_. + g_.*Log[c_.*(d_ + e_.*x_)^n_.]), x_Symbol] := x^(m + 1)*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n])/(m + 1) - e*n/(m + 1)* Int[(x^(m + 1)*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f, g, n, m}, x] && NeQ[m, -1]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])*(f_. + g_.*Log[h_.*(i_. + j_.*x_)^m_.])/x_, x_Symbol] := Log[x]*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]) - e*g*m*Int[Log[x]*(a + b*Log[c*(d + e*x)^n])/(d + e*x), x] - b*j*n*Int[Log[x]*(f + g*Log[h*(i + j*x)^m])/(i + j*x), x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && EqQ[e*i - d*j, 0]
+Int[Log[a_ + b_.*x_]*Log[c_ + d_.*x_]/x_, x_Symbol] := Log[-b*x/a]*Log[a + b*x]*Log[c + d*x] - 1/2*(Log[-b*x/a] - Log[-d*x/c])*(Log[a + b*x] + Log[a*(c + d*x)/(c*(a + b*x))])^2 + 1/2*(Log[-b*x/a] - Log[-(b*c - a*d)*x/(a*(c + d*x))] + Log[(b*c - a*d)/(b*(c + d*x))])* Log[a*(c + d*x)/(c*(a + b*x))]^2 + (Log[c + d*x] - Log[a*(c + d*x)/(c*(a + b*x))])* PolyLog[2, 1 + b*x/a] + (Log[a + b*x] + Log[a*(c + d*x)/(c*(a + b*x))])* PolyLog[2, 1 + d*x/c] - Log[a*(c + d*x)/(c*(a + b*x))]* PolyLog[2, d*(a + b*x)/(b*(c + d*x))] + Log[a*(c + d*x)/(c*(a + b*x))]* PolyLog[2, c*(a + b*x)/(a*(c + d*x))] - PolyLog[3, 1 + b*x/a] - PolyLog[3, 1 + d*x/c] - PolyLog[3, d*(a + b*x)/(b*(c + d*x))] + PolyLog[3, c*(a + b*x)/(a*(c + d*x))] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
+Int[Log[v_]*Log[w_]/x_, x_Symbol] := Int[Log[ExpandToSum[v, x]]*Log[ExpandToSum[w, x]]/x, x] /; LinearQ[{v, w}, x] && Not[LinearMatchQ[{v, w}, x]]
+Int[Log[c_.*(d_ + e_.*x_)^n_.]*Log[h_.*(i_. + j_.*x_)^m_.]/x_, x_Symbol] := m*Int[Log[i + j*x]*Log[c*(d + e*x)^n]/x, x] - (m*Log[i + j*x] - Log[h*(i + j*x)^m])* Int[Log[c*(d + e*x)^n]/x, x] /; FreeQ[{c, d, e, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0] && NeQ[i + j*x, h*(i + j*x)^m]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])*(f_ + g_.*Log[h_.*(i_. + j_.*x_)^m_.])/x_, x_Symbol] := f*Int[(a + b*Log[c*(d + e*x)^n])/x, x] + g*Int[Log[h*(i + j*x)^m]*(a + b*Log[c*(d + e*x)^n])/x, x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0]
+Int[x_^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^ p_.*(f_. + g_.*Log[h_.*(i_. + j_.*x_)^m_.]), x_Symbol] := x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^ p*(f + g*Log[h*(i + j*x)^m])/(r + 1) - g*j*m/(r + 1)* Int[x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^p/(i + j*x), x] - b*e*n*p/(r + 1)* Int[x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1)*(f + g*Log[h*(i + j*x)^m])/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0] && IntegerQ[r] && (EqQ[p, 1] || GtQ[r, 0]) && NeQ[r, -1]
+Int[(k_ + l_.*x_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])*(f_. + g_.*Log[h_.*(i_. + j_.*x_)^m_.]), x_Symbol] := 1/l*Subst[ Int[x^r*(a + b*Log[c*(-(e*k - d*l)/l + e*x/l)^n])*(f + g*Log[h*(-(j*k - i*l)/l + j*x/l)^m]), x], x, k + l*x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, m, n}, x] && IntegerQ[r]
+Int[(k_. + l_.*x_)^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^ p_.*(f_. + g_.*Log[h_.*(i_. + j_.*x_)^m_.])^q_., x_Symbol] := Unintegrable[(k + l*x)^r*(a + b*Log[c*(d + e*x)^n])^ p*(f + g*Log[h*(i + j*x)^m])^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, m, n, p, q, r}, x]
+Int[PolyLog[k_, h_ + i_.*x_]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^ p_./(f_ + g_.*x_), x_Symbol] := 1/g*Subst[Int[PolyLog[k, h*x/d]*(a + b*Log[c*x^n])^p/x, x], x, d + e*x] /; FreeQ[{a, b, c, d, e, f, g, h, i, k, n}, x] && EqQ[e*f - d*g, 0] && EqQ[g*h - f*i, 0] && IGtQ[p, 0]
+Int[Px_.* F_[f_.*(g_. + h_.*x_)]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.]), x_Symbol] := With[{u = IntHide[Px*F[f*(g + h*x)], x]}, Dist[(a + b*Log[c*(d + e*x)^n]), u, x] - b*e*n*Int[SimplifyIntegrand[u/(d + e*x), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h, n}, x] && PolynomialQ[Px, x] && MemberQ[{ArcSin, ArcCos, ArcTan, ArcCot, ArcSinh, ArcCosh, ArcTanh, ArcCoth}, F]
+Int[u_.*(a_. + b_.*Log[c_.*v_^n_.])^p_., x_Symbol] := Int[u*(a + b*Log[c*ExpandToSum[v, x]^n])^p, x] /; FreeQ[{a, b, c, n, p}, x] && LinearQ[v, x] && Not[LinearMatchQ[v, x]] && Not[EqQ[n, 1] && MatchQ[c*v, e_.*(f_ + g_.*x) /; FreeQ[{e, f, g}, x]]]
+Int[u_.*(a_. + b_.*Log[c_.*(d_.*(e_. + f_. x_)^m_.)^n_])^p_., x_Symbol] := Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[n]] && Not[EqQ[d, 1] && EqQ[m, 1]] && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
+Int[AFx_*(a_. + b_.*Log[c_.*(d_.*(e_. + f_. x_)^m_.)^n_])^p_., x_Symbol] := Unintegrable[AFx*(a + b*Log[c*(d*(e + f*x)^m)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && AlgebraicFunctionQ[AFx, x, True]
diff --git a/IntegrationRules/3 Logarithms/3.4 u (a+b log(c (d+e x^m)^n))^p.m b/IntegrationRules/3 Logarithms/3.4 u (a+b log(c (d+e x^m)^n))^p.m
new file mode 100755
index 0000000..a53d568
--- /dev/null
+++ b/IntegrationRules/3 Logarithms/3.4 u (a+b log(c (d+e x^m)^n))^p.m	
@@ -0,0 +1,43 @@
+
+(* ::Subsection::Closed:: *)
+(* 3.4 u (a+b log(c (d+e x^m)^n))^p *)
+Int[Pq_^m_.*Log[u_], x_Symbol] := With[{C = FullSimplify[Pq^m*(1 - u)/D[u, x]]}, C*PolyLog[2, 1 - u] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
+Int[Log[c_.*(d_ + e_.*x_^n_)^p_.], x_Symbol] := x*Log[c*(d + e*x^n)^p] - e*n*p*Int[x^n/(d + e*x^n), x] /; FreeQ[{c, d, e, n, p}, x]
+Int[(a_. + b_.*Log[c_.*(d_ + e_./x_)^p_.])^q_, x_Symbol] := (e + d*x)*(a + b*Log[c*(d + e/x)^p])^q/d + b*e*p*q/d*Int[(a + b*Log[c*(d + e/x)^p])^(q - 1)/x, x] /; FreeQ[{a, b, c, d, e, p}, x] && IGtQ[q, 0]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_, x_Symbol] := x*(a + b*Log[c*(d + e*x^n)^p])^q - b*e*n*p*q* Int[x^n*(a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && IGtQ[q, 0] && (EqQ[q, 1] || IntegerQ[n])
+(* Int[(a_.+b_.*Log[c_.*(d_+e_.*x_^n_)^p_.])^q_,x_Symbol] := With[{k=Denominator[n]}, k*Subst[Int[x^(k-1)*(a+b*Log[c*(d+e*x^(k*n))^p])^q,x],x,x^(1/k)]] /; FreeQ[{a,b,c,d,e,p,q},x] && LtQ[-1,n,1] && (GtQ[n,0] || IGtQ[q,0]) *)
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_, x_Symbol] := With[{k = Denominator[n]}, k*Subst[Int[x^(k - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)]] /; FreeQ[{a, b, c, d, e, p, q}, x] && FractionQ[n]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_, x_Symbol] := Unintegrable[(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, n, p, q}, x]
+Int[(a_. + b_.*Log[c_.*v_^p_.])^q_., x_Symbol] := Int[(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, p, q}, x] && BinomialQ[v, x] && Not[BinomialMatchQ[v, x]]
+Int[x_^m_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[ Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) && Not[EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0]]
+Int[(f_.*x_)^m_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.]), x_Symbol] := (f*x)^(m + 1)*(a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1)) - b*e*n*p/(f*(m + 1))*Int[x^(n - 1)*(f*x)^(m + 1)/(d + e*x^n), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]
+Int[(f_*x_)^m_*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol] := (f*x)^m/x^m*Int[x^m*(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])
+Int[(f_.*x_)^m_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_, x_Symbol] := (f*x)^(m + 1)*(a + b*Log[c*(d + e*x^n)^p])^q/(f*(m + 1)) - b*e*n*p*q/(f^n*(m + 1))* Int[(f*x)^(m + n)*(a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[q, 1] && IntegerQ[n] && NeQ[m, -1]
+Int[x_^m_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_, x_Symbol] := With[{k = Denominator[n]}, k*Subst[ Int[x^(k*(m + 1) - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)]] /; FreeQ[{a, b, c, d, e, m, p, q}, x] && FractionQ[n]
+Int[(f_*x_)^m_*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol] := (f*x)^m/x^m*Int[x^m*(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && FractionQ[n]
+Int[(f_.*x_)^m_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol] := Unintegrable[(f*x)^m*(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x]
+Int[(f_.*x_)^m_.*(a_. + b_.*Log[c_.*v_^p_.])^q_., x_Symbol] := Int[(f*x)^m*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, f, m, p, q}, x] && BinomialQ[v, x] && Not[BinomialMatchQ[v, x]]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])/(f_. + g_.*x_), x_Symbol] := Log[f + g*x]*(a + b*Log[c*(d + e*x^n)^p])/g - b*e*n*p/g*Int[x^(n - 1)*Log[f + g*x]/(d + e*x^n), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && RationalQ[n]
+Int[(f_. + g_.*x_)^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.]), x_Symbol] := (f + g*x)^(r + 1)*(a + b*Log[c*(d + e*x^n)^p])/(g*(r + 1)) - b*e*n*p/(g*(r + 1))* Int[x^(n - 1)*(f + g*x)^(r + 1)/(d + e*x^n), x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x] && (IGtQ[r, 0] || RationalQ[n]) && NeQ[r, -1]
+Int[(f_. + g_.*x_)^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol] := Unintegrable[(f + g*x)^r*(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r}, x]
+Int[u_^r_.*(a_. + b_.*Log[c_.*v_^p_.])^q_., x_Symbol] := Int[ExpandToSum[u, x]^r*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, p, q, r}, x] && LinearQ[u, x] && BinomialQ[v, x] && Not[LinearMatchQ[u, x] && BinomialMatchQ[v, x]]
+Int[x_^m_.*(f_. + g_.*x_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol] := Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x)^r, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x] && IntegerQ[m] && IntegerQ[r]
+Int[(h_.*x_)^m_*(f_. + g_.*x_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_.)^p_.])^q_., x_Symbol] := With[{k = Denominator[m]}, k/h* Subst[Int[ x^(k*(m + 1) - 1)*(f + g*x^k/h)^ r*(a + b*Log[c*(d + e*x^(k*n)/h^n)^p])^q, x], x, (h*x)^(1/k)]] /; FreeQ[{a, b, c, d, e, f, g, h, p, r}, x] && FractionQ[m] && IntegerQ[n] && IntegerQ[r]
+Int[(h_.*x_)^m_.*(f_. + g_.*x_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol] := Unintegrable[(h*x)^m*(f + g*x)^r*(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q, r}, x]
+Int[(h_.*x_)^m_.*u_^r_.*(a_. + b_.*Log[c_.*v_^p_.])^q_., x_Symbol] := Int[(h*x)^m* ExpandToSum[u, x]^r*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, h, m, p, q, r}, x] && LinearQ[u, x] && BinomialQ[v, x] && Not[LinearMatchQ[u, x] && BinomialMatchQ[v, x]]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])/(f_ + g_.*x_^2), x_Symbol] := With[{u = IntHide[1/(f + g*x^2), x]}, u*(a + b*Log[c*(d + e*x^n)^p]) - b*e*n*p*Int[u*x^(n - 1)/(d + e*x^n), x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]
+Int[(f_ + g_.*x_^s_)^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^ q_., x_Symbol] := With[{t = ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[q, 1] || GtQ[r, 0] && GtQ[s, 1] || LtQ[s, 0] && LtQ[r, 0])
+Int[(f_ + g_.*x_^s_)^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^ q_., x_Symbol] := With[{k = Denominator[n]}, k*Subst[ Int[x^(k - 1)*(f + g*x^(k*s))^ r*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)] /; IntegerQ[k*s]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && FractionQ[n]
+Int[(f_ + g_.*x_^s_)^r_. (a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^ q_., x_Symbol] := Unintegrable[(f + g*x^s)^r*(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x]
+Int[u_^r_.*(a_. + b_.*Log[c_.*v_^p_.])^q_., x_Symbol] := Int[ExpandToSum[u, x]^r*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, p, q, r}, x] && BinomialQ[{u, v}, x] && Not[BinomialMatchQ[{u, v}, x]]
+Int[x_^m_.*(f_ + g_.*x_^s_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^ r*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])
+Int[x_^m_.*(f_ + g_.*x_^s_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol] := Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]
+Int[(f_ + g_.*x_^s_)^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^ q_., x_Symbol] := With[{k = Denominator[n]}, k*Subst[ Int[x^(k - 1)*(f + g*x^(k*s))^ r*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)] /; IntegerQ[k*s]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && FractionQ[n]
+Int[x_^m_.*(f_ + g_.*x_^s_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol] := 1/n*Subst[ Int[x^(m + 1/n - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^ q, x], x, x^n] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && FractionQ[n] && IntegerQ[1/n] && IntegerQ[s/n]
+Int[(h_.*x_)^m_*(f_. + g_.*x_^s_.)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_.)^p_.])^q_., x_Symbol] := With[{k = Denominator[m]}, k/h* Subst[Int[ x^(k*(m + 1) - 1)*(f + g*x^(k*s)/h^s)^ r*(a + b*Log[c*(d + e*x^(k*n)/h^n)^p])^q, x], x, (h*x)^(1/k)]] /; FreeQ[{a, b, c, d, e, f, g, h, p, r}, x] && FractionQ[m] && IntegerQ[n] && IntegerQ[s]
+Int[(h_.*x_)^m_.*(f_ + g_.*x_^s_)^ r_. (a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol] := Unintegrable[(h*x)^m*(f + g*x^s)^r*(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q, r, s}, x]
+Int[(h_.*x_)^m_.*u_^r_.*(a_. + b_.*Log[c_.*v_^p_.])^q_., x_Symbol] := Int[(h*x)^m* ExpandToSum[u, x]^r*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, h, m, p, q, r}, x] && BinomialQ[{u, v}, x] && Not[BinomialMatchQ[{u, v}, x]]
+Int[Log[f_.*x_^q_.]^m_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])/x_, x_Symbol] := Log[f*x^q]^(m + 1)*(a + b*Log[c*(d + e*x^n)^p])/(q*(m + 1)) - b*e*n*p/(q*(m + 1))* Int[x^(n - 1)*Log[f*x^q]^(m + 1)/(d + e*x^n), x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && NeQ[m, -1]
+Int[F_[f_.*x_]^m_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.]), x_Symbol] := With[{u = IntHide[F[f*x]^m, x]}, Dist[a + b*Log[c*(d + e*x^n)^p], u, x] - b*e*n*p*Int[SimplifyIntegrand[u*x^(n - 1)/(d + e*x^n), x], x]] /; FreeQ[{a, b, c, d, e, f, p}, x] && MemberQ[{ArcSin, ArcCos, ArcSinh, ArcCosh}, F] && IGtQ[m, 0] && IGtQ[n, 1]
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*(f_. + g_.*x_)^n_)^p_.])^q_., x_Symbol] := 1/g*Subst[Int[(a + b*Log[c*(d + e*x^n)^p])^q, x], x, f + g*x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IGtQ[q, 0] && (EqQ[q, 1] || IntegerQ[n])
+Int[(a_. + b_.*Log[c_.*(d_ + e_.*(f_. + g_.*x_)^n_)^p_.])^q_., x_Symbol] := Unintegrable[(a + b*Log[c*(d + e*(f + g*x)^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
diff --git a/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m b/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m
new file mode 100755
index 0000000..ca08008
--- /dev/null
+++ b/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m	
@@ -0,0 +1,48 @@
+
+(* ::Subsection::Closed:: *)
+(* 3.5 Miscellaneous logarithms *)
+Int[u_*Log[v_], x_Symbol] := With[{w = DerivativeDivides[v, u*(1 - v), x]}, w*PolyLog[2, 1 - v] /; Not[FalseQ[w]]]
+Int[(a_. + b_.*Log[u_])*Log[v_]*w_, x_Symbol] := With[{z = DerivativeDivides[v, w*(1 - v), x]}, z*(a + b*Log[u])*PolyLog[2, 1 - v] - b*Int[SimplifyIntegrand[z*PolyLog[2, 1 - v]*D[u, x]/u, x], x] /; Not[FalseQ[z]]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x]
+Int[Log[c_.*Log[d_.*x_^n_.]^p_.], x_Symbol] := x*Log[c*Log[d*x^n]^p] - n*p*Int[1/Log[d*x^n], x] /; FreeQ[{c, d, n, p}, x]
+Int[(a_. + b_.*Log[c_.*Log[d_.*x_^n_.]^p_.])/x_, x_Symbol] := Log[d*x^n]*(a + b*Log[c*Log[d*x^n]^p])/n - b*p*Log[x] /; FreeQ[{a, b, c, d, n, p}, x]
+Int[(e_.*x_)^m_.*(a_. + b_.*Log[c_.*Log[d_.*x_^n_.]^p_.]), x_Symbol] := (e*x)^(m + 1)*(a + b*Log[c*Log[d*x^n]^p])/(e*(m + 1)) - b*n*p/(m + 1)*Int[(e*x)^m/Log[d*x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[m, -1]
+Int[(a_. + b_.*Log[c_.*RFx_^p_.])^n_., x_Symbol] := x*(a + b*Log[c*RFx^p])^n - b*n*p* Int[SimplifyIntegrand[ x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x]/RFx, x], x] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
+Int[(a_. + b_.*Log[c_.*RFx_^p_.])^n_./(d_. + e_.*x_), x_Symbol] := Log[d + e*x]*(a + b*Log[c*RFx^p])^n/e - b*n*p/e* Int[Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x]/RFx, x] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*Log[c_.*RFx_^p_.])^n_., x_Symbol] := (d + e*x)^(m + 1)*(a + b*Log[c*RFx^p])^n/(e*(m + 1)) - b*n*p/(e*(m + 1))* Int[SimplifyIntegrand[(d + e*x)^(m + 1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x]/RFx, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]
+Int[Log[c_.*RFx_^n_.]/(d_ + e_.*x_^2), x_Symbol] := With[{u = IntHide[1/(d + e*x^2), x]}, u*Log[c*RFx^n] - n*Int[SimplifyIntegrand[u*D[RFx, x]/RFx, x], x]] /; FreeQ[{c, d, e, n}, x] && RationalFunctionQ[RFx, x] && Not[PolynomialQ[RFx, x]]
+Int[Log[c_.*Px_^n_.]/Qx_, x_Symbol] := With[{u = IntHide[1/Qx, x]}, u*Log[c*Px^n] - n*Int[SimplifyIntegrand[u*D[Px, x]/Px, x], x]] /; FreeQ[{c, n}, x] && QuadraticQ[{Qx, Px}, x] && EqQ[D[Px/Qx, x], 0]
+Int[RGx_*(a_. + b_.*Log[c_.*RFx_^p_.])^n_., x_Symbol] := With[{u = ExpandIntegrand[(a + b*Log[c*RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalFunctionQ[RGx, x] && IGtQ[n, 0]
+Int[RGx_*(a_. + b_.*Log[c_.*RFx_^p_.])^n_., x_Symbol] := With[{u = ExpandIntegrand[RGx*(a + b*Log[c*RFx^p])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalFunctionQ[RGx, x] && IGtQ[n, 0]
+Int[RFx_*(a_. + b_.*Log[u_]), x_Symbol] := With[{lst = SubstForFractionalPowerOfLinear[RFx*(a + b*Log[u]), x]}, lst[[2]]*lst[[4]]* Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])] /; Not[FalseQ[lst]]] /; FreeQ[{a, b}, x] && RationalFunctionQ[RFx, x]
+Int[(f_. + g_.*x_)^m_.*Log[1 + e_.*(F_^(c_.*(a_. + b_.*x_)))^n_.], x_Symbol] := -(f + g*x)^m*PolyLog[2, -e*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F]) + g*m/(b*c*n*Log[F])* Int[(f + g*x)^(m - 1)*PolyLog[2, -e*(F^(c*(a + b*x)))^n], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]
+Int[(f_. + g_.*x_)^m_.*Log[d_ + e_.*(F_^(c_.*(a_. + b_.*x_)))^n_.], x_Symbol] := (f + g*x)^(m + 1)*Log[d + e*(F^(c*(a + b*x)))^n]/(g*(m + 1)) - (f + g*x)^(m + 1)* Log[1 + e/d*(F^(c*(a + b*x)))^n]/(g*(m + 1)) + Int[(f + g*x)^m*Log[1 + e/d*(F^(c*(a + b*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && NeQ[d, 1]
+Int[Log[d_. + e_.*x_ + f_.*Sqrt[a_. + b_.*x_ + c_.*x_^2]], x_Symbol] := x*Log[d + e*x + f*Sqrt[a + b*x + c*x^2]] + f^2*(b^2 - 4*a*c)/2* Int[x/((2*d*e - b*f^2)*(a + b*x + c*x^2) - f*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e^2 - c*f^2, 0]
+Int[Log[d_. + e_.*x_ + f_.*Sqrt[a_. + c_.*x_^2]], x_Symbol] := x*Log[d + e*x + f*Sqrt[a + c*x^2]] - a*c*f^2* Int[x/(d*e*(a + c*x^2) + f*(a*e - c*d*x)*Sqrt[a + c*x^2]), x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[e^2 - c*f^2, 0]
+Int[(g_.*x_)^m_.* Log[d_. + e_.*x_ + f_.*Sqrt[a_. + b_.*x_ + c_.*x_^2]], x_Symbol] := (g*x)^(m + 1)* Log[d + e*x + f*Sqrt[a + b*x + c*x^2]]/(g*(m + 1)) + f^2*(b^2 - 4*a*c)/(2*g*(m + 1))* Int[(g*x)^(m + 1)/((2*d*e - b*f^2)*(a + b*x + c*x^2) - f*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[e^2 - c*f^2, 0] && NeQ[m, -1] && IntegerQ[2*m]
+Int[(g_.*x_)^m_.*Log[d_. + e_.*x_ + f_.*Sqrt[a_. + c_.*x_^2]], x_Symbol] := (g*x)^(m + 1)*Log[d + e*x + f*Sqrt[a + c*x^2]]/(g*(m + 1)) - a*c*f^2/(g*(m + 1))* Int[(g*x)^(m + 1)/(d*e*(a + c*x^2) + f*(a*e - c*d*x)*Sqrt[a + c*x^2]), x] /; FreeQ[{a, c, d, e, f, g, m}, x] && EqQ[e^2 - c*f^2, 0] && NeQ[m, -1] && IntegerQ[2*m]
+Int[v_.*Log[d_. + e_.*x_ + f_.*Sqrt[u_]], x_Symbol] := Int[v*Log[d + e*x + f*Sqrt[ExpandToSum[u, x]]], x] /; FreeQ[{d, e, f}, x] && QuadraticQ[u, x] && Not[QuadraticMatchQ[u, x]] && (EqQ[v, 1] || MatchQ[v, (g_.*x)^m_. /; FreeQ[{g, m}, x]])
+Int[Log[c_.*x_^n_.]^r_./(x_*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)), x_Symbol] := Log[a*x^m + b*Log[c*x^n]^q]/(b*n*q) - a*m/(b*n*q)*Int[x^(m - 1)/(a*x^m + b*Log[c*x^n]^q), x] /; FreeQ[{a, b, c, m, n, q, r}, x] && EqQ[r, q - 1]
+Int[Log[c_.*x_^n_.]^r_.*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)^p_./x_, x_Symbol] := Int[ExpandIntegrand[Log[c*x^n]^r/x, (a*x^m + b*Log[c*x^n]^q)^p, x], x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && EqQ[r, q - 1] && IGtQ[p, 0]
+Int[Log[c_.*x_^n_.]^r_.*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)^p_./x_, x_Symbol] := (a*x^m + b*Log[c*x^n]^q)^(p + 1)/(b*n*q*(p + 1)) - a*m/(b*n*q)*Int[x^(m - 1)*(a*x^m + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1]
+Int[(d_.*x_^m_. + e_.*Log[c_.*x_^n_.]^r_.)/(x_*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)), x_Symbol] := e*Log[a*x^m + b*Log[c*x^n]^q]/(b*n*q) /; FreeQ[{a, b, c, d, e, m, n, q, r}, x] && EqQ[r, q - 1] && EqQ[a*e*m - b*d*n*q, 0]
+Int[(u_ + d_.*x_^m_. + e_.*Log[c_.*x_^n_.]^r_.)/(x_*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)), x_Symbol] := e*Log[a*x^m + b*Log[c*x^n]^q]/(b*n*q) + Int[u/(x*(a*x^m + b*Log[c*x^n]^q)), x] /; FreeQ[{a, b, c, d, e, m, n, q, r}, x] && EqQ[r, q - 1] && EqQ[a*e*m - b*d*n*q, 0]
+Int[(d_.*x_^m_. + e_.*Log[c_.*x_^n_.]^r_.)/(x_*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)), x_Symbol] := e*Log[a*x^m + b*Log[c*x^n]^q]/(b*n*q) - (a*e*m - b*d*n*q)/(b*n*q)* Int[x^(m - 1)/(a*x^m + b*Log[c*x^n]^q), x] /; FreeQ[{a, b, c, d, e, m, n, q, r}, x] && EqQ[r, q - 1] && NeQ[a*e*m - b*d*n*q, 0]
+Int[(d_.*x_^m_. + e_.*Log[c_.*x_^n_.]^r_.)*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)^ p_./x_, x_Symbol] := e*(a*x^m + b*Log[c*x^n]^q)^(p + 1)/(b*n*q*(p + 1)) /; FreeQ[{a, b, c, d, e, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1] && EqQ[a*e*m - b*d*n*q, 0]
+Int[(d_.*x_^m_. + e_.*Log[c_.*x_^n_.]^r_.)*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)^ p_./x_, x_Symbol] := e*(a*x^m + b*Log[c*x^n]^q)^(p + 1)/(b*n*q*(p + 1)) - (a*e*m - b*d*n*q)/(b*n*q)* Int[x^(m - 1)*(a*x^m + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, d, e, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1] && NeQ[a*e*m - b*d*n*q, 0]
+Int[(d_.*x_^m_. + e_.*x_^m_.*Log[c_.*x_^n_.] + f_.*Log[c_.*x_^n_.]^ q_.)/(x_*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)^2), x_Symbol] := d*Log[c*x^n]/(a*n*(a*x^m + b*Log[c*x^n]^q)) /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[e*n + d*m, 0] && EqQ[a*f + b*d*(q - 1), 0]
+Int[(d_ + e_.*Log[c_.*x_^n_.])/(a_.*x_ + b_.*Log[c_.*x_^n_.]^q_)^2, x_Symbol] := -e*Log[c*x^n]/(a*(a*x + b*Log[c*x^n]^q)) + (d + e*n)/a* Int[1/(x*(a*x + b*Log[c*x^n]^q)), x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[d + e*n*q, 0]
+Int[Log[u_], x_Symbol] := x*Log[u] - Int[SimplifyIntegrand[x*D[u, x]/u, x], x] /; InverseFunctionFreeQ[u, x]
+Int[Log[u_], x_Symbol] := x*Log[u] - Int[SimplifyIntegrand[x*Simplify[D[u, x]/u], x], x] /; ProductQ[u]
+Int[Log[u_]/(a_. + b_.*x_), x_Symbol] := Log[a + b*x]*Log[u]/b - 1/b*Int[SimplifyIntegrand[Log[a + b*x]*D[u, x]/u, x], x] /; FreeQ[{a, b}, x] && RationalFunctionQ[D[u, x]/u, x] && (NeQ[a, 0] || Not[BinomialQ[u, x] && EqQ[BinomialDegree[u, x]^2, 1]])
+Int[(a_. + b_.*x_)^m_.*Log[u_], x_Symbol] := (a + b*x)^(m + 1)*Log[u]/(b*(m + 1)) - 1/(b*(m + 1))* Int[SimplifyIntegrand[(a + b*x)^(m + 1)*D[u, x]/u, x], x] /; FreeQ[{a, b, m}, x] && InverseFunctionFreeQ[u, x] && NeQ[m, -1] (* && Not[FunctionOfQ[x^(m+1),u,x]] && FalseQ[PowerVariableExpn[u,m+1, x]] *)
+Int[Log[u_]/Qx_, x_Symbol] := With[{v = IntHide[1/Qx, x]}, v*Log[u] - Int[SimplifyIntegrand[v*D[u, x]/u, x], x]] /; QuadraticQ[Qx, x] && InverseFunctionFreeQ[u, x]
+Int[u_^(a_.*x_)*Log[u_], x_Symbol] := u^(a*x)/a - Int[SimplifyIntegrand[x*u^(a*x - 1)*D[u, x], x], x] /; FreeQ[a, x] && InverseFunctionFreeQ[u, x]
+Int[v_*Log[u_], x_Symbol] := With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*D[u, x]/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]
+Int[v_*Log[u_], x_Symbol] := With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*Simplify[D[u, x]/u], x], x] /; InverseFunctionFreeQ[w, x]] /; ProductQ[u]
+Int[Log[v_]*Log[w_], x_Symbol] := x*Log[v]*Log[w] - Int[SimplifyIntegrand[x*Log[w]*D[v, x]/v, x], x] - Int[SimplifyIntegrand[x*Log[v]*D[w, x]/w, x], x] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]
+Int[u_*Log[v_]*Log[w_], x_Symbol] := With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] - Int[SimplifyIntegrand[z*Log[w]*D[v, x]/v, x], x] - Int[SimplifyIntegrand[z*Log[v]*D[w, x]/w, x], x] /; InverseFunctionFreeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]
+Int[f_^(a_.*Log[u_]), x_Symbol] := Int[u^(a*Log[f]), x] /; FreeQ[{a, f}, x]
+(* If[TrueQ[$LoadShowSteps], Int[u_/x_,x_Symbol] := With[{lst=FunctionOfLog[u,x]}, ShowStep["","Int[F[Log[a*x^n]]/x,x]","Subst[Int[F[x],x],x,Log[a*x^n] ]/n",Hold[ 1/lst[[3]]*Subst[Int[lst[[1]],x],x,Log[lst[[2]]]]]] /; Not[FalseQ[lst]]] /; SimplifyFlag && NonsumQ[u], Int[u_/x_,x_Symbol] := With[{lst=FunctionOfLog[u,x]}, 1/lst[[3]]*Subst[Int[lst[[1]],x],x,Log[lst[[2]]]] /; Not[FalseQ[lst]]] /; NonsumQ[u]] *)
+If[TrueQ[$LoadShowSteps], Int[u_, x_Symbol] := With[{lst = FunctionOfLog[Cancel[x*u], x]}, ShowStep["", "Int[F[Log[a*x^n]]/x,x]", "Subst[Int[F[x],x],x,Log[a*x^n]]/n", Hold[ 1/lst[[3]]*Subst[Int[lst[[1]], x], x, Log[lst[[2]]]]]] /; Not[FalseQ[lst]]] /; SimplifyFlag && NonsumQ[u], Int[u_, x_Symbol] := With[{lst = FunctionOfLog[Cancel[x*u], x]}, 1/lst[[3]]*Subst[Int[lst[[1]], x], x, Log[lst[[2]]]] /; Not[FalseQ[lst]]] /; NonsumQ[u]]
+Int[u_.*Log[Gamma[v_]], x_Symbol] := (Log[Gamma[v]] - LogGamma[v])*Int[u, x] + Int[u*LogGamma[v], x]
+Int[u_.*(a_.*x_^m_. + b_.*x_^r_.*Log[c_.*x_^n_.]^q_.)^p_., x_Symbol] := Int[u*x^(p*r)*(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m
new file mode 100755
index 0000000..6d3a357
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m	
@@ -0,0 +1,33 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.0.1 (a sin)^m (b trg)^n *)
+If[TrueQ[$LoadShowSteps], Int[u_, x_Symbol] := Int[DeactivateTrig[u, x], x] /; SimplifyFlag && FunctionOfTrigOfLinearQ[u, x], Int[u_, x_Symbol] := Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinearQ[u, x]]
+Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*cos[e_. + f_.*x_])^n_., x_Symbol] := (a*Sin[e + f*x])^(m + 1)*(b*Cos[e + f*x])^(n + 1)/(a*b*f*(m + 1)) /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
+Int[(a_.*sin[e_. + f_.*x_])^m_.*cos[e_. + f_.*x_]^n_., x_Symbol] := 1/(a*f)* Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && Not[IntegerQ[(m - 1)/2] && LtQ[0, m, n]]
+Int[(a_.*cos[e_. + f_.*x_])^m_.*sin[e_. + f_.*x_]^n_., x_Symbol] := -1/(a*f)* Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && Not[IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n]]
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*cos[e_. + f_.*x_])^n_, x_Symbol] := -a*(a*Sin[e + f*x])^(m - 1)*(b*Cos[e + f*x])^(n + 1)/(b* f*(n + 1)) + a^2*(m - 1)/(b^2*(n + 1))* Int[(a*Sin[e + f*x])^(m - 2)*(b*Cos[e + f*x])^(n + 2), x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
+Int[(a_.*cos[e_. + f_.*x_])^m_*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol] := a*(a*Cos[e + f*x])^(m - 1)*(b*Sin[e + f*x])^(n + 1)/(b*f*(n + 1)) + a^2*(m - 1)/(b^2*(n + 1))* Int[(a*Cos[e + f*x])^(m - 2)*(b*Sin[e + f*x])^(n + 2), x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*cos[e_. + f_.*x_])^n_, x_Symbol] := -a*(b*Cos[e + f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1)/(b* f*(m + n)) + a^2*(m - 1)/(m + n)* Int[(b*Cos[e + f*x])^n*(a*Sin[e + f*x])^(m - 2), x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
+Int[(a_.*cos[e_. + f_.*x_])^m_*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol] := a*(b*Sin[e + f*x])^(n + 1)*(a*Cos[e + f*x])^(m - 1)/(b*f*(m + n)) + a^2*(m - 1)/(m + n)* Int[(b*Sin[e + f*x])^n*(a*Cos[e + f*x])^(m - 2), x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*cos[e_. + f_.*x_])^n_, x_Symbol] := (b*Cos[e + f*x])^(n + 1)*(a*Sin[e + f*x])^(m + 1)/(a*b*f*(m + 1)) + (m + n + 2)/(a^2*(m + 1))* Int[(b*Cos[e + f*x])^n*(a*Sin[e + f*x])^(m + 2), x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
+Int[(a_.*cos[e_. + f_.*x_])^m_*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -(b*Sin[e + f*x])^(n + 1)*(a*Cos[e + f*x])^(m + 1)/(a*b* f*(m + 1)) + (m + n + 2)/(a^2*(m + 1))* Int[(b*Sin[e + f*x])^n*(a*Cos[e + f*x])^(m + 2), x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
+Int[Sqrt[a_.*sin[e_. + f_.*x_]]*Sqrt[b_.*cos[e_. + f_.*x_]], x_Symbol] := Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]* Int[Sqrt[Sin[2*e + 2*f*x]], x] /; FreeQ[{a, b, e, f}, x]
+Int[1/(Sqrt[a_.*sin[e_. + f_.*x_]]*Sqrt[b_.*cos[e_. + f_.*x_]]), x_Symbol] := Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]])* Int[1/Sqrt[Sin[2*e + 2*f*x]], x] /; FreeQ[{a, b, e, f}, x]
+(* Int[(a_.*sin[e_.+f_.*x_])^m_*(b_.*cos[e_.+f_.*x_])^n_,x_Symbol] := (a*Sin[e+f*x])^m*(b*Cos[e+f*x])^n/(a*Tan[e+f*x])^m*Int[(a*Tan[e+f*x] )^m,x] /; FreeQ[{a,b,e,f,m,n},x] && EqQ[m+n,0] *)
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*cos[e_. + f_.*x_])^n_, x_Symbol] := With[{k = Denominator[m]}, k*a*b/f* Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos[e + f*x])^(1/k)]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
+Int[(a_.*cos[e_. + f_.*x_])^m_*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol] := With[{k = Denominator[m]}, -k*a*b/f* Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Cos[e + f*x])^(1/k)/(b*Sin[e + f*x])^(1/k)]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
+(* Int[(a_.*sin[e_.+f_.*x_])^m_*(b_.*cos[e_.+f_.*x_])^n_,x_Symbol] := b^(2*IntPart[(n-1)/2]+1)*(b*Cos[e+f*x])^(2*FracPart[(n-1)/2])/(a*f*( Cos[e+f*x]^2)^FracPart[(n-1)/2])* Subst[Int[x^m*(1-x^2/a^2)^((n-1)/2),x],x,a*Sin[e+f*x]] /; FreeQ[{a,b,e,f,m,n},x] && (RationalQ[n] || Not[RationalQ[m]] &&  (EqQ[b,1] || NeQ[a,1])) *)
+(* Int[(a_.*cos[e_.+f_.*x_])^m_*(b_.*sin[e_.+f_.*x_])^n_,x_Symbol] := -b^(2*IntPart[(n-1)/2]+1)*(b*Sin[e+f*x])^(2*FracPart[(n-1)/2])/(a*f* (Sin[e+f*x]^2)^FracPart[(n-1)/2])* Subst[Int[x^m*(1-x^2/a^2)^((n-1)/2),x],x,a*Cos[e+f*x]] /; FreeQ[{a,b,e,f,m,n},x] *)
+Int[(a_.*cos[e_. + f_.*x_])^m_*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -b^(2*IntPart[(n - 1)/2] + 1)*(b*Sin[e + f*x])^(2* FracPart[(n - 1)/2])*(a*Cos[e + f*x])^(m + 1)/(a* f*(m + 1)*(Sin[e + f*x]^2)^FracPart[(n - 1)/2])* Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Cos[e + f*x]^2] /; FreeQ[{a, b, e, f, m, n}, x] && SimplerQ[n, m]
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*cos[e_. + f_.*x_])^n_, x_Symbol] := b^(2*IntPart[(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2* FracPart[(n - 1)/2])*(a*Sin[e + f*x])^(m + 1)/(a* f*(m + 1)*(Cos[e + f*x]^2)^FracPart[(n - 1)/2])* Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2] /; FreeQ[{a, b, e, f, m, n}, x]
+Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*sec[e_. + f_.*x_])^n_., x_Symbol] := b*(a*Sin[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n - 1)/(a*f*(m + 1)) /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m - n + 2, 0] && NeQ[m, -1]
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := a*b*(a*Sin[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1)/(f*(n - 1)) - a^2*b^2*(m - 1)/(n - 1)* Int[(a*Sin[e + f*x])^(m - 2)*(b*Sec[e + f*x])^(n - 2), x] /; FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && GtQ[m, 1] && IntegersQ[2*m, 2*n]
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := (a*Sin[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n + 1)/(a*b*f*(m - n)) - (n + 1)/(b^2*(m - n))* Int[(a*Sin[e + f*x])^m*(b*Sec[e + f*x])^(n + 2), x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n]
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := (a*Sin[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n + 1)/(a*b*f*(m + 1)) - (n + 1)/(a^2*b^2*(m + 1))* Int[(a*Sin[e + f*x])^(m + 2)*(b*Sec[e + f*x])^(n + 2), x] /; FreeQ[{a, b, e, f}, x] && LtQ[n, -1] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := (a*Sin[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n + 1)/(a*b*f*(m - n)) - (n + 1)/(b^2*(m - n))* Int[(a*Sin[e + f*x])^m*(b*Sec[e + f*x])^(n + 2), x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && NeQ[m - n, 0] && IntegersQ[2*m, 2*n]
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := -a*b*(a*Sin[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1)/(f*(m - n)) + a^2*(m - 1)/(m - n)* Int[(a*Sin[e + f*x])^(m - 2)*(b*Sec[e + f*x])^n, x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m - n, 0] && IntegersQ[2*m, 2*n]
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := b*(a*Sin[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n - 1)/(a*f*(m + 1)) + (m - n + 2)/(a^2*(m + 1))* Int[(a*Sin[e + f*x])^(m + 2)*(b*Sec[e + f*x])^n, x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := (b*Cos[e + f*x])^n*(b*Sec[e + f*x])^n* Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x] /; FreeQ[{a, b, e, f, m, n}, x] && IntegerQ[m - 1/2] && IntegerQ[n - 1/2]
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := 1/b^2*(b*Cos[e + f*x])^(n + 1)*(b*Sec[e + f*x])^(n + 1)* Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x] /; FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && LtQ[n, 1]
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := b^2*(b*Cos[e + f*x])^(n - 1)*(b*Sec[e + f*x])^(n - 1)* Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x] /; FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]]
+Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*csc[e_. + f_.*x_])^n_, x_Symbol] := (a*b)^IntPart[n]*(a*Sin[e + f*x])^FracPart[n]*(b*Csc[e + f*x])^ FracPart[n]*Int[(a*Sin[e + f*x])^(m - n), x] /; FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m
new file mode 100755
index 0000000..dea5cc6
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m	
@@ -0,0 +1,34 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.0.2 (a trg)^m (b tan)^n *)
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := -b*(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n - 1)/(f*m) /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 1, 0]
+Int[sin[e_. + f_.*x_]^m_.*tan[e_. + f_.*x_]^n_., x_Symbol] := -1/f*Subst[Int[(1 - x^2)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]
+Int[sin[e_. + f_.*x_]^m_*(b_.*tan[e_. + f_.*x_])^n_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, b*ff/f* Subst[Int[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*Tan[e + f*x]/ff]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]
+Int[(a_.*sin[e_. + f_.*x_])^m_.*tan[e_. + f_.*x_]^n_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*Sin[e + f*x]/ff]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b*(a*Sin[e + f*x])^(m + 2)*(b*Tan[e + f*x])^(n - 1)/(a^2* f*(n - 1)) - b^2*(m + 2)/(a^2*(n - 1))* Int[(a*Sin[e + f*x])^(m + 2)*(b*Tan[e + f*x])^(n - 2), x] /; FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && (LtQ[m, -1] || EqQ[m, -1] && EqQ[n, 3/2]) && IntegersQ[2*m, 2*n]
+Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b*(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n - 1)/(f*(n - 1)) - b^2*(m + n - 1)/(n - 1)* Int[(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n - 2), x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n] && Not[GtQ[m, 1] && Not[IntegerQ[(m - 1)/2]]]
+Int[Sqrt[a_.*sin[e_. + f_.*x_]]/(b_.*tan[e_. + f_.*x_])^(3/2), x_Symbol] := 2*Sqrt[a*Sin[e + f*x]]/(b*f*Sqrt[b*Tan[e + f*x]]) + a^2/b^2*Int[Sqrt[b*Tan[e + f*x]]/(a*Sin[e + f*x])^(3/2), x] /; FreeQ[{a, b, e, f}, x]
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := (a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n + 1)/(b*f*m) - a^2*(n + 1)/(b^2*m)* Int[(a*Sin[e + f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 2), x] /; FreeQ[{a, b, e, f}, x] && LtQ[n, -1] && GtQ[m, 1] && IntegersQ[2*m, 2*n]
+Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := (a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n + 1)/(b*f*(m + n + 1)) - (n + 1)/(b^2*(m + n + 1))* Int[(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n + 2), x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && NeQ[m + n + 1, 0] && IntegersQ[2*m, 2*n] && Not[EqQ[n, -3/2] && EqQ[m, 1]]
+Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_., x_Symbol] := -b*(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n - 1)/(f*m) + a^2*(m + n - 1)/m* Int[(a*Sin[e + f*x])^(m - 2)*(b*Tan[e + f*x])^n, x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || EqQ[m, 1] && EqQ[n, 1/2]) && IntegersQ[2*m, 2*n]
+Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_., x_Symbol] := b*(a*Sin[e + f*x])^(m + 2)*(b*Tan[e + f*x])^(n - 1)/(a^2* f*(m + n + 1)) + (m + 2)/(a^2*(m + n + 1))* Int[(a*Sin[e + f*x])^(m + 2)*(b*Tan[e + f*x])^n, x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && NeQ[m + n + 1, 0] && IntegersQ[2*m, 2*n]
+Int[(a_.*sin[e_. + f_.*x_])^m_*tan[e_. + f_.*x_]^n_, x_Symbol] := 1/a^n*Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[n] && Not[IntegerQ[m]]
+Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := Cos[e + f*x]^n*(b*Tan[e + f*x])^n/(a*Sin[e + f*x])^n* Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x] /; FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[n]] && (ILtQ[m, 0] || EqQ[m, 1] && EqQ[n, -1/2] || IntegersQ[m - 1/2, n - 1/2])
+Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := a*Cos[ e + f*x]^(n + 1)*(b*Tan[e + f*x])^(n + 1)/(b*(a*Sin[e + f*x])^(n + 1))* Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x] /; FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[n]]
+Int[(a_.*cos[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := (a*Cos[e + f*x])^FracPart[m]*(Sec[e + f*x]/a)^FracPart[m]* Int[(b*Tan[e + f*x])^n/(Sec[e + f*x]/a)^m, x] /; FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]]
+Int[(a_.*cot[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := (a*Cot[e + f*x])^m*(b*Tan[e + f*x])^m* Int[(b*Tan[e + f*x])^(n - m), x] /; FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]]
+Int[(a_.*sec[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_., x_Symbol] := -(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n + 1)/(b*f*m) /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 1, 0]
+Int[(a_.*sec[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_., x_Symbol] := a/f*Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && Not[IntegerQ[m/2] && LtQ[0, m, n + 1]]
+Int[sec[e_. + f_.*x_]^m_*(b_.*tan[e_. + f_.*x_])^n_., x_Symbol] := 1/f*Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && Not[IntegerQ[(n - 1)/2] && LtQ[0, n, m - 1]]
+Int[(a_.*sec[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := a^2*(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 1)/(b* f*(n + 1)) - a^2*(m - 2)/(b^2*(n + 1))* Int[(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 2), x] /; FreeQ[{a, b, e, f}, x] && LtQ[n, -1] && (GtQ[m, 1] || EqQ[m, 1] && EqQ[n, -3/2]) && IntegersQ[2*m, 2*n]
+Int[(a_.*sec[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := (a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n + 1)/(b*f*(n + 1)) - (m + n + 1)/(b^2*(n + 1))* Int[(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n + 2), x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && IntegersQ[2*m, 2*n]
+Int[(a_.*sec[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b*(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n - 1)/(f*m) - b^2*(n - 1)/(a^2*m)* Int[(a*Sec[e + f*x])^(m + 2)*(b*Tan[e + f*x])^(n - 2), x] /; FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && (LtQ[m, -1] || EqQ[m, -1] && EqQ[n, 3/2]) && IntegersQ[2*m, 2*n]
+Int[(a_.*sec[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b*(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1)) - b^2*(n - 1)/(m + n - 1)* Int[(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n - 2), x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
+Int[(a_.*sec[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := -(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n + 1)/(b*f*m) + (m + n + 1)/(a^2*m)* Int[(a*Sec[e + f*x])^(m + 2)*(b*Tan[e + f*x])^n, x] /; FreeQ[{a, b, e, f, n}, x] && (LtQ[m, -1] || EqQ[m, -1] && EqQ[n, -1/2]) && IntegersQ[2*m, 2*n]
+Int[(a_.*sec[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := a^2*(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 1)/(b* f*(m + n - 1)) + a^2*(m - 2)/(m + n - 1)* Int[(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^n, x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || EqQ[m, 1] && EqQ[n, 1/2]) && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
+Int[sec[e_. + f_.*x_]/Sqrt[b_.*tan[e_. + f_.*x_]], x_Symbol] := Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])* Int[1/(Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]]), x] /; FreeQ[{b, e, f}, x]
+Int[Sqrt[b_.*tan[e_. + f_.*x_]]/sec[e_. + f_.*x_], x_Symbol] := Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]/Sqrt[Sin[e + f*x]]* Int[Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]], x] /; FreeQ[{b, e, f}, x]
+Int[(a_.*sec[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := a^(m + n)*(b*Tan[e + f*x])^ n/((a*Sec[e + f*x])^n*(b*Sin[e + f*x])^n)* Int[(b*Sin[e + f*x])^n/Cos[e + f*x]^(m + n), x] /; FreeQ[{a, b, e, f, m, n}, x] && IntegerQ[n + 1/2] && IntegerQ[m + 1/2]
+(* Int[(a_.*sec[e_.+f_.*x_])^m_.*(b_.*tan[e_.+f_.*x_])^n_,x_Symbol]:= (a*Sec[e+f*x])^m*(b*Tan[e+f*x])^(n+1)*(Cos[e+f*x]^2)^((m+n+1)/2)/(b* f*(b*Sin[e+f*x])^(n+1))* Subst[Int[x^n/(1-x^2/b^2)^((m+n+1)/2),x],x,b*Sin[e+f*x]] /; FreeQ[{a,b,e,f,m,n},x] && Not[IntegerQ[(n-1)/2]] &&  Not[IntegerQ[m/2]] *)
+Int[(a_.*sec[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := (a*Sec[e + f*x])^ m*(b*Tan[e + f*x])^(n + 1)*(Cos[e + f*x]^2)^((m + n + 1)/2)/(b* f*(n + 1))* Hypergeometric2F1[(n + 1)/2, (m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2] /; FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[(n - 1)/2]] && Not[IntegerQ[m/2]]
+Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := (a*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/a)^FracPart[m]* Int[(b*Tan[e + f*x])^n/(Sin[e + f*x]/a)^m, x] /; FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.3 (a csc)^m (b sec)^n.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.3 (a csc)^m (b sec)^n.m
new file mode 100755
index 0000000..fba0e52
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.3 (a csc)^m (b sec)^n.m	
@@ -0,0 +1,17 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.0.3 (a csc)^m (b sec)^n *)
+Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := a*b*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1)/(f*(n - 1)) /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 2, 0] && NeQ[n, 1]
+Int[csc[e_. + f_.*x_]^m_.*sec[e_. + f_.*x_]^n_., x_Symbol] := 1/f*Subst[Int[(1 + x^2)^((m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
+Int[(a_.*csc[e_. + f_.*x_])^m_*sec[e_. + f_.*x_]^n_., x_Symbol] := -1/(f*a^n)* Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2] && Not[IntegerQ[(m + 1)/2] && LtQ[0, m, n]]
+Int[(a_.*sec[e_. + f_.*x_])^m_*csc[e_. + f_.*x_]^n_., x_Symbol] := 1/(f*a^n)* Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2] && Not[IntegerQ[(m + 1)/2] && LtQ[0, m, n]]
+Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := -a*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n + 1)/(f* b*(m - 1)) + a^2*(n + 1)/(b^2*(m - 1))* Int[(a*Csc[e + f*x])^(m - 2)*(b*Sec[e + f*x])^(n + 2), x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && IntegersQ[2*m, 2*n]
+Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := b*(a*Csc[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n - 1)/(f*a*(n - 1)) + b^2*(m + 1)/(a^2*(n - 1))* Int[(a*Csc[e + f*x])^(m + 2)*(b*Sec[e + f*x])^(n - 2), x] /; FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
+Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_., x_Symbol] := -a*b*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1)/(f*(m - 1)) + a^2*(m + n - 2)/(m - 1)* Int[(a*Csc[e + f*x])^(m - 2)*(b*Sec[e + f*x])^n, x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && IntegersQ[2*m, 2*n] && Not[GtQ[n, m]]
+Int[(a_.*csc[e_. + f_.*x_])^m_.*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := a*b*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1)/(f*(n - 1)) + b^2*(m + n - 2)/(n - 1)* Int[(a*Csc[e + f*x])^m*(b*Sec[e + f*x])^(n - 2), x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n]
+Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_., x_Symbol] := b*(a*Csc[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n - 1)/(a*f*(m + n)) + (m + 1)/(a^2*(m + n))* Int[(a*Csc[e + f*x])^(m + 2)*(b*Sec[e + f*x])^n, x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
+Int[(a_.*csc[e_. + f_.*x_])^m_.*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := -a*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n + 1)/(b* f*(m + n)) + (n + 1)/(b^2*(m + n))* Int[(a*Csc[e + f*x])^m*(b*Sec[e + f*x])^(n + 2), x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
+Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := (a*Csc[e + f*x])^m*(b*Sec[e + f*x])^n/Tan[e + f*x]^n* Int[Tan[e + f*x]^n, x] /; FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[n]] && EqQ[m + n, 0]
+Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := (a*Csc[e + f*x])^m*(b*Sec[e + f*x])^n*(a*Sin[e + f*x])^ m*(b*Cos[e + f*x])^n* Int[(a*Sin[e + f*x])^(-m)*(b*Cos[e + f*x])^(-n), x] /; FreeQ[{a, b, e, f, m, n}, x] && IntegerQ[m - 1/2] && IntegerQ[n - 1/2]
+Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol] := a^2/b^2*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n + 1)*(a* Sin[e + f*x])^(m - 1)*(b*Cos[e + f*x])^(n + 1)* Int[(a*Sin[e + f*x])^(-m)*(b*Cos[e + f*x])^(-n), x] /; FreeQ[{a, b, e, f, m, n}, x] && Not[SimplerQ[-m, -n]]
+Int[(a_.*sec[e_. + f_.*x_])^m_*(b_.*csc[e_. + f_.*x_])^n_, x_Symbol] := a^2/b^2*(a*Sec[e + f*x])^(m - 1)*(b*Csc[e + f*x])^(n + 1)*(a* Cos[e + f*x])^(m - 1)*(b*Sin[e + f*x])^(n + 1)* Int[(a*Cos[e + f*x])^(-m)*(b*Sin[e + f*x])^(-n), x] /; FreeQ[{a, b, e, f, m, n}, x]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m
new file mode 100755
index 0000000..488354c
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m	
@@ -0,0 +1,39 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.1.1 (a+b sin)^n *)
+Int[sin[c_. + d_.*x_]^n_, x_Symbol] := -1/d*Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
+Int[sin[c_. + d_.*x_/2]^2, x_Symbol] := x/2 - Sin[2*c + d*x]/(2*d) /; FreeQ[{c, d}, x]
+Int[(b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := (* -Cot[c+d*x]*(c*Sin[c+d*x])^n/(d*n) +  b^2*(n-1)/n*Int[(b*Sin[c+d*x])^(n-2),x] *) -b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1)/(d*n) + b^2*(n - 1)/n*Int[(b*Sin[c + d*x])^(n - 2), x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
+Int[(b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)) + (n + 2)/(b^2*(n + 1))*Int[(b*Sin[c + d*x])^(n + 2), x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
+Int[sin[c_. + Pi/2 + d_.*x_], x_Symbol] := Sin[c + d*x]/d /; FreeQ[{c, d}, x]
+Int[sin[c_. + d_.*x_], x_Symbol] := -Cos[c + d*x]/d /; FreeQ[{c, d}, x]
+(* Int[1/sin[c_.+d_.*x_],x_Symbol] := Int[Csc[c+d*x],x] /; FreeQ[{c,d},x] *)
+Int[Sqrt[sin[c_. + d_.*x_]], x_Symbol] := 2/d*EllipticE[1/2*(c - Pi/2 + d*x), 2] /; FreeQ[{c, d}, x]
+Int[1/Sqrt[sin[c_. + d_.*x_]], x_Symbol] := 2/d*EllipticF[1/2*(c - Pi/2 + d*x), 2] /; FreeQ[{c, d}, x]
+Int[(b_*sin[c_. + d_.*x_])^n_, x_Symbol] := (b*Sin[c + d*x])^n/Sin[c + d*x]^n*Int[Sin[c + d*x]^n, x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]
+(* Int[(b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := Cos[c+d*x]/(b*d*Sqrt[Cos[c+d*x]^2])*Subst[Int[x^n/Sqrt[1-x^2/b^2],x] ,x,b*Sin[c+d*x]] /; FreeQ[{b,c,d,n},x] && Not[IntegerQ[2*n] || IntegerQ[3*n]] *)
+Int[(b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2])* Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2] /; FreeQ[{b, c, d, n}, x] && Not[IntegerQ[2*n]]
+Int[(a_ + b_.*sin[c_. + d_.*x_])^2, x_Symbol] := (2*a^2 + b^2)*x/2 - 2*a*b*Cos[c + d*x]/d - b^2*Cos[c + d*x]*Sin[c + d*x]/(2*d) /; FreeQ[{a, b, c, d}, x]
+Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := Int[ExpandTrig[(a + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]
+Int[Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol] := -2*b*Cos[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]]) /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
+Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := -b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n - 1)/(d*n) + a*(2*n - 1)/n*Int[(a + b*Sin[c + d*x])^(n - 1), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 0]
+Int[1/(a_ + b_.*sin[c_. + d_.*x_]), x_Symbol] := -Cos[c + d*x]/(d*(b + a*Sin[c + d*x])) /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
+Int[1/Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol] := -2/d*Subst[Int[1/(2*a - x^2), x], x, b*Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]]] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
+Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := b*Cos[c + d*x]*(a + b*Sin[c + d*x])^n/(a*d*(2*n + 1)) + (n + 1)/(a*(2*n + 1))*Int[(a + b*Sin[c + d*x])^(n + 1), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
+(* Int[(a_+b_.*sin[c_.+d_.*x_])^n_,x_Symbol] := a^2*Cos[c+d*x]/(d*Sqrt[a+b*Sin[c+d*x]]*Sqrt[a-b*Sin[c+d*x]])*Subst[ Int[(a+b*x)^(n-1/2)/Sqrt[a-b*x],x],x,Sin[c+d*x]] /; FreeQ[{a,b,c,d,n},x] && EqQ[a^2-b^2,0] && Not[IntegerQ[2*n]] *)
+Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := -2^(n + 1/2)*a^(n - 1/2)*b* Cos[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])* Hypergeometric2F1[1/2, 1/2 - n, 3/2, 1/2*(1 - b*Sin[c + d*x]/a)] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[2*n]] && GtQ[a, 0]
+Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := a^IntPart[n]*(a + b*Sin[c + d*x])^ FracPart[n]/(1 + b/a*Sin[c + d*x])^FracPart[n]* Int[(1 + b/a*Sin[c + d*x])^n, x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[2*n]] && Not[GtQ[a, 0]]
+Int[Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol] := 2*Sqrt[a + b]/d*EllipticE[1/2*(c - Pi/2 + d*x), 2*b/(a + b)] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
+Int[Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol] := 2*Sqrt[a - b]/d*EllipticE[1/2*(c + Pi/2 + d*x), -2*b/(a - b)] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a - b, 0]
+Int[Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol] := Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]* Int[Sqrt[a/(a + b) + b/(a + b)*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && Not[GtQ[a + b, 0]]
+Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := -b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n - 1)/(d*n) + 1/n* Int[(a + b*Sin[c + d*x])^(n - 2)* Simp[a^2*n + b^2*(n - 1) + a*b*(2*n - 1)*Sin[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
+Int[1/(a_ + b_.*sin[c_. + d_.*x_]), x_Symbol] := With[{q = Rt[a^2 - b^2, 2]}, x/q + 2/(d*q)*ArcTan[b*Cos[c + d*x]/(a + q + b*Sin[c + d*x])]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2, 0] && PosQ[a]
+Int[1/(a_ + b_.*sin[c_. + d_.*x_]), x_Symbol] := With[{q = Rt[a^2 - b^2, 2]}, -x/q - 2/(d*q)*ArcTan[b*Cos[c + d*x]/(a - q + b*Sin[c + d*x])]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2, 0] && NegQ[a]
+Int[1/(a_ + b_.*sin[c_. + Pi/2 + d_.*x_]), x_Symbol] := With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, 2*e/d* Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
+Int[1/(a_ + b_.*sin[c_. + d_.*x_]), x_Symbol] := With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, 2*e/d* Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
+Int[1/Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol] := 2/(d*Sqrt[a + b])*EllipticF[1/2*(c - Pi/2 + d*x), 2*b/(a + b)] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
+Int[1/Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol] := 2/(d*Sqrt[a - b])*EllipticF[1/2*(c + Pi/2 + d*x), -2*b/(a - b)] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a - b, 0]
+Int[1/Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol] := Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]* Int[1/Sqrt[a/(a + b) + b/(a + b)*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && Not[GtQ[a + b, 0]]
+Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := -b*Cos[ c + d*x]*(a + b*Sin[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 - b^2)) + 1/((n + 1)*(a^2 - b^2))* Int[(a + b*Sin[c + d*x])^(n + 1)* Simp[a*(n + 1) - b*(n + 2)*Sin[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
+Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := Cos[c + d*x]/(d*Sqrt[1 + Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]])* Subst[Int[(a + b*x)^n/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Sin[c + d*x]] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && Not[IntegerQ[2*n]]
+Int[(a_ + b_.*sin[c_. + d_.*x_]*cos[c_. + d_.*x_])^n_, x_Symbol] := Int[(a + b*Sin[2*c + 2*d*x]/2)^n, x] /; FreeQ[{a, b, c, d, n}, x]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m
new file mode 100755
index 0000000..1bd06d5
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m	
@@ -0,0 +1,42 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.1.2 (g cos)^p (a+b sin)^m *)
+Int[cos[e_. + f_.*x_]^p_.*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol] := 1/(b^p*f)* Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || Not[IntegerQ[m + 1/2]])
+Int[cos[e_. + f_.*x_]^p_.*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol] := 1/(b^p*f)* Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := -b*(g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1)) + a*Int[(g*Cos[e + f*x])^p, x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := (a/g)^(2*m)* Int[(g*Cos[e + f*x])^(2*m + p)/(a - b*Sin[e + f*x])^m, x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && LtQ[p, -1] && GeQ[2*m + p, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m/(a*f*g*m) /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[Simplify[m + p + 1], 0] && Not[ILtQ[p, 0]]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(a*f*g*Simplify[2*m + p + 1]) + Simplify[m + p + 1]/(a*Simplify[2*m + p + 1])* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && Not[IGtQ[m, 0]]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)/(f* g*(m - 1)) /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := -b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)/(f* g*(m + p)) + a*(2*m + p - 1)/(m + p)* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := -b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(a*f*g*(p + 1)) + a*(m + p + 1)/(g^2*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[p, -2*m] && IntegersQ[m + 1/2, 2*p]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := -2*b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)/(f* g*(p + 1)) + b^2*(2*m + p - 1)/(g^2*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2), x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && IntegersQ[2*m, 2*p]
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/Sqrt[g_.*cos[e_. + f_.*x_]], x_Symbol] := a*Sqrt[1 + Cos[e + f*x]]* Sqrt[a + b*Sin[e + f*x]]/(a + a*Cos[e + f*x] + b*Sin[e + f*x])* Int[Sqrt[1 + Cos[e + f*x]]/Sqrt[g*Cos[e + f*x]], x] + b*Sqrt[1 + Cos[e + f*x]]* Sqrt[a + b*Sin[e + f*x]]/(a + a*Cos[e + f*x] + b*Sin[e + f*x])* Int[Sin[e + f*x]/(Sqrt[g*Cos[e + f*x]]*Sqrt[1 + Cos[e + f*x]]), x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := -b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)/(f* g*(m + p)) + a*(2*m + p - 1)/(m + p)* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)/(b* f*(m + p)) + g^2*(p - 1)/(a*(m + p))* Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1), x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && (GtQ[m, -2] || EqQ[2*m + p + 1, 0] || EqQ[m, -2] && IntegerQ[p]) && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := 2*g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)/(b* f*(2*m + p + 1)) + g^2*(p - 1)/(b^2*(2*m + p + 1))* Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] && Not[ILtQ[m + p + 1, 0]] && IntegersQ[2*m, 2*p]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(a*f*g*(2*m + p + 1)) + (m + p + 1)/(a*(2*m + p + 1))* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] && IntegersQ[2*m, 2*p]
+Int[(g_.*cos[e_. + f_.*x_])^p_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := g*(g*Cos[e + f*x])^(p - 1)/(b*f*(p - 1)) + g^2/a*Int[(g*Cos[e + f*x])^(p - 2), x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]
+Int[(g_.*cos[e_. + f_.*x_])^p_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := b*(g*Cos[e + f*x])^(p + 1)/(a*f*g*(p - 1)*(a + b*Sin[e + f*x])) + p/(a*(p - 1))*Int[(g*Cos[e + f*x])^p, x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && Not[GeQ[p, 1]] && IntegerQ[2*p]
+Int[Sqrt[g_.*cos[e_. + f_.*x_]]/Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol] := g*Sqrt[1 + Cos[e + f*x]]* Sqrt[a + b*Sin[e + f*x]]/(a + a*Cos[e + f*x] + b*Sin[e + f*x])* Int[Sqrt[1 + Cos[e + f*x]]/Sqrt[g*Cos[e + f*x]], x] - g*Sqrt[1 + Cos[e + f*x]]* Sqrt[a + b*Sin[e + f*x]]/(b + b*Cos[e + f*x] + a*Sin[e + f*x])* Int[Sin[e + f*x]/(Sqrt[g*Cos[e + f*x]]*Sqrt[1 + Cos[e + f*x]]), x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]
+Int[(g_.*cos[e_. + f_.*x_])^(3/2)/Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol] := g*Sqrt[g*Cos[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]/(b*f) + g^2/(2*a)*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[g*Cos[e + f*x]], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_/Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol] := -2*b*(g*Cos[e + f*x])^(p + 1)/(f* g*(2*p - 1)*(a + b*Sin[e + f*x])^(3/2)) + 2*a*(p - 2)/(2*p - 1)* Int[(g*Cos[e + f*x])^p/(a + b*Sin[e + f*x])^(3/2), x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 2] && IntegerQ[2*p]
+Int[(g_.*cos[e_. + f_.*x_])^p_/Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol] := -b*(g*Cos[e + f*x])^(p + 1)/(a*f*g*(p + 1)* Sqrt[a + b*Sin[e + f*x]]) + a*(2*p + 1)/(2*g^2*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)/(a + b*Sin[e + f*x])^(3/2), x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol] := a^m*(g*Cos[e + f*x])^(p + 1)/(f* g*(1 + Sin[e + f*x])^((p + 1)/2)*(1 - Sin[e + f*x])^((p + 1)/2))* Subst[ Int[(1 + b/a*x)^(m + (p - 1)/2)*(1 - b/a*x)^((p - 1)/2), x], x, Sin[e + f*x]] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol] := a^2*(g*Cos[e + f*x])^(p + 1)/(f* g*(a + b*Sin[e + f*x])^((p + 1)/2)*(a - b*Sin[e + f*x])^((p + 1)/2))* Subst[Int[(a + b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := -(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m* Sin[e + f*x]/(f*g*(p + 1)) + 1/(g^2*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1)*(a*(p + 2) + b*(m + p + 2)*Sin[e + f*x]), x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[0, m, 1] && LtQ[p, -1] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := -(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(b + a*Sin[e + f*x])/(f*g*(p + 1)) + 1/(g^2*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*Sin[e + f*x]), x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := -b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)/(f* g*(m + p)) + 1/(m + p)* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1)*Sin[e + f*x]), x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)/(b* f*(m + 1)) + g^2*(p - 1)/(b*(m + 1))* Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)* Sin[e + f*x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && IntegersQ[2*m, 2*p]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := -b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)/(f* g*(a^2 - b^2)*(m + 1)) + 1/((a^2 - b^2)*(m + 1))* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + p + 2)*Sin[e + f*x]), x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)/(b* f*(m + p)) + g^2*(p - 1)/(b*(m + p))* Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^ m*(b + a*Sin[e + f*x]), x] /; FreeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := (g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*(b - a*Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1)) + 1/(g^2*(a^2 - b^2)*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^ m*(a^2*(p + 2) - b^2*(m + p + 2) + a*b*(m + p + 3)*Sin[e + f*x]), x] /; FreeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegersQ[2*m, 2*p]
+Int[1/(Sqrt[g_.*cos[e_. + f_.*x_]]*Sqrt[a_ + b_.*sin[e_. + f_.*x_]]), x_Symbol] := 2*Sqrt[2]*Sqrt[g*Cos[e + f*x]]* Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Sin[e + f*x]))]/ (f*g*Sqrt[a + b*Sin[e + f*x]]* Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])])* Subst[Int[1/Sqrt[1 + (a + b)*x^4/(a - b)], x], x, Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])]] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := g*(g*Cos[e + f*x])^(p - 1)*(1 - Sin[e + f*x])*(a + b*Sin[e + f*x])^(m + 1)*(-(a - b)*(1 - Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x])))^(m/2)/ (f*(a + b)*(m + 1))* Hypergeometric2F1[m + 1, m/2 + 1, m + 2, 2*(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x]))] /; FreeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && EqQ[m + p + 1, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := (g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)/(f* g*(a - b)*(p + 1)) + a/(g^2*(a - b))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^ m/(1 - Sin[e + f*x]), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && EqQ[m + p + 2, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := (g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)/(f* g*(a - b)*(p + 1)) - b*(m + p + 2)/(g^2*(a - b)*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m, x] + a/(g^2*(a - b))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^ m/(1 - Sin[e + f*x]), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m + p + 2, 0]
+Int[Sqrt[g_.*cos[e_. + f_.*x_]]/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := With[{q = Rt[-a^2 + b^2, 2]}, a*g/(2*b)*Int[1/(Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x] - a*g/(2*b)* Int[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x] + b*g/f* Subst[Int[Sqrt[x]/(g^2*(a^2 - b^2) + b^2*x^2), x], x, g*Cos[e + f*x]]] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
+Int[1/(Sqrt[g_.*cos[e_. + f_.*x_]]*(a_ + b_.*sin[e_. + f_.*x_])), x_Symbol] := With[{q = Rt[-a^2 + b^2, 2]}, -a/(2*q)* Int[1/(Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x] - a/(2*q)* Int[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x] + b*g/f* Subst[Int[1/(Sqrt[x]*(g^2*(a^2 - b^2) + b^2*x^2)), x], x, g*Cos[e + f*x]]] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)/ (b* f*(m + p)*(-b*(1 - Sin[e + f*x])/(a + b*Sin[e + f*x]))^((p - 1)/ 2)*(b*(1 + Sin[e + f*x])/(a + b*Sin[e + f*x]))^((p - 1)/2))* AppellF1[-p - m, (1 - p)/2, (1 - p)/2, 1 - p - m, (a + b)/(a + b*Sin[e + f*x]), (a - b)/(a + b*Sin[e + f*x])] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m, 0] && Not[IGtQ[m + p + 1, 0]]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := g*(g*Cos[e + f*x])^(p - 1)/(f*(1 - (a + b*Sin[e + f*x])/(a - b))^((p - 1)/ 2)*(1 - (a + b*Sin[e + f*x])/(a + b))^((p - 1)/2))* Subst[ Int[(-b/(a - b) - b*x/(a - b))^((p - 1)/2)*(b/(a + b) - b*x/(a + b))^((p - 1)/2)*(a + b*x)^m, x], x, Sin[e + f*x]] /; FreeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && Not[IGtQ[m, 0]]
+Int[(g_.*sec[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol] := g^(2*IntPart[p])*(g*Cos[e + f*x])^FracPart[p]*(g*Sec[e + f*x])^ FracPart[p]*Int[(a + b*Sin[e + f*x])^m/(g*Cos[e + f*x])^p, x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Not[IntegerQ[p]]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m
new file mode 100755
index 0000000..9ffdb4e
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m	
@@ -0,0 +1,32 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.1.3 (g tan)^p (a+b sin)^m *)
+Int[(g_.*tan[e_. + f_.*x_])^p_./(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := 1/a*Int[Sec[e + f*x]^2*(g*Tan[e + f*x])^p, x] - 1/(b*g)*Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]
+Int[tan[e_. + f_.*x_]^p_.*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol] := 1/f*Subst[Int[x^p*(a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]
+Int[tan[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol] := a^p*Int[Sin[e + f*x]^p/(a - b*Sin[e + f*x])^m, x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[p, 2*m]
+Int[tan[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := a^p*Int[ ExpandIntegrand[ Sin[e + f*x]^ p*(a + b*Sin[e + f*x])^(m - p/2)/(a - b*Sin[e + f*x])^(p/2), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])
+Int[(g_.*tan[e_. + f_.*x_])^p_.*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol] := Int[ExpandIntegrand[(g*Tan[e + f*x])^p, (a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
+Int[(g_.*tan[e_. + f_.*x_])^p_.*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := a^(2*m)* Int[ExpandIntegrand[(g*Tan[e + f*x])^p* Sec[e + f*x]^(-m), (a*Sec[e + f*x] - b*Tan[e + f*x])^(-m), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
+Int[tan[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := b*(a + b*Sin[e + f*x])^m/(a*f*(2*m - 1)*Cos[e + f*x]) - 1/(a^2*(2*m - 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(a*m - b*(2*m - 1)*Sin[e + f*x])/ Cos[e + f*x]^2, x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && LtQ[m, 0]
+Int[tan[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := -(a + b*Sin[e + f*x])^(m + 1)/(b*f*m*Cos[e + f*x]) + 1/(b*m)* Int[(a + b*Sin[e + f*x])^m*(b*(m + 1) + a*Sin[e + f*x])/ Cos[e + f*x]^2, x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && Not[LtQ[m, 0]]
+Int[tan[e_. + f_.*x_]^4*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := Int[(a + b*Sin[e + f*x])^m, x] - Int[(a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2)/Cos[e + f*x]^4, x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_/tan[e_. + f_.*x_]^2, x_Symbol] := -(a + b*Sin[e + f*x])^(m + 1)/(a*f*Tan[e + f*x]) + 1/b^2* Int[(a + b*Sin[e + f*x])^(m + 1)*(b*m - a*(m + 1)*Sin[e + f*x])/ Sin[e + f*x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && LtQ[m, -1]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_./tan[e_. + f_.*x_]^2, x_Symbol] := -(a + b*Sin[e + f*x])^m/(f*Tan[e + f*x]) + 1/a* Int[(a + b*Sin[e + f*x])^m*(b*m - a*(m + 1)*Sin[e + f*x])/ Sin[e + f*x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && Not[LtQ[m, -1]]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_/tan[e_. + f_.*x_]^4, x_Symbol] := -2/(a*b)*Int[(a + b*Sin[e + f*x])^(m + 2)/Sin[e + f*x]^3, x] + 1/a^2* Int[(a + b*Sin[e + f*x])^(m + 2)*(1 + Sin[e + f*x]^2)/ Sin[e + f*x]^4, x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && LtQ[m, -1]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_/tan[e_. + f_.*x_]^4, x_Symbol] := Int[(a + b*Sin[e + f*x])^m, x] + Int[(a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2)/Sin[e + f*x]^4, x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && Not[LtQ[m, -1]]
+Int[tan[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := Sqrt[a + b*Sin[e + f*x]]* Sqrt[a - b*Sin[e + f*x]]/(b*f*Cos[e + f*x])* Subst[Int[x^p*(a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && IntegerQ[p/2]
+Int[(g_.*tan[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := (g*Tan[e + f*x])^(p + 1)*(a - b*Sin[e + f*x])^((p + 1)/ 2)*(a + b*Sin[e + f*x])^((p + 1)/2)/(f* g*(b*Sin[e + f*x])^(p + 1))* Subst[Int[x^p*(a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && Not[IntegerQ[p]]
+Int[tan[e_. + f_.*x_]^p_.*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol] := 1/f*Subst[Int[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]
+Int[(g_.*tan[e_. + f_.*x_])^p_.*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol] := Int[ExpandIntegrand[(g*Tan[e + f*x])^p, (a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_/tan[e_. + f_.*x_]^2, x_Symbol] := Int[(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2)/Sin[e + f*x]^2, x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_/tan[e_. + f_.*x_]^4, x_Symbol] := -Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(3*a*f*Sin[e + f*x]^3) - (3*a^2 + b^2*(m - 2))* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(3*a^2*b*f*(m + 1)* Sin[e + f*x]^2) - 1/(3*a^2*b*(m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)/Sin[e + f*x]^3* Simp[6*a^2 - b^2*(m - 1)*(m - 2) + a*b*(m + 1)*Sin[e + f*x] - (3*a^2 - b^2*m*(m - 2))* Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
+(* Int[(a_+b_.*sin[e_.+f_.*x_])^m_/tan[e_.+f_.*x_]^4,x_Symbol] := -Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(3*a*f*Sin[e+f*x]^3) - Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(b*f*m*Sin[e+f*x]^2) - 1/(3*a*b*m)*Int[(a+b*Sin[e+f*x])^m/Sin[e+f*x]^3* Simp[6*a^2-b^2*m*(m-2)+a*b*(m+3)*Sin[e+f*x]-(3*a^2-b^2*m*(m-1))* Sin[e+f*x]^2,x],x] /; FreeQ[{a,b,e,f,m},x] && NeQ[a^2-b^2,0] && Not[LtQ[m,-1]] &&  IntegerQ[2*m] *)
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_/tan[e_. + f_.*x_]^4, x_Symbol] := -Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(3*a*f*Sin[e + f*x]^3) - b*(m - 2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(6*a^2*f* Sin[e + f*x]^2) - 1/(6*a^2)*Int[(a + b*Sin[e + f*x])^m/Sin[e + f*x]^2* Simp[8*a^2 - b^2*(m - 1)*(m - 2) + a*b*m*Sin[e + f*x] - (6*a^2 - b^2*m*(m - 2))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] && Not[LtQ[m, -1]] && IntegerQ[2*m]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_/tan[e_. + f_.*x_]^6, x_Symbol] := -Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(5*a*f*Sin[e + f*x]^5) - b*(m - 4)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(20*a^2*f* Sin[e + f*x]^4) + a*Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b^2*f*m*(m - 1)* Sin[e + f*x]^3) + Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*m*Sin[e + f*x]^2) + 1/(20*a^2*b^2*m*(m - 1))* Int[(a + b*Sin[e + f*x])^m/Sin[e + f*x]^4* Simp[60*a^4 - 44*a^2*b^2*(m - 1)*m + b^4*m*(m - 1)*(m - 3)*(m - 4) + a*b*m*(20*a^2 - b^2*m*(m - 1))*Sin[e + f*x] - (40*a^4 + b^4*m*(m - 1)*(m - 2)*(m - 4) - 20*a^2*b^2*(m - 1)*(2*m + 1))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] && NeQ[m, 1] && IntegerQ[2*m]
+Int[(g_.*tan[e_. + f_.*x_])^p_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := a/(a^2 - b^2)*Int[(g*Tan[e + f*x])^p/Sin[e + f*x]^2, x] - b*g/(a^2 - b^2)*Int[(g*Tan[e + f*x])^(p - 1)/Cos[e + f*x], x] - a^2*g^2/(a^2 - b^2)* Int[(g*Tan[e + f*x])^(p - 2)/(a + b*Sin[e + f*x]), x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*p] && GtQ[p, 1]
+Int[(g_.*tan[e_. + f_.*x_])^p_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := 1/a*Int[(g*Tan[e + f*x])^p/Cos[e + f*x]^2, x] - b/(a^2*g)*Int[(g*Tan[e + f*x])^(p + 1)/Cos[e + f*x], x] - (a^2 - b^2)/(a^2*g^2)* Int[(g*Tan[e + f*x])^(p + 2)/(a + b*Sin[e + f*x]), x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*p] && LtQ[p, -1]
+Int[Sqrt[g_.*tan[e_. + f_.*x_]]/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := Sqrt[Cos[e + f*x]]*Sqrt[g*Tan[e + f*x]]/Sqrt[Sin[e + f*x]]* Int[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*(a + b*Sin[e + f*x])), x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
+Int[1/(Sqrt[g_*tan[e_. + f_.*x_]]*(a_ + b_.*sin[e_. + f_.*x_])), x_Symbol] := Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[g*Tan[e + f*x]])* Int[Sqrt[Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])), x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
+Int[tan[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := Int[ExpandIntegrand[ Sin[e + f*x]^p*(a + b*Sin[e + f*x])^m/(1 - Sin[e + f*x]^2)^(p/2), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, p/2]
+Int[(g_.*tan[e_. + f_.*x_])^p_.*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol] := Unintegrable[(g*Tan[e + f*x])^p*(a + b*Sin[e + f*x])^m, x] /; FreeQ[{a, b, e, f, g, m, p}, x]
+Int[(g_.*cot[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol] := g^(2*IntPart[p])*(g*Cot[e + f*x])^FracPart[p]*(g*Tan[e + f*x])^ FracPart[p]*Int[(a + b*Sin[e + f*x])^m/(g*Tan[e + f*x])^p, x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Not[IntegerQ[p]]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m
new file mode 100755
index 0000000..f8f2ccf
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m	
@@ -0,0 +1,40 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.10 (c+d x)^m (a+b sin)^n *)
+Int[(c_. + d_.*x_)^m_.*sin[e_. + f_.*x_], x_Symbol] := -(c + d*x)^m*Cos[e + f*x]/f + d*m/f*Int[(c + d*x)^(m - 1)*Cos[e + f*x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
+Int[(c_. + d_.*x_)^m_*sin[e_. + f_.*x_], x_Symbol] := (c + d*x)^(m + 1)*Sin[e + f*x]/(d*(m + 1)) - f/(d*(m + 1))*Int[(c + d*x)^(m + 1)*Cos[e + f*x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, -1]
+Int[sin[e_. + f_.*Complex[0, fz_]*x_]/(c_. + d_.*x_), x_Symbol] := I*SinhIntegral[c*f*fz/d + f*fz*x]/d /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
+Int[sin[e_. + f_.*x_]/(c_. + d_.*x_), x_Symbol] := SinIntegral[e + f*x]/d /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
+Int[sin[e_. + f_.*Complex[0, fz_]*x_]/(c_. + d_.*x_), x_Symbol] := CoshIntegral[-c*f*fz/d - f*fz*x]/d /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0] && NegQ[c*f*fz/d, 0]
+Int[sin[e_. + f_.*Complex[0, fz_]*x_]/(c_. + d_.*x_), x_Symbol] := CoshIntegral[c*f*fz/d + f*fz*x]/d /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
+Int[sin[e_. + f_.*x_]/(c_. + d_.*x_), x_Symbol] := CosIntegral[e - Pi/2 + f*x]/d /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
+Int[sin[e_. + f_.*x_]/(c_. + d_.*x_), x_Symbol] := Cos[(d*e - c*f)/d]*Int[Sin[c*f/d + f*x]/(c + d*x), x] + Sin[(d*e - c*f)/d]*Int[Cos[c*f/d + f*x]/(c + d*x), x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
+Int[sin[e_. + Pi/2 + f_.*x_]/Sqrt[c_. + d_.*x_], x_Symbol] := 2/d*Subst[Int[Cos[f*x^2/d], x], x, Sqrt[c + d*x]] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
+Int[sin[e_. + f_.*x_]/Sqrt[c_. + d_.*x_], x_Symbol] := 2/d*Subst[Int[Sin[f*x^2/d], x], x, Sqrt[c + d*x]] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
+Int[sin[e_. + f_.*x_]/Sqrt[c_. + d_.*x_], x_Symbol] := Cos[(d*e - c*f)/d]*Int[Sin[c*f/d + f*x]/Sqrt[c + d*x], x] + Sin[(d*e - c*f)/d]*Int[Cos[c*f/d + f*x]/Sqrt[c + d*x], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]
+Int[(c_. + d_.*x_)^m_.*sin[e_. + k_.*Pi + f_.*x_], x_Symbol] := I/2*Int[(c + d*x)^m*E^(-I*k*Pi)*E^(-I*(e + f*x)), x] - I/2*Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x] /; FreeQ[{c, d, e, f, m}, x] && IntegerQ[2*k]
+Int[(c_. + d_.*x_)^m_.*sin[e_. + f_.*x_], x_Symbol] := I/2*Int[(c + d*x)^m*E^(-I*(e + f*x)), x] - I/2*Int[(c + d*x)^m*E^(I*(e + f*x)), x] /; FreeQ[{c, d, e, f, m}, x]
+Int[(c_. + d_.*x_)^m_.*sin[e_. + f_.*x_/2]^2, x_Symbol] := 1/2*Int[(c + d*x)^m, x] - 1/2*Int[(c + d*x)^m*Cos[2*e + f*x], x] /; FreeQ[{c, d, e, f, m}, x]
+Int[(c_. + d_.*x_)*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol] := d*(b*Sin[e + f*x])^n/(f^2*n^2) - b*(c + d*x)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1)/(f*n) + b^2*(n - 1)/n*Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
+Int[(c_. + d_.*x_)^m_*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol] := d*m*(c + d*x)^(m - 1)*(b*Sin[e + f*x])^n/(f^2*n^2) - b*(c + d*x)^m*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1)/(f*n) + b^2*(n - 1)/n*Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x] - d^2*m*(m - 1)/(f^2*n^2)* Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
+Int[(c_. + d_.*x_)^m_*sin[e_. + f_.*x_]^n_, x_Symbol] := Int[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && (Not[RationalQ[m]] || GeQ[m, -1] && LtQ[m, 1])
+Int[(c_. + d_.*x_)^m_*sin[e_. + f_.*x_]^n_, x_Symbol] := (c + d*x)^(m + 1)*Sin[e + f*x]^n/(d*(m + 1)) - f*n/(d*(m + 1))* Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n - 1), x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -1]
+Int[(c_. + d_.*x_)^m_*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol] := (c + d*x)^(m + 1)*(b*Sin[e + f*x])^n/(d*(m + 1)) - b*f*n*(c + d*x)^(m + 2)* Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1)*(m + 2)) - f^2*n^2/(d^2*(m + 1)*(m + 2))* Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x] + b^2*f^2*n*(n - 1)/(d^2*(m + 1)*(m + 2))* Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
+Int[(c_. + d_.*x_)*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol] := (c + d*x)*Cos[e + f*x]*(b*Sin[e + f*x])^(n + 1)/(b*f*(n + 1)) - d*(b*Sin[e + f*x])^(n + 2)/(b^2*f^2*(n + 1)*(n + 2)) + (n + 2)/(b^2*(n + 1))* Int[(c + d*x)*(b*Sin[e + f*x])^(n + 2), x] /; FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1] && NeQ[n, -2]
+Int[(c_. + d_.*x_)^m_.*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol] := (c + d*x)^m*Cos[e + f*x]*(b*Sin[e + f*x])^(n + 1)/(b*f*(n + 1)) - d*m*(c + d*x)^(m - 1)*(b*Sin[e + f*x])^(n + 2)/(b^2* f^2*(n + 1)*(n + 2)) + (n + 2)/(b^2*(n + 1))* Int[(c + d*x)^m*(b*Sin[e + f*x])^(n + 2), x] + d^2*m*(m - 1)/(b^2*f^2*(n + 1)*(n + 2))* Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^(n + 2), x] /; FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1] && NeQ[n, -2] && GtQ[m, 1]
+Int[(c_. + d_.*x_)^m_.*(a_ + b_.*sin[e_. + f_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[m, 0] || NeQ[a^2 - b^2, 0])
+Int[(c_. + d_.*x_)^m_.*(a_ + b_.*sin[e_. + f_.*x_])^n_., x_Symbol] := (2*a)^n* Int[(c + d*x)^m*Sin[1/2*(e + Pi*a/(2*b)) + f*x/2]^(2*n), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
+Int[(c_. + d_.*x_)^m_.*(a_ + b_.*sin[e_. + f_.*x_])^n_, x_Symbol] := (2*a)^IntPart[n]*(a + b*Sin[e + f*x])^FracPart[n]/ Sin[e/2 + a*Pi/(4*b) + f*x/2]^(2*FracPart[n])* Int[(c + d*x)^m*Sin[e/2 + a*Pi/(4*b) + f*x/2]^(2*n), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
+(* Int[(c_.+d_.*x_)^m_.*(a_+b_.*sin[e_.+f_.*x_])^n_.,x_Symbol] := (2*a)^n*Int[(c+d*x)^m*Cos[1/2*(e-Pi*a/(2*b))+f*x/2]^(2*n),x] /; FreeQ[{a,b,c,d,e,f,m},x] && EqQ[a^2-b^2,0] && IntegerQ[n] &&  (GtQ[n,0] || IGtQ[m,0]) *)
+(* Int[(c_.+d_.*x_)^m_.*(a_+b_.*sin[e_.+f_.*x_])^n_,x_Symbol] := (2*a)^IntPart[n]*(a+b*Sin[e+f*x])^FracPart[n]/Cos[1/2*(e-Pi*a/(2*b)) +f*x/2]^(2*FracPart[n])* Int[(c+d*x)^m*Cos[1/2*(e-Pi*a/(2*b))+f*x/2]^(2*n),x] /; FreeQ[{a,b,c,d,e,f,m},x] && EqQ[a^2-b^2,0] && IntegerQ[n+1/2] &&  (GtQ[n,0] || IGtQ[m,0]) *)
+Int[(c_. + d_.*x_)^ m_./(a_ + b_.*sin[e_. + k_.*Pi + f_.*Complex[0, fz_]*x_]), x_Symbol] := 2*Int[(c + d*x)^m*E^(-I*Pi*(k - 1/2))* E^(-I*e + f*fz*x)/(b + 2*a*E^(-I*Pi*(k - 1/2))*E^(-I*e + f*fz*x) - b*E^(-2*I*k*Pi)*E^(2*(-I*e + f*fz*x))), x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_./(a_ + b_.*sin[e_. + k_.*Pi + f_.*x_]), x_Symbol] := 2*Int[(c + d*x)^m*E^(I*Pi*(k - 1/2))* E^(I*(e + f*x))/(b + 2*a*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)) - b*E^(2*I*k*Pi)*E^(2*I*(e + f*x))), x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
+(* Int[(c_.+d_.*x_)^m_./(a_+b_.*sin[e_.+f_.*Complex[0,fz_]*x_]),x_ Symbol] := 2*I*Int[(c+d*x)^m*E^(-I*e+f*fz*x)/(b+2*I*a*E^(-I*e+f*fz*x)-b*E^(2*(- I*e+f*fz*x))),x] /; FreeQ[{a,b,c,d,e,f,fz},x] && NeQ[a^2-b^2,0] && IGtQ[m,0] *)
+(* Int[(c_.+d_.*x_)^m_./(a_+b_.*sin[e_.+f_.*x_]),x_Symbol] := -2*I*Int[(c+d*x)^m*E^(I*(e+f*x))/(b-2*I*a*E^(I*(e+f*x))-b*E^(2*I*(e+ f*x))),x] /; FreeQ[{a,b,c,d,e,f},x] && NeQ[a^2-b^2,0] && IGtQ[m,0] *)
+Int[(c_. + d_.*x_)^m_./(a_ + b_.*sin[e_. + f_.*Complex[0, fz_]*x_]), x_Symbol] := 2*Int[(c + d*x)^m* E^(-I*e + f*fz*x)/(-I*b + 2*a*E^(-I*e + f*fz*x) + I*b*E^(2*(-I*e + f*fz*x))), x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_./(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := 2*Int[(c + d*x)^m* E^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x))), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_./(a_ + b_.*sin[e_. + f_.*x_])^2, x_Symbol] := b*(c + d*x)^m*Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])) + a/(a^2 - b^2)*Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] - b*d*m/(f*(a^2 - b^2))* Int[(c + d*x)^(m - 1)*Cos[e + f*x]/(a + b*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*(a_ + b_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -b*(c + d*x)^m* Cos[e + f* x]*(a + b*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(a^2 - b^2)) + a/(a^2 - b^2)* Int[(c + d*x)^m*(a + b*Sin[e + f*x])^(n + 1), x] + b*d*m/(f*(n + 1)*(a^2 - b^2))* Int[(c + d*x)^(m - 1)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(n + 1), x] - b*(n + 2)/((n + 1)*(a^2 - b^2))* Int[(c + d*x)^m*Sin[e + f*x]*(a + b*Sin[e + f*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && ILtQ[n, -2] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*sin[e_. + f_.*x_])^n_., x_Symbol] := Unintegrable[(c + d*x)^m*(a + b*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
+Int[u_^m_.*(a_. + b_.*Sin[v_])^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*(a + b*Sin[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
+Int[u_^m_.*(a_. + b_.*Cos[v_])^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*(a + b*Cos[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m
new file mode 100755
index 0000000..ef994c0
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m	
@@ -0,0 +1,25 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.11 (e x)^m (a+b x^n)^p sin *)
+Int[(a_ + b_.*x_^n_)^p_.*Sin[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Sin[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
+Int[(a_ + b_.*x_^n_)^p_.*Cos[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Cos[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
+Int[(a_ + b_.*x_^n_)^p_*Sin[c_. + d_.*x_], x_Symbol] := x^(-n + 1)*(a + b*x^n)^(p + 1)*Sin[c + d*x]/(b*n*(p + 1)) - (-n + 1)/(b*n*(p + 1))* Int[x^(-n)*(a + b*x^n)^(p + 1)*Sin[c + d*x], x] - d/(b*n*(p + 1))* Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Cos[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1] && IGtQ[n, 2]
+Int[(a_ + b_.*x_^n_)^p_*Cos[c_. + d_.*x_], x_Symbol] := x^(-n + 1)*(a + b*x^n)^(p + 1)*Cos[c + d*x]/(b*n*(p + 1)) - (-n + 1)/(b*n*(p + 1))* Int[x^(-n)*(a + b*x^n)^(p + 1)*Cos[c + d*x], x] + d/(b*n*(p + 1))* Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Sin[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1] && IGtQ[n, 2]
+Int[(a_ + b_.*x_^n_)^p_*Sin[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Sin[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
+Int[(a_ + b_.*x_^n_)^p_*Cos[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Cos[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
+Int[(a_ + b_.*x_^n_)^p_*Sin[c_. + d_.*x_], x_Symbol] := Int[x^(n*p)*(b + a*x^(-n))^p*Sin[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && ILtQ[n, 0]
+Int[(a_ + b_.*x_^n_)^p_*Cos[c_. + d_.*x_], x_Symbol] := Int[x^(n*p)*(b + a*x^(-n))^p*Cos[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && ILtQ[n, 0]
+Int[(a_ + b_.*x_^n_)^p_*Sin[c_. + d_.*x_], x_Symbol] := Unintegrable[(a + b*x^n)^p*Sin[c + d*x], x] /; FreeQ[{a, b, c, d, n, p}, x]
+Int[(a_ + b_.*x_^n_)^p_*Cos[c_. + d_.*x_], x_Symbol] := Unintegrable[(a + b*x^n)^p*Cos[c + d*x], x] /; FreeQ[{a, b, c, d, n, p}, x]
+Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*Sin[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Sin[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
+Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*Cos[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Cos[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
+Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*Sin[c_. + d_.*x_], x_Symbol] := e^m*(a + b*x^n)^(p + 1)*Sin[c + d*x]/(b*n*(p + 1)) - d*e^m/(b*n*(p + 1))*Int[(a + b*x^n)^(p + 1)*Cos[c + d*x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, -1] && EqQ[m, n - 1] && (IntegerQ[n] || GtQ[e, 0])
+Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*Cos[c_. + d_.*x_], x_Symbol] := e^m*(a + b*x^n)^(p + 1)*Cos[c + d*x]/(b*n*(p + 1)) + d*e^m/(b*n*(p + 1))*Int[(a + b*x^n)^(p + 1)*Sin[c + d*x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, -1] && EqQ[m, n - 1] && (IntegerQ[n] || GtQ[e, 0])
+Int[x_^m_.*(a_ + b_.*x_^n_)^p_*Sin[c_. + d_.*x_], x_Symbol] := x^(m - n + 1)*(a + b*x^n)^(p + 1)*Sin[c + d*x]/(b*n*(p + 1)) - (m - n + 1)/(b*n*(p + 1))* Int[x^(m - n)*(a + b*x^n)^(p + 1)*Sin[c + d*x], x] - d/(b*n*(p + 1))* Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Cos[c + d*x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, -1] && IGtQ[n, 0] && (GtQ[m - n + 1, 0] || GtQ[n, 2]) && RationalQ[m]
+Int[x_^m_.*(a_ + b_.*x_^n_)^p_*Cos[c_. + d_.*x_], x_Symbol] := x^(m - n + 1)*(a + b*x^n)^(p + 1)*Cos[c + d*x]/(b*n*(p + 1)) - (m - n + 1)/(b*n*(p + 1))* Int[x^(m - n)*(a + b*x^n)^(p + 1)*Cos[c + d*x], x] + d/(b*n*(p + 1))* Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Sin[c + d*x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, -1] && IGtQ[n, 0] && (GtQ[m - n + 1, 0] || GtQ[n, 2]) && RationalQ[m]
+Int[x_^m_.*(a_ + b_.*x_^n_)^p_*Sin[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Sin[c + d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1]) && IntegerQ[m]
+Int[x_^m_.*(a_ + b_.*x_^n_)^p_*Cos[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Cos[c + d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1]) && IntegerQ[m]
+Int[x_^m_.*(a_ + b_.*x_^n_)^p_*Sin[c_. + d_.*x_], x_Symbol] := Int[x^(m + n*p)*(b + a*x^(-n))^p*Sin[c + d*x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && ILtQ[n, 0]
+Int[x_^m_.*(a_ + b_.*x_^n_)^p_*Cos[c_. + d_.*x_], x_Symbol] := Int[x^(m + n*p)*(b + a*x^(-n))^p*Cos[c + d*x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && ILtQ[n, 0]
+Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*Sin[c_. + d_.*x_], x_Symbol] := Unintegrable[(e*x)^m*(a + b*x^n)^p*Sin[c + d*x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
+Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*Cos[c_. + d_.*x_], x_Symbol] := Unintegrable[(e*x)^m*(a + b*x^n)^p*Cos[c + d*x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m
new file mode 100755
index 0000000..3fff50e
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m	
@@ -0,0 +1,99 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.12 (e x)^m (a+b sin(c+d x^n))^p *)
+Int[Sin[d_.*(e_. + f_.*x_)^2], x_Symbol] := Sqrt[Pi/2]/(f*Rt[d, 2])*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)] /; FreeQ[{d, e, f}, x]
+Int[Cos[d_.*(e_. + f_.*x_)^2], x_Symbol] := Sqrt[Pi/2]/(f*Rt[d, 2])*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)] /; FreeQ[{d, e, f}, x]
+Int[Sin[c_ + d_.*(e_. + f_.*x_)^2], x_Symbol] := Sin[c]*Int[Cos[d*(e + f*x)^2], x] + Cos[c]*Int[Sin[d*(e + f*x)^2], x] /; FreeQ[{c, d, e, f}, x]
+Int[Cos[c_ + d_.*(e_. + f_.*x_)^2], x_Symbol] := Cos[c]*Int[Cos[d*(e + f*x)^2], x] - Sin[c]*Int[Sin[d*(e + f*x)^2], x] /; FreeQ[{c, d, e, f}, x]
+Int[Sin[c_. + d_.*(e_. + f_.*x_)^n_], x_Symbol] := I/2*Int[E^(-c*I - d*I*(e + f*x)^n), x] - I/2*Int[E^(c*I + d*I*(e + f*x)^n), x] /; FreeQ[{c, d, e, f}, x] && IGtQ[n, 2]
+Int[Cos[c_. + d_.*(e_. + f_.*x_)^n_], x_Symbol] := 1/2*Int[E^(-c*I - d*I*(e + f*x)^n), x] + 1/2*Int[E^(c*I + d*I*(e + f*x)^n), x] /; FreeQ[{c, d, e, f}, x] && IGtQ[n, 2]
+Int[(a_. + b_.*Sin[c_. + d_.*(e_. + f_.*x_)^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(a + b*Sin[c + d*(e + f*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 1] && IGtQ[n, 1]
+Int[(a_. + b_.*Cos[c_. + d_.*(e_. + f_.*x_)^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(a + b*Cos[c + d*(e + f*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 1] && IGtQ[n, 1]
+Int[(a_. + b_.*Sin[c_. + d_.*(e_. + f_.*x_)^n_])^p_., x_Symbol] := -1/f*Subst[Int[(a + b*Sin[c + d*x^(-n)])^p/x^2, x], x, 1/(e + f*x)] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[n, 0] && EqQ[n, -2]
+Int[(a_. + b_.*Cos[c_. + d_.*(e_. + f_.*x_)^n_])^p_., x_Symbol] := -1/f*Subst[Int[(a + b*Cos[c + d*x^(-n)])^p/x^2, x], x, 1/(e + f*x)] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[n, 0] && EqQ[n, -2]
+Int[(a_. + b_.*Sin[c_. + d_.*(e_. + f_.*x_)^n_])^p_., x_Symbol] := 1/(n*f)* Subst[Int[x^(1/n - 1)*(a + b*Sin[c + d*x])^p, x], x, (e + f*x)^n] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[1/n]
+Int[(a_. + b_.*Cos[c_. + d_.*(e_. + f_.*x_)^n_])^p_., x_Symbol] := 1/(n*f)* Subst[Int[x^(1/n - 1)*(a + b*Cos[c + d*x])^p, x], x, (e + f*x)^n] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[1/n]
+Int[(a_. + b_.*Sin[c_. + d_.*(e_. + f_.*x_)^n_])^p_., x_Symbol] := Module[{k = Denominator[n]}, k/f* Subst[Int[x^(k - 1)*(a + b*Sin[c + d*x^(k*n)])^p, x], x, (e + f*x)^(1/k)]] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && FractionQ[n]
+Int[(a_. + b_.*Cos[c_. + d_.*(e_. + f_.*x_)^n_])^p_., x_Symbol] := Module[{k = Denominator[n]}, k/f* Subst[Int[x^(k - 1)*(a + b*Cos[c + d*x^(k*n)])^p, x], x, (e + f*x)^(1/k)]] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && FractionQ[n]
+Int[Sin[c_. + d_.*(e_. + f_.*x_)^n_], x_Symbol] := I/2*Int[E^(-c*I - d*I*(e + f*x)^n), x] - I/2*Int[E^(c*I + d*I*(e + f*x)^n), x] /; FreeQ[{c, d, e, f, n}, x]
+Int[Cos[c_. + d_.*(e_. + f_.*x_)^n_], x_Symbol] := 1/2*Int[E^(-c*I - d*I*(e + f*x)^n), x] + 1/2*Int[E^(c*I + d*I*(e + f*x)^n), x] /; FreeQ[{c, d, e, f, n}, x]
+Int[(a_. + b_.*Sin[c_. + d_.*(e_. + f_.*x_)^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(a + b*Sin[c + d*(e + f*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[p, 1]
+Int[(a_. + b_.*Cos[c_. + d_.*(e_. + f_.*x_)^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(a + b*Cos[c + d*(e + f*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[p, 1]
+Int[(a_. + b_.*Sin[c_. + d_.*(e_. + f_.*x_)^n_])^p_, x_Symbol] := Unintegrable[(a + b*Sin[c + d*(e + f*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, n, p}, x]
+Int[(a_. + b_.*Cos[c_. + d_.*(e_. + f_.*x_)^n_])^p_, x_Symbol] := Unintegrable[(a + b*Cos[c + d*(e + f*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, n, p}, x]
+Int[(a_. + b_.*Sin[c_. + d_.*u_^n_])^p_., x_Symbol] := Int[(a + b*Sin[c + d*ExpandToSum[u, x]^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && Not[LinearMatchQ[u, x]]
+Int[(a_. + b_.*Cos[c_. + d_.*u_^n_])^p_., x_Symbol] := Int[(a + b*Cos[c + d*ExpandToSum[u, x]^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && Not[LinearMatchQ[u, x]]
+Int[(a_. + b_.*Sin[u_])^p_., x_Symbol] := Int[(a + b*Sin[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[(a_. + b_.*Cos[u_])^p_., x_Symbol] := Int[(a + b*Cos[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[Sin[d_.*x_^n_]/x_, x_Symbol] := SinIntegral[d*x^n]/n /; FreeQ[{d, n}, x]
+Int[Cos[d_.*x_^n_]/x_, x_Symbol] := CosIntegral[d*x^n]/n /; FreeQ[{d, n}, x]
+Int[Sin[c_ + d_.*x_^n_]/x_, x_Symbol] := Sin[c]*Int[Cos[d*x^n]/x, x] + Cos[c]*Int[Sin[d*x^n]/x, x] /; FreeQ[{c, d, n}, x]
+Int[Cos[c_ + d_.*x_^n_]/x_, x_Symbol] := Cos[c]*Int[Cos[d*x^n]/x, x] - Sin[c]*Int[Sin[d*x^n]/x, x] /; FreeQ[{c, d, n}, x]
+Int[x_^m_.*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[ Simplify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0])
+Int[x_^m_.*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[ Simplify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0])
+Int[(e_*x_)^m_*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Sin[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
+Int[(e_*x_)^m_*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Cos[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
+Int[x_^m_.*Sin[a_. + b_.*x_^n_], x_Symbol] := 2/n*Subst[Int[Sin[a + b*x^2], x], x, x^(n/2)] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n/2 - 1]
+Int[x_^m_.*Cos[a_. + b_.*x_^n_], x_Symbol] := 2/n*Subst[Int[Cos[a + b*x^2], x], x, x^(n/2)] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n/2 - 1]
+Int[(e_.*x_)^m_.*Sin[c_. + d_.*x_^n_], x_Symbol] := -e^(n - 1)*(e*x)^(m - n + 1)*Cos[c + d*x^n]/(d*n) + e^n*(m - n + 1)/(d*n)*Int[(e*x)^(m - n)*Cos[c + d*x^n], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[n, m + 1]
+Int[(e_.*x_)^m_.*Cos[c_. + d_.*x_^n_], x_Symbol] := e^(n - 1)*(e*x)^(m - n + 1)*Sin[c + d*x^n]/(d*n) - e^n*(m - n + 1)/(d*n)*Int[(e*x)^(m - n)*Sin[c + d*x^n], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[n, m + 1]
+Int[(e_.*x_)^m_*Sin[c_. + d_.*x_^n_], x_Symbol] := (e*x)^(m + 1)*Sin[c + d*x^n]/(e*(m + 1)) - d*n/(e^n*(m + 1))*Int[(e*x)^(m + n)*Cos[c + d*x^n], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[m, -1]
+Int[(e_.*x_)^m_*Cos[c_. + d_.*x_^n_], x_Symbol] := (e*x)^(m + 1)*Cos[c + d*x^n]/(e*(m + 1)) + d*n/(e^n*(m + 1))*Int[(e*x)^(m + n)*Sin[c + d*x^n], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[m, -1]
+Int[(e_.*x_)^m_.*Sin[c_. + d_.*x_^n_], x_Symbol] := I/2*Int[(e*x)^m*E^(-c*I - d*I*x^n), x] - I/2*Int[(e*x)^m*E^(c*I + d*I*x^n), x] /; FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]
+Int[(e_.*x_)^m_.*Cos[c_. + d_.*x_^n_], x_Symbol] := 1/2*Int[(e*x)^m*E^(-c*I - d*I*x^n), x] + 1/2*Int[(e*x)^m*E^(c*I + d*I*x^n), x] /; FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]
+Int[x_^m_.*Sin[a_. + b_.*x_^n_/2]^2, x_Symbol] := 1/2*Int[x^m, x] - 1/2*Int[x^m*Cos[2*a + b*x^n], x] /; FreeQ[{a, b, m, n}, x]
+Int[x_^m_.*Cos[a_. + b_.*x_^n_/2]^2, x_Symbol] := 1/2*Int[x^m, x] + 1/2*Int[x^m*Cos[2*a + b*x^n], x] /; FreeQ[{a, b, m, n}, x]
+Int[x_^m_.*Sin[a_. + b_.*x_^n_]^p_, x_Symbol] := x^(m + 1)*Sin[a + b*x^n]^p/(m + 1) - b*n*p/(m + 1)*Int[Sin[a + b*x^n]^(p - 1)*Cos[a + b*x^n], x] /; FreeQ[{a, b}, x] && IGtQ[p, 1] && EqQ[m + n, 0] && NeQ[n, 1] && IntegerQ[n]
+Int[x_^m_.*Cos[a_. + b_.*x_^n_]^p_, x_Symbol] := x^(m + 1)*Cos[a + b*x^n]^p/(m + 1) + b*n*p/(m + 1)*Int[Cos[a + b*x^n]^(p - 1)*Sin[a + b*x^n], x] /; FreeQ[{a, b}, x] && IGtQ[p, 1] && EqQ[m + n, 0] && NeQ[n, 1] && IntegerQ[n]
+Int[x_^m_.*Sin[a_. + b_.*x_^n_]^p_, x_Symbol] := n*Sin[a + b*x^n]^p/(b^2*n^2*p^2) - x^n*Cos[a + b*x^n]*Sin[a + b*x^n]^(p - 1)/(b*n*p) + (p - 1)/p*Int[x^m*Sin[a + b*x^n]^(p - 2), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && GtQ[p, 1]
+Int[x_^m_.*Cos[a_. + b_.*x_^n_]^p_, x_Symbol] := n*Cos[a + b*x^n]^p/(b^2*n^2*p^2) + x^n*Sin[a + b*x^n]*Cos[a + b*x^n]^(p - 1)/(b*n*p) + (p - 1)/p*Int[x^m*Cos[a + b*x^n]^(p - 2), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && GtQ[p, 1]
+Int[x_^m_.*Sin[a_. + b_.*x_^n_]^p_, x_Symbol] := (m - n + 1)*x^(m - 2*n + 1)*Sin[a + b*x^n]^p/(b^2*n^2*p^2) - x^(m - n + 1)*Cos[a + b*x^n]*Sin[a + b*x^n]^(p - 1)/(b*n*p) + (p - 1)/p*Int[x^m*Sin[a + b*x^n]^(p - 2), x] - (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*p^2)* Int[x^(m - 2*n)*Sin[a + b*x^n]^p, x] /; FreeQ[{a, b}, x] && GtQ[p, 1] && IGtQ[n, 0] && IGtQ[m, 2*n - 1]
+Int[x_^m_.*Cos[a_. + b_.*x_^n_]^p_, x_Symbol] := (m - n + 1)*x^(m - 2*n + 1)*Cos[a + b*x^n]^p/(b^2*n^2*p^2) + x^(m - n + 1)*Sin[a + b*x^n]*Cos[a + b*x^n]^(p - 1)/(b*n*p) + (p - 1)/p*Int[x^m*Cos[a + b*x^n]^(p - 2), x] - (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*p^2)* Int[x^(m - 2*n)*Cos[a + b*x^n]^p, x] /; FreeQ[{a, b}, x] && GtQ[p, 1] && IGtQ[n, 0] && IGtQ[m, 2*n - 1]
+Int[x_^m_.*Sin[a_. + b_.*x_^n_]^p_, x_Symbol] := x^(m + 1)*Sin[a + b*x^n]^p/(m + 1) - b*n*p*x^(m + n + 1)*Cos[a + b*x^n]* Sin[a + b*x^n]^(p - 1)/((m + 1)*(m + n + 1)) - b^2*n^2*p^2/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Sin[a + b*x^n]^p, x] + b^2*n^2*p*(p - 1)/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Sin[a + b*x^n]^(p - 2), x] /; FreeQ[{a, b}, x] && GtQ[p, 1] && IGtQ[n, 0] && ILtQ[m, -2*n + 1] && NeQ[m + n + 1, 0]
+Int[x_^m_.*Cos[a_. + b_.*x_^n_]^p_, x_Symbol] := x^(m + 1)*Cos[a + b*x^n]^p/(m + 1) + b*n*p*x^(m + n + 1)*Sin[a + b*x^n]* Cos[a + b*x^n]^(p - 1)/((m + 1)*(m + n + 1)) - b^2*n^2*p^2/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Cos[a + b*x^n]^p, x] + b^2*n^2*p*(p - 1)/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Cos[a + b*x^n]^(p - 2), x] /; FreeQ[{a, b}, x] && GtQ[p, 1] && IGtQ[n, 0] && ILtQ[m, -2*n + 1] && NeQ[m + n + 1, 0]
+Int[(e_.*x_)^m_*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol] := With[{k = Denominator[m]}, k/e* Subst[Int[x^(k*(m + 1) - 1)*(a + b*Sin[c + d*x^(k*n)/e^n])^p, x], x, (e*x)^(1/k)]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && IGtQ[n, 0] && FractionQ[m]
+Int[(e_.*x_)^m_*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol] := With[{k = Denominator[m]}, k/e* Subst[Int[x^(k*(m + 1) - 1)*(a + b*Cos[c + d*x^(k*n)/e^n])^p, x], x, (e*x)^(1/k)]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && IGtQ[n, 0] && FractionQ[m]
+Int[(e_.*x_)^m_.*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]
+Int[(e_.*x_)^m_.*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Cos[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]
+Int[x_^m_.*Sin[a_. + b_.*x_^n_]^p_, x_Symbol] := x^n*Cos[a + b*x^n]*Sin[a + b*x^n]^(p + 1)/(b*n*(p + 1)) - n*Sin[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) + (p + 2)/(p + 1)*Int[x^m*Sin[a + b*x^n]^(p + 2), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && LtQ[p, -1] && NeQ[p, -2]
+Int[x_^m_.*Cos[a_. + b_.*x_^n_]^p_, x_Symbol] := -x^n*Sin[a + b*x^n]*Cos[a + b*x^n]^(p + 1)/(b*n*(p + 1)) - n*Cos[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) + (p + 2)/(p + 1)*Int[x^m*Cos[a + b*x^n]^(p + 2), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && LtQ[p, -1] && NeQ[p, -2]
+Int[x_^m_.*Sin[a_. + b_.*x_^n_]^p_, x_Symbol] := x^(m - n + 1)*Cos[a + b*x^n]* Sin[a + b*x^n]^(p + 1)/(b*n*(p + 1)) - (m - n + 1)*x^(m - 2*n + 1)* Sin[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) + (p + 2)/(p + 1)*Int[x^m*Sin[a + b*x^n]^(p + 2), x] + (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*(p + 1)*(p + 2))* Int[x^(m - 2*n)*Sin[a + b*x^n]^(p + 2), x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && NeQ[p, -2] && IGtQ[n, 0] && IGtQ[m, 2*n - 1]
+Int[x_^m_.*Cos[a_. + b_.*x_^n_]^p_, x_Symbol] := -x^(m - n + 1)*Sin[a + b*x^n]*Cos[a + b*x^n]^(p + 1)/(b*n*(p + 1)) - (m - n + 1)*x^(m - 2*n + 1)* Cos[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) + (p + 2)/(p + 1)*Int[x^m*Cos[a + b*x^n]^(p + 2), x] + (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*(p + 1)*(p + 2))* Int[x^(m - 2*n)*Cos[a + b*x^n]^(p + 2), x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && NeQ[p, -2] && IGtQ[n, 0] && IGtQ[m, 2*n - 1]
+Int[x_^m_.*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol] := -Subst[Int[(a + b*Sin[c + d*x^(-n)])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m] && EqQ[n, -2]
+Int[x_^m_.*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol] := -Subst[Int[(a + b*Cos[c + d*x^(-n)])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m] && EqQ[n, -2]
+Int[(e_.*x_)^m_*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol] := With[{k = Denominator[m]}, -k/e* Subst[Int[(a + b*Sin[c + d/(e^n*x^(k*n))])^p/x^(k*(m + 1) + 1), x], x, 1/(e*x)^(1/k)]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && ILtQ[n, 0] && FractionQ[m]
+Int[(e_.*x_)^m_*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol] := With[{k = Denominator[m]}, -k/e* Subst[Int[(a + b*Cos[c + d/(e^n*x^(k*n))])^p/x^(k*(m + 1) + 1), x], x, 1/(e*x)^(1/k)]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && ILtQ[n, 0] && FractionQ[m]
+Int[(e_.*x_)^m_*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol] := -(e*x)^m*(x^(-1))^m* Subst[Int[(a + b*Sin[c + d*x^(-n)])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0] && ILtQ[n, 0] && Not[RationalQ[m]]
+Int[(e_.*x_)^m_*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol] := -(e*x)^m*(x^(-1))^m* Subst[Int[(a + b*Cos[c + d*x^(-n)])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0] && ILtQ[n, 0] && Not[RationalQ[m]]
+Int[x_^m_.*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol] := Module[{k = Denominator[n]}, k*Subst[Int[x^(k*(m + 1) - 1)*(a + b*Sin[c + d*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p] && FractionQ[n]
+Int[x_^m_.*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol] := Module[{k = Denominator[n]}, k*Subst[Int[x^(k*(m + 1) - 1)*(a + b*Cos[c + d*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p] && FractionQ[n]
+Int[(e_*x_)^m_*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Sin[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[p] && FractionQ[n]
+Int[(e_*x_)^m_*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Cos[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[p] && FractionQ[n]
+Int[x_^m_.*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/(m + 1)* Subst[Int[(a + b*Sin[c + d*x^Simplify[n/(m + 1)]])^p, x], x, x^(m + 1)] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && Not[IntegerQ[n]]
+Int[x_^m_.*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/(m + 1)* Subst[Int[(a + b*Cos[c + d*x^Simplify[n/(m + 1)]])^p, x], x, x^(m + 1)] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && Not[IntegerQ[n]]
+Int[(e_*x_)^m_*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Sin[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && Not[IntegerQ[n]]
+Int[(e_*x_)^m_*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Cos[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && Not[IntegerQ[n]]
+Int[(e_.*x_)^m_.*Sin[c_. + d_.*x_^n_], x_Symbol] := I/2*Int[(e*x)^m*E^(-c*I - d*I*x^n), x] - I/2*Int[(e*x)^m*E^(c*I + d*I*x^n), x] /; FreeQ[{c, d, e, m, n}, x]
+Int[(e_.*x_)^m_.*Cos[c_. + d_.*x_^n_], x_Symbol] := 1/2*Int[(e*x)^m*E^(-c*I - d*I*x^n), x] + 1/2*Int[(e*x)^m*E^(c*I + d*I*x^n), x] /; FreeQ[{c, d, e, m, n}, x]
+Int[(e_.*x_)^m_.*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
+Int[(e_.*x_)^m_.*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Cos[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
+Int[(e_.*x_)^m_.*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol] := Unintegrable[(e*x)^m*(a + b*Sin[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
+Int[(e_.*x_)^m_.*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol] := Unintegrable[(e*x)^m*(a + b*Cos[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
+Int[(e_*x_)^m_.*(a_. + b_.*Sin[u_])^p_., x_Symbol] := Int[(e*x)^m*(a + b*Sin[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[(e_*x_)^m_.*(a_. + b_.*Cos[u_])^p_., x_Symbol] := Int[(e*x)^m*(a + b*Cos[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[(g_. + h_.*x_)^m_.*(a_. + b_.*Sin[c_. + d_.*(e_. + f_.*x_)^n_])^ p_., x_Symbol] := 1/(n*f)* Subst[Int[ ExpandIntegrand[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - e*h/f + h*x^(1/n)/f)^m, x], x], x, (e + f*x)^n] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]
+Int[(g_. + h_.*x_)^m_.*(a_. + b_.*Cos[c_. + d_.*(e_. + f_.*x_)^n_])^ p_., x_Symbol] := 1/(n*f)* Subst[Int[ ExpandIntegrand[(a + b*Cos[c + d*x])^p, x^(1/n - 1)*(g - e*h/f + h*x^(1/n)/f)^m, x], x], x, (e + f*x)^n] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]
+(* Int[(g_.+h_.*x_)^m_.*(a_.+b_.*Sin[c_.+d_.*(e_.+f_.*x_)^n_])^p_.,x_ Symbol] := 1/(n*f^(m+1))*Subst[Int[ExpandIntegrand[(a+b*Sin[c+d*x])^p,x^(1/n-1) *(f*g-e*h+h*x^(1/n))^m,x],x],x,(e+f*x)^n] /; FreeQ[{a,b,c,d,e,f,g,h},x] && IGtQ[p,0] && IntegerQ[m] &&  IntegerQ[1/n] *)
+(* Int[(g_.+h_.*x_)^m_.*(a_.+b_.*Cos[c_.+d_.*(e_.+f_.*x_)^n_])^p_.,x_ Symbol] := 1/(n*f^(m+1))*Subst[Int[ExpandIntegrand[(a+b*Cos[c+d*x])^p,x^(1/n-1) *(f*g-e*h+h*x^(1/n))^m,x],x],x,(e+f*x)^n] /; FreeQ[{a,b,c,d,e,f,g,h},x] && IGtQ[p,0] && IntegerQ[m] &&  IntegerQ[1/n] *)
+Int[(g_. + h_.*x_)^m_.*(a_. + b_.*Sin[c_. + d_.*(e_. + f_.*x_)^n_])^ p_., x_Symbol] := Module[{k = If[FractionQ[n], Denominator[n], 1]}, k/f^(m + 1)* Subst[Int[ ExpandIntegrand[(a + b*Sin[c + d*x^(k*n)])^p, x^(k - 1)*(f*g - e*h + h*x^k)^m, x], x], x, (e + f*x)^(1/k)]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[p, 0] && IGtQ[m, 0]
+Int[(g_. + h_.*x_)^m_.*(a_. + b_.*Cos[c_. + d_.*(e_. + f_.*x_)^n_])^ p_., x_Symbol] := Module[{k = If[FractionQ[n], Denominator[n], 1]}, k/f^(m + 1)* Subst[Int[ ExpandIntegrand[(a + b*Cos[c + d*x^(k*n)])^p, x^(k - 1)*(f*g - e*h + h*x^k)^m, x], x], x, (e + f*x)^(1/k)]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[p, 0] && IGtQ[m, 0]
+Int[(g_. + h_.*x_)^m_.*(a_. + b_.*Sin[c_. + d_.*(e_. + f_.*x_)^n_])^ p_., x_Symbol] := 1/f*Subst[Int[(h*x/f)^m*(a + b*Sin[c + d*x^n])^p, x], x, e + f*x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && EqQ[f*g - e*h, 0]
+Int[(g_. + h_.*x_)^m_.*(a_. + b_.*Cos[c_. + d_.*(e_. + f_.*x_)^n_])^ p_., x_Symbol] := 1/f*Subst[Int[(h*x/f)^m*(a + b*Cos[c + d*x^n])^p, x], x, e + f*x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && EqQ[f*g - e*h, 0]
+Int[(g_. + h_.*x_)^m_.*(a_. + b_.*Sin[c_. + d_.*(e_. + f_.*x_)^n_])^ p_., x_Symbol] := Unintegrable[(g + h*x)^m*(a + b*Sin[c + d*(e + f*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x]
+Int[(g_. + h_.*x_)^m_.*(a_. + b_.*Cos[c_. + d_.*(e_. + f_.*x_)^n_])^ p_., x_Symbol] := Unintegrable[(g + h*x)^m*(a + b*Cos[c + d*(e + f*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x]
+Int[v_^m_.*(a_. + b_.*Sin[c_. + d_.*u_^n_])^p_., x_Symbol] := Int[ExpandToSum[v, x]^m*(a + b*Sin[c + d*ExpandToSum[u, x]^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && LinearQ[u, x] && LinearQ[v, x] && Not[LinearMatchQ[u, x] && LinearMatchQ[v, x]]
+Int[v_^m_.*(a_. + b_.*Cos[c_. + d_.*u_^n_])^p_., x_Symbol] := Int[ExpandToSum[v, x]^m*(a + b*Cos[c + d*ExpandToSum[u, x]^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && LinearQ[u, x] && LinearQ[v, x] && Not[LinearMatchQ[u, x] && LinearMatchQ[v, x]]
+Int[x_^m_.*Sin[a_. + b_.*x_^n_.]^p_.*Cos[a_. + b_.*x_^n_.], x_Symbol] := Sin[a + b*x^n]^(p + 1)/(b*n*(p + 1)) /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
+Int[x_^m_.*Cos[a_. + b_.*x_^n_.]^p_.*Sin[a_. + b_.*x_^n_.], x_Symbol] := -Cos[a + b*x^n]^(p + 1)/(b*n*(p + 1)) /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
+Int[x_^m_.*Sin[a_. + b_.*x_^n_.]^p_.*Cos[a_. + b_.*x_^n_.], x_Symbol] := x^(m - n + 1)*Sin[a + b*x^n]^(p + 1)/(b*n*(p + 1)) - (m - n + 1)/(b*n*(p + 1))* Int[x^(m - n)*Sin[a + b*x^n]^(p + 1), x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
+Int[x_^m_.*Cos[a_. + b_.*x_^n_.]^p_.*Sin[a_. + b_.*x_^n_.], x_Symbol] := -x^(m - n + 1)*Cos[a + b*x^n]^(p + 1)/(b*n*(p + 1)) + (m - n + 1)/(b*n*(p + 1))* Int[x^(m - n)*Cos[a + b*x^n]^(p + 1), x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m
new file mode 100755
index 0000000..95bb02b
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m	
@@ -0,0 +1,31 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.13 (d+e x)^m sin(a+b x+c x^2)^n *)
+Int[Sin[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := Int[Sin[(b + 2*c*x)^2/(4*c)], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]
+Int[Cos[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := Int[Cos[(b + 2*c*x)^2/(4*c)], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]
+Int[Sin[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := Cos[(b^2 - 4*a*c)/(4*c)]*Int[Sin[(b + 2*c*x)^2/(4*c)], x] - Sin[(b^2 - 4*a*c)/(4*c)]*Int[Cos[(b + 2*c*x)^2/(4*c)], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
+Int[Cos[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := Cos[(b^2 - 4*a*c)/(4*c)]*Int[Cos[(b + 2*c*x)^2/(4*c)], x] + Sin[(b^2 - 4*a*c)/(4*c)]*Int[Sin[(b + 2*c*x)^2/(4*c)], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
+Int[Sin[a_. + b_.*x_ + c_.*x_^2]^n_, x_Symbol] := Int[ExpandTrigReduce[Sin[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 1]
+Int[Cos[a_. + b_.*x_ + c_.*x_^2]^n_, x_Symbol] := Int[ExpandTrigReduce[Cos[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 1]
+Int[Sin[a_. + b_.*x_ + c_.*x_^2]^n_., x_Symbol] := Unintegrable[Sin[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, n}, x]
+Int[Cos[a_. + b_.*x_ + c_.*x_^2]^n_., x_Symbol] := Unintegrable[Cos[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, n}, x]
+Int[Sin[v_]^n_., x_Symbol] := Int[Sin[ExpandToSum[v, x]]^n, x] /; IGtQ[n, 0] && QuadraticQ[v, x] && Not[QuadraticMatchQ[v, x]]
+Int[Cos[v_]^n_., x_Symbol] := Int[Cos[ExpandToSum[v, x]]^n, x] /; IGtQ[n, 0] && QuadraticQ[v, x] && Not[QuadraticMatchQ[v, x]]
+Int[(d_ + e_.*x_)*Sin[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := -e*Cos[a + b*x + c*x^2]/(2*c) /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
+Int[(d_ + e_.*x_)*Cos[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*Sin[a + b*x + c*x^2]/(2*c) /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
+Int[(d_. + e_.*x_)^m_*Sin[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := -e*(d + e*x)^(m - 1)*Cos[a + b*x + c*x^2]/(2*c) + e^2*(m - 1)/(2*c)* Int[(d + e*x)^(m - 2)*Cos[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && GtQ[m, 1]
+Int[(d_. + e_.*x_)^m_*Cos[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*(d + e*x)^(m - 1)*Sin[a + b*x + c*x^2]/(2*c) - e^2*(m - 1)/(2*c)* Int[(d + e*x)^(m - 2)*Sin[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && GtQ[m, 1]
+Int[(d_. + e_.*x_)^m_*Sin[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := (d + e*x)^(m + 1)*Sin[a + b*x + c*x^2]/(e*(m + 1)) - 2*c/(e^2*(m + 1))* Int[(d + e*x)^(m + 2)*Cos[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && LtQ[m, -1]
+Int[(d_. + e_.*x_)^m_*Cos[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := (d + e*x)^(m + 1)*Cos[a + b*x + c*x^2]/(e*(m + 1)) + 2*c/(e^2*(m + 1))* Int[(d + e*x)^(m + 2)*Sin[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && LtQ[m, -1]
+Int[(d_. + e_.*x_)*Sin[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := -e*Cos[a + b*x + c*x^2]/(2*c) + (2*c*d - b*e)/(2*c)*Int[Sin[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0]
+Int[(d_. + e_.*x_)*Cos[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*Sin[a + b*x + c*x^2]/(2*c) + (2*c*d - b*e)/(2*c)*Int[Cos[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0]
+Int[(d_. + e_.*x_)^m_*Sin[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := -e*(d + e*x)^(m - 1)*Cos[a + b*x + c*x^2]/(2*c) - (b*e - 2*c*d)/(2*c)* Int[(d + e*x)^(m - 1)*Sin[a + b*x + c*x^2], x] + e^2*(m - 1)/(2*c)* Int[(d + e*x)^(m - 2)*Cos[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]
+Int[(d_. + e_.*x_)^m_*Cos[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*(d + e*x)^(m - 1)*Sin[a + b*x + c*x^2]/(2*c) - (b*e - 2*c*d)/(2*c)* Int[(d + e*x)^(m - 1)*Cos[a + b*x + c*x^2], x] - e^2*(m - 1)/(2*c)* Int[(d + e*x)^(m - 2)*Sin[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]
+Int[(d_. + e_.*x_)^m_*Sin[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := (d + e*x)^(m + 1)*Sin[a + b*x + c*x^2]/(e*(m + 1)) - (b*e - 2*c*d)/(e^2*(m + 1))* Int[(d + e*x)^(m + 1)*Cos[a + b*x + c*x^2], x] - 2*c/(e^2*(m + 1))* Int[(d + e*x)^(m + 2)*Cos[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && LtQ[m, -1]
+Int[(d_. + e_.*x_)^m_*Cos[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := (d + e*x)^(m + 1)*Cos[a + b*x + c*x^2]/(e*(m + 1)) + (b*e - 2*c*d)/(e^2*(m + 1))* Int[(d + e*x)^(m + 1)*Sin[a + b*x + c*x^2], x] + 2*c/(e^2*(m + 1))* Int[(d + e*x)^(m + 2)*Sin[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && LtQ[m, -1]
+Int[(d_. + e_.*x_)^m_.*Sin[a_. + b_.*x_ + c_.*x_^2]^n_, x_Symbol] := Int[ExpandTrigReduce[(d + e*x)^m, Sin[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]
+Int[(d_. + e_.*x_)^m_.*Cos[a_. + b_.*x_ + c_.*x_^2]^n_, x_Symbol] := Int[ExpandTrigReduce[(d + e*x)^m, Cos[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]
+Int[(d_. + e_.*x_)^m_.*Sin[a_. + b_.*x_ + c_.*x_^2]^n_., x_Symbol] := Unintegrable[(d + e*x)^m*Sin[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[(d_. + e_.*x_)^m_.*Cos[a_. + b_.*x_ + c_.*x_^2]^n_., x_Symbol] := Unintegrable[(d + e*x)^m*Cos[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[u_^m_.*Sin[v_]^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*Sin[ExpandToSum[v, x]]^n, x] /; FreeQ[m, x] && IGtQ[n, 0] && LinearQ[u, x] && QuadraticQ[v, x] && Not[LinearMatchQ[u, x] && QuadraticMatchQ[v, x]]
+Int[u_^m_.*Cos[v_]^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*Cos[ExpandToSum[v, x]]^n, x] /; FreeQ[m, x] && IGtQ[n, 0] && LinearQ[u, x] && QuadraticQ[v, x] && Not[LinearMatchQ[u, x] && QuadraticMatchQ[v, x]]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m
new file mode 100755
index 0000000..277c7d2
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m	
@@ -0,0 +1,107 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.2.1 (a+b sin)^m (c+d sin)^n *)
+Int[(a_ + b_.*sin[e_. + f_.*x_])*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := (2*a*c + b*d)*x/2 - (b*c + a*d)*Cos[e + f*x]/f - b*d*Cos[e + f*x]*Sin[e + f*x]/(2*f) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])/(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := b*x/d - (b*c - a*d)/d*Int[1/(c + d*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := a^m*c^m*Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && Not[IntegerQ[ n] && (LtQ[m, 0] && GtQ[n, 0] || LtQ[0, n, m] || LtQ[m, n, 0])]
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/Sqrt[c_ + d_.*sin[e_. + f_.*x_]], x_Symbol] := a*c*Cos[ e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])* Int[Cos[e + f*x]/(c + d*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -2*b*Cos[ e + f*x]*(c + d*Sin[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]]) /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -1/2]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -2*b*Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^ n/(f*(2*n + 1)) - b*(2*m - 1)/(d*(2*n + 1))* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0] && LtQ[n, -1] && Not[ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0]]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -b*Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^ n/(f*(m + n)) + a*(2*m - 1)/(m + n)* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0] && Not[LtQ[n, -1]] && Not[IGtQ[n - 1/2, 0] && LtQ[n, m]] && Not[ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0]]
+Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])* Int[1/Cos[e + f*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := b*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*(2*m + 1)) /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && NeQ[m, -1/2]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := b*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*(2*m + 1)) + (m + n + 1)/(a*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + n + 1], 0] && NeQ[m, -1/2] && (SumSimplerQ[m, 1] || Not[SumSimplerQ[n, 1]])
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := b*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*(2*m + 1)) + (m + n + 1)/(a*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && Not[LtQ[m, n, -1]] && IntegersQ[2*m, 2*n]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := a^IntPart[m]* c^IntPart[m]*(a + b*Sin[e + f*x])^ FracPart[m]*(c + d*Sin[e + f*x])^FracPart[m]/ Cos[e + f*x]^(2*FracPart[m])* Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (FractionQ[m] || Not[FractionQ[n]])
+Int[(a_. + b_.*sin[e_. + f_.*x_])^2/(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := -b^2*Cos[e + f*x]/(d*f) + 1/d*Int[Simp[a^2*d - b*(b*c - 2*a*d)*Sin[e + f*x], x]/(c + d*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
+Int[1/((a_. + b_.*sin[e_. + f_.*x_])*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol] := b/(b*c - a*d)*Int[1/(a + b*Sin[e + f*x]), x] - d/(b*c - a*d)*Int[1/(c + d*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
+Int[(b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_]), x_Symbol] := c*Int[(b*Sin[e + f*x])^m, x] + d/b*Int[(b*Sin[e + f*x])^(m + 1), x] /; FreeQ[{b, c, d, e, f, m}, x]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_]), x_Symbol] := -d*Cos[e + f*x]*(a + b*Sin[e + f*x])^m/(f*(m + 1)) /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[a*d*m + b*c*(m + 1), 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := (b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m/(a*f*(2*m + 1)) + (a*d*m + b*c*(m + 1))/(a*b*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := -d*Cos[e + f*x]*(a + b*Sin[e + f*x])^m/(f*(m + 1)) + (a*d*m + b*c*(m + 1))/(b*(m + 1))* Int[(a + b*Sin[e + f*x])^m, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]
+Int[(c_. + d_.*sin[e_. + f_.*x_])/Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol] := (b*c - a*d)/b*Int[1/Sqrt[a + b*Sin[e + f*x]], x] + d/b*Int[Sqrt[a + b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := -d*Cos[e + f*x]*(a + b*Sin[e + f*x])^m/(f*(m + 1)) + 1/(m + 1)* Int[(a + b*Sin[e + f*x])^(m - 1)* Simp[b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := -(b*c - a*d)* Cos[e + f* x]*(a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_]), x_Symbol] := c*Cos[e + f*x]/(f*Sqrt[1 + Sin[e + f*x]]*Sqrt[1 - Sin[e + f*x]])* Subst[Int[(a + b*x)^m*Sqrt[1 + d/c*x]/Sqrt[1 - d/c*x], x], x, Sin[e + f*x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && Not[IntegerQ[2*m]] && EqQ[c^2 - d^2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := (b*c - a*d)/b*Int[(a + b*Sin[e + f*x])^m, x] + d/b*Int[(a + b*Sin[e + f*x])^(m + 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := Int[ExpandTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && RationalQ[n]
+Int[sin[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m/(a*f*(2*m + 1)) - 1/(a^2*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(a*m - b*(2*m + 1)*Sin[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]
+Int[sin[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := -Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[(a + b*Sin[e + f*x])^m*(b*(m + 1) - a*Sin[e + f*x]), x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^2, x_Symbol] := (b*c - a*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])/(a*f*(2*m + 1)) + 1/(a*b*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[a*c*d*(m - 1) + b*(d^2 + c^2*(m + 1)) + d*(a*d*(m - 1) + b*c*(m + 2))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^2, x_Symbol] := -d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[(a + b*Sin[e + f*x])^m* Simp[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1]]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -b^2*(b*c - a*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d)) + b^2/(d*(n + 1)*(b*c + a*d))* Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)* Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))* Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || IntegerQ[m] && EqQ[c, 0])
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -b^2*Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n)) + 1/(d*(m + n))* Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n* Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && Not[LtQ[n, -1]] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || IntegerQ[m] && EqQ[c, 0])
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := b*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*(2*m + 1)) - 1/(a*b*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)* Simp[a*d*n - b*c*(m + 1) - b*d*(m + n + 1)*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && (IntegersQ[2*m, 2*n] || IntegerQ[m] && EqQ[c, 0])
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := (b*c - a*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1)) + 1/(a*b*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)* Simp[b*(c^2*(m + 1) + d^2*(n - 1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n] || IntegerQ[m] && EqQ[c, 0])
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d)) + 1/(a*(2*m + 1)*(b*c - a*d))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[b*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && Not[GtQ[n, 0]] && (IntegersQ[2*m, 2*n] || IntegerQ[m] && EqQ[c, 0])
+Int[(c_. + d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := -(b*c - a*d)* Cos[e + f*x]*(c + d*Sin[e + f*x])^(n - 1)/(a* f*(a + b*Sin[e + f*x])) - d/(a*b)* Int[(c + d*Sin[e + f*x])^(n - 2)* Simp[b*d*(n - 1) - a*c*n + (b*c*(n - 1) - a*d*n)*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && (IntegerQ[2*n] || EqQ[c, 0])
+Int[(c_. + d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := -b^2*Cos[ e + f*x]*(c + d*Sin[e + f*x])^(n + 1)/(a* f*(b*c - a*d)*(a + b*Sin[e + f*x])) + d/(a*(b*c - a*d))* Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, 0] && (IntegerQ[2*n] || EqQ[c, 0])
+Int[(c_. + d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := -b*Cos[e + f*x]*(c + d*Sin[e + f*x])^n/(a*f*(a + b*Sin[e + f*x])) + d*n/(a*b)* Int[(c + d*Sin[e + f*x])^(n - 1)*(a - b*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && (IntegerQ[2*n] || EqQ[c, 0])
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -2*b*Cos[ e + f*x]*(c + d*Sin[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]]) + 2*n*(b*c + a*d)/(b*(2*n + 1))* Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && IntegerQ[2*n]
+Int[Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]/(c_. + d_.*sin[e_. + f_.*x_])^(3/2), x_Symbol] := -2*b^2* Cos[e + f*x]/(f*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]* Sqrt[c + d*Sin[e + f*x]]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := (b*c - a*d)* Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)* Sqrt[a + b*Sin[e + f*x]]) + (2*n + 3)*(b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))* Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := -2*b/f* Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/Sqrt[d_.*sin[e_. + f_.*x_]], x_Symbol] := -2/f*Subst[Int[1/Sqrt[1 - x^2/a], x], x, b*Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]]] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[d, a/b]
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/ Sqrt[c_. + d_.*sin[e_. + f_.*x_]], x_Symbol] := -2*b/f* Subst[Int[1/(b + d*x^2), x], x, b*Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]* Sqrt[c + d*Sin[e + f*x]])] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := a^2*Cos[ e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])* Subst[Int[(c + d*x)^n/Sqrt[a - b*x], x], x, Sin[e + f*x]] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[IntegerQ[2*n]]
+Int[Sqrt[c_. + d_.*sin[e_. + f_.*x_]]/ Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol] := d/b*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + (b*c - a*d)/b* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(c_. + d_.*sin[e_. + f_.*x_])^n_/Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol] := -2*d*Cos[ e + f*x]*(c + d*Sin[e + f*x])^(n - 1)/(f*(2*n - 1)* Sqrt[a + b*Sin[e + f*x]]) - 1/(b*(2*n - 1))* Int[(c + d*Sin[e + f*x])^(n - 2)/Sqrt[a + b*Sin[e + f*x]]* Simp[a*c*d - b*(2*d^2*(n - 1) + c^2*(2*n - 1)) + d*(a*d - b*c*(4*n - 3))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
+Int[(c_. + d_.*sin[e_. + f_.*x_])^n_/Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol] := -d*Cos[ e + f*x]*(c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)* Sqrt[a + b*Sin[e + f*x]]) - 1/(2*b*(n + 1)*(c^2 - d^2))* Int[(c + d*Sin[e + f*x])^(n + 1)* Simp[a*d - 2*b*c*(n + 1) + b*d*(2*n + 3)*Sin[e + f*x], x]/ Sqrt[a + b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
+Int[1/(Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol] := b/(b*c - a*d)*Int[1/Sqrt[a + b*Sin[e + f*x]], x] - d/(b*c - a*d)* Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol] := -Sqrt[2]/(Sqrt[a]*f)* Subst[Int[1/Sqrt[1 - x^2], x], x, b*Cos[e + f*x]/(a + b*Sin[e + f*x])] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[d, a/b] && GtQ[a, 0]
+Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol] := -2*a/f* Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]* Sqrt[c + d*Sin[e + f*x]])] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -d*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n - 1)/(f*(m + n)) + 1/(b*(m + n))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 2)* Simp[d*(a*c*m + b*d*(n - 1)) + b*c^2*(m + n) + d*(a*d*m + b*c*(m + 2*n - 1))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && IntegerQ[n]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := a^m*Cos[e + f*x]/(f*Sqrt[1 + Sin[e + f*x]]*Sqrt[1 - Sin[e + f*x]])* Subst[Int[(1 + b/a*x)^(m - 1/2)*(c + d*x)^n/Sqrt[1 - b/a*x], x], x, Sin[e + f*x]] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && IntegerQ[m]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -b*(d/b)^n* Cos[e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]* Sqrt[a - b*Sin[e + f*x]])* Subst[Int[(a - x)^n*(2*a - x)^(m - 1/2)/Sqrt[x], x], x, a - b*Sin[e + f*x]] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && GtQ[a, 0] && GtQ[d/b, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := (d/b)^ IntPart[n]*(d*Sin[e + f*x])^FracPart[n]/(b*Sin[e + f*x])^ FracPart[n]*Int[(a + b*Sin[e + f*x])^m*(b*Sin[e + f*x])^n, x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && GtQ[a, 0] && Not[GtQ[d/b, 0]]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := a^IntPart[m]*(a + b*Sin[e + f*x])^ FracPart[m]/(1 + b/a*Sin[e + f*x])^FracPart[m]* Int[(1 + b/a*Sin[e + f*x])^m*(d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && Not[GtQ[a, 0]]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := a^2*Cos[ e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])* Subst[Int[(a + b*x)^(m - 1/2)*(c + d*x)^n/Sqrt[a - b*x], x], x, Sin[e + f*x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[IntegerQ[m]]
+Int[(b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^2, x_Symbol] := 2*c*d/b*Int[(b*Sin[e + f*x])^(m + 1), x] + Int[(b*Sin[e + f*x])^m*(c^2 + d^2*Sin[e + f*x]^2), x] /; FreeQ[{b, c, d, e, f, m}, x]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^2, x_Symbol] := -(b^2*c^2 - 2*a*b*c*d + a^2*d^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 - b^2)) - 1/(b*(m + 1)*(a^2 - b^2))*Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[b*(m + 1)*(2*b*c*d - a*(c^2 + d^2)) + (a^2*d^2 - 2*a*b*c*d*(m + 2) + b^2*(d^2*(m + 1) + c^2*(m + 2)))* Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^2, x_Symbol] := -d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[(a + b*Sin[e + f*x])^m* Simp[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && Not[LtQ[m, -1]]
+(* Int[(a_+b_.*sin[e_.+f_.*x_])^m_.*(d_.*sin[e_.+f_.*x_])^n_.,x_ Symbol] := Int[ExpandTrig[(a+b*sin[e+f*x])^m*(d*sin[e+f*x])^n,x],x] /; FreeQ[{a,b,d,e,f,n},x] && NeQ[a^2-b^2,0] && IGtQ[m,0] *)
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -(b^2*c^2 - 2*a*b*c*d + a^2*d^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2)) + 1/(d*(n + 1)*(c^2 - d^2))* Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)* Simp[b*(m - 2)*(b*c - a*d)^2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -b^2*Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n)) + 1/(d*(m + n))* Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n* Simp[a^3*d*(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))* Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && Not[IGtQ[n, 2] && (Not[IntegerQ[m]] || EqQ[a, 0] && NeQ[c, 0])]
+Int[Sqrt[d_.*sin[e_. + f_.*x_]]/(a_ + b_.*sin[e_. + f_.*x_])^(3/2), x_Symbol] := -2*a*d* Cos[e + f*x]/(f*(a^2 - b^2)*Sqrt[a + b*Sin[e + f*x]]* Sqrt[d*Sin[e + f*x]]) - d^2/(a^2 - b^2)* Int[Sqrt[a + b*Sin[e + f*x]]/(d*Sin[e + f*x])^(3/2), x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[Sqrt[ c_ + d_.*sin[e_. + f_.*x_]]/(a_. + b_.*sin[e_. + f_.*x_])^(3/2), x_Symbol] := (c - d)/(a - b)* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] - (b*c - a*d)/(a - b)* Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -b*Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^ n/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)* Simp[a*c*(m + 1) + b*d*n + (a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(m + n + 2)*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && IntegersQ[2*m, 2*n]
+Int[(d_.*sin[e_. + f_.*x_])^(3/2)/(a_ + b_.*sin[e_. + f_.*x_])^(3/2), x_Symbol] := d/b*Int[Sqrt[d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x] - a*d/b*Int[Sqrt[d*Sin[e + f*x]]/(a + b*Sin[e + f*x])^(3/2), x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[(c_ + d_.*sin[e_. + f_.*x_])^(3/2)/(a_. + b_.*sin[e_. + f_.*x_])^(3/2), x_Symbol] := d^2/b^2*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + (b*c - a*d)/b^2* Int[Simp[b*c + a*d + 2*b*d*Sin[e + f*x], x]/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -(b*c - a*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)* Simp[c*(a*c - b*d)*(m + 1) + d*(b*c - a*d)*(n - 1) + (d*(a*c - b*d)*(m + 1) - c*(b*c - a*d)*(m + 2))*Sin[e + f*x] - d*(b*c - a*d)*(m + n + 1)*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegersQ[2*m, 2*n]
+Int[1/((a_ + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol] := 2*b*Cos[ e + f*x]/(f*(a^2 - b^2)*Sqrt[a + b*Sin[e + f*x]]* Sqrt[d*Sin[e + f*x]]) + d/(a^2 - b^2)* Int[(b + a*Sin[e + f*x])/(Sqrt[ a + b*Sin[e + f*x]]*(d*Sin[e + f*x])^(3/2)), x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[1/((a_. + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol] := 1/(a - b)* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] - b/(a - b)* Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -b^2*Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)) + 1/((m + 1)*(b*c - a*d)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))* Sin[e + f*x] - b^2*d*(m + n + 3)*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && (EqQ[a, 0] && IntegerQ[m] && Not[IntegerQ[n]] || Not[IntegerQ[2*n] && LtQ[n, -1] && (IntegerQ[n] && Not[IntegerQ[m]] || EqQ[a, 0])])
+Int[Sqrt[c_. + d_.*sin[e_. + f_.*x_]]/(a_. + b_.*sin[e_. + f_.*x_]), x_Symbol] := d/b*Int[1/Sqrt[c + d*Sin[e + f*x]], x] + (b*c - a*d)/b* Int[1/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^(3/2)/(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := b/d*Int[Sqrt[a + b*Sin[e + f*x]], x] - (b*c - a*d)/d* Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[1/((a_. + b_.*sin[e_. + f_.*x_])* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol] := 2/(f*(a + b)*Sqrt[c + d])* EllipticPi[2*b/(a + b), 1/2*(e - Pi/2 + f*x), 2*d/(c + d)] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
+Int[1/((a_. + b_.*sin[e_. + f_.*x_])* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol] := 2/(f*(a - b)*Sqrt[c - d])* EllipticPi[-2*b/(a - b), 1/2*(e + Pi/2 + f*x), -2*d/(c - d)] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c - d, 0]
+Int[1/((a_. + b_.*sin[e_. + f_.*x_])* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol] := Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]]* Int[1/((a + b*Sin[e + f*x])* Sqrt[c/(c + d) + d/(c + d)*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[GtQ[c + d, 0]]
+Int[Sqrt[b_.*sin[e_. + f_.*x_]]/Sqrt[c_ + d_.*sin[e_. + f_.*x_]], x_Symbol] := 2*c*Rt[b*(c + d), 2]*Tan[e + f*x]*Sqrt[1 + Csc[e + f*x]]* Sqrt[1 - Csc[e + f*x]]/(d*f*Sqrt[c^2 - d^2])* EllipticPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/ Rt[(c + d)/b, 2]], -(c + d)/(c - d)] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c^2 - d^2, 0] && PosQ[(c + d)/b] && GtQ[c^2, 0]
+Int[Sqrt[b_.*sin[e_. + f_.*x_]]/Sqrt[c_ + d_.*sin[e_. + f_.*x_]], x_Symbol] := 2*b*Tan[e + f*x]/(d*f)*Rt[(c + d)/b, 2]* Sqrt[c*(1 + Csc[e + f*x])/(c - d)]* Sqrt[c*(1 - Csc[e + f*x])/(c + d)]* EllipticPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/ Rt[(c + d)/b, 2]], -(c + d)/(c - d)] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/b]
+Int[Sqrt[b_.*sin[e_. + f_.*x_]]/Sqrt[c_ + d_.*sin[e_. + f_.*x_]], x_Symbol] := Sqrt[b*Sin[e + f*x]]/Sqrt[-b*Sin[e + f*x]]* Int[Sqrt[-b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && NegQ[(c + d)/b]
+(* Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/Sqrt[d_.*sin[e_.+f_.*x_]],x_ Symbol] := a*Int[1/(Sqrt[a+b*Sin[e+f*x]]*Sqrt[d*Sin[e+f*x]]),x] + b/d*Int[Sqrt[d*Sin[e+f*x]]/Sqrt[a+b*Sin[e+f*x]],x] /; FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2,0] *)
+(* Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/Sqrt[d_.*sin[e_.+f_.*x_]],x_ Symbol] := 2*(a+b*Sin[e+f*x])/(d*f*Rt[(a+b)/d,2]*Cos[e+f*x])*Sqrt[a*(1-Sin[e+f* x])/(a+b*Sin[e+f*x])]*Sqrt[a*(1+Sin[e+f*x])/(a+b*Sin[e+f*x])]* EllipticPi[b/(a+b),ArcSin[Rt[(a+b)/d,2]*(Sqrt[d*Sin[e+f*x]]/Sqrt[ a+b*Sin[e+f*x]])],-(a-b)/(a+b)] /; FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2,0] && PosQ[(a+b)/d] *)
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/ Sqrt[c_. + d_.*sin[e_. + f_.*x_]], x_Symbol] := 2*(a + b*Sin[e + f*x])/(d*f*Rt[(a + b)/(c + d), 2]*Cos[e + f*x])* Sqrt[(b*c - a*d)*(1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x]))]* Sqrt[-(b*c - a*d)*(1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x]))]* EllipticPi[b*(c + d)/(d*(a + b)), ArcSin[Rt[(a + b)/(c + d), 2]* Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]]], (a - b)*(c + d)/((a + b)*(c - d))] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(a + b)/(c + d)]
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/ Sqrt[c_. + d_.*sin[e_. + f_.*x_]], x_Symbol] := Sqrt[-c - d*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]]* Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[-c - d*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NegQ[(a + b)/(c + d)]
+Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol] := -2*d/(f*Sqrt[a + b*d])* EllipticF[ ArcSin[Cos[e + f*x]/(1 + d*Sin[e + f*x])], -(a - b*d)/(a + b*d)] /; FreeQ[{a, b, d, e, f}, x] && LtQ[a^2 - b^2, 0] && EqQ[d^2, 1] && GtQ[b*d, 0]
+Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol] := Sqrt[Sign[b]*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[Sign[b]*Sin[e + f*x]]), x] /; FreeQ[{a, b, d, e, f}, x] && LtQ[a^2 - b^2, 0] && GtQ[b^2, 0] && Not[EqQ[d^2, 1] && GtQ[b*d, 0]]
+Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol] := -2*Sqrt[a^2]* Sqrt[-Cot[e + f*x]^2]/(a*f*Sqrt[a^2 - b^2]*Cot[e + f*x])* Rt[(a + b)/d, 2]* EllipticF[ ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/ Rt[(a + b)/d, 2]], -(a + b)/(a - b)] /; FreeQ[{a, b, d, e, f}, x] && GtQ[a^2 - b^2, 0] && PosQ[(a + b)/d] && GtQ[a^2, 0]
+Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol] := -2*Tan[e + f*x]/(a*f)*Rt[(a + b)/d, 2]* Sqrt[a*(1 - Csc[e + f*x])/(a + b)]* Sqrt[a*(1 + Csc[e + f*x])/(a - b)]* EllipticF[ ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/ Rt[(a + b)/d, 2]], -(a + b)/(a - b)] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
+Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol] := Sqrt[-d*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[-d*Sin[e + f*x]]), x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && NegQ[(a + b)/d]
+Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := 2*(c + d*Sin[e + f*x])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]* Cos[e + f*x])* Sqrt[(b*c - a*d)*(1 - Sin[e + f*x])/((a + b)*(c + d*Sin[e + f*x]))]* Sqrt[-(b*c - a*d)*(1 + Sin[e + f*x])/((a - b)*(c + d*Sin[e + f*x]))]* EllipticF[ ArcSin[Rt[(c + d)/(a + b), 2]*(Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*(c - d)/((a - b)*(c + d))] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/(a + b)]
+Int[1/(Sqrt[a_. + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := Sqrt[-a - b*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]]* Int[1/(Sqrt[-a - b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NegQ[(c + d)/(a + b)]
+Int[(d_.*sin[e_. + f_.*x_])^(3/2)/Sqrt[a_. + b_.*sin[e_. + f_.*x_]], x_Symbol] := -a*d/(2*b)* Int[Sqrt[d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x] + d/(2*b)* Int[Sqrt[d*Sin[e + f*x]]*(a + 2*b*Sin[e + f*x])/ Sqrt[a + b*Sin[e + f*x]], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -b*Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^ n/(f*(m + n)) + 1/(d*(m + n))* Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n - 1)* Simp[a^2*c*d*(m + n) + b*d*(b*c*(m - 1) + a*d*n) + (a*d*(2*b*c + a*d)*(m + n) - b*d*(a*c - b*d*(m + n - 1)))*Sin[e + f*x] + b*d*(b*c*n + a*d*(2*m + n - 1))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[0, m, 2] && LtQ[-1, n, 2] && NeQ[m + n, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := b/d*Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x] - (b*c - a*d)/d* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0]
+Int[(d_.*sin[e_. + f_.*x_])^n_./(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := a*Int[(d*Sin[e + f*x])^n/(a^2 - b^2*Sin[e + f*x]^2), x] - b/d*Int[(d*Sin[e + f*x])^(n + 1)/(a^2 - b^2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := Int[ExpandTrig[(d*sin[e + f*x])^ n*(a - b*sin[e + f*x])^(-m)/(a^2 - b^2*sin[e + f*x]^2)^(-m), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m, -1]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := Unintegrable[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+(* Int[(a_.+b_.*sin[e_.+f_.*x_])^m_.*(d_./sin[e_.+f_.*x_])^n_,x_ Symbol] := d^m*Int[(d*Csc[e+f*x])^(n-m)*(b+a*Csc[e+f*x])^m,x] /; FreeQ[{a,b,d,e,f,n},x] && Not[IntegerQ[n]] && IntegerQ[m] *)
+(* Int[(a_.+b_.*cos[e_.+f_.*x_])^m_.*(d_./cos[e_.+f_.*x_])^n_,x_ Symbol] := d^m*Int[(d*Sec[e+f*x])^(n-m)*(b+a*Sec[e+f*x])^m,x] /; FreeQ[{a,b,d,e,f,n},x] && Not[IntegerQ[n]] && IntegerQ[m] *)
+Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(c_.*(d_.*sin[e_. + f_.*x_])^p_)^n_, x_Symbol] := c^IntPart[n]*(c*(d*Sin[e + f*x])^p)^ FracPart[n]/(d*Sin[e + f*x])^(p*FracPart[n])* Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n*p), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[n]]
+Int[(a_. + b_.*cos[e_. + f_.*x_])^ m_.*(c_.*(d_.*cos[e_. + f_.*x_])^p_)^n_, x_Symbol] := c^IntPart[n]*(c*(d*Cos[e + f*x])^p)^ FracPart[n]/(d*Cos[e + f*x])^(p*FracPart[n])* Int[(a + b*Cos[e + f*x])^m*(d*Cos[e + f*x])^(n*p), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[n]]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := Int[(a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/Sin[e + f*x]^n, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[n]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := Int[(b + a*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n/Csc[e + f*x]^m, x] /; FreeQ[{a, b, c, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]
+Int[(a_ + b_.*cos[e_. + f_.*x_])^m_.*(c_ + d_.*sec[e_. + f_.*x_])^n_, x_Symbol] := Int[(b + a*Sec[e + f*x])^m*(c + d*Sec[e + f*x])^n/Sec[e + f*x]^m, x] /; FreeQ[{a, b, c, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := Sin[e + f*x]^n*(c + d*Csc[e + f*x])^n/(d + c*Sin[e + f*x])^n* Int[(a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/Sin[e + f*x]^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && Not[IntegerQ[n]] && Not[IntegerQ[m]]
+Int[(a_ + b_.*cos[e_. + f_.*x_])^m_*(c_ + d_.*sec[e_. + f_.*x_])^n_, x_Symbol] := Cos[e + f*x]^n*(c + d*Sec[e + f*x])^n/(d + c*Cos[e + f*x])^n* Int[(a + b*Cos[e + f*x])^m*(d + c*Cos[e + f*x])^n/Cos[e + f*x]^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && Not[IntegerQ[n]] && Not[IntegerQ[m]]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m
new file mode 100755
index 0000000..f044e5e
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m	
@@ -0,0 +1,100 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n *)
+Int[cos[e_. + f_.*x_]*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := 1/(b*f)*Subst[Int[(a + x)^m*(c + d/b*x)^n, x], x, b*Sin[e + f*x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
+Int[cos[e_. + f_.*x_]^p_*(d_.*sin[e_. + f_.*x_])^ n_.*(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := a*Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^n, x] + b/d*Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^(n + 1), x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && IntegerQ[ n] && (LtQ[p, 0] && NeQ[a^2 - b^2, 0] || LtQ[0, n, p - 1] || LtQ[p + 1, -n, 2*p + 1])
+Int[cos[e_. + f_.*x_]^ p_*(d_.*sin[e_. + f_.*x_])^n_./(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := 1/a*Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x] - 1/(b*d)*Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^(n + 1), x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[ n] && (LtQ[0, n, (p + 1)/2] || LeQ[p, -n] && LtQ[-n, 2*p - 3] || GtQ[n, 0] && LeQ[n, -p])
+Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := 1/(b^p*f)* Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + d/b*x)^ n, x], x, b*Sin[e + f*x]] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
+Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := 1/(b^p*f)* Subst[Int[(a + x)^m*(c + d/b*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_.*(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := a*Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x] + b/d*Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
+Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_./(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := g^2/a*Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x] - g^2/(b*d)* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := a^m*c^m/g^(2*m)* Int[(g*Cos[e + f*x])^(2*m + p)*(c + d*Sin[e + f*x])^(n - m), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && Not[IntegerQ[n] && LtQ[n^2, m^2]]
+Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := 1/(a^(p/2)*c^(p/2))* Int[(a + b*Sin[e + f*x])^(m + p/2)*(c + d*Sin[e + f*x])^(n + p/2), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2]
+Int[(g_.*cos[e_. + f_.*x_])^ p_/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := g*Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])* Int[(g*Cos[e + f*x])^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := a^IntPart[m]* c^IntPart[m]*(a + b*Sin[e + f*x])^ FracPart[m]*(c + d*Sin[e + f*x])^FracPart[m]/ (g^(2*IntPart[m])*(g*Cos[e + f*x])^(2*FracPart[m]))* Int[(g*Cos[e + f*x])^(2*m + p)/(c + d*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && EqQ[m - n - 1, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n/(f*g*(m - n - 1)) /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m - n - 1, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -2*b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n/(f*g*(2*n + p + 1)) - b*(2*m + p - 1)/(d*(2*n + p + 1))* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[m + p/2 - 1/2], 0] && LtQ[n, -1] && NeQ[2*n + p + 1, 0] && Not[ILtQ[Simplify[m + n + p], 0] && GtQ[Simplify[2*m + n + 3*p/2 + 1], 0]]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n/(f*g*(m + n + p)) + a*(2*m + p - 1)/(m + n + p)* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[m + p/2 - 1/2], 0] && Not[LtQ[n, -1]] && Not[IGtQ[Simplify[n + p/2 - 1/2], 0] && GtQ[m - n, 0]] && Not[ILtQ[Simplify[m + n + p], 0] && GtQ[Simplify[2*m + n + 3*p/2 + 1], 0]]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^m_, x_Symbol] := a^IntPart[m]* c^IntPart[m]*(a + b*Sin[e + f*x])^ FracPart[m]*(c + d*Sin[e + f*x])^FracPart[m]/ (g^(2*IntPart[m])*(g*Cos[e + f*x])^(2*FracPart[m]))* Int[(g*Cos[e + f*x])^(2*m + p), x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p + 1, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*g*(m - n)) /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + p + 1, 0] && NeQ[m, n]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*g*(2*m + p + 1)) + (m + n + p + 1)/(a*(2*m + p + 1))* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + n + p + 1], 0] && NeQ[2*m + p + 1, 0] && (SumSimplerQ[m, 1] || Not[SumSimplerQ[n, 1]])
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -2*b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n/(f*g*(2*n + p + 1)) - b*(2*m + p - 1)/(d*(2*n + p + 1))* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LtQ[n, -1] && NeQ[2*n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := -b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n/(f*g*(m + n + p)) + a*(2*m + p - 1)/(m + n + p)* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + n + p, 0] && Not[LtQ[0, n, m]] && IntegersQ[2*m, 2*n, 2*p]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*g*(2*m + p + 1)) + (m + n + p + 1)/(a*(2*m + p + 1))* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] && Not[LtQ[m, n, -1]] && IntegersQ[2*m, 2*n, 2*p]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := a^IntPart[m]* c^IntPart[m]*(a + b*Sin[e + f*x])^ FracPart[m]*(c + d*Sin[e + f*x])^FracPart[m]/ (g^(2*IntPart[m])*(g*Cos[e + f*x])^(2*FracPart[m]))* Int[(g*Cos[e + f*x])^(2*m + p)*(c + d*Sin[e + f*x])^(n - m), x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (FractionQ[m] || Not[FractionQ[n]])
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := -d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(f*g*(m + p + 1)) /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[a*d*m + b*c*(m + p + 1), 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := -(b*c + a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(a*f*g*(p + 1)) + b*(a*d*m + b*c*(m + p + 1))/(a*g^2*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, -1] && LtQ[p, -1]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := -d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(f*g*(m + p + 1)) + (a*d*m + b*c*(m + p + 1))/(b*(m + p + 1))* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p + 1)/2], 0] && NeQ[m + p + 1, 0]
+Int[cos[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := 2*(b*c - a*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(2*m + 3)) + 1/(b^3*(2*m + 3))* Int[(a + b*Sin[e + f*x])^(m + 2)*(b*c + 2*a*d*(m + 1) - b*d*(2*m + 3)*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -3/2]
+Int[cos[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := d*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 2)/(b^2*f*(m + 3)) - 1/(b^2*(m + 3))* Int[(a + b*Sin[e + f*x])^(m + 1)*(b*d*(m + 2) - a*c*(m + 3) + (b*c*(m + 3) - a*d*(m + 4))*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GeQ[m, -3/2] && LtQ[m, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := (b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(a*f*g*(2*m + p + 1)) + (a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1))* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[m + p], 0]) && NeQ[2*m + p + 1, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := -d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(f*g*(m + p + 1)) + (a*d*m + b*c*(m + p + 1))/(b*(m + p + 1))* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[m + p + 1, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := -(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m*(d + c*Sin[e + f*x])/(f*g*(p + 1)) + 1/(g^2*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1)* Simp[a*c*(p + 2) + b*d*m + b*c*(m + p + 2)*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*m] && Not[EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x]]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := -d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(f*g*(m + p + 1)) + 1/(m + p + 1)* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)* Simp[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && Not[LtQ[p, -1]] && IntegerQ[2*m] && Not[EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x]]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c*(m + p + 1) - a*d*p + b*d*(m + 1)*Sin[e + f*x])/(b^2* f*(m + 1)*(m + p + 1)) + g^2*(p - 1)/(b^2*(m + 1)*(m + p + 1))* Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)* Simp[b*d*(m + 1) + (b*c*(m + p + 1) - a*d*p)*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && NeQ[m + p + 1, 0] && IntegerQ[2*m]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := -(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1)) + 1/((a^2 - b^2)*(m + 1))* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)* Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c*(m + p + 1) - a*d*p + b*d*(m + p)*Sin[e + f*x])/(b^2* f*(m + p)*(m + p + 1)) + g^2*(p - 1)/(b^2*(m + p)*(m + p + 1))* Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m* Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1) - d*(a^2*p - b^2*(m + p)))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*m]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := (g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c - a*d - (a*c - b*d)*Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1)) + 1/(g^2*(a^2 - b^2)*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m* Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegerQ[2*m]
+Int[(g_.*cos[e_. + f_.*x_])^ p_*(c_. + d_.*sin[e_. + f_.*x_])/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := d/b*Int[(g*Cos[e + f*x])^p, x] + (b*c - a*d)/b* Int[(g*Cos[e + f*x])^p/(a + b*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_]), x_Symbol] := c*g*(g*Cos[e + f*x])^(p - 1)/(f*(1 + Sin[e + f*x])^((p - 1)/2)*(1 - Sin[e + f*x])^((p - 1)/2))* Subst[ Int[(1 + d/c*x)^((p + 1)/2)*(1 - d/c*x)^((p - 1)/2)*(a + b*x)^m, x], x, Sin[e + f*x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
+Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := a^(2*m)*Int[(d*Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_* sin[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := -(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)/(2*b*f* g*(m + 1)) + a/(2*g^2)* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[m - p, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_* sin[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m/(a*f*g*m) - 1/g^2*Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m, x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + p + 1, 0]
+Int[cos[e_. + f_.*x_]^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := 1/a^p* Int[ExpandTrig[(d*sin[e + f*x])^ n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p/2] && (GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1] || GtQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0])
+Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := Int[ExpandTrig[(g*cos[e + f*x])^ p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
+Int[cos[e_. + f_.*x_]^2*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := 1/b^2* Int[(d*Sin[e + f*x])^ n*(a + b*Sin[e + f*x])^(m + 1)*(a - b*Sin[e + f*x]), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && (ILtQ[m, 0] || Not[IGtQ[n, 0]])
+Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := (a/g)^(2*m)* Int[(g*Cos[e + f*x])^(2*m + p)*(d*Sin[e + f*x])^ n/(a - b*Sin[e + f*x])^m, x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := (a/g)^(2*m)* Int[(g*Cos[e + f*x])^(2*m + p)*(d*Sin[e + f*x])^ n/(a - b*Sin[e + f*x])^m, x] /; FreeQ[{a, b, d, e, f, g, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[p] && (EqQ[2*m + p, 0] || GtQ[2*m + p, 0] && LtQ[p, -1])
+Int[(g_.*cos[e_. + f_.*x_])^p_* sin[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(a*f*g*(2*m + p + 1)) - 1/(a^2*(2*m + p + 1))* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m + 1)*(a*m - b*(2*m + p + 1)*Sin[e + f*x]), x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -1/2] && NeQ[2*m + p + 1, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_* sin[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := -(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)/(b*f* g*(m + p + 2)) + 1/(b*(m + p + 2))* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^ m*(b*(m + 1) - a*(p + 1)*Sin[e + f*x]), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[m + p + 2, 0]
+Int[cos[e_. + f_.*x_]^2*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := 1/b^2* Int[(d*Sin[e + f*x])^ n*(a + b*Sin[e + f*x])^(m + 1)*(a - b*Sin[e + f*x]), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n]
+Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := -2/(a*b*d)* Int[(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 2), x] + 1/a^2* Int[(d*Sin[e + f*x])^ n*(a + b*Sin[e + f*x])^(m + 2)*(1 + Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1]
+Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := 1/d^4*Int[(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^m, x] + Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^ m*(1 - 2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IGtQ[m, 0]]
+Int[cos[e_. + f_.*x_]^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := a^m*Cos[ e + f*x]/(f*Sqrt[1 + Sin[e + f*x]]*Sqrt[1 - Sin[e + f*x]])* Subst[ Int[(d*x)^n*(1 + b/a*x)^(m + (p - 1)/2)*(1 - b/a*x)^((p - 1)/2), x], x, Sin[e + f*x]] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2] && IntegerQ[m]
+Int[cos[e_. + f_.*x_]^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := Cos[e + f*x]/(a^(p - 2)*f*Sqrt[a + b*Sin[e + f*x]]* Sqrt[a - b*Sin[e + f*x]])* Subst[ Int[(d*x)^n (a + b*x)^(m + p/2 - 1/2)*(a - b*x)^(p/2 - 1/2), x], x, Sin[e + f*x]] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2] && Not[IntegerQ[m]]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := Int[ExpandTrig[(g*cos[e + f*x])^ p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && (IntegerQ[p] || IGtQ[n, 0])
+Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := a^m*g*(g*Cos[e + f*x])^(p - 1)/(f*(1 + Sin[e + f*x])^((p - 1)/2)*(1 - Sin[e + f*x])^((p - 1)/2))* Subst[ Int[(d*x)^n*(1 + b/a*x)^(m + (p - 1)/2)*(1 - b/a*x)^((p - 1)/2), x], x, Sin[e + f*x]] /; FreeQ[{a, b, d, e, f, n, p}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := g*(g*Cos[e + f*x])^(p - 1)/(f*(a + b*Sin[e + f*x])^((p - 1)/2)*(a - b*Sin[e + f*x])^((p - 1)/2))* Subst[ Int[(d*x)^n*(a + b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_/ Sqrt[d_.*sin[e_. + f_.*x_]], x_Symbol] := -g*(g*Cos[e + f*x])^(p - 1)* Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(m + 1)/(a*d* f*(m + 1)) + g^2*(2*m + 3)/(2*a*(m + 1))* Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)/ Sqrt[d*Sin[e + f*x]], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && EqQ[m + p + 1/2, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_/ Sqrt[d_.*sin[e_. + f_.*x_]], x_Symbol] := 2*(g*Cos[e + f*x])^(p + 1)* Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^m/(d*f*g*(2*m + 1)) + 2*a*m/(g^2*(2*m + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1)/ Sqrt[d*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && EqQ[m + p + 3/2, 0]
+Int[cos[e_. + f_.*x_]^2*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
+(* Int[cos[e_.+f_.*x_]^4*sin[e_.+f_.*x_]^n_*(a_+b_.*sin[e_.+f_.*x_])^ m_,x_Symbol] := (a^2-b^2)*Cos[e+f*x]*Sin[e+f*x]^(n+1)*(a+b*Sin[e+f*x])^(m+1)/(a*b^2* d*(m+1)) - (a^2*(n+1)-b^2*(m+n+2))*Cos[e+f*x]*Sin[e+f*x]^(n+1)*(a+b*Sin[e+f*x]) ^(m+2)/(a^2*b^2*d*(n+1)*(m+1)) + 1/(a^2*b*(n+1)*(m+1))*Int[Sin[e+f*x]^(n+1)*(a+b*Sin[e+f*x])^(m+1)* Simp[a^2*(n+1)*(n+2)-b^2*(m+n+2)*(m+n+3)+a*b*(m+1)*Sin[e+f*x]-(a^ 2*(n+1)*(n+3)-b^2*(m+n+2)*(m+n+4))*Sin[e+f*x]^2,x],x] /; FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2,0] && IntegersQ[2*m,2*n] &&  LtQ[m,-1] && LtQ[n,-1] *)
+Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1)/(a*d* f*(n + 1)) - (a^2*(n + 1) - b^2*(m + n + 2))* Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^(m + 1)/(a^2*b*d^2*f*(n + 1)*(m + 1)) + 1/(a^2*b*d*(n + 1)*(m + 1))* Int[(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1)* Simp[a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 1)* Sin[e + f*x] - (a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && LtQ[n, -1]
+Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := (a^2 - b^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f*(m + 1)) + (a^2*(n - m + 1) - b^2*(m + n + 2))* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^(n + 1)/(a^2*b^2*d*f*(m + 1)*(m + 2)) - 1/(a^2*b^2*(m + 1)*(m + 2))* Int[(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^n* Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 2)* Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && Not[LtQ[n, -1]] && (LtQ[m, -2] || EqQ[m + n + 4, 0])
+Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := (a^2 - b^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f*(m + 1)) - Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^(n + 1)/(b^2*d*f*(m + n + 4)) - 1/(a*b^2*(m + 1)*(m + n + 4))* Int[(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^n* Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 4) + a*b*(m + 1)* Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 3)*(m + n + 4))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && Not[LtQ[n, -1]] && NeQ[m + n + 4, 0]
+Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1)/(a*d* f*(n + 1)) - b*(m + n + 2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 2)/(a^2*d^2*f*(n + 1)*(n + 2)) - 1/(a^2*d^2*(n + 1)*(n + 2))* Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 2)* Simp[a^2*n*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*m*Sin[ e + f*x] - (a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && Not[m < -1] && LtQ[n, -1] && (LtQ[n, -2] || EqQ[m + n + 4, 0])
+Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1)/(a*d* f*(n + 1)) - Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 2)/(b*d^2*f*(m + n + 4)) + 1/(a*b*d*(n + 1)*(m + n + 4))* Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 1)* Simp[a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4) + a*b*(m + 3)* Sin[e + f*x] - (a^2*(n + 1)*(n + 3) - b^2*(m + n + 3)*(m + n + 4))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && Not[m < -1] && LtQ[n, -1] && NeQ[m + n + 4, 0]
+Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := a*(n + 3)* Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1)/(b^2*d* f*(m + n + 3)*(m + n + 4)) - Cos[ e + f*x]*(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^(m + 1)/(b*d^2*f*(m + n + 4)) - 1/(b^2*(m + n + 3)*(m + n + 4))* Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m* Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 3)*(m + n + 4) + a*b*m*Sin[ e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 3)*(m + n + 5))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && Not[m < -1] && Not[LtQ[n, -1]] && NeQ[m + n + 3, 0] && NeQ[m + n + 4, 0]
+Int[cos[e_. + f_.*x_]^6*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1)/(a*d* f*(n + 1)) - b*(m + n + 2)* Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^(m + 1)/(a^2*d^2*f*(n + 1)*(n + 2)) - a*(n + 5)* Cos[e + f*x]*(d*Sin[e + f*x])^(n + 3)*(a + b*Sin[e + f*x])^(m + 1)/(b^2*d^3* f*(m + n + 5)*(m + n + 6)) + Cos[ e + f*x]*(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^(m + 1)/(b*d^4*f*(m + n + 6)) + 1/(a^2*b^2*d^2*(n + 1)*(n + 2)*(m + n + 5)*(m + n + 6))* Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^m* Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2*n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m + n + 2)*(m + n + 3)*(m + n + 5)*(m + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))* Sin[e + f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4)*(m + n + 5)*(m + n + 6) - a^2*b^2*(n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m + 13))* Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m + n + 5, 0] && NeQ[m + n + 6, 0] && Not[IGtQ[m, 0]]
+Int[cos[e_. + f_.*x_]^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := Int[ExpandTrig[(d*sin[e + f*x])^n*(a + b*sin[e + f*x])^ m*(1 - sin[e + f*x]^2)^(p/2), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && (LtQ[m, -1] || EqQ[m, -1] && GtQ[p, 0])
+Int[(g_.*cos[e_. + f_.*x_])^p_* sin[e_. + f_.*x_]^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := Int[ExpandTrig[(g*cos[e + f*x])^p, sin[e + f*x]^n/(a + b*sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[n] && (LtQ[n, 0] || IGtQ[p + 1/2, 0])
+Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := g^2/a*Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x] - b*g^2/(a^2*d)* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x] - g^2*(a^2 - b^2)/(a^2*d^2)* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 2)/(a + b*Sin[e + f*x]), x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && GtQ[p, 1] && (LeQ[n, -2] || EqQ[n, -3/2] && EqQ[p, 3/2])
+Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := g^2/(a*b)* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^ n*(b - a*Sin[e + f*x]), x] + g^2*(a^2 - b^2)/(a*b*d)* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1)/(a + b*Sin[e + f*x]), x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && GtQ[p, 1] && (LtQ[n, -1] || EqQ[p, 3/2] && EqQ[n, -1/2])
+Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := g^2/b^2* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^ n*(a - b*Sin[e + f*x]), x] - g^2*(a^2 - b^2)/b^2* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^ n/(a + b*Sin[e + f*x]), x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && GtQ[p, 1]
+(* Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_/(a_+b_.*sin[ e_.+f_.*x_]),x_Symbol] := g^2*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^n/(a+b*Sin[e+f*x]),x] -   g^2/d^2*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^(n+2)/(a+b*Sin[e+f* x]),x] /; FreeQ[{a,b,d,e,f,g},x] && NeQ[a^2-b^2,0] && IntegersQ[2*n,2*p] &&  GtQ[p,1] *)
+Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := a*d^2/(a^2 - b^2)* Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 2), x] - b*d/(a^2 - b^2)* Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x] - a^2*d^2/(g^2*(a^2 - b^2))* Int[(g*Cos[e + f*x])^(p + 2)*(d*Sin[e + f*x])^(n - 2)/(a + b*Sin[e + f*x]), x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1] && GtQ[n, 1]
+Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := -d/(a^2 - b^2)* Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n - 1)*(b - a*Sin[e + f*x]), x] + a*b*d/(g^2*(a^2 - b^2))* Int[(g*Cos[e + f*x])^(p + 2)*(d*Sin[e + f*x])^(n - 1)/(a + b*Sin[e + f*x]), x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1] && GtQ[n, 0]
+Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := 1/(a^2 - b^2)* Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n*(a - b*Sin[e + f*x]), x] - b^2/(g^2*(a^2 - b^2))* Int[(g*Cos[e + f*x])^(p + 2)*(d*Sin[e + f*x])^ n/(a + b*Sin[e + f*x]), x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1]
+Int[Sqrt[ g_.*cos[e_. + f_.*x_]]/(Sqrt[ sin[e_. + f_.*x_]]*(a_ + b_.*sin[e_. + f_.*x_])), x_Symbol] := -4*Sqrt[2]*g/f* Subst[Int[x^2/(((a + b)*g^2 + (a - b)*x^4)*Sqrt[1 - x^4/g^2]), x], x, Sqrt[g*Cos[e + f*x]]/Sqrt[1 + Sin[e + f*x]]] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
+Int[Sqrt[ g_.*cos[e_. + f_.*x_]]/(Sqrt[ d_*sin[e_. + f_.*x_]]*(a_ + b_.*sin[e_. + f_.*x_])), x_Symbol] := Sqrt[Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]* Int[Sqrt[g*Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])), x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
+Int[Sqrt[ d_.*sin[e_. + f_.*x_]]/(Sqrt[ cos[e_. + f_.*x_]]*(a_ + b_.*sin[e_. + f_.*x_])), x_Symbol] := With[{q = Rt[-a^2 + b^2, 2]}, 2*Sqrt[2]*d*(b + q)/(f*q)* Subst[Int[1/((d*(b + q) + a*x^2)*Sqrt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]] - 2*Sqrt[2]*d*(b - q)/(f*q)* Subst[Int[1/((d*(b - q) + a*x^2)*Sqrt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]]] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[Sqrt[ d_.*sin[e_. + f_.*x_]]/(Sqrt[ g_.*cos[e_. + f_.*x_]]*(a_ + b_.*sin[e_. + f_.*x_])), x_Symbol] := Sqrt[Cos[e + f*x]]/Sqrt[g*Cos[e + f*x]]* Int[Sqrt[d*Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*(a + b*Sin[e + f*x])), x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
+Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := d/b*Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x] - a*d/b* Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n - 1)/(a + b*Sin[e + f*x]), x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[-1, p, 1] && GtQ[n, 0]
+Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := 1/a*Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x] - b/(a*d)* Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1)/(a + b*Sin[e + f*x]), x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[-1, p, 1] && LtQ[n, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^2, x_Symbol] := 2*a*b/d*Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x] + Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^ n*(a^2 + b^2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := Int[ExpandTrig[(g*cos[e + f*x])^ p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[m] && (GtQ[m, 0] || IntegerQ[n])
+Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol] := g^2/a* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^ n*(a + b*Sin[e + f*x])^(m + 1), x] - b*g^2/(a^2*d)* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1), x] - g^2*(a^2 - b^2)/(a^2*d^2)* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^m, x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, 2*p] && LtQ[m, 0] && GtQ[p, 1] && (LeQ[n, -2] || EqQ[m, -1] && EqQ[n, -3/2] && EqQ[p, 3/2])
+Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := a^(2*m)*Int[(c + d*Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := (a/g)^(2*m)* Int[(g*Cos[e + f*x])^(2*m + p)*(c + d*Sin[e + f*x])^ n/(a - b*Sin[e + f*x])^m, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && (EqQ[2*m + p, 0] || GtQ[2*m + p, 0] && LtQ[p, -1])
+Int[cos[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := 1/b^2* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^ n*(a - b*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n]
+Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := a^m*Cos[ e + f*x]/(f*Sqrt[1 + Sin[e + f*x]]*Sqrt[1 - Sin[e + f*x]])* Subst[ Int[(1 + b/a*x)^(m + (p - 1)/2)*(1 - b/a*x)^((p - 1)/2)*(c + d*x)^ n, x], x, Sin[e + f*x]] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2] && IntegerQ[m]
+Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := Cos[e + f*x]/(a^(p - 2)*f*Sqrt[a + b*Sin[e + f*x]]* Sqrt[a - b*Sin[e + f*x]])* Subst[ Int[(a + b*x)^(m + p/2 - 1/2)*(a - b*x)^(p/2 - 1/2)*(c + d*x)^n, x], x, Sin[e + f*x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2] && Not[IntegerQ[m]]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := Int[ExpandTrig[(g*cos[e + f*x])^ p, (a + b*sin[e + f*x])^m*(c + d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && (IntegerQ[p] || IGtQ[n, 0])
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := a^m*g*(g*Cos[e + f*x])^(p - 1)/(f*(1 + Sin[e + f*x])^((p - 1)/2)*(1 - Sin[e + f*x])^((p - 1)/2))* Subst[ Int[(1 + b/a*x)^(m + (p - 1)/2)*(1 - b/a*x)^((p - 1)/2)*(c + d*x)^ n, x], x, Sin[e + f*x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := g*(g*Cos[e + f*x])^(p - 1)/(f*(a + b*Sin[e + f*x])^((p - 1)/2)*(a - b*Sin[e + f*x])^((p - 1)/2))* Subst[ Int[(a + b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2)*(c + d*x)^n, x], x, Sin[e + f*x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]]
+Int[cos[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^ n*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
+Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := Int[ExpandTrig[(a + b*sin[e + f*x])^m*(c + d*sin[e + f*x])^ n*(1 - sin[e + f*x]^2)^(p/2), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && IGtQ[p/2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := Int[ExpandTrig[(g*cos[e + f*x])^p*(a + b*sin[e + f*x])^ m*(c + d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n]
+Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := Unintegrable[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[a^2 - b^2, 0]
+Int[(g_.*sec[e_. + f_.*x_])^p_*(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := g^(2*IntPart[p])*(g*Cos[e + f*x])^FracPart[p]*(g*Sec[e + f*x])^ FracPart[p]* Int[(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(g*Cos[e + f*x])^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && Not[IntegerQ[p]]
+Int[(g_.*csc[e_. + f_.*x_])^p_*(a_. + b_.*cos[e_. + f_.*x_])^ m_.*(c_. + d_.*cos[e_. + f_.*x_])^n_., x_Symbol] := g^(2*IntPart[p])*(g*Sin[e + f*x])^FracPart[p]*(g*Csc[e + f*x])^ FracPart[p]* Int[(a + b*Cos[e + f*x])^ m*(c + d*Cos[e + f*x])^n/(g*Sin[e + f*x])^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && Not[IntegerQ[p]]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m
new file mode 100755
index 0000000..578df11
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m	
@@ -0,0 +1,41 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n *)
+Int[Sqrt[g_.*sin[e_. + f_.*x_]]* Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(c_ + d_.*sin[e_. + f_.*x_]), x_Symbol] := g/d*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[g*Sin[e + f*x]], x] - c*g/d* Int[Sqrt[ a + b*Sin[e + f*x]]/(Sqrt[ g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
+Int[Sqrt[g_.*sin[e_. + f_.*x_]]* Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(c_ + d_.*sin[e_. + f_.*x_]), x_Symbol] := b/d*Int[Sqrt[g*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x] - (b*c - a*d)/d* Int[Sqrt[ g*Sin[e + f*x]]/(Sqrt[ a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]/(Sqrt[ g_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol] := -2*b/f* Subst[Int[1/(b*c + a*d + c*g*x^2), x], x, b*Cos[e + f*x]/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]/(Sqrt[ sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol] := -Sqrt[a + b]/(c*f)* EllipticE[ ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -(a - b)/(a + b)] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[d, c] && GtQ[b^2 - a^2, 0] && GtQ[b, 0]
+Int[Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]/(Sqrt[ g_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol] := -Sqrt[a + b*Sin[e + f*x]]*Sqrt[d*Sin[e + f*x]/(c + d*Sin[e + f*x])]/ (d*f*Sqrt[g*Sin[e + f*x]]* Sqrt[c^2*(a + b*Sin[e + f*x])/((a*c + b*d)*(c + d*Sin[e + f*x]))])* EllipticE[ ArcSin[c* Cos[e + f*x]/(c + d*Sin[e + f*x])], (b*c - a*d)/(b*c + a*d)] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
+Int[Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]/(Sqrt[ g_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol] := a/c*Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x] + (b*c - a*d)/(c*g)* Int[Sqrt[ g*Sin[e + f*x]]/(Sqrt[ a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]/(sin[ e_. + f_.*x_]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol] := 1/c*Int[Sqrt[a + b*Sin[e + f*x]]/Sin[e + f*x], x] - d/c*Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]/(sin[ e_. + f_.*x_]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol] := a/c*Int[1/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x] + (b*c - a*d)/c* Int[1/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
+Int[Sqrt[ g_.*sin[e_. + f_.*x_]]/(Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol] := -a*g/(b*c - a*d)* Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x] + c*g/(b*c - a*d)* Int[Sqrt[ a + b*Sin[e + f*x]]/(Sqrt[ g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
+Int[Sqrt[ g_.*sin[e_. + f_.*x_]]/(Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol] := 2*Sqrt[-Cot[e + f*x]^2]* Sqrt[g*Sin[e + f*x]]/(f*(c + d)*Cot[e + f*x]* Sqrt[a + b*Sin[e + f*x]])*Sqrt[(b + a*Csc[e + f*x])/(a + b)]* EllipticPi[2*c/(c + d), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], 2*a/(a + b)] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[1/(Sqrt[g_.*sin[e_. + f_.*x_]]* Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol] := b/(b*c - a*d)* Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x] - d/(b*c - a*d)* Int[Sqrt[ a + b*Sin[e + f*x]]/(Sqrt[ g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
+Int[1/(Sqrt[g_.*sin[e_. + f_.*x_]]* Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol] := 1/c*Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x] - d/(c*g)* Int[Sqrt[ g*Sin[e + f*x]]/(Sqrt[ a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[1/(sin[e_. + f_.*x_]* Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol] := d^2/(c*(b*c - a*d))* Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x] + 1/(c*(b*c - a*d))* Int[(b*c - a*d - b*d*Sin[e + f*x])/(Sin[e + f*x]* Sqrt[a + b*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[1/(sin[e_. + f_.*x_]* Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol] := 1/c*Int[1/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x] - d/c*Int[1/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
+Int[Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]/(sin[e_. + f_.*x_]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := -d/c*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + 1/c* Int[Sqrt[a + b*Sin[e + f*x]]* Sqrt[c + d*Sin[e + f*x]]/Sin[e + f*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[b*c + a*d, 0]
+Int[Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]/(sin[e_. + f_.*x_]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := -2*a/f* Subst[Int[1/(1 - a*c*x^2), x], x, Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]* Sqrt[c + d*Sin[e + f*x]])] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[b*c + a*d, 0]
+Int[Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]/(sin[e_. + f_.*x_]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := (b*c - a*d)/c* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] + a/c* Int[Sqrt[ c + d*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
+Int[Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]/(sin[e_. + f_.*x_]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := -2*(a + b*Sin[e + f*x])/(c*f*Rt[(a + b)/(c + d), 2]*Cos[e + f*x])* Sqrt[-(b*c - a*d)*(1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x]))]* Sqrt[(b*c - a*d)*(1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x]))]* EllipticPi[a*(c + d)/(c*(a + b)), ArcSin[Rt[(a + b)/(c + d), 2]* Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]]], (a - b)*(c + d)/((a + b)*(c - d))] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[1/(sin[e_. + f_.*x_]*Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])* Int[1/(Cos[e + f*x]*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
+Int[1/(sin[e_. + f_.*x_]*Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := -b/a*Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] + 1/a* Int[Sqrt[ a + b*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (NeQ[a^2 - b^2, 0] || NeQ[c^2 - d^2, 0])
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]/sin[e_. + f_.*x_], x_Symbol] := Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]/Cos[e + f*x]* Int[Cot[e + f*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]/sin[e_. + f_.*x_], x_Symbol] := d*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + c*Int[ Sqrt[a + b*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (NeQ[a^2 - b^2, 0] || NeQ[c^2 - d^2, 0])
+Int[sin[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := a^n*c^n*Int[Tan[e + f*x]^p*(a + b*Sin[e + f*x])^(m - n), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[p + 2*n, 0] && IntegerQ[n]
+Int[(g_.*sin[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := Sqrt[a - b*Sin[e + f*x]]* Sqrt[a + b*Sin[e + f*x]]/(f*Cos[e + f*x])* Subst[ Int[(g*x)^p*(a + b*x)^(m - 1/2)*(c + d*x)^n/Sqrt[a - b*x], x], x, Sin[e + f*x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && IntegerQ[m - 1/2]
+Int[(g_.*sin[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := Int[ExpandTrig[(g*sin[e + f*x])^p*(a + b*sin[e + f*x])^ m*(c + d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[b*c - a*d, 0] && (IntegersQ[m, n] || IntegersQ[m, p] || IntegersQ[n, p]) && NeQ[p, 2]
+Int[(g_.*sin[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol] := Unintegrable[(g*Sin[e + f*x])^p*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, 2]
+Int[(g_.*sin[e_. + f_.*x_])^p_.*(a_. + b_.*csc[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := g^(m + n)* Int[(g*Sin[e + f*x])^(p - m - n)*(b + a*Sin[e + f*x])^ m*(d + c*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && Not[IntegerQ[p]] && IntegerQ[m] && IntegerQ[n]
+Int[(g_.*sin[e_. + f_.*x_])^p_.*(a_. + b_.*csc[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := (g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p* Int[(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^n/(g*Csc[e + f*x])^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && Not[IntegerQ[p]] && Not[IntegerQ[m] && IntegerQ[n]]
+Int[(g_.*sin[e_. + f_.*x_])^p_.*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := g^n*Int[(g*Sin[e + f*x])^(p - n)*(a + b*Sin[e + f*x])^ m*(d + c*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IntegerQ[n]
+Int[sin[e_. + f_.*x_]^p_.*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := Int[(b + a*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n/ Csc[e + f*x]^(m + p), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m] && IntegerQ[p]
+Int[(g_.*sin[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := Csc[e + f*x]^p*(g*Sin[e + f*x])^p* Int[(b + a*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n/ Csc[e + f*x]^(m + p), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && Not[IntegerQ[n]] && IntegerQ[m] && Not[IntegerQ[p]]
+Int[(g_.*sin[e_. + f_.*x_])^p_.*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := (g*Sin[e + f*x])^n*(c + d*Csc[e + f*x])^n/(d + c*Sin[e + f*x])^n* Int[(g*Sin[e + f*x])^(p - n)*(a + b*Sin[e + f*x])^ m*(d + c*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && Not[IntegerQ[n]] && Not[IntegerQ[m]]
+Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := g^(m + n)* Int[(g*Csc[e + f*x])^(p - m - n)*(b + a*Csc[e + f*x])^ m*(d + c*Csc[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && Not[IntegerQ[p]] && IntegerQ[m] && IntegerQ[n]
+Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_., x_Symbol] := (g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p* Int[(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(g*Sin[e + f*x])^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && Not[IntegerQ[p]] && Not[IntegerQ[m] && IntegerQ[n]]
+Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := g^m*Int[(g*Csc[e + f*x])^(p - m)*(b + a*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[m]
+Int[csc[e_. + f_.*x_]^p_.*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := Int[(a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/ Sin[e + f*x]^(n + p), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && Not[IntegerQ[m]] && IntegerQ[n] && IntegerQ[p]
+Int[(g_.*csc[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := Sin[e + f*x]^p*(g*Csc[e + f*x])^p* Int[(a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/ Sin[e + f*x]^(n + p), x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && Not[IntegerQ[m]] && IntegerQ[n] && Not[IntegerQ[p]]
+Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := (a + b*Sin[e + f*x])^m*(g*Csc[e + f*x])^m/(b + a*Csc[e + f*x])^m* Int[(g*Csc[e + f*x])^(p - m)*(b + a*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m
new file mode 100755
index 0000000..6b578f1
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m	
@@ -0,0 +1,49 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin) *)
+Int[sin[e_. + f_.*x_]^n_.*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := Int[ExpandTrig[ sin[e + f*x]^n*(a + b*sin[e + f*x])^m*(A + B*sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, A, B}, x] && EqQ[A*b + a*B, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && IntegerQ[n]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_ + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := a^m*c^m* Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && Not[IntegerQ[ n] && (LtQ[m, 0] && GtQ[n, 0] || LtQ[0, n, m] || LtQ[m, n, 0])]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_])*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := Int[(a + b*Sin[e + f*x])^ m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
+Int[(A_. + B_.*sin[e_. + f_.*x_])/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := (A*b + a*B)/(2*a*b)* Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + (B*c + A*d)/(2*c*d)* Int[Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := -B*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(f*(m + n + 1)) /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[A*b*(m + n + 1) + a*B*(m - n), 0] && NeQ[m, -1/2]
+Int[Sqrt[a_. + b_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := B/d*Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x] - (B*c - A*d)/d* Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := (A*b - a*B)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*(2*m + 1)) + (a*B*(m - n) + A*b*(m + n + 1))/(a*b*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1/2] || ILtQ[m + n, 0] && Not[SumSimplerQ[n, 1]]) && NeQ[2*m + 1, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_ + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := -B*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(f*(m + n + 1)) - (B*c*(m - n) - A*d*(m + n + 1))/(d*(m + n + 1))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]] && NeQ[m + n + 1, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := (B*c - A*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)) /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[m + n + 2, 0] && EqQ[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)), 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := -b^2*(B*c - A*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d)) - b/(d*(n + 1)*(b*c + a*d))* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)* Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := -b*B*Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1)) + 1/(d*(m + n + 1))* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n* Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && Not[LtQ[n, -1]] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := (A*b - a*B)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*(2*m + 1)) - 1/(a*b*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)* Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1/2] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := b*(A*b - a*B)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d)) + 1/(a*(2*m + 1)*(b*c - a*d))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[B*(a*c*m + b*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1/2] && Not[GtQ[n, 0]] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := -2*b*B* Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)* Sqrt[a + b*Sin[e + f*x]]) /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)), 0]
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := -b^2*(B*c - A*d)* Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1)/(d* f*(n + 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]) + (A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2* d*(n + 1)*(b*c + a*d))* Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := -2*b*B* Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)* Sqrt[a + b*Sin[e + f*x]]) + (A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3))* Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[LtQ[n, -1]]
+Int[(A_. + B_.*sin[e_. + f_.*x_])/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol] := (A*b - a*B)/b* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] + B/b*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := -B*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(f*(m + n + 1)) + 1/(b*(m + n + 1))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1)* Simp[A*b*c*(m + n + 1) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + 1) + B*(a*d*m + b*c*n))* Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && (IntegerQ[n] || EqQ[m + 1/2, 0])
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := (B*c - A*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)) + 1/(b*(n + 1)*(c^2 - d^2))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)* Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 0])
+Int[(A_. + B_.*sin[e_. + f_.*x_])/(Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol] := (A*b - a*B)/(b*c - a*d)*Int[1/Sqrt[a + b*Sin[e + f*x]], x] + (B*c - A*d)/(b*c - a*d)* Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^ m_*(A_. + B_.*sin[e_. + f_.*x_])/(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := B/d*Int[(a + b*Sin[e + f*x])^m, x] - (B*c - A*d)/d* Int[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[m + 1/2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := (A*b - a*B)/b* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x] + B/b*Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A*b + a*B, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^2*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := (B*c - A*d)*(b*c - a*d)^2* Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1)/(f* d^2*(n + 1)*(c^2 - d^2)) - 1/(d^2*(n + 1)*(c^2 - d^2))*Int[(c + d*Sin[e + f*x])^(n + 1)* Simp[d*(n + 1)*(B*(b*c - a*d)^2 - A*d*(a^2*c + b^2*c - 2*a*b*d)) - ((B*c - A*d)*(a^2*d^2*(n + 2) + b^2*(c^2 + d^2*(n + 1))) + 2*a*b*d*(A*c*d*(n + 2) - B*(c^2 + d^2*(n + 1))))* Sin[e + f*x] - b^2*B*d*(n + 1)*(c^2 - d^2)*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := -(b*c - a*d)*(B*c - A*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2)) + 1/(d*(n + 1)*(c^2 - d^2))* Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)* Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a*B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := -b*B*Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1)) + 1/(d*(m + n + 1))* Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n* Simp[a^2*A*d*(m + n + 1) + b*B*(b*c*(m - 1) + a*d*(n + 1)) + (a*d*(2*A*b + a*B)*(m + n + 1) - b*B*(a*c - b*d*(m + n)))*Sin[e + f*x] + b*(A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))* Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && Not[IGtQ[n, 1] && (Not[IntegerQ[m]] || EqQ[a, 0] && NeQ[c, 0])]
+Int[Sqrt[ c_ + d_.*sin[e_. + f_.*x_]]*(A_. + B_.*sin[e_. + f_.*x_])/(b_.*sin[e_. + f_.*x_])^(3/2), x_Symbol] := B*d/b^2*Int[Sqrt[b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + Int[(A*c + (B*c + A*d)*Sin[e + f*x])/((b*Sin[e + f*x])^(3/2)* Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0]
+Int[Sqrt[ c_. + d_.*sin[e_. + f_.*x_]]*(A_. + B_.*sin[e_. + f_.*x_])/(a_ + b_.*sin[e_. + f_.*x_])^(3/2), x_Symbol] := B/b*Int[Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x] + (A*b - a*B)/b* Int[Sqrt[c + d*Sin[e + f*x]]/(a + b*Sin[e + f*x])^(3/2), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(A_. + B_.*sin[e_. + f_.*x_])/((a_ + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol] := 2*(A*b - a*B)* Cos[e + f*x]/(f*(a^2 - b^2)*Sqrt[a + b*Sin[e + f*x]]* Sqrt[d*Sin[e + f*x]]) + d/(a^2 - b^2)* Int[(A*b - a*B + (a*A - b*B)*Sin[e + f*x])/(Sqrt[ a + b*Sin[e + f*x]]*(d*Sin[e + f*x])^(3/2)), x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[a^2 - b^2, 0]
+Int[(A_ + B_.*sin[e_. + f_.*x_])/((b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := -2*A*(c - d)*Tan[e + f*x]/(f*b*c^2)*Rt[(c + d)/b, 2]* Sqrt[c*(1 + Csc[e + f*x])/(c - d)]* Sqrt[c*(1 - Csc[e + f*x])/(c + d)]* EllipticE[ ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/ Rt[(c + d)/b, 2]], -(c + d)/(c - d)] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]
+Int[(A_ + B_.*sin[e_. + f_.*x_])/((b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := -Sqrt[-b*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]* Int[(A + B*Sin[e + f*x])/((-b*Sin[e + f*x])^(3/2)* Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && NegQ[(c + d)/b]
+Int[(A_ + B_.*sin[e_. + f_.*x_])/((a_ + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := -2*A*(c - d)*(a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2]* Cos[e + f*x])* Sqrt[(b*c - a*d)*(1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x]))]* Sqrt[-(b*c - a*d)*(1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x]))]* EllipticE[ ArcSin[Rt[(a + b)/(c + d), 2]* Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]]], (a - b)*(c + d)/((a + b)*(c - d))] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]
+Int[(A_ + B_.*sin[e_. + f_.*x_])/((a_ + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := Sqrt[-c - d*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]]* Int[(A + B*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[-c - d*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && NegQ[(a + b)/(c + d)]
+Int[(A_. + B_.*sin[e_. + f_.*x_])/((a_. + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := (A - B)/(a - b)* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] - (A*b - a*B)/(a - b)* Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := (B*a - A*b)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^ n/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)* Simp[c*(a*A - b*B)*(m + 1) + d*n*(A*b - a*B) + (d*(a*A - b*B)*(m + 1) - c*(A*b - a*B)*(m + 2))*Sin[e + f*x] - d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := -(A*b^2 - a*b*B)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(1 + n)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)) + 1/((m + 1)*(b*c - a*d)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[(a*A - b*B)*(b*c - a*d)*(m + 1) + b*d*(A*b - a*B)*(m + n + 2) + (A*b - a*B)*(a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(A*b - a*B)*(m + n + 3)*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && RationalQ[m] && m < -1 && (EqQ[a, 0] && IntegerQ[m] && Not[IntegerQ[n]] || Not[IntegerQ[2*n] && LtQ[n, -1] && (IntegerQ[n] && Not[IntegerQ[m]] || EqQ[a, 0])])
+Int[(A_. + B_.*sin[e_. + f_.*x_])/((a_. + b_.*sin[e_. + f_.*x_])*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol] := (A*b - a*B)/(b*c - a*d)* Int[1/(a + b*Sin[e + f*x]), x] + (B*c - A*d)/(b*c - a*d)* Int[1/(c + d*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_*(A_. + B_.*sin[e_. + f_.*x_])/(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol] := B/d*Int[(a + b*Sin[e + f*x])^m, x] - (B*c - A*d)/d* Int[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[Sqrt[a_. + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := -2*B*Cos[e + f*x]* Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n/(f*(2*n + 3)) + 1/(2*n + 3)* Int[(c + d*Sin[e + f*x])^(n - 1)/Sqrt[a + b*Sin[e + f*x]]* Simp[a*A*c*(2*n + 3) + B*(b*c + 2*a*d*n) + (B*(a*c + b*d)*(2*n + 1) + A*(b*c + a*d)*(2*n + 3))* Sin[e + f*x] + (A*b*d*(2*n + 3) + B*(a*d + 2*b*c*n))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[n^2, 1/4]
+Int[(A_ + B_.*sin[e_. + f_.*x_])/(Sqrt[sin[e_. + f_.*x_]]* Sqrt[a_ + b_.*sin[e_. + f_.*x_]]), x_Symbol] := 4*A/(f*Sqrt[a + b])* EllipticPi[-1, -ArcSin[ Cos[e + f*x]/(1 + Sin[e + f*x])], -(a - b)/(a + b)] /; FreeQ[{a, b, e, f, A, B}, x] && GtQ[b, 0] && GtQ[b^2 - a^2, 0] && EqQ[A, B]
+Int[(A_ + B_.*sin[e_. + f_.*x_])/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[d_*sin[e_. + f_.*x_]]), x_Symbol] := Sqrt[Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]* Int[(A + B*Sin[e + f*x])/(Sqrt[Sin[e + f*x]]* Sqrt[a + b*Sin[e + f*x]]), x] /; FreeQ[{a, b, e, f, d, A, B}, x] && GtQ[b, 0] && GtQ[b^2 - a^2, 0] && EqQ[A, B]
+Int[(A_. + B_.*sin[e_. + f_.*x_])/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol] := B/d*Int[Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x] - (B*c - A*d)/d* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol] := Unintegrable[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^ n*(A + B*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+(* Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_ .*sin[e_.+f_.*x_])^p_,x_Symbol] := a^m*c^m*Int[Cos[e+f*x]^(2*m)*(c+d*Sin[e+f*x])^(n-m)*(A+B*Sin[e+f*x]) ^p,x] /; FreeQ[{a,b,c,d,e,f,A,B,n,p},x] && EqQ[b*c+a*d,0] && EqQ[a^2-b^2,0] &&  IntegerQ[m] && Not[IntegerQ[n] && (LtQ[m,0] && GtQ[n,0] || LtQ[0,n,m] ||  LtQ[m,n,0])] *)
+(* Int[(a_+b_.*cos[e_.+f_.*x_])^m_*(c_+d_.*cos[e_.+f_.*x_])^n_*(A_.+B_ .*cos[e_.+f_.*x_])^p_,x_Symbol] := a^m*c^m*Int[Sin[e+f*x]^(2*m)*(c+d*Cos[e+f*x])^(n-m)*(A+B*Cos[e+f*x]) ^p,x] /; FreeQ[{a,b,c,d,e,f,A,B,n,p},x] && EqQ[b*c+a*d,0] && EqQ[a^2-b^2,0] &&  IntegerQ[m] && Not[IntegerQ[n] && (LtQ[m,0] && GtQ[n,0] || LtQ[0,n,m] ||  LtQ[m,n,0])] *)
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_ + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_])^p_, x_Symbol] := Sqrt[a + b*Sin[e + f*x]]* Sqrt[c + d*Sin[e + f*x]]/(f*Cos[e + f*x])* Subst[ Int[(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2)*(A + B*x)^p, x], x, Sin[e + f*x]] /; FreeQ[{a, b, c, d, e, f, A, B, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[(a_ + b_.*cos[e_. + f_.*x_])^m_.*(c_ + d_.*cos[e_. + f_.*x_])^ n_.*(A_. + B_.*cos[e_. + f_.*x_])^p_, x_Symbol] := -Sqrt[a + b*Cos[e + f*x]]*Sqrt[c + d*Cos[e + f*x]]/(f*Sin[e + f*x])* Subst[ Int[(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2)*(A + B*x)^p, x], x, Cos[e + f*x]] /; FreeQ[{a, b, c, d, e, f, A, B, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m
new file mode 100755
index 0000000..d8aa61a
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m	
@@ -0,0 +1,22 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.4.1 (a+b sin)^m (A+B sin+C sin^2) *)
+Int[(b_.*sin[e_. + f_.*x_])^ m_.*(B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := 1/b*Int[(b*Sin[e + f*x])^(m + 1)*(B + C*Sin[e + f*x]), x] /; FreeQ[{b, e, f, B, C, m}, x]
+Int[(b_.*sin[e_. + f_.*x_])^m_.*(A_ + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := A*Cos[e + f*x]*(b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)) /; FreeQ[{b, e, f, A, C, m}, x] && EqQ[A*(m + 2) + C*(m + 1), 0]
+Int[(b_.*sin[e_. + f_.*x_])^m_*(A_ + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := A*Cos[ e + f*x]*(b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)) + (A*(m + 2) + C*(m + 1))/(b^2*(m + 1))*Int[(b*Sin[e + f*x])^(m + 2), x] /; FreeQ[{b, e, f, A, C}, x] && LtQ[m, -1]
+Int[sin[e_. + f_.*x_]^m_.*(A_ + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -1/f*Subst[Int[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2), x], x, Cos[e + f*x]] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2, 0]
+Int[(b_.*sin[e_. + f_.*x_])^m_.*(A_ + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -C*Cos[ e + f*x]*(b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2)) + (A*(m + 2) + C*(m + 1))/(m + 2)*Int[(b*Sin[e + f*x])^m, x] /; FreeQ[{b, e, f, A, C, m}, x] && Not[LtQ[m, -1]]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := 1/b^2* Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[b*B - a*C + b*C*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := C/b^2* Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[-a + b*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A*b^2 + a^2*C, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := (A - C)*Int[(a + b*Sin[e + f*x])^m*(1 + Sin[e + f*x]), x] + C*Int[(a + b*Sin[e + f*x])^m*(1 + Sin[e + f*x])^2, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A - B + C, 0] && Not[IntegerQ[2*m]]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := (A - C)*Int[(a + b*Sin[e + f*x])^m*(1 + Sin[e + f*x]), x] + C*Int[(a + b*Sin[e + f*x])^m*(1 + Sin[e + f*x])^2, x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A + C, 0] && Not[IntegerQ[2*m]]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^ m_*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := (A*b - a*B + b*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^m/(a*f*(2*m + 1)) + 1/(a^2*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[a*A*(m + 1) + m*(b*B - a*C) + b*C*(2*m + 1)*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := b*(A + C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m/(a*f*(2*m + 1)) + 1/(a^2*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[a*A*(m + 1) - a*C*m + b*C*(2*m + 1)*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -(A*b^2 - a*b*B + a^2*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 - b^2)) + 1/(b*(m + 1)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -(A*b^2 + a^2*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 - b^2)) + 1/(b*(m + 1)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[a*b*(A + C)*(m + 1) - (A*b^2 + a^2*C + b^2*(A + C)*(m + 1))*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[(a + b*Sin[e + f*x])^m* Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && Not[LtQ[m, -1]]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[(a + b*Sin[e + f*x])^m* Simp[A*b*(m + 2) + b*C*(m + 1) - a*C*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && Not[LtQ[m, -1]]
+Int[(b_.*sin[e_. + f_.*x_]^p_)^ m_*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := (b*Sin[e + f*x]^p)^m/(b*Sin[e + f*x])^(m*p)* Int[(b*Sin[e + f*x])^(m*p)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x] /; FreeQ[{b, e, f, A, B, C, m, p}, x] && Not[IntegerQ[m]]
+Int[(b_.*cos[e_. + f_.*x_]^p_)^ m_*(A_. + B_.*cos[e_. + f_.*x_] + C_.*cos[e_. + f_.*x_]^2), x_Symbol] := (b*Cos[e + f*x]^p)^m/(b*Cos[e + f*x])^(m*p)* Int[(b*Cos[e + f*x])^(m*p)*(A + B*Cos[e + f*x] + C*Cos[e + f*x]^2), x] /; FreeQ[{b, e, f, A, B, C, m, p}, x] && Not[IntegerQ[m]]
+Int[(b_.*sin[e_. + f_.*x_]^p_)^m_*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := (b*Sin[e + f*x]^p)^m/(b*Sin[e + f*x])^(m*p)* Int[(b*Sin[e + f*x])^(m*p)*(A + C*Sin[e + f*x]^2), x] /; FreeQ[{b, e, f, A, C, m, p}, x] && Not[IntegerQ[m]]
+Int[(b_.*cos[e_. + f_.*x_]^p_)^m_*(A_. + C_.*cos[e_. + f_.*x_]^2), x_Symbol] := (b*Cos[e + f*x]^p)^m/(b*Cos[e + f*x])^(m*p)* Int[(b*Cos[e + f*x])^(m*p)*(A + C*Cos[e + f*x]^2), x] /; FreeQ[{b, e, f, A, C, m, p}, x] && Not[IntegerQ[m]]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m
new file mode 100755
index 0000000..f7f59f8
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m	
@@ -0,0 +1,45 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2) *)
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := 1/b^2* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^ n*(b*B - a*C + b*C*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -C/b^2* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^ n*(a - b*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 + a^2*C, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_*(c_. + d_.*sin[e_. + f_.*x_])*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -(b*c - a*d)*(A*b^2 - a*b*B + a^2*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b^2* f*(m + 1)*(a^2 - b^2)) - 1/(b^2*(m + 1)*(a^2 - b^2))*Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b* B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))* Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_*(c_. + d_.*sin[e_. + f_.*x_])*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -(b*c - a*d)*(A*b^2 + a^2*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b^2* f*(m + 1)*(a^2 - b^2)) + 1/(b^2*(m + 1)*(a^2 - b^2))*Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[b*(m + 1)*(a*C*(b*c - a*d) + A*b*(a*c - b*d)) - ((b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))* Sin[e + f*x] + b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -C*d*Cos[e + f*x]* Sin[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3)) + 1/(b*(m + 3))*Int[(a + b*Sin[e + f*x])^m* Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))* Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))* Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && Not[LtQ[m, -1]]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -C*d*Cos[e + f*x]* Sin[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3)) + 1/(b*(m + 3))*Int[(a + b*Sin[e + f*x])^m* Simp[a*C*d + A*b*c*(m + 3) + b*d*(C*(m + 2) + A*(m + 3))* Sin[e + f*x] - (2*a*C*d - b*c*C*(m + 3))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && Not[LtQ[m, -1]]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := (a*A - b*B + a*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(2*b*c*f*(2*m + 1)) - 1/(2*b*c*d*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[A*(c^2*(m + 1) + d^2*(2*m + n + 2)) - B*c*d*(m - n - 1) - C*(c^2*m - d^2*(n + 1)) + d*((A*c + B*d)*(m + n + 2) - c*C*(3*m - n))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1/2] || EqQ[m + n + 2, 0] && NeQ[2*m + 1, 0])
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := (a*A + a*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(2*b*c*f*(2*m + 1)) - 1/(2*b*c*d*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[A*(c^2*(m + 1) + d^2*(2*m + n + 2)) - C*(c^2*m - d^2*(n + 1)) + d*(A*c*(m + n + 2) - c*C*(3*m - n))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1/2] || EqQ[m + n + 2, 0] && NeQ[2*m + 1, 0])
+Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2)/ Sqrt[c_. + d_.*sin[e_. + f_.*x_]], x_Symbol] := -2*C*Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(2*m + 3)* Sqrt[c + d*Sin[e + f*x]]) + Int[(a + b*Sin[e + f*x])^m* Simp[A + C + B*Sin[e + f*x], x]/Sqrt[c + d*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_.*(A_. + C_.*sin[e_. + f_.*x_]^2)/ Sqrt[c_. + d_.*sin[e_. + f_.*x_]], x_Symbol] := -2*C*Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(2*m + 3)* Sqrt[c + d*Sin[e + f*x]]) + (A + C)*Int[(a + b*Sin[e + f*x])^m/Sqrt[c + d*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -C*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2)) + 1/(b*d*(m + n + 2))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n* Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (b*B*d*(m + n + 2) - b*c*C*(2*m + 1))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]] && NeQ[m + n + 2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -C*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2)) + 1/(b*d*(m + n + 2))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n* Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) - b*c*C*(2*m + 1)*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]] && NeQ[m + n + 2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := (a*A - b*B + a*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(f*(b*c - a*d)*(2*m + 1)) + 1/(b*(b*c - a*d)*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) + B*(b*c*m + a*d*(n + 1)) - C*(a*c*m + b*d*(n + 1)) + (d*(a*A - b*B)*(m + n + 2) + C*(b*c*(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1/2]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := a*(A + C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(f*(b*c - a*d)*(2*m + 1)) + 1/(b*(b*c - a*d)*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) - C*(a*c*m + b*d*(n + 1)) + (a*A*d*(m + n + 2) + C*(b*c*(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1/2]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -(c^2*C - B*c*d + A*d^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2)) + 1/(b*d*(n + 1)*(c^2 - d^2))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)* Simp[A*d*(a*d*m + b*c*(n + 1)) + (c*C - B*d)*(a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))* Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[LtQ[m, -1/2]] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -(c^2*C + A*d^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2)) + 1/(b*d*(n + 1)*(c^2 - d^2))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)* Simp[A*d*(a*d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))* Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[LtQ[m, -1/2]] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -C*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2)) + 1/(b*d*(m + n + 2))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n* Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[LtQ[m, -1/2]] && NeQ[m + n + 2, 0]
+Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -C*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2)) + 1/(b*d*(m + n + 2))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n* Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[LtQ[m, -1/2]] && NeQ[m + n + 2, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -(c^2*C - B*c*d + A*d^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2)) + 1/(d*(n + 1)*(c^2 - d^2))* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)* Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))* Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -(c^2*C + A*d^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2)) + 1/(d*(n + 1)*(c^2 - d^2))* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)* Simp[A*d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))* Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -C*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2)) + 1/(d*(m + n + 2))* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n* Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))* Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))* Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && Not[IGtQ[n, 0] && (Not[IntegerQ[m]] || EqQ[a, 0] && NeQ[c, 0])]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -C*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2)) + 1/(d*(m + n + 2))* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n* Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && Not[IGtQ[n, 0] && (Not[IntegerQ[m]] || EqQ[a, 0] && NeQ[c, 0])]
+Int[(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2)/((a_ + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol] := C/(b*d)*Int[Sqrt[d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x] + 1/b* Int[(A*b + (b*B - a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[d*Sin[e + f*x]]), x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
+Int[(A_. + C_.*sin[e_. + f_.*x_]^2)/((a_ + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol] := C/(b*d)*Int[Sqrt[d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x] + 1/b* Int[(A*b - a*C*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[d*Sin[e + f*x]]), x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0]
+Int[(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2)/((a_. + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol] := C/b^2*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + 1/b^2* Int[(A*b^2 - a^2*C + b*(b*B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(A_. + C_.*sin[e_. + f_.*x_]^2)/((a_. + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol] := C/b^2*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + 1/b^2* Int[(A*b^2 - a^2*C - 2*a*b*C*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -(A*b^2 - a*b*B + a^2*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)) + 1/((m + 1)*(b*c - a*d)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[(m + 1)*(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && (EqQ[a, 0] && IntegerQ[m] && Not[IntegerQ[n]] || Not[IntegerQ[2*n] && LtQ[n, -1] && (IntegerQ[n] && Not[IntegerQ[m]] || EqQ[a, 0])])
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := -(A*b^2 + a^2*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)) + 1/((m + 1)*(b*c - a*d)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))* Sin[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && (EqQ[a, 0] && IntegerQ[m] && Not[IntegerQ[n]] || Not[IntegerQ[2*n] && LtQ[n, -1] && (IntegerQ[n] && Not[IntegerQ[m]] || EqQ[a, 0])])
+Int[(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2)/((a_ + b_.*sin[e_. + f_.*x_])*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol] := C*x/(b*d) + (A*b^2 - a*b*B + a^2*C)/(b*(b*c - a*d))* Int[1/(a + b*Sin[e + f*x]), x] - (c^2*C - B*c*d + A*d^2)/(d*(b*c - a*d))* Int[1/(c + d*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(A_. + C_.*sin[e_. + f_.*x_]^2)/((a_ + b_.*sin[e_. + f_.*x_])*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol] := C*x/(b*d) + (A*b^2 + a^2*C)/(b*(b*c - a*d))* Int[1/(a + b*Sin[e + f*x]), x] - (c^2*C + A*d^2)/(d*(b*c - a*d))*Int[1/(c + d*Sin[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2)/(Sqrt[ a_. + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol] := C/(b*d)*Int[Sqrt[a + b*Sin[e + f*x]], x] - 1/(b*d)* Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(A_. + C_.*sin[e_. + f_.*x_]^2)/(Sqrt[ a_. + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol] := C/(b*d)*Int[Sqrt[a + b*Sin[e + f*x]], x] - 1/(b*d)* Int[Simp[a*c*C - A*b*d + (b*c*C + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2)/(Sqrt[a_. + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := -C*Cos[e + f*x]* Sqrt[c + d*Sin[e + f*x]]/(d*f*Sqrt[a + b*Sin[e + f*x]]) + 1/(2*d)* Int[1/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]])* Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - d*(A*b + a*B))* Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d))*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(A_. + C_.*sin[e_. + f_.*x_]^2)/(Sqrt[a_. + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol] := -C*Cos[e + f*x]* Sqrt[c + d*Sin[e + f*x]]/(d*f*Sqrt[a + b*Sin[e + f*x]]) + 1/(2*d)* Int[1/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]])* Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - A*b*d)*Sin[e + f*x] - C*(b*c + a*d)*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2)/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := (b*B - a*C)/b^2*Int[(d*Sin[e + f*x])^n, x] + C/(b*d)*Int[(d*Sin[e + f*x])^(n + 1), x] + (A*b^2 - a*b*B + a^2*C)/b^2* Int[(d*Sin[e + f*x])^n/(a + b*Sin[e + f*x]), x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0]
+Int[(d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2)/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol] := -a*C/b^2*Int[(d*Sin[e + f*x])^n, x] + C/(b*d)*Int[(d*Sin[e + f*x])^(n + 1), x] + (A*b^2 + a^2*C)/b^2* Int[(d*Sin[e + f*x])^n/(a + b*Sin[e + f*x]), x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && NeQ[a^2 - b^2, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := Unintegrable[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^ n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := Unintegrable[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^ n*(A + C*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(b_.*sin[e_. + f_.*x_]^p_)^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := (b*Sin[e + f*x]^p)^m/(b*Sin[e + f*x])^(m*p)* Int[(b*Sin[e + f*x])^(m*p)*(c + d*Sin[e + f*x])^ n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x] /; FreeQ[{b, c, d, e, f, A, B, C, m, n, p}, x] && Not[IntegerQ[m]]
+Int[(b_.*cos[e_. + f_.*x_]^p_)^m_*(c_. + d_.*cos[e_. + f_.*x_])^ n_.*(A_. + B_.*cos[e_. + f_.*x_] + C_.*cos[e_. + f_.*x_]^2), x_Symbol] := (b*Cos[e + f*x]^p)^m/(b*Cos[e + f*x])^(m*p)* Int[(b*Cos[e + f*x])^(m*p)*(c + d*Cos[e + f*x])^ n*(A + B*Cos[e + f*x] + C*Cos[e + f*x]^2), x] /; FreeQ[{b, c, d, e, f, A, B, C, m, n, p}, x] && Not[IntegerQ[m]]
+Int[(b_.*sin[e_. + f_.*x_]^p_)^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol] := (b*Sin[e + f*x]^p)^m/(b*Sin[e + f*x])^(m*p)* Int[(b*Sin[e + f*x])^(m*p)*(c + d*Sin[e + f*x])^ n*(A + C*Sin[e + f*x]^2), x] /; FreeQ[{b, c, d, e, f, A, C, m, n, p}, x] && Not[IntegerQ[m]]
+Int[(b_.*cos[e_. + f_.*x_]^p_)^m_*(c_. + d_.*cos[e_. + f_.*x_])^ n_.*(A_. + C_.*cos[e_. + f_.*x_]^2), x_Symbol] := (b*Cos[e + f*x]^p)^m/(b*Cos[e + f*x])^(m*p)* Int[(b*Cos[e + f*x])^(m*p)*(c + d*Cos[e + f*x])^ n*(A + C*Cos[e + f*x]^2), x] /; FreeQ[{b, c, d, e, f, A, C, m, n, p}, x] && Not[IntegerQ[m]]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m
new file mode 100755
index 0000000..1f13fd8
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m	
@@ -0,0 +1,46 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.5 trig^m (a cos+b sin)^n *)
+Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := a*(a*Cos[c + d*x] + b*Sin[c + d*x])^n/(b*d*n) /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 + b^2, 0]
+Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := -1/d*Subst[Int[(a^2 + b^2 - x^2)^((n - 1)/2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[(n - 1)/2, 0]
+Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := -(b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/(d* n) + (n - 1)*(a^2 + b^2)/n* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && Not[IntegerQ[(n - 1)/2]] && GtQ[n, 1]
+Int[1/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol] := -1/d*Subst[Int[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
+Int[1/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^2, x_Symbol] := Sin[c + d*x]/(a*d*(a*Cos[c + d*x] + b*Sin[c + d*x])) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
+Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := (b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2)) + (n + 2)/((n + 1)*(a^2 + b^2))* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] && NeQ[n, -2]
+Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := (a^2 + b^2)^(n/2)*Int[(Cos[c + d*x - ArcTan[a, b]])^n, x] /; FreeQ[{a, b, c, d, n}, x] && Not[GeQ[n, 1] || LeQ[n, -1]] && GtQ[a^2 + b^2, 0]
+Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := (a*Cos[c + d*x] + b*Sin[c + d*x])^ n/((a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2])^n* Int[Cos[c + d*x - ArcTan[a, b]]^n, x] /; FreeQ[{a, b, c, d, n}, x] && Not[GeQ[n, 1] || LeQ[n, -1]] && Not[GtQ[a^2 + b^2, 0] || EqQ[a^2 + b^2, 0]]
+Int[sin[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := -a*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/(d*(n - 1)* Sin[c + d*x]^(n - 1)) + 2*b* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/ Sin[c + d*x]^(n - 1), x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && GtQ[n, 1]
+Int[cos[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := b*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/(d*(n - 1)* Cos[c + d*x]^(n - 1)) + 2*a* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/ Cos[c + d*x]^(n - 1), x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && GtQ[n, 1]
+Int[sin[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := a*(a*Cos[c + d*x] + b*Sin[c + d*x])^n/(2*b*d*n*Sin[c + d*x]^n) + 1/(2*b)* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1)/ Sin[c + d*x]^(n + 1), x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]
+Int[cos[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := -b*(a*Cos[c + d*x] + b*Sin[c + d*x])^n/(2*a*d*n*Cos[c + d*x]^n) + 1/(2*a)* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1)/ Cos[c + d*x]^(n + 1), x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]
+Int[sin[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := a*(a*Cos[c + d*x] + b*Sin[c + d*x])^n/(2*b*d*n*Sin[c + d*x]^n)* Hypergeometric2F1[1, n, n + 1, (b + a*Cot[c + d*x])/(2*b)] /; FreeQ[{a, b, c, d, n}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && Not[IntegerQ[n]]
+Int[cos[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := -b*(a*Cos[c + d*x] + b*Sin[c + d*x])^n/(2*a*d*n*Cos[c + d*x]^n)* Hypergeometric2F1[1, n, n + 1, (a + b*Tan[c + d*x])/(2*a)] /; FreeQ[{a, b, c, d, n}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && Not[IntegerQ[n]]
+Int[sin[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_., x_Symbol] := Int[(b + a*Cot[c + d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b^2, 0]
+Int[cos[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_., x_Symbol] := Int[(a + b*Tan[c + d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b^2, 0]
+Int[sin[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := 1/d*Subst[Int[x^m*(a + b*x)^n/(1 + x^2)^((m + n + 2)/2), x], x, Tan[c + d*x]] /; FreeQ[{a, b, c, d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] && Not[GtQ[n, 0] && GtQ[m, 1]]
+Int[cos[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := -1/d*Subst[Int[x^m*(b + a*x)^n/(1 + x^2)^((m + n + 2)/2), x], x, Cot[c + d*x]] /; FreeQ[{a, b, c, d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] && Not[GtQ[n, 0] && GtQ[m, 1]]
+Int[sin[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_., x_Symbol] := Int[ExpandTrig[sin[c + d*x]^m*(a*cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[m] && IGtQ[n, 0]
+Int[cos[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_., x_Symbol] := Int[ExpandTrig[cos[c + d*x]^m*(a*cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[m] && IGtQ[n, 0]
+Int[sin[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := a^n*b^n* Int[Sin[c + d*x]^m*(b*Cos[c + d*x] + a*Sin[c + d*x])^(-n), x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[a^2 + b^2, 0] && ILtQ[n, 0]
+Int[cos[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := a^n*b^n* Int[Cos[c + d*x]^m*(b*Cos[c + d*x] + a*Sin[c + d*x])^(-n), x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[a^2 + b^2, 0] && ILtQ[n, 0]
+Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_/ sin[c_. + d_.*x_], x_Symbol] := a*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/(d*(n - 1)) + b*Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1), x] + a^2* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2)/Sin[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
+Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_/ cos[c_. + d_.*x_], x_Symbol] := -b*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/(d*(n - 1)) + a*Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1), x] + b^2* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2)/Cos[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
+Int[sin[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := -(a^2 + b^2)* Int[Sin[c + d*x]^(m + 2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x] + 2*b* Int[Sin[c + d*x]^(m + 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1), x] + a^2* Int[Sin[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[n, 1] && LtQ[m, -1]
+Int[cos[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := -(a^2 + b^2)* Int[Cos[c + d*x]^(m + 2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x] + 2*a* Int[Cos[c + d*x]^(m + 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1), x] + b^2* Int[Cos[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[n, 1] && LtQ[m, -1]
+Int[sin[c_. + d_.*x_]/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol] := b*x/(a^2 + b^2) - a/(a^2 + b^2)* Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
+Int[cos[c_. + d_.*x_]/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol] := a*x/(a^2 + b^2) + b/(a^2 + b^2)* Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
+Int[sin[c_. + d_.*x_]^ m_/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol] := -a*Sin[c + d*x]^(m - 1)/(d*(a^2 + b^2)*(m - 1)) + b/(a^2 + b^2)*Int[Sin[c + d*x]^(m - 1), x] + a^2/(a^2 + b^2)* Int[Sin[c + d*x]^(m - 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[m, 1]
+Int[cos[c_. + d_.*x_]^ m_/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol] := b*Cos[c + d*x]^(m - 1)/(d*(a^2 + b^2)*(m - 1)) + a/(a^2 + b^2)*Int[Cos[c + d*x]^(m - 1), x] + b^2/(a^2 + b^2)* Int[Cos[c + d*x]^(m - 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[m, 1]
+Int[1/(sin[ c_. + d_.*x_]*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])), x_Symbol] := 1/a*Int[Cot[c + d*x], x] - 1/a* Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
+Int[1/(cos[ c_. + d_.*x_]*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])), x_Symbol] := 1/b*Int[Tan[c + d*x], x] + 1/b* Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
+Int[sin[c_. + d_.*x_]^ m_/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol] := Sin[c + d*x]^(m + 1)/(a*d*(m + 1)) - b/a^2*Int[Sin[c + d*x]^(m + 1), x] + (a^2 + b^2)/a^2* Int[Sin[c + d*x]^(m + 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]
+Int[cos[c_. + d_.*x_]^ m_/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol] := -Cos[c + d*x]^(m + 1)/(b*d*(m + 1)) - a/b^2*Int[Cos[c + d*x]^(m + 1), x] + (a^2 + b^2)/b^2* Int[Cos[c + d*x]^(m + 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]
+Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_/ sin[c_. + d_.*x_], x_Symbol] := -(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1)/(a*d*(n + 1)) - b/a^2*Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1), x] + 1/a^2* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2)/Sin[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
+Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_/ cos[c_. + d_.*x_], x_Symbol] := (a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)) - a/b^2*Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1), x] + 1/b^2* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2)/Cos[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
+Int[sin[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := (a^2 + b^2)/a^2* Int[Sin[c + d*x]^(m + 2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^n, x] - 2*b/a^2* Int[Sin[c + d*x]^(m + 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1), x] + 1/a^2* Int[Sin[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] && LtQ[m, -1]
+Int[cos[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol] := (a^2 + b^2)/b^2* Int[Cos[c + d*x]^(m + 2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^n, x] - 2*a/b^2* Int[Cos[c + d*x]^(m + 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1), x] + 1/b^2* Int[Cos[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] && LtQ[m, -1]
+Int[cos[c_. + d_.*x_]^m_.* sin[c_. + d_.*x_]^ n_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^p_., x_Symbol] := Int[ExpandTrig[ cos[c + d*x]^m*sin[c + d*x]^n*(a*cos[c + d*x] + b*sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0]
+Int[cos[c_. + d_.*x_]^m_.* sin[c_. + d_.*x_]^ n_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^p_, x_Symbol] := a^p*b^p* Int[Cos[c + d*x]^m* Sin[c + d*x]^n*(b*Cos[c + d*x] + a*Sin[c + d*x])^(-p), x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a^2 + b^2, 0] && ILtQ[p, 0]
+Int[cos[c_. + d_.*x_]^m_.* sin[c_. + d_.*x_]^ n_./(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol] := b/(a^2 + b^2)*Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x] + a/(a^2 + b^2)*Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x] - a*b/(a^2 + b^2)* Int[Cos[c + d*x]^(m - 1)* Sin[c + d*x]^(n - 1)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[cos[c_. + d_.*x_]^m_.* sin[c_. + d_.*x_]^ n_./(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol] := Int[ExpandTrig[ cos[c + d*x]^m*sin[c + d*x]^n/(a*cos[c + d*x] + b*sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegersQ[m, n]
+Int[cos[c_. + d_.*x_]^m_.* sin[c_. + d_.*x_]^ n_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^p_, x_Symbol] := b/(a^2 + b^2)* Int[Cos[c + d*x]^m* Sin[c + d*x]^(n - 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(p + 1), x] + a/(a^2 + b^2)* Int[Cos[c + d*x]^(m - 1)* Sin[c + d*x]^n*(a*Cos[c + d*x] + b*Sin[c + d*x])^(p + 1), x] - a*b/(a^2 + b^2)* Int[Cos[c + d*x]^(m - 1)* Sin[c + d*x]^(n - 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^p, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0] && IGtQ[n, 0] && ILtQ[p, 0]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m
new file mode 100755
index 0000000..0f78355
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m	
@@ -0,0 +1,62 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.6 (a+b cos+c sin)^n *)
+Int[Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol] := -2*(c*Cos[d + e*x] - b*Sin[d + e*x])/(e* Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]
+Int[(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_, x_Symbol] := -(c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)/(e*n) + a*(2*n - 1)/n* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0] && GtQ[n, 0]
+Int[1/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol] := -(c - a*Sin[d + e*x])/(c*e*(c*Cos[d + e*x] - b*Sin[d + e*x])) /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]
+Int[1/Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol] := Int[1/Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]
+Int[(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_, x_Symbol] := (c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(2*n + 1)) + (n + 1)/(a*(2*n + 1))* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1]
+Int[Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol] := b/(c*e)* Subst[Int[Sqrt[a + x]/x, x], x, b*Cos[d + e*x] + c*Sin[d + e*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[b^2 + c^2, 0]
+Int[Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol] := Int[Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 + c^2, 0] && GtQ[a + Sqrt[b^2 + c^2], 0]
+Int[Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol] := Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/ Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]* Int[ Sqrt[a/(a + Sqrt[b^2 + c^2]) + Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2])* Cos[d + e*x - ArcTan[b, c]]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && Not[GtQ[a + Sqrt[b^2 + c^2], 0]]
+Int[(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_, x_Symbol] := -(c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)/(e*n) + 1/n* Int[Simp[ n*a^2 + (n - 1)*(b^2 + c^2) + a*b*(2*n - 1)*Cos[d + e*x] + a*c*(2*n - 1)*Sin[d + e*x], x]* (a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 2), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && GtQ[n, 1]
+(* Int[1/(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := x/Sqrt[a^2-b^2-c^2] + 2/(e*Sqrt[a^2-b^2-c^2])*ArcTan[(c*Cos[d+e*x]-b*Sin[d+e*x])/(a+Sqrt[ a^2-b^2-c^2]+b*Cos[d+e*x]+c*Sin[d+e*x])] /; FreeQ[{a,b,c,d,e},x] && GtQ[a^2-b^2-c^2,0] *)
+(* Int[1/(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol] := Log[RemoveContent[b^2+c^2+(a*b-c*Rt[-a^2+b^2+c^2,2])*Cos[d+e*x]+(a* c+b*Sqrt[-a^2+b^2+c^2])*Sin[d+e*x],x]]/ (2*e*Rt[-a^2+b^2+c^2,2]) - Log[RemoveContent[b^2+c^2+(a*b+c*Rt[-a^2+b^2+c^2,2])*Cos[d+e*x]+(a* c-b*Sqrt[-a^2+b^2+c^2])*Sin[d+e*x],x]]/ (2*e*Rt[-a^2+b^2+c^2,2]) /; FreeQ[{a,b,c,d,e},x] && LtQ[a^2-b^2-c^2,0] *)
+Int[1/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol] := Module[{f = FreeFactors[Cot[(d + e*x)/2], x]}, -f/e*Subst[Int[1/(a + c*f*x), x], x, Cot[(d + e*x)/2]/f]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a + b, 0]
+Int[1/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol] := Module[{f = FreeFactors[Tan[(d + e*x)/2 + Pi/4], x]}, f/e*Subst[Int[1/(a + b*f*x), x], x, Tan[(d + e*x)/2 + Pi/4]/f]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a + c, 0]
+Int[1/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol] := Module[{f = FreeFactors[Cot[(d + e*x)/2 + Pi/4], x]}, -f/e* Subst[Int[1/(a + b*f*x), x], x, Cot[(d + e*x)/2 + Pi/4]/f]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a - c, 0] && NeQ[a - b, 0]
+Int[1/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol] := Module[{f = FreeFactors[Tan[(d + e*x)/2], x]}, 2*f/e* Subst[Int[1/(a + b + 2*c*f*x + (a - b)*f^2*x^2), x], x, Tan[(d + e*x)/2]/f]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]
+Int[1/Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol] := b/(c*e)* Subst[Int[1/(x*Sqrt[a + x]), x], x, b*Cos[d + e*x] + c*Sin[d + e*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[b^2 + c^2, 0]
+Int[1/Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol] := Int[1/Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 + c^2, 0] && GtQ[a + Sqrt[b^2 + c^2], 0]
+Int[1/Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol] := Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]/ Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]* Int[ 1/Sqrt[a/(a + Sqrt[b^2 + c^2]) + Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2])* Cos[d + e*x - ArcTan[b, c]]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && Not[GtQ[a + Sqrt[b^2 + c^2], 0]]
+Int[1/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^(3/2), x_Symbol] := 2*(c*Cos[d + e*x] - b*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)* Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]) + 1/(a^2 - b^2 - c^2)* Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]
+Int[(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_, x_Symbol] := (-c*Cos[d + e*x] + b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(a^2 - b^2 - c^2)) + 1/((n + 1)*(a^2 - b^2 - c^2))* Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c*(n + 2)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1] && NeQ[n, -3/2]
+Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol] := (2*a*A - b*B - c*C)* x/(2*a^2) - (b*B + c*C)*(b*Cos[d + e*x] - c*Sin[d + e*x])/(2*a*b*c*e) + (a^2*(b*B - c*C) - 2*a*A*b^2 + b^2*(b*B + c*C))* Log[RemoveContent[a + b*Cos[d + e*x] + c*Sin[d + e*x], x]]/(2*a^2* b*c*e) /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[b^2 + c^2, 0]
+Int[(A_. + C_.*sin[d_. + e_.*x_])/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol] := (2*a*A - c*C)*x/(2*a^2) - C*Cos[d + e*x]/(2*a*e) + c*C*Sin[d + e*x]/(2*a*b*e) + (-a^2*C + 2*a*c*A + b^2*C)* Log[RemoveContent[a + b*Cos[d + e*x] + c*Sin[d + e*x], x]]/(2*a^2* b*e) /; FreeQ[{a, b, c, d, e, A, C}, x] && EqQ[b^2 + c^2, 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_])/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol] := (2*a*A - b*B)*x/(2*a^2) - b*B*Cos[d + e*x]/(2*a*c*e) + B*Sin[d + e*x]/(2*a*e) + (a^2*B - 2*a*b*A + b^2*B)* Log[RemoveContent[a + b*Cos[d + e*x] + c*Sin[d + e*x], x]]/(2*a^2* c*e) /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 + c^2, 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol] := (b*B + c*C)*x/(b^2 + c^2) + (c*B - b*C)* Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2)) /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2 + c^2, 0] && EqQ[A*(b^2 + c^2) - a*(b*B + c*C), 0]
+Int[(A_. + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol] := c*C*x/(b^2 + c^2) - b*C*Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2)) /; FreeQ[{a, b, c, d, e, A, C}, x] && NeQ[b^2 + c^2, 0] && EqQ[A*(b^2 + c^2) - a*c*C, 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol] := b*B*x/(b^2 + c^2) + c*B*Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2)) /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 + c^2, 0] && EqQ[A*(b^2 + c^2) - a*b*B, 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol] := (b*B + c*C)*x/(b^2 + c^2) + (c*B - b*C)* Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2)) + (A*(b^2 + c^2) - a*(b*B + c*C))/(b^2 + c^2)* Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2 + c^2, 0] && NeQ[A*(b^2 + c^2) - a*(b*B + c*C), 0]
+Int[(A_. + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol] := c*C*(d + e*x)/(e*(b^2 + c^2)) - b*C*Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2)) + (A*(b^2 + c^2) - a*c*C)/(b^2 + c^2)* Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x] /; FreeQ[{a, b, c, d, e, A, C}, x] && NeQ[b^2 + c^2, 0] && NeQ[A*(b^2 + c^2) - a*c*C, 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol] := b*B*(d + e*x)/(e*(b^2 + c^2)) + c*B*Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2)) + (A*(b^2 + c^2) - a*b*B)/(b^2 + c^2)* Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 + c^2, 0] && NeQ[A*(b^2 + c^2) - a*b*B, 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol] := (B*c - b*C - a*C*Cos[d + e*x] + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1)) /; FreeQ[{a, b, c, d, e, A, B, C, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && EqQ[(b*B + c*C)*n + a*A*(n + 1), 0]
+Int[(A_. + C_.*sin[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol] := -(b*C + a*C*Cos[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1)) /; FreeQ[{a, b, c, d, e, A, C, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && EqQ[c*C*n + a*A*(n + 1), 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol] := (B*c + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1)) /; FreeQ[{a, b, c, d, e, A, B, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && EqQ[b*B*n + a*A*(n + 1), 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol] := (B*c - b*C - a*C*Cos[d + e*x] + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1)) + ((b*B + c*C)*n + a*A*(n + 1))/(a*(n + 1))* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e, A, B, C, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && NeQ[(b*B + c*C)*n + a*A*(n + 1), 0]
+Int[(A_. + C_.*sin[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol] := -(b*C + a*C*Cos[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1)) + (c*C*n + a*A*(n + 1))/(a*(n + 1))* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e, A, C, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && NeQ[c*C*n + a*A*(n + 1), 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol] := (B*c + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1)) + (b*B*n + a*A*(n + 1))/(a*(n + 1))* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && NeQ[b*B*n + a*A*(n + 1), 0]
+Int[(B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])*(b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol] := (c*B - b*C)*(b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(b^2 + c^2)) /; FreeQ[{b, c, d, e, B, C}, x] && NeQ[n, -1] && NeQ[b^2 + c^2, 0] && EqQ[b*B + c*C, 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol] := (B*c - b*C - a*C*Cos[d + e*x] + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1)) + 1/(a*(n + 1))*Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)* Simp[a*(b*B + c*C)*n + a^2*A*(n + 1) + (n*(a^2*B - B*c^2 + b*c*C) + a*b*A*(n + 1))* Cos[d + e*x] + (n*(b*B*c + a^2*C - b^2*C) + a*c*A*(n + 1))* Sin[d + e*x], x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && GtQ[n, 0] && NeQ[a^2 - b^2 - c^2, 0]
+Int[(A_. + C_.*sin[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol] := -(b*C + a*C*Cos[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1)) + 1/(a*(n + 1))*Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)* Simp[a*c*C*n + a^2*A*(n + 1) + (c*b*C*n + a*b*A*(n + 1))* Cos[d + e*x] + (a^2*C*n - b^2*C*n + a*c*A*(n + 1))* Sin[d + e*x], x], x] /; FreeQ[{a, b, c, d, e, A, C}, x] && GtQ[n, 0] && NeQ[a^2 - b^2 - c^2, 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol] := (B*c + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1)) + 1/(a*(n + 1))*Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)* Simp[a*b*B*n + a^2*A*(n + 1) + (a^2*B*n - c^2*B*n + a*b*A*(n + 1))* Cos[d + e*x] + (b*c*B*n + a*c*A*(n + 1))*Sin[d + e*x], x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && GtQ[n, 0] && NeQ[a^2 - b^2 - c^2, 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])/ Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol] := B/b*Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] + (A*b - a*B)/b* Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[A*b - a*B, 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^2, x_Symbol] := (c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)*Sin[d + e*x])/ (e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])) /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[a^2 - b^2 - c^2, 0] && EqQ[a*A - b*B - c*C, 0]
+Int[(A_. + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^2, x_Symbol] := -(b*C + (a*C - c*A)*Cos[d + e*x] + b*A*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])) /; FreeQ[{a, b, c, d, e, A, C}, x] && NeQ[a^2 - b^2 - c^2, 0] && EqQ[a*A - c*C, 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^2, x_Symbol] := (c*B + c*A*Cos[d + e*x] + (a*B - b*A)* Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])) /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[a^2 - b^2 - c^2, 0] && EqQ[a*A - b*B, 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^2, x_Symbol] := (c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)*Sin[d + e*x])/ (e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])) + (a*A - b*B - c*C)/(a^2 - b^2 - c^2)* Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[a*A - b*B - c*C, 0]
+Int[(A_. + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^2, x_Symbol] := -(b*C + (a*C - c*A)*Cos[d + e*x] + b*A*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])) + (a*A - c*C)/(a^2 - b^2 - c^2)* Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x] /; FreeQ[{a, b, c, d, e, A, C}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[a*A - c*C, 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^2, x_Symbol] := (c*B + c*A*Cos[d + e*x] + (a*B - b*A)* Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])) + (a*A - b*B)/(a^2 - b^2 - c^2)* Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[a*A - b*B, 0]
+Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])*(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_, x_Symbol] := -(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/ (e*(n + 1)*(a^2 - b^2 - c^2)) + 1/((n + 1)*(a^2 - b^2 - c^2))* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)* Simp[(n + 1)*(a*A - b*B - c*C) + (n + 2)*(a*B - b*A)* Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]
+Int[(A_. + C_.*sin[d_. + e_.*x_])*(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_, x_Symbol] := (b*C + (a*C - c*A)*Cos[d + e*x] + b*A*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/ (e*(n + 1)*(a^2 - b^2 - c^2)) + 1/((n + 1)*(a^2 - b^2 - c^2))* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)* Simp[(n + 1)*(a*A - c*C) - (n + 2)*b*A* Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x] /; FreeQ[{a, b, c, d, e, A, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]
+Int[(A_. + B_.*cos[d_. + e_.*x_])*(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_, x_Symbol] := -(c*B + c*A*Cos[d + e*x] + (a*B - b*A)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/ (e*(n + 1)*(a^2 - b^2 - c^2)) + 1/((n + 1)*(a^2 - b^2 - c^2))* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)* Simp[(n + 1)*(a*A - b*B) + (n + 2)*(a*B - b*A)* Cos[d + e*x] - (n + 2)*c*A*Sin[d + e*x], x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]
+Int[1/(a_. + b_.*sec[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_]), x_Symbol] := Int[Cos[d + e*x]/(b + a*Cos[d + e*x] + c*Sin[d + e*x]), x] /; FreeQ[{a, b, c, d, e}, x]
+Int[1/(a_. + b_.*csc[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_]), x_Symbol] := Int[Sin[d + e*x]/(b + a*Sin[d + e*x] + c*Cos[d + e*x]), x] /; FreeQ[{a, b, c, d, e}, x]
+Int[cos[d_. + e_.*x_]^ n_.*(a_. + b_.*sec[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_])^n_., x_Symbol] := Int[(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[n]
+Int[sin[d_. + e_.*x_]^ n_.*(a_. + b_.*csc[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_])^n_., x_Symbol] := Int[(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[n]
+Int[cos[d_. + e_.*x_]^ n_*(a_. + b_.*sec[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_])^n_, x_Symbol] := Cos[d + e*x]^ n*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^ n/(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n* Int[(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && Not[IntegerQ[n]]
+Int[sin[d_. + e_.*x_]^ n_*(a_. + b_.*csc[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_])^n_, x_Symbol] := Sin[d + e*x]^ n*(a + b*Csc[d + e*x] + c*Cot[d + e*x])^ n/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n* Int[(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && Not[IntegerQ[n]]
+Int[sec[d_. + e_.*x_]^ n_.*(a_. + b_.*sec[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_])^m_, x_Symbol] := Int[1/(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0] && IntegerQ[n]
+Int[csc[d_. + e_.*x_]^ n_.*(a_. + b_.*csc[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_])^m_, x_Symbol] := Int[1/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0] && IntegerQ[n]
+Int[sec[d_. + e_.*x_]^ n_.*(a_. + b_.*sec[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_])^m_, x_Symbol] := Sec[d + e*x]^ n*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a + b*Sec[d + e*x] + c*Tan[d + e*x])^n* Int[1/(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0] && Not[IntegerQ[n]]
+Int[csc[d_. + e_.*x_]^ n_.*(a_. + b_.*csc[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_])^m_, x_Symbol] := Csc[d + e*x]^ n*(b + a*Sin[d + e*x] + c*Cos[d + e*x])^ n/(a + b*Csc[d + e*x] + c*Cot[d + e*x])^n* Int[1/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0] && Not[IntegerQ[n]]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m
new file mode 100755
index 0000000..66df6e4
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m	
@@ -0,0 +1,79 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.7 (d trig)^m (a+b (c sin)^n)^p *)
+Int[(a_ + b_.*sin[e_. + f_.*x_]^2)*(A_. + B_.*sin[e_. + f_.*x_]^2), x_Symbol] := (4*A*(2*a + b) + B*(4*a + 3*b))*x/8 - (4*A*b + B*(4*a + 3*b))*Cos[e + f*x]*Sin[e + f*x]/(8*f) - b*B*Cos[e + f*x]*Sin[e + f*x]^3/(4*f) /; FreeQ[{a, b, e, f, A, B}, x]
+Int[(a_ + b_.*sin[e_. + f_.*x_]^2)^p_*(A_. + B_.*sin[e_. + f_.*x_]^2), x_Symbol] := -B*Cos[e + f*x]* Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p/(2*f*(p + 1)) + 1/(2*(p + 1))*Int[(a + b*Sin[e + f*x]^2)^(p - 1)* Simp[a*B + 2*a*A*(p + 1) + (2*A*b*(p + 1) + B*(b + 2*a*p + 2*b*p))* Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, A, B}, x] && GtQ[p, 0]
+Int[(A_. + B_.*sin[e_. + f_.*x_]^2)/(a_ + b_.*sin[e_. + f_.*x_]^2), x_Symbol] := B*x/b + (A*b - a*B)/b*Int[1/(a + b*Sin[e + f*x]^2), x] /; FreeQ[{a, b, e, f, A, B}, x]
+Int[(A_. + B_.*sin[e_. + f_.*x_]^2)/ Sqrt[a_ + b_.*sin[e_. + f_.*x_]^2], x_Symbol] := B/b*Int[Sqrt[a + b*Sin[e + f*x]^2], x] + (A*b - a*B)/b* Int[1/Sqrt[a + b*Sin[e + f*x]^2], x] /; FreeQ[{a, b, e, f, A, B}, x]
+Int[(a_ + b_.*sin[e_. + f_.*x_]^2)^p_*(A_. + B_.*sin[e_. + f_.*x_]^2), x_Symbol] := -(A*b - a*B)*Cos[e + f*x]* Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(p + 1)/(2*a* f*(a + b)*(p + 1)) - 1/(2*a*(a + b)*(p + 1))*Int[(a + b*Sin[e + f*x]^2)^(p + 1)* Simp[a*B - A*(2*a*(p + 1) + b*(2*p + 3)) + 2*(A*b - a*B)*(p + 2)*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, A, B}, x] && LtQ[p, -1] && NeQ[a + b, 0]
+Int[(a_. + b_.*sin[e_. + f_.*x_]^2)^ p_*(A_. + B_.*sin[e_. + f_.*x_]^2), x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff*(a + b*Sin[e + f*x]^2)^ p*(Sec[e + f*x]^2)^p/(f*(a + (a + b)*Tan[e + f*x]^2)^p)* Subst[ Int[(a + (a + b)*ff^2*x^2)^ p*(A + (A + B)*ff^2*x^2)/(1 + ff^2*x^2)^(p + 2), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f, A, B}, x] && Not[IntegerQ[p]]
+Int[u_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_, x_Symbol] := a^p*Int[ActivateTrig[u*cos[e + f*x]^(2*p)], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]
+Int[u_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_, x_Symbol] := Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]^2], x_Symbol] := Sqrt[a]/f*EllipticE[e + f*x, -b/a] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
+Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]^2], x_Symbol] := Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[1 + b*Sin[e + f*x]^2/a]* Int[Sqrt[1 + (b*Sin[e + f*x]^2)/a], x] /; FreeQ[{a, b, e, f}, x] && Not[GtQ[a, 0]]
+Int[(a_ + b_.*sin[e_. + f_.*x_]^2)^2, x_Symbol] := (8*a^2 + 8*a*b + 3*b^2)*x/8 - b*(8*a + 3*b)*Cos[e + f*x]*Sin[e + f*x]/(8*f) - b^2*Cos[e + f*x]*Sin[e + f*x]^3/(4*f) /; FreeQ[{a, b, e, f}, x]
+Int[(a_ + b_.*sin[e_. + f_.*x_]^2)^p_, x_Symbol] := -b*Cos[e + f*x]* Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(p - 1)/(2*f*p) + 1/(2*p)* Int[(a + b*Sin[e + f*x]^2)^(p - 2)* Simp[a*(b + 2*a*p) + b*(2*a + b)*(2*p - 1)*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && GtQ[p, 1]
+Int[1/(a_ + b_.*sin[e_. + f_.*x_]^2), x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f}, x]
+Int[1/Sqrt[a_ + b_.*sin[e_. + f_.*x_]^2], x_Symbol] := 1/(Sqrt[a]*f)*EllipticF[e + f*x, -b/a] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
+Int[1/Sqrt[a_ + b_.*sin[e_. + f_.*x_]^2], x_Symbol] := Sqrt[1 + b*Sin[e + f*x]^2/a]/Sqrt[a + b*Sin[e + f*x]^2]* Int[1/Sqrt[1 + (b*Sin[e + f*x]^2)/a], x] /; FreeQ[{a, b, e, f}, x] && Not[GtQ[a, 0]]
+Int[(a_ + b_.*sin[e_. + f_.*x_]^2)^p_, x_Symbol] := -b*Cos[e + f*x]* Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(p + 1)/(2*a* f*(p + 1)*(a + b)) + 1/(2*a*(p + 1)*(a + b))* Int[(a + b*Sin[e + f*x]^2)^(p + 1)* Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && LtQ[p, -1]
+Int[(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff*Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])* Subst[Int[(a + b*ff^2*x^2)^p/Sqrt[1 - ff^2*x^2], x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && Not[IntegerQ[p]]
+Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Cos[e + f*x], x]}, -ff/f* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
+Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff^(m + 1)/f* Subst[Int[ x^m*(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
+Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff^(m + 1)*Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])* Subst[Int[x^m*(a + b*ff^2*x^2)^p/Sqrt[1 - ff^2*x^2], x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && Not[IntegerQ[p]]
+Int[(d_.*sin[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Cos[e + f*x], x]}, -ff* d^(2*IntPart[(m - 1)/2] + 1)*(d*Sin[e + f*x])^(2* FracPart[(m - 1)/2])/(f*(Sin[e + f*x]^2)^ FracPart[(m - 1)/2])* Subst[ Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff]] /; FreeQ[{a, b, d, e, f, m, p}, x] && Not[IntegerQ[m]]
+Int[cos[e_. + f_.*x_]^m_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
+Int[cos[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
+Int[cos[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff*Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && Not[IntegerQ[p]]
+Int[(d_.*cos[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff*d^(2*IntPart[(m - 1)/2] + 1)*(d*Cos[e + f*x])^(2* FracPart[(m - 1)/2])/(f*(Cos[e + f*x]^2)^ FracPart[(m - 1)/2])* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, d, e, f, m, p}, x] && Not[IntegerQ[m]]
+Int[tan[e_. + f_.*x_]^m_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, ff^((m + 1)/2)/(2*f)* Subst[Int[x^((m - 1)/2)*(a + b*ff*x)^p/(1 - ff*x)^((m + 1)/2), x], x, Sin[e + f*x]^2/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
+Int[(d_.*tan[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(d*ff*x)^ m*(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(p + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, d, e, f, m}, x] && IntegerQ[p]
+Int[tan[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff^(m + 1)*Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])* Subst[ Int[x^m*(a + b*ff^2*x^2)^p/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && Not[IntegerQ[p]]
+Int[(d_.*tan[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff*(d*Tan[e + f*x])^(m + 1)*(Cos[e + f*x]^2)^((m + 1)/2)/(d*f* Sin[e + f*x]^(m + 1))* Subst[ Int[(ff*x)^m*(a + b*ff^2*x^2)^p/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, d, e, f, m, p}, x] && Not[IntegerQ[m]]
+Int[cos[e_. + f_.*x_]^m_.*(d_.*sin[e_. + f_.*x_])^ n_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[(d*ff*x)^n*(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^ p, x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(m - 1)/2]
+Int[(c_.*sin[e_. + f_.*x_])^m_* sin[e_. + f_.*x_]^n_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Cos[e + f*x], x]}, -ff/f* Subst[Int[(c*ff*x)^ m*(1 - ff^2*x^2)^((n - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff]] /; FreeQ[{a, b, c, e, f, m, p}, x] && IntegerQ[(n - 1)/2]
+Int[cos[e_. + f_.*x_]^m_* sin[e_. + f_.*x_]^n_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff^(n + 1)/f* Subst[Int[ x^n*(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^((m + n)/2 + p + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[n/2] && IntegerQ[p]
+Int[cos[e_. + f_.*x_]^m_*(d_.*sin[e_. + f_.*x_])^ n_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff*Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])* Subst[Int[(d*ff*x)^n*(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^ p, x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[m/2]
+Int[(c_.*cos[e_. + f_.*x_])^m_*(d_.*sin[e_. + f_.*x_])^ n_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff*c^(2*IntPart[(m - 1)/2] + 1)*(c*Cos[e + f*x])^(2* FracPart[(m - 1)/2])/(f*(Cos[e + f*x]^2)^ FracPart[(m - 1)/2])* Subst[ Int[(d*ff*x)^n*(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]
+Int[(b_.*sin[e_. + f_.*x_]^2)^p_, x_Symbol] := -Cot[e + f*x]*(b*Sin[e + f*x]^2)^p/(2*f*p) + b*(2*p - 1)/(2*p)*Int[(b*Sin[e + f*x]^2)^(p - 1), x] /; FreeQ[{b, e, f}, x] && Not[IntegerQ[p]] && GtQ[p, 1]
+Int[(b_.*sin[e_. + f_.*x_]^2)^p_, x_Symbol] := Cot[e + f*x]*(b*Sin[e + f*x]^2)^(p + 1)/(b*f*(2*p + 1)) + 2*(p + 1)/(b*(2*p + 1))*Int[(b*Sin[e + f*x]^2)^(p + 1), x] /; FreeQ[{b, e, f}, x] && Not[IntegerQ[p]] && LtQ[p, -1]
+Int[tan[e_. + f_.*x_]^m_.*(b_.*sin[e_. + f_.*x_]^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, ff^((m + 1)/2)/(2*f)* Subst[Int[ x^((m - 1)/2)*(b*ff^(n/2)*x^(n/2))^p/(1 - ff*x)^((m + 1)/2), x], x, Sin[e + f*x]^2/ff]] /; FreeQ[{b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2]
+Int[tan[e_. + f_.*x_]^m_.*(b_.*(c_.*sin[e_. + f_.*x_])^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff^(m + 1)/f* Subst[Int[x^m*(b*(c*ff*x)^n)^p/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff]] /; FreeQ[{b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]
+Int[u_.*(b_.*sin[e_. + f_.*x_]^n_)^p_, x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, (b*ff^n)^ IntPart[p]*(b*Sin[e + f*x]^n)^ FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])* Int[ActivateTrig[u]*(Sin[e + f*x]/ff)^(n*p), x]] /; FreeQ[{b, e, f, n, p}, x] && Not[IntegerQ[p]] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, (d_.*trig_[e + f*x])^m_. /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
+Int[u_.*(b_.*(c_.*sin[e_. + f_.*x_])^n_)^p_, x_Symbol] := b^IntPart[p]*(b*(c*Sin[e + f*x])^n)^ FracPart[p]/(c*Sin[e + f*x])^(n*FracPart[p])* Int[ActivateTrig[u]*(c*Sin[e + f*x])^(n*p), x] /; FreeQ[{b, c, e, f, n, p}, x] && Not[IntegerQ[p]] && Not[IntegerQ[n]] && (EqQ[u, 1] || MatchQ[u, (d_.*trig_[e + f*x])^m_. /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
+(* Int[(a_+b_.*sin[e_.+f_.*x_]^4)^p_.,x_Symbol] := With[{ff=FreeFactors[Tan[e+f*x],x]}, -ff/f*Subst[Int[(a+b+2*a*ff^2*x^2+a*ff^4*x^4)^p/(1+ff^2*x^2)^(2*p+1) ,x],x,Cot[e+f*x]/ff]] /; FreeQ[{a,b,e,f},x] && IntegerQ[p] *)
+Int[(a_ + b_.*sin[e_. + f_.*x_]^4)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^ p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[p]
+Int[(a_ + b_.*sin[e_. + f_.*x_]^4)^p_, x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff*(a + b*Sin[e + f*x]^4)^ p*(Sec[e + f*x]^2)^(2* p)/(f*(a + 2*a*Tan[e + f*x]^2 + (a + b)*Tan[e + f*x]^4)^p)* Subst[ Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^ p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[p - 1/2]
+Int[1/(a_ + b_.*sin[e_. + f_.*x_]^n_), x_Symbol] := Module[{k}, Dist[2/(a*n), Sum[Int[1/(1 - Sin[e + f*x]^2/((-1)^(4*k/n)*Rt[-a/b, n/2])), x], {k, 1, n/2}], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/2]
+(* Int[(a_+b_.*sin[e_.+f_.*x_]^n_)^p_,x_Symbol] := With[{ff=FreeFactors[Tan[e+f*x],x]}, -ff/f*Subst[Int[(b+a*(1+ff^2*x^2)^(n/2))^p/(1+ff^2*x^2)^(n*p/2+1),x] ,x,Cot[e+f*x]/ff]] /; FreeQ[{a,b,e,f},x] && IntegerQ[n/2] && IGtQ[p,0] *)
+Int[(a_ + b_.*sin[e_. + f_.*x_]^n_)^p_, x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(b*ff^n*x^n + a*(1 + ff^2*x^2)^(n/2))^ p/(1 + ff^2*x^2)^(n*p/2 + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/2] && IGtQ[p, 0]
+Int[(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^p_, x_Symbol] := Int[ExpandTrig[(a + b*(c*sin[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, e, f, n}, x] && (IGtQ[p, 0] || EqQ[p, -1] && IntegerQ[n])
+Int[(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^p_, x_Symbol] := Unintegrable[(a + b*(c*Sin[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, e, f, n, p}, x]
+Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*sin[e_. + f_.*x_]^4)^p_., x_Symbol] := With[{ff = FreeFactors[Cos[e + f*x], x]}, -ff/f* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, x], x, Cos[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
+Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*sin[e_. + f_.*x_]^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Cos[e + f*x], x]}, -ff/f* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(1 - ff^2*x^2)^(n/2))^ p, x], x, Cos[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2]
+Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^4)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff^(m + 1)/f* Subst[Int[ x^m*(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^ p/(1 + ff^2*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
+Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff^(m + 1)/f* Subst[Int[ x^m*(a*(1 + ff^2*x^2)^(n/2) + b*ff^n*x^n)^ p/(1 + ff^2*x^2)^(m/2 + n*p/2 + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[n/2] && IntegerQ[p]
+Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^4)^p_, x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff^(m + 1)*(a + b*Sin[e + f*x]^4)^ p*(Sec[e + f*x]^2)^(2*p)/(f* Apart[a*(1 + Tan[e + f*x]^2)^2 + b*Tan[e + f*x]^4]^p)* Subst[ Int[x^m*ExpandToSum[a*(1 + ff^2*x^2)^2 + b*ff^4*x^4, x]^ p/(1 + ff^2*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[p - 1/2]
+Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*sin[e_. + f_.*x_]^n_)^p_., x_Symbol] := Int[ExpandTrig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4] || GtQ[p, 0] || EqQ[p, -1] && IntegerQ[n])
+Int[(d_.*sin[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_., x_Symbol] := Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*sin[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
+Int[(d_.*sin[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_., x_Symbol] := Unintegrable[(d*Sin[e + f*x])^m*(a + b*(c*Sin[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
+Int[cos[e_. + f_.*x_]^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0] || IGtQ[p, 0] || IntegersQ[m, p])
+Int[cos[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^4)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^ p/(1 + ff^2*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
+Int[cos[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(b*ff^n*x^n + a*(1 + ff^2*x^2)^(n/2))^ p/(1 + ff^2*x^2)^(m/2 + n*p/2 + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[n/2] && IntegerQ[p]
+Int[cos[e_. + f_.*x_]^m_/(a_ + b_.*sin[e_. + f_.*x_]^n_), x_Symbol] := Int[Expand[(1 - Sin[e + f*x]^2)^(m/2)/(a + b*Sin[e + f*x]^n), x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m/2, 0] && IntegerQ[(n - 1)/2]
+(* Int[cos[e_.+f_.*x_]^m_*(a_+b_.*sin[e_.+f_.*x_]^n_)^p_,x_Symbol] := Int[ExpandTrig[(1-sin[e+f*x]^2)^(m/2)*(a+b*sin[e+f*x]^n)^p,x],x] /; FreeQ[{a,b,e,f},x] && IntegerQ[m/2] && IntegerQ[(n-1)/2] &&  ILtQ[p,-1] && LtQ[m,0] *)
+Int[(d_.*cos[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_., x_Symbol] := Int[ExpandTrig[(d*cos[e + f*x])^m*(a + b*(c*sin[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
+Int[(d_.*cos[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_., x_Symbol] := Unintegrable[(d*Cos[e + f*x])^m*(a + b*(c*Sin[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
+Int[tan[e_. + f_.*x_]^m_.*(a_ + b_.*sin[e_. + f_.*x_]^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, ff^((m + 1)/2)/(2*f)* Subst[Int[ x^((m - 1)/2)*(a + b*ff^(n/2)*x^(n/2))^p/(1 - ff*x)^((m + 1)/2), x], x, Sin[e + f*x]^2/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2]
+Int[tan[e_. + f_.*x_]^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff^(m + 1)/f* Subst[Int[x^m*(a + b*(c*ff*x)^n)^p/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]
+Int[(d_.*tan[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^4)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(d*ff*x)^m* ExpandToSum[a*(1 + ff^2*x^2)^2 + b*ff^4*x^4, x]^ p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, d, e, f, m}, x] && IntegerQ[p]
+Int[(d_.*tan[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^4)^p_, x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff*(a + b*Sin[e + f*x]^4)^ p*(Sec[e + f*x]^2)^(2*p)/(f* Apart[a*(1 + Tan[e + f*x]^2)^2 + b*Tan[e + f*x]^4]^p)* Subst[ Int[(d*ff*x)^m* ExpandToSum[a*(1 + ff^2*x^2)^2 + b*ff^4*x^4, x]^ p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, d, e, f, m}, x] && IntegerQ[p - 1/2]
+Int[(d_.*tan[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff^(m + 1)/f* Subst[Int[(d*x)^ m*(b*ff^n*x^n + a*(1 + ff^2*x^2)^(n/2))^ p/(1 + ff^2*x^2)^(n*p/2 + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, d, e, f, m}, x] && IntegerQ[n/2] && IGtQ[p, 0]
+Int[(d_.*tan[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_., x_Symbol] := Int[ExpandTrig[(d*tan[e + f*x])^m*(a + b*(c*sin[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
+Int[(d_.*tan[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_., x_Symbol] := Unintegrable[(d*Tan[e + f*x])^m*(a + b*(c*Sin[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
+Int[(d_.*cot[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_, x_Symbol] := (d*Cot[e + f*x])^FracPart[m]*(Tan[e + f*x]/d)^FracPart[m]* Int[(Tan[e + f*x]/d)^(-m)*(a + b*(c*Sin[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]
+Int[(d_.*sec[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_, x_Symbol] := (d*Sec[e + f*x])^FracPart[m]*(Cos[e + f*x]/d)^FracPart[m]* Int[(Cos[e + f*x]/d)^(-m)*(a + b*(c*Sin[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]
+Int[(d_.*csc[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^n_.)^p_., x_Symbol] := d^(n*p)* Int[(d*Csc[e + f*x])^(m - n*p)*(b + a*Csc[e + f*x]^n)^p, x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && Not[IntegerQ[m]] && IntegersQ[n, p]
+Int[(d_.*csc[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_, x_Symbol] := (d*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/d)^FracPart[m]* Int[(Sin[e + f*x]/d)^(-m)*(a + b*(c*Sin[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]
+Int[(a_ + b_.*(c_.*sin[e_. + f_.*x_] + d_.*cos[e_. + f_.*x_])^2)^p_, x_Symbol] := Int[(a + b*(Sqrt[c^2 + d^2]*Sin[ArcTan[c, d] + e + f*x])^2)^p, x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[p^2, 1/4] && GtQ[a, 0]
+Int[(a_ + b_.*(c_.*sin[e_. + f_.*x_] + d_.*cos[e_. + f_.*x_])^2)^p_, x_Symbol] := (a + b*(c*Sin[e + f*x] + d*Cos[e + f*x])^2)^ p/(1 + (b*(c*Sin[e + f*x] + d*Cos[e + f*x])^2)/a)^p* Int[(1 + (b*(c*Sin[e + f*x] + d*Cos[e + f*x])^2)/a)^p, x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[p^2, 1/4] && Not[GtQ[a, 0]]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.8 trig^m (a+b cos^p+c sin^q)^n.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.8 trig^m (a+b cos^p+c sin^q)^n.m
new file mode 100755
index 0000000..8a279fb
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.8 trig^m (a+b cos^p+c sin^q)^n.m	
@@ -0,0 +1,11 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.8 trig^m (a+b cos^p+c sin^q)^n *)
+Int[sin[d_. + e_.*x_]^ m_*(a_ + b_.*cos[d_. + e_.*x_]^p_ + c_.*sin[d_. + e_.*x_]^q_)^n_, x_Symbol] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -f/e* Subst[Int[ ExpandToSum[ c + b*(1 + f^2*x^2)^(q/2 - p/2) + a*(1 + f^2*x^2)^(q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*q/2 + 1), x], x, Cot[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && GtQ[p, 0] && LeQ[p, q]
+Int[cos[d_. + e_.*x_]^ m_*(a_ + b_.*sin[d_. + e_.*x_]^p_ + c_.*cos[d_. + e_.*x_]^q_)^n_, x_Symbol] := Module[{f = FreeFactors[Tan[d + e*x], x]}, f/e* Subst[Int[ ExpandToSum[ c + b*(1 + f^2*x^2)^(q/2 - p/2) + a*(1 + f^2*x^2)^(q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*q/2 + 1), x], x, Tan[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && GtQ[p, 0] && LeQ[p, q]
+Int[sin[d_. + e_.*x_]^ m_*(a_ + b_.*cos[d_. + e_.*x_]^p_ + c_.*sin[d_. + e_.*x_]^q_)^n_, x_Symbol] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -f/e* Subst[Int[ ExpandToSum[ a*(1 + f^2*x^2)^(p/2) + b*f^p*x^p + c*(1 + f^2*x^2)^(p/2 - q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*p/2 + 1), x], x, Cot[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && LtQ[0, q, p]
+Int[cos[d_. + e_.*x_]^ m_*(a_ + b_.*sin[d_. + e_.*x_]^p_ + c_.*cos[d_. + e_.*x_]^q_)^n_, x_Symbol] := Module[{f = FreeFactors[Tan[d + e*x], x]}, f/e* Subst[Int[ ExpandToSum[ a*(1 + f^2*x^2)^(p/2) + b*f^p*x^p + c*(1 + f^2*x^2)^(p/2 - q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*p/2 + 1), x], x, Tan[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && LtQ[0, q, p]
+Int[sin[d_. + e_.*x_]^ m_*(a_ + b_.*cos[d_. + e_.*x_]^p_ + c_.*sin[d_. + e_.*x_]^q_)^n_, x_Symbol] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -f/e* Subst[Int[ ExpandToSum[ c + b*(1 + f^2*x^2)^(q/2 - p/2) + a*(1 + f^2*x^2)^(q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*q/2 + 1), x], x, Cot[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && GtQ[p, 0] && LeQ[p, q]
+Int[cos[d_. + e_.*x_]^ m_*(a_ + b_.*sin[d_. + e_.*x_]^p_ + c_.*cos[d_. + e_.*x_]^q_)^n_, x_Symbol] := Module[{f = FreeFactors[Tan[d + e*x], x]}, f/e* Subst[Int[ ExpandToSum[ c + b*(1 + f^2*x^2)^(q/2 - p/2) + a*(1 + f^2*x^2)^(q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*q/2 + 1), x], x, Tan[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && GtQ[p, 0] && LeQ[p, q]
+Int[sin[d_. + e_.*x_]^ m_*(a_ + b_.*cos[d_. + e_.*x_]^p_ + c_.*sin[d_. + e_.*x_]^q_)^n_, x_Symbol] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -f/e* Subst[Int[ ExpandToSum[ a*(1 + f^2*x^2)^(p/2) + b*f^p*x^p + c*(1 + f^2*x^2)^(p/2 - q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*p/2 + 1), x], x, Cot[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && LtQ[0, q, p]
+Int[cos[d_. + e_.*x_]^ m_*(a_ + b_.*sin[d_. + e_.*x_]^p_ + c_.*cos[d_. + e_.*x_]^q_)^n_, x_Symbol] := Module[{f = FreeFactors[Tan[d + e*x], x]}, f/e* Subst[Int[ ExpandToSum[ a*(1 + f^2*x^2)^(p/2) + b*f^p*x^p + c*(1 + f^2*x^2)^(p/2 - q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*p/2 + 1), x], x, Tan[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && LtQ[0, q, p]
diff --git a/IntegrationRules/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m
new file mode 100755
index 0000000..9a36efa
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m	
@@ -0,0 +1,55 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p *)
+Int[(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := 1/(4^p*c^p)*Int[(b + 2*c*Sin[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := 1/(4^p*c^p)*Int[(b + 2*c*Cos[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := (a + b*Sin[d + e*x]^n + c*Sin[d + e*x]^(2*n))^ p/(b + 2*c*Sin[d + e*x]^n)^(2*p)* Int[u*(b + 2*c*Sin[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := (a + b*Cos[d + e*x]^n + c*Cos[d + e*x]^(2*n))^ p/(b + 2*c*Cos[d + e*x]^n)^(2*p)* Int[u*(b + 2*c*Cos[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[1/(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.), x_Symbol] := Module[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[1/(b - q + 2*c*Sin[d + e*x]^n), x] - 2*c/q*Int[1/(b + q + 2*c*Sin[d + e*x]^n), x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
+Int[1/(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.), x_Symbol] := Module[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[1/(b - q + 2*c*Cos[d + e*x]^n), x] - 2*c/q*Int[1/(b + q + 2*c*Cos[d + e*x]^n), x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
+Int[sin[d_. + e_.*x_]^ m_.*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := 1/(4^p*c^p)*Int[Sin[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[cos[d_. + e_.*x_]^ m_.*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := 1/(4^p*c^p)*Int[Cos[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[sin[d_. + e_.*x_]^ m_.*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := (a + b*Sin[d + e*x]^n + c*Sin[d + e*x]^(2*n))^ p/(b + 2*c*Sin[d + e*x]^n)^(2*p)* Int[Sin[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[cos[d_. + e_.*x_]^ m_.*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := (a + b*Cos[d + e*x]^n + c*Cos[d + e*x]^(2*n))^ p/(b + 2*c*Cos[d + e*x]^n)^(2*p)* Int[Cos[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[sin[d_. + e_.*x_]^ m_*(a_. + b_.*sin[d_. + e_.*x_]^n_ + c_.*sin[d_. + e_.*x_]^n2_)^ p_, x_Symbol] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -f/e* Subst[Int[ ExpandToSum[c + b*(1 + x^2)^(n/2) + a*(1 + x^2)^n, x]^ p/(1 + f^2*x^2)^(m/2 + n*p + 1), x], x, Cot[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2] && IntegerQ[p]
+Int[cos[d_. + e_.*x_]^ m_.*(a_. + b_.*cos[d_. + e_.*x_]^n_ + c_.*cos[d_. + e_.*x_]^n2_)^ p_, x_Symbol] := Module[{f = FreeFactors[Tan[d + e*x], x]}, f/e* Subst[Int[ ExpandToSum[c + b*(1 + x^2)^(n/2) + a*(1 + x^2)^n, x]^ p/(1 + f^2*x^2)^(m/2 + n*p + 1), x], x, Tan[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2] && IntegerQ[p]
+Int[sin[d_. + e_.*x_]^ m_.*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := Int[ExpandTrig[ sin[d + e*x]^m*(a + b*sin[d + e*x]^n + c*sin[d + e*x]^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[m, n, p]
+Int[cos[d_. + e_.*x_]^ m_.*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := Int[ExpandTrig[ cos[d + e*x]^m*(a + b*cos[d + e*x]^n + c*cos[d + e*x]^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[m, n, p]
+Int[cos[d_. + e_.*x_]^ m_.*(a_. + b_.*(f_.*sin[d_. + e_.*x_])^n_. + c_.*(f_.*sin[d_. + e_.*x_])^n2_.)^p_., x_Symbol] := Module[{g = FreeFactors[Sin[d + e*x], x]}, g/e* Subst[Int[(1 - g^2*x^2)^((m - 1)/2)*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^p, x], x, Sin[d + e*x]/g]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2]
+Int[sin[d_. + e_.*x_]^ m_.*(a_. + b_.*(f_.*cos[d_. + e_.*x_])^n_. + c_.*(f_.*cos[d_. + e_.*x_])^n2_.)^p_., x_Symbol] := Module[{g = FreeFactors[Cos[d + e*x], x]}, -g/e* Subst[Int[(1 - g^2*x^2)^((m - 1)/2)*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^p, x], x, Cos[d + e*x]/g]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2]
+Int[cos[d_. + e_.*x_]^ m_*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := 1/(4^p*c^p)*Int[Cos[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[sin[d_. + e_.*x_]^ m_*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := 1/(4^p*c^p)*Int[Sin[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[cos[d_. + e_.*x_]^ m_*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^ p_, x_Symbol] := (a + b*Sin[d + e*x]^n + c*Sin[d + e*x]^(2*n))^ p/(b + 2*c*Sin[d + e*x]^n)^(2*p)* Int[Cos[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[sin[d_. + e_.*x_]^ m_*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^ p_, x_Symbol] := (a + b*Cos[d + e*x]^n + c*Cos[d + e*x]^(2*n))^ p/(b + 2*c*Cos[d + e*x]^n)^(2*p)* Int[Sin[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[cos[d_. + e_.*x_]^ m_*(a_. + b_.*sin[d_. + e_.*x_]^n_ + c_.*sin[d_. + e_.*x_]^n2_)^ p_., x_Symbol] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[c + b*(1 + x^2)^(n/2) + a*(1 + x^2)^n, x]^ p/(1 + f^2*x^2)^(m/2 + n*p + 1), x], x, Cot[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2] && IntegerQ[p]
+Int[sin[d_. + e_.*x_]^ m_.*(a_. + b_.*cos[d_. + e_.*x_]^n_ + c_.*cos[d_. + e_.*x_]^n2_)^ p_., x_Symbol] := Module[{f = FreeFactors[Tan[d + e*x], x]}, f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[c + b*(1 + x^2)^(n/2) + a*(1 + x^2)^n, x]^ p/(1 + f^2*x^2)^(m/2 + n*p + 1), x], x, Tan[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2] && IntegerQ[p]
+Int[cos[d_. + e_.*x_]^ m_.*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^p_., x_Symbol] := Int[ExpandTrig[(1 - sin[d + e*x]^2)^(m/2)*(a + b*sin[d + e*x]^n + c*sin[d + e*x]^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]
+Int[sin[d_. + e_.*x_]^ m_.*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^p_., x_Symbol] := Int[ExpandTrig[(1 - cos[d + e*x]^2)^(m/2)*(a + b*cos[d + e*x]^n + c*cos[d + e*x]^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]
+Int[tan[d_. + e_.*x_]^ m_.*(a_ + b_.*(f_.*sin[d_. + e_.*x_])^n_ + c_.*(f_.*sin[d_. + e_.*x_])^n2_.)^p_., x_Symbol] := Module[{g = FreeFactors[Sin[d + e*x], x]}, g^(m + 1)/e* Subst[Int[ x^m*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^ p/(1 - g^2*x^2)^((m + 1)/2), x], x, Sin[d + e*x]/g]] /; FreeQ[{a, b, c, d, e, f, n}, x] && IntegerQ[(m - 1)/2] && IntegerQ[2*p]
+Int[cot[d_. + e_.*x_]^ m_.*(a_ + b_.*(f_.*cos[d_. + e_.*x_])^n_ + c_.*(f_.*cos[d_. + e_.*x_])^n2_.)^p_., x_Symbol] := Module[{g = FreeFactors[Cos[d + e*x], x]}, -g^(m + 1)/e* Subst[Int[ x^m*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^ p/(1 - g^2*x^2)^((m + 1)/2), x], x, Cos[d + e*x]/g]] /; FreeQ[{a, b, c, d, e, f, n}, x] && IntegerQ[(m - 1)/2] && IntegerQ[2*p]
+Int[tan[d_. + e_.*x_]^ m_*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := 1/(4^p*c^p)*Int[Tan[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[cot[d_. + e_.*x_]^ m_*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := 1/(4^p*c^p)*Int[Cot[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[tan[d_. + e_.*x_]^ m_*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^ p_, x_Symbol] := (a + b*Sin[d + e*x]^n + c*Sin[d + e*x]^(2*n))^ p/(b + 2*c*Sin[d + e*x]^n)^(2*p)* Int[Tan[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[cot[d_. + e_.*x_]^ m_*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^ p_, x_Symbol] := (a + b*Cos[d + e*x]^n + c*Cos[d + e*x]^(2*n))^ p/(b + 2*c*Cos[d + e*x]^n)^(2*p)* Int[Cot[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*sin[d_. + e_.*x_]^n_ + c_.*sin[d_. + e_.*x_]^n2_)^ p_., x_Symbol] := Module[{f = FreeFactors[Tan[d + e*x], x]}, f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[ c*x^(2*n) + b*x^n*(1 + x^2)^(n/2) + a*(1 + x^2)^n, x]^ p/(1 + f^2*x^2)^(n*p + 1), x], x, Tan[d + e*x]/f]] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2] && IntegerQ[p]
+Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*cos[d_. + e_.*x_]^n_ + c_.*cos[d_. + e_.*x_]^n2_)^ p_., x_Symbol] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[ c*x^(2*n) + b*x^n*(1 + x^2)^(n/2) + a*(1 + x^2)^n, x]^ p/(1 + f^2*x^2)^(n*p + 1), x], x, Cot[d + e*x]/f]] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2] && IntegerQ[p]
+Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^p_., x_Symbol] := Int[ExpandTrig[ sin[d + e*x]^ m*(a + b*sin[d + e*x]^n + c*sin[d + e*x]^(2*n))^ p/(1 - sin[d + e*x]^2)^(m/2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]
+Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^p_., x_Symbol] := Int[ExpandTrig[ cos[d + e*x]^ m*(a + b*cos[d + e*x]^n + c*cos[d + e*x]^(2*n))^ p/(1 - cos[d + e*x]^2)^(m/2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]
+Int[cot[d_. + e_.*x_]^ m_.*(a_ + b_.*(f_.*sin[d_. + e_.*x_])^n_ + c_.*(f_.*sin[d_. + e_.*x_])^n2_.)^p_., x_Symbol] := Module[{g = FreeFactors[Sin[d + e*x], x]}, g^(m + 1)/e* Subst[Int[(1 - g^2*x^2)^((m - 1)/ 2)*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^p/x^m, x], x, Sin[d + e*x]/g]] /; FreeQ[{a, b, c, d, e, f, n}, x] && IntegerQ[(m - 1)/2] && IntegerQ[2*p]
+Int[tan[d_. + e_.*x_]^ m_.*(a_ + b_.*(f_.*cos[d_. + e_.*x_])^n_ + c_.*(f_.*cos[d_. + e_.*x_])^n2_.)^p_., x_Symbol] := Module[{g = FreeFactors[Cos[d + e*x], x]}, -g^(m + 1)/e* Subst[Int[(1 - g^2*x^2)^((m - 1)/ 2)*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^p/x^m, x], x, Cos[d + e*x]/g]] /; FreeQ[{a, b, c, d, e, f, n}, x] && IntegerQ[(m - 1)/2] && IntegerQ[2*p]
+Int[cot[d_. + e_.*x_]^ m_*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := 1/(4^p*c^p)*Int[Cot[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[tan[d_. + e_.*x_]^ m_*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := 1/(4^p*c^p)*Int[Tan[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[cot[d_. + e_.*x_]^ m_*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^ p_, x_Symbol] := (a + b*Sin[d + e*x]^n + c*Sin[d + e*x]^(2*n))^ p/(b + 2*c*Sin[d + e*x]^n)^(2*p)* Int[Cot[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[tan[d_. + e_.*x_]^ m_*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^ p_, x_Symbol] := (a + b*Cos[d + e*x]^n + c*Cos[d + e*x]^(2*n))^ p/(b + 2*c*Cos[d + e*x]^n)^(2*p)* Int[Tan[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[cot[d_. + e_.*x_]^ m_.*(a_ + b_.*sin[d_. + e_.*x_]^n_ + c_.*sin[d_. + e_.*x_]^n2_)^ p_., x_Symbol] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[c + b*(1 + f^2*x^2)^(n/2) + a*(1 + f^2*x^2)^n, x]^p/(1 + f^2*x^2)^(n*p + 1), x], x, Cot[d + e*x]/f]] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[n2, 2*n] && IntegerQ[n/2] && IntegerQ[p]
+Int[tan[d_. + e_.*x_]^ m_.*(a_ + b_.*cos[d_. + e_.*x_]^n_ + c_.*cos[d_. + e_.*x_]^n2_)^ p_., x_Symbol] := Module[{f = FreeFactors[Tan[d + e*x], x]}, f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[c + b*(1 + f^2*x^2)^(n/2) + a*(1 + f^2*x^2)^n, x]^p/(1 + f^2*x^2)^(n*p + 1), x], x, Tan[d + e*x]/f]] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[n2, 2*n] && IntegerQ[n/2] && IntegerQ[p]
+Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^p_., x_Symbol] := Int[ExpandTrig[(1 - sin[d + e*x]^2)^(m/ 2)*(a + b*sin[d + e*x]^n + c*sin[d + e*x]^(2*n))^p/ sin[d + e*x]^m, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]
+Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^p_., x_Symbol] := Int[ExpandTrig[(1 - cos[d + e*x]^2)^(m/ 2)*(a + b*cos[d + e*x]^n + c*cos[d + e*x]^(2*n))^p/ cos[d + e*x]^m, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]
+Int[(A_ + B_.*sin[d_. + e_.*x_])*(a_ + b_.*sin[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]^2)^n_, x_Symbol] := 1/(4^n*c^n)* Int[(A + B*Sin[d + e*x])*(b + 2*c*Sin[d + e*x])^(2*n), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]
+Int[(A_ + B_.*cos[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*cos[d_. + e_.*x_]^2)^n_, x_Symbol] := 1/(4^n*c^n)* Int[(A + B*Cos[d + e*x])*(b + 2*c*Cos[d + e*x])^(2*n), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]
+Int[(A_ + B_.*sin[d_. + e_.*x_])*(a_ + b_.*sin[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]^2)^n_, x_Symbol] := (a + b*Sin[d + e*x] + c*Sin[d + e*x]^2)^ n/(b + 2*c*Sin[d + e*x])^(2*n)* Int[(A + B*Sin[d + e*x])*(b + 2*c*Sin[d + e*x])^(2*n), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[n]]
+Int[(A_ + B_.*cos[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*cos[d_. + e_.*x_]^2)^n_, x_Symbol] := (a + b*Cos[d + e*x] + c*Cos[d + e*x]^2)^ n/(b + 2*c*Cos[d + e*x])^(2*n)* Int[(A + B*Cos[d + e*x])*(b + 2*c*Cos[d + e*x])^(2*n), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[n]]
+Int[(A_ + B_.*sin[d_. + e_.*x_])/(a_. + b_.*sin[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]^2), x_Symbol] := Module[{q = Rt[b^2 - 4*a*c, 2]}, (B + (b*B - 2*A*c)/q)*Int[1/(b + q + 2*c*Sin[d + e*x]), x] + (B - (b*B - 2*A*c)/q)*Int[1/(b - q + 2*c*Sin[d + e*x]), x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]
+Int[(A_ + B_.*cos[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*cos[d_. + e_.*x_]^2), x_Symbol] := Module[{q = Rt[b^2 - 4*a*c, 2]}, (B + (b*B - 2*A*c)/q)*Int[1/(b + q + 2*c*Cos[d + e*x]), x] + (B - (b*B - 2*A*c)/q)*Int[1/(b - q + 2*c*Cos[d + e*x]), x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]
+Int[(A_ + B_.*sin[d_. + e_.*x_])*(a_. + b_.*sin[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]^2)^n_, x_Symbol] := Int[ExpandTrig[(A + B*sin[d + e*x])*(a + b*sin[d + e*x] + c*sin[d + e*x]^2)^n, x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]
+Int[(A_ + B_.*cos[d_. + e_.*x_])*(a_. + b_.*cos[d_. + e_.*x_] + c_.*cos[d_. + e_.*x_]^2)^n_, x_Symbol] := Int[ExpandTrig[(A + B*cos[d + e*x])*(a + b*cos[d + e*x] + c*cos[d + e*x]^2)^n, x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]
diff --git a/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m
new file mode 100755
index 0000000..09d905f
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m	
@@ -0,0 +1,18 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.3.1.1 (a+b tan)^n *)
+Int[(b_.*tan[c_. + d_.*x_])^n_, x_Symbol] := b*(b*Tan[c + d*x])^(n - 1)/(d*(n - 1)) - b^2*Int[(b*Tan[c + d*x])^(n - 2), x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
+Int[(b_.*tan[c_. + d_.*x_])^n_, x_Symbol] := (b*Tan[c + d*x])^(n + 1)/(b*d*(n + 1)) - 1/b^2*Int[(b*Tan[c + d*x])^(n + 2), x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]
+Int[tan[c_. + d_.*x_], x_Symbol] := -Log[RemoveContent[Cos[c + d*x], x]]/d /; FreeQ[{c, d}, x]
+(* Int[1/tan[c_.+d_.*x_],x_Symbol] := Log[RemoveContent[Sin[c+d*x],x]]/d /; FreeQ[{c,d},x] *)
+Int[(b_.*tan[c_. + d_.*x_])^n_, x_Symbol] := b/d*Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]] /; FreeQ[{b, c, d, n}, x] && Not[IntegerQ[n]]
+Int[(a_ + b_.*tan[c_. + d_.*x_])^2, x_Symbol] := (a^2 - b^2)*x + b^2*Tan[c + d*x]/d + 2*a*b*Int[Tan[c + d*x], x] /; FreeQ[{a, b, c, d}, x]
+(* Int[(a_+b_.*tan[c_.+d_.*x_])^n_,x_Symbol] := Int[ExpandIntegrand[(a+b*Tan[c+d*x])^n,x],x] /; FreeQ[{a,b,c,d},x] && IGtQ[n,0] *)
+Int[(a_ + b_.*tan[c_. + d_.*x_])^n_, x_Symbol] := b*(a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1)) + 2*a*Int[(a + b*Tan[c + d*x])^(n - 1), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 1]
+Int[(a_ + b_.*tan[c_. + d_.*x_])^n_, x_Symbol] := a*(a + b*Tan[c + d*x])^n/(2*b*d*n) + 1/(2*a)*Int[(a + b*Tan[c + d*x])^(n + 1), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]
+Int[Sqrt[a_ + b_.*tan[c_. + d_.*x_]], x_Symbol] := -2*b/d*Subst[Int[1/(2*a - x^2), x], x, Sqrt[a + b*Tan[c + d*x]]] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]
+Int[(a_ + b_.*tan[c_. + d_.*x_])^n_, x_Symbol] := -b/d*Subst[Int[(a + x)^(n - 1)/(a - x), x], x, b*Tan[c + d*x]] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 + b^2, 0]
+Int[(a_ + b_.*tan[c_. + d_.*x_])^n_, x_Symbol] := b*(a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1)) + Int[(a^2 - b^2 + 2*a*b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n - 2), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[n, 1]
+Int[(a_ + b_.*tan[c_. + d_.*x_])^n_, x_Symbol] := b*(a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2)) + 1/(a^2 + b^2)* Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
+Int[1/(a_ + b_.*tan[c_. + d_.*x_]), x_Symbol] := a*x/(a^2 + b^2) + b/(a^2 + b^2)*Int[(b - a*Tan[c + d*x])/(a + b*Tan[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
+Int[(a_ + b_.*tan[c_. + d_.*x_])^n_, x_Symbol] := b/d*Subst[Int[(a + x)^n/(b^2 + x^2), x], x, b*Tan[c + d*x]] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 + b^2, 0]
diff --git a/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m
new file mode 100755
index 0000000..3c1794e
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m	
@@ -0,0 +1,35 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.3.1.2 (d sec)^m (a+b tan)^n *)
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol] := b*(d*Sec[e + f*x])^m/(f*m) + a*Int[(d*Sec[e + f*x])^m, x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] || NeQ[a^2 + b^2, 0])
+Int[sec[e_. + f_.*x_]^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := 1/(a^(m - 2)*b*f)* Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]] /; FreeQ[{a, b, e, f, n}, x] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n/(a*f*m) /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && EqQ[Simplify[m + n], 0]
+Int[sec[e_. + f_.*x_]/Sqrt[a_ + b_.*tan[e_. + f_.*x_]], x_Symbol] := -2*a/(b*f)* Subst[Int[1/(2 - a*x^2), x], x, Sec[e + f*x]/Sqrt[a + b*Tan[e + f*x]]] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 + b^2, 0]
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n/(a*f*m) + a/(2*d^2)* Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 1), x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && EqQ[m/2 + n, 0] && GtQ[n, 0]
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := 2*d^2*(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1)/(b* f*(m - 2)) + 2*d^2/a* Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1), x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && EqQ[m/2 + n, 0] && LtQ[n, -1]
+Int[(d_.*sec[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := (a/d)^(2*IntPart[n])*(a + b*Tan[e + f*x])^ FracPart[n]*(a - b*Tan[e + f*x])^ FracPart[n]/(d*Sec[e + f*x])^(2*FracPart[n])* Int[1/(a - b*Tan[e + f*x])^n, x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && EqQ[Simplify[m/2 + n], 0]
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := 2*b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1)/(f*m) /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && EqQ[Simplify[m/2 + n - 1], 0]
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1)) + a*(m + 2*n - 2)/(m + n - 1)* Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && IGtQ[Simplify[m/2 + n - 1], 0] && Not[IntegerQ[n]]
+Int[Sqrt[d_.*sec[e_. + f_.*x_]]*Sqrt[a_ + b_.*tan[e_. + f_.*x_]], x_Symbol] := -4*b*d^2/f* Subst[Int[x^2/(a^2 + d^2*x^4), x], x, Sqrt[a + b*Tan[e + f*x]]/Sqrt[d*Sec[e + f*x]]] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]
+Int[(d_.*sec[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := 2*b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1)/(f*m) - b^2*(m + 2*n - 2)/(d^2*m)* Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 2), x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 1] && (IGtQ[n/2, 0] && ILtQ[m - 1/2, 0] || EqQ[n, 2] && LtQ[m, 0] || LeQ[m, -1] && GtQ[m + n, 0] || ILtQ[m, 0] && LtQ[m/2 + n - 1, 0] && IntegerQ[n] || EqQ[n, 3/2] && EqQ[m, -1/2]) && IntegerQ[2*m]
+Int[(d_.*sec[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n/(a*f*m) + a*(m + n)/(m*d^2)* Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 1), x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1)) + a*(m + 2*n - 2)/(m + n - 1)* Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
+Int[(d_.*sec[e_. + f_.*x_])^(3/2)/Sqrt[a_ + b_.*tan[e_. + f_.*x_]], x_Symbol] := d*Sec[e + f*x]/(Sqrt[a - b*Tan[e + f*x]]*Sqrt[a + b*Tan[e + f*x]])* Int[Sqrt[d*Sec[e + f*x]]*Sqrt[a - b*Tan[e + f*x]], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]
+Int[(d_.*sec[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := 2*d^2*(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1)/(b* f*(m + 2*n)) - d^2*(m - 2)/(b^2*(m + 2*n))* Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 2), x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, -1] && (ILtQ[n/2, 0] && IGtQ[m - 1/2, 0] || EqQ[n, -2] || IGtQ[m + n, 0] || IntegersQ[n, m + 1/2] && GtQ[2*m + n + 1, 0]) && IntegerQ[2*m]
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := d^2*(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1)/(b* f*(m + n - 1)) + d^2*(m - 2)/(a*(m + n - 1))* Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1), x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0] && GtQ[m, 1] && Not[ILtQ[m + n, 0]] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := a*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n/(b*f*(m + 2*n)) + Simplify[m + n]/(a*(m + 2*n))* Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2*n]
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b*(d*Sec[e + f*x])^ m*(a + b*Tan[e + f*x])^(n - 1)/(f*Simplify[m + n - 1]) + a*(m + 2*n - 2)/Simplify[m + n - 1]* Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && IGtQ[Simplify[m + n - 1], 0] && RationalQ[n]
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := a*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n/(b*f*(m + 2*n)) + Simplify[m + n]/(a*(m + 2*n))* Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && ILtQ[Simplify[m + n], 0] && NeQ[m + 2*n, 0]
+(* Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol]  := a^n*(d*Sec[e+f*x])^m/(b*f*(Sec[e+f*x]^2)^(m/2))*Subst[Int[(1+x/a)^( n+m/2-1)*(1-x/a)^(m/2-1),x],x,b*Tan[e+f*x]] /; FreeQ[{a,b,d,e,f,m},x] && EqQ[a^2+b^2,0] && IntegerQ[n] *)
+(* Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol]  := (d*Sec[e+f*x])^m/(b*f*(Sec[e+f*x]^2)^(m/2))*Subst[Int[(a+x)^n*(1+x^ 2/b^2)^(m/2-1),x],x,b*Tan[e+f*x]] /; FreeQ[{a,b,d,e,f,m,n},x] && EqQ[a^2+b^2,0] *)
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_., x_Symbol] := (d*Sec[e + f*x])^ m/((a + b*Tan[e + f*x])^(m/2)*(a - b*Tan[e + f*x])^(m/2))* Int[(a + b*Tan[e + f*x])^(m/2 + n)*(a - b*Tan[e + f*x])^(m/2), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0]
+Int[sec[e_. + f_.*x_]^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := 1/(b*f)* Subst[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b^2, 0] && IntegerQ[m/2]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^2/sec[e_. + f_.*x_], x_Symbol] := b^2*ArcTanh[Sin[e + f*x]]/f - 2*a*b*Cos[e + f*x]/f + (a^2 - b^2)*Sin[e + f*x]/f /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 + b^2, 0]
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^2, x_Symbol] := b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])/(f*(m + 1)) + 1/(m + 1)* Int[(d*Sec[e + f*x])^ m*(a^2*(m + 1) - b^2 + a*b*(m + 2)*Tan[e + f*x]), x] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 + b^2, 0] && NeQ[m, -1]
+Int[sec[e_. + f_.*x_]/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol] := -1/f*Subst[Int[1/(a^2 + b^2 - x^2), x], x, (b - a*Tan[e + f*x])/Sec[e + f*x]] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 + b^2, 0]
+Int[(d_.*sec[e_. + f_.*x_])^m_/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol] := -d^2/b^2*Int[(d*Sec[e + f*x])^(m - 2)*(a - b*Tan[e + f*x]), x] + d^2*(a^2 + b^2)/b^2* Int[(d*Sec[e + f*x])^(m - 2)/(a + b*Tan[e + f*x]), x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 1]
+Int[(d_.*sec[e_. + f_.*x_])^m_/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol] := 1/(a^2 + b^2)*Int[(d*Sec[e + f*x])^m*(a - b*Tan[e + f*x]), x] + b^2/(d^2*(a^2 + b^2))* Int[(d*Sec[e + f*x])^(m + 2)/(a + b*Tan[e + f*x]), x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[m, 0]
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := d^(2*IntPart[m/2])*(d*Sec[e + f*x])^(2*FracPart[m/2])/(b* f*(Sec[e + f*x]^2)^FracPart[m/2])* Subst[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]] /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 + b^2, 0] && Not[IntegerQ[m/2]]
+Int[Sqrt[a_ + b_.*tan[e_. + f_.*x_]]/Sqrt[d_. cos[e_. + f_.*x_]], x_Symbol] := -4*b/f* Subst[Int[x^2/(a^2*d^2 + x^4), x], x, Sqrt[d*Cos[e + f*x]]*Sqrt[a + b*Tan[e + f*x]]] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]
+Int[1/((d_. cos[e_. + f_.*x_])^(3/2)* Sqrt[a_ + b_.*tan[e_. + f_.*x_]]), x_Symbol] := 1/(d*Cos[e + f*x]*Sqrt[a - b*Tan[e + f*x]]* Sqrt[a + b*Tan[e + f*x]])* Int[Sqrt[a - b*Tan[e + f*x]]/Sqrt[d*Cos[e + f*x]], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]
+Int[(d_.*cos[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_., x_Symbol] := (d*Cos[e + f*x])^m*(d*Sec[e + f*x])^m* Int[(a + b*Tan[e + f*x])^n/(d*Sec[e + f*x])^m, x] /; FreeQ[{a, b, d, e, f, m, n}, x] && Not[IntegerQ[m]]
diff --git a/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.3 (d sin)^m (a+b tan)^n.m b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.3 (d sin)^m (a+b tan)^n.m
new file mode 100755
index 0000000..401894d
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.3 (d sin)^m (a+b tan)^n.m	
@@ -0,0 +1,9 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.3.1.3 (d sin)^m (a+b tan)^n *)
+Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b/f*Subst[Int[x^m*(a + x)^n/(b^2 + x^2)^(m/2 + 1), x], x, b*Tan[e + f*x]] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/2]
+Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_., x_Symbol] := Int[Expand[Sin[e + f*x]^m*(a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IGtQ[n, 0]
+Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_., x_Symbol] := Int[Sin[e + f*x]^m*(a*Cos[e + f*x] + b*Sin[e + f*x])^n/ Cos[e + f*x]^n, x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && ILtQ[n, 0] && (LtQ[m, 5] && GtQ[n, -4] || EqQ[m, 5] && EqQ[n, -1])
+Int[(d_.*csc[e_. + f_.*x_])^m_*(a_. + b_.*tan[e_. + f_.*x_])^n_., x_Symbol] := (d*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/d)^FracPart[m]* Int[(a + b*Tan[e + f*x])^n/(Sin[e + f*x]/d)^m, x] /; FreeQ[{a, b, d, e, f, m, n}, x] && Not[IntegerQ[m]]
+Int[cos[e_. + f_.*x_]^m_.* sin[e_. + f_.*x_]^p_.*(a_ + b_.*tan[e_. + f_.*x_])^n_., x_Symbol] := Int[Cos[e + f*x]^(m - n)* Sin[e + f*x]^p*(a*Cos[e + f*x] + b*Sin[e + f*x])^n, x] /; FreeQ[{a, b, e, f, m, p}, x] && IntegerQ[n]
+Int[sin[e_. + f_.*x_]^m_.* cos[e_. + f_.*x_]^p_.*(a_ + b_.*cot[e_. + f_.*x_])^n_., x_Symbol] := Int[Sin[e + f*x]^(m - n)* Cos[e + f*x]^p*(a*Sin[e + f*x] + b*Cos[e + f*x])^n, x] /; FreeQ[{a, b, e, f, m, p}, x] && IntegerQ[n]
diff --git a/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m
new file mode 100755
index 0000000..76ef286
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m	
@@ -0,0 +1,31 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.3.10 (c+d x)^m (a+b tan)^n *)
+Int[(c_. + d_.*x_)^m_.*tan[e_. + k_.*Pi + f_.*Complex[0, fz_]*x_], x_Symbol] := -I*(c + d*x)^(m + 1)/(d*(m + 1)) + 2*I*Int[(c + d*x)^m*E^(-2*I*k*Pi)* E^(2*(-I*e + f*fz*x))/(1 + E^(-2*I*k*Pi)*E^(2*(-I*e + f*fz*x))), x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*tan[e_. + k_.*Pi + f_.*x_], x_Symbol] := I*(c + d*x)^(m + 1)/(d*(m + 1)) - 2*I*Int[(c + d*x)^m*E^(2*I*k*Pi)* E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))), x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*tan[e_. + f_.*Complex[0, fz_]*x_], x_Symbol] := -I*(c + d*x)^(m + 1)/(d*(m + 1)) + 2*I*Int[(c + d*x)^m* E^(2*(-I*e + f*fz*x))/(1 + E^(2*(-I*e + f*fz*x))), x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*tan[e_. + f_.*x_], x_Symbol] := I*(c + d*x)^(m + 1)/(d*(m + 1)) - 2*I*Int[(c + d*x)^m*E^(2*I*(e + f*x))/(1 + E^(2*I*(e + f*x))), x] /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b*(c + d*x)^m*(b*Tan[e + f*x])^(n - 1)/(f*(n - 1)) - b*d*m/(f*(n - 1))* Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] - b^2*Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := (c + d*x)^m*(b*Tan[e + f*x])^(n + 1)/(b*f*(n + 1)) - d*m/(b*f*(n + 1))* Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n + 1), x] - 1/b^2*Int[(c + d*x)^m*(b*Tan[e + f*x])^(n + 2), x] /; FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1] && GtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(c_. + d_.*x_)^m_./(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol] := (c + d*x)^(m + 1)/(2*a*d*(m + 1)) - a*(c + d*x)^m/(2*b*f*(a + b*Tan[e + f*x])) + a*d*m/(2*b*f)*Int[(c + d*x)^(m - 1)/(a + b*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[m, 0]
+Int[1/((c_. + d_.*x_)^2*(a_ + b_.*tan[e_. + f_.*x_])), x_Symbol] := -1/(d*(c + d*x)*(a + b*Tan[e + f*x])) + f/(b*d)*Int[Cos[2*e + 2*f*x]/(c + d*x), x] - f/(a*d)*Int[Sin[2*e + 2*f*x]/(c + d*x), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0]
+Int[(c_. + d_.*x_)^m_/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol] := f*(c + d*x)^(m + 2)/(b*d^2*(m + 1)*(m + 2)) + (c + d*x)^(m + 1)/(d*(m + 1)*(a + b*Tan[e + f*x])) + 2*b*f/(a*d*(m + 1))* Int[(c + d*x)^(m + 1)/(a + b*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && LtQ[m, -1] && NeQ[m, -2]
+(* Int[(c_.+d_.*x_)^m_/(a_+b_.*tan[e_.+f_.*x_]),x_Symbol] := (c+d*x)^(m+1)/(d*(m+1)*(a+b*Tan[e+f*x])) + f/(b*d*(m+1))*Int[(c+d*x)^(m+1),x] + 2*b*f/(a*d*(m+1))*Int[(c+d*x)^(m+1)/(a+b*Tan[e+f*x]),x] /; FreeQ[{a,b,c,d,e,f},x] && EqQ[a^2+b^2,0] && LtQ[m,-1] *)
+Int[1/((c_. + d_.*x_)*(a_ + b_.*tan[e_. + f_.*x_])), x_Symbol] := Log[c + d*x]/(2*a*d) + 1/(2*a)*Int[Cos[2*e + 2*f*x]/(c + d*x), x] + 1/(2*b)*Int[Sin[2*e + 2*f*x]/(c + d*x), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0]
+Int[(c_. + d_.*x_)^m_/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol] := (c + d*x)^(m + 1)/(2*a*d*(m + 1)) + 1/(2*a)*Int[(c + d*x)^m*E^(2*a/b*(e + f*x)), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && Not[IntegerQ[m]]
+Int[(c_. + d_.*x_)^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[(c + d*x)^ m, (1/(2*a) + Cos[2*e + 2*f*x]/(2*a) + Sin[2*e + 2*f*x]/(2*b))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && ILtQ[m, 0] && ILtQ[n, 0]
+Int[(c_. + d_.*x_)^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[(c + d*x)^ m, (1/(2*a) + E^(2*a/b*(e + f*x))/(2*a))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && ILtQ[n, 0]
+Int[(c_. + d_.*x_)^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := With[{u = IntHide[(a + b*Tan[e + f*x])^n, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[Dist[(c + d*x)^(m - 1), u, x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && ILtQ[n, -1] && GtQ[m, 0]
+Int[(c_. + d_.*x_)^m_./(a_ + b_.*tan[e_. + k_.*Pi + f_.*x_]), x_Symbol] := (c + d*x)^(m + 1)/(d*(m + 1)*(a + I*b)) + 2*I*b* Int[(c + d*x)^m*E^(2*I*k*Pi)* E^Simp[2*I*(e + f*x), x]/((a + I*b)^2 + (a^2 + b^2)*E^(2*I*k*Pi)* E^Simp[2*I*(e + f*x), x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[4*k] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_./(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol] := (c + d*x)^(m + 1)/(d*(m + 1)*(a + I*b)) + 2*I*b* Int[(c + d*x)^m* E^Simp[2*I*(e + f*x), x]/((a + I*b)^2 + (a^2 + b^2)*E^Simp[2*I*(e + f*x), x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)/(a_ + b_.*tan[e_. + f_.*x_])^2, x_Symbol] := -(c + d*x)^2/(2*d*(a^2 + b^2)) - b*(c + d*x)/(f*(a^2 + b^2)*(a + b*Tan[e + f*x])) + 1/(f*(a^2 + b^2))* Int[(b*d + 2*a*c*f + 2*a*d*f*x)/(a + b*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0]
+Int[(c_. + d_.*x_)^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[(c + d*x)^ m, (1/(a - I*b) - 2*I*b/(a^2 + b^2 + (a - I*b)^2*E^(2*I*(e + f*x))))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[n, 0] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)*Sqrt[a_ + b_.*tan[e_. + f_.*x_]], x_Symbol] := -Sqrt[2]*b*(c + d*x)* ArcTanh[Sqrt[a + b*Tan[e + f*x]]/(Sqrt[2]*Rt[a, 2])]/(Rt[a, 2]* f) + Sqrt[2]*b*d/(Rt[a, 2]*f)* Int[ArcTanh[Sqrt[a + b*Tan[e + f*x]]/(Sqrt[2]*Rt[a, 2])], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0]
+Int[(c_. + d_.*x_)*Sqrt[a_. + b_.*tan[e_. + f_.*x_]], x_Symbol] := -I*Rt[a - I*b, 2]*(c + d*x)/f* ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[a - I*b, 2]] + I*Rt[a + I*b, 2]*(c + d*x)/f* ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[a + I*b, 2]] + I*d*Rt[a - I*b, 2]/f* Int[ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[a - I*b, 2]], x] - I*d*Rt[a + I*b, 2]/f* Int[ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[a + I*b, 2]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0]
+Int[(c_. + d_.*x_)/Sqrt[a_ + b_.*tan[e_. + f_.*x_]], x_Symbol] := 1/(2*a)*Int[(c + d*x)*Sqrt[a + b*Tan[e + f*x]], x] + a/2*Int[(c + d*x)*Sec[e + f*x]^2/(a + b*Tan[e + f*x])^(3/2), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0]
+Int[(c_. + d_.*x_)/Sqrt[a_. + b_.*tan[e_. + f_.*x_]], x_Symbol] := -I*(c + d*x)/(f*Rt[a - I*b, 2])* ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[a - I*b, 2]] + I*(c + d*x)/(f*Rt[a + I*b, 2])* ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[a + I*b, 2]] + I*d/(f*Rt[a - I*b, 2])* Int[ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[a - I*b, 2]], x] - I*d/(f*Rt[a + I*b, 2])* Int[ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[a + I*b, 2]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0]
+Int[(c_. + d_.*x_)^m_.*tan[e_. + f_.*x_]^n_., x_Symbol] := If[MatchQ[f, f1_.*Complex[0, j_]], If[MatchQ[e, e1_. + Pi/2], I^n*Unintegrable[(c + d*x)^m*Coth[-I*(e - Pi/2) - I*f*x]^n, x], I^n*Unintegrable[(c + d*x)^m*Tanh[-I*e - I*f*x]^n, x]], If[MatchQ[e, e1_. + Pi/2], (-1)^n*Unintegrable[(c + d*x)^m*Cot[e - Pi/2 + f*x]^n, x], Unintegrable[(c + d*x)^m*Tan[e + f*x]^n, x]]] /; FreeQ[{c, d, e, f, m, n}, x] && IntegerQ[n]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*tan[e_. + f_.*x_])^n_., x_Symbol] := Unintegrable[(c + d*x)^m*(a + b*Tan[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
+Int[u_^m_.*(a_. + b_.*Tan[v_])^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*(a + b*Tan[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
+Int[u_^m_.*(a_. + b_.*Cot[v_])^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*(a + b*Cot[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
diff --git a/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m
new file mode 100755
index 0000000..13c9cc8
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m	
@@ -0,0 +1,27 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.3.11 (e x)^m (a+b tan(c+d x^n))^p *)
+Int[(a_. + b_.*Tan[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[Int[x^(1/n - 1)*(a + b*Tan[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
+Int[(a_. + b_.*Cot[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[Int[x^(1/n - 1)*(a + b*Cot[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
+Int[(a_. + b_.*Tan[c_. + d_.*x_^n_])^p_., x_Symbol] := Unintegrable[(a + b*Tan[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
+Int[(a_. + b_.*Cot[c_. + d_.*x_^n_])^p_., x_Symbol] := Unintegrable[(a + b*Cot[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
+Int[(a_. + b_.*Tan[c_. + d_.*u_^n_])^p_., x_Symbol] := 1/Coefficient[u, x, 1]* Subst[Int[(a + b*Tan[c + d*x^n])^p, x], x, u] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
+Int[(a_. + b_.*Cot[c_. + d_.*u_^n_])^p_., x_Symbol] := 1/Coefficient[u, x, 1]* Subst[Int[(a + b*Cot[c + d*x^n])^p, x], x, u] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
+Int[(a_. + b_.*Tan[u_])^p_., x_Symbol] := Int[(a + b*Tan[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[(a_. + b_.*Cot[u_])^p_., x_Symbol] := Int[(a + b*Cot[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[x_^m_.*(a_. + b_.*Tan[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Tan[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
+Int[x_^m_.*(a_. + b_.*Cot[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cot[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
+Int[x_^m_.*Tan[c_. + d_.*x_^n_]^2, x_Symbol] := x^(m - n + 1)*Tan[c + d*x^n]/(d*n) - Int[x^m, x] - (m - n + 1)/(d*n)*Int[x^(m - n)*Tan[c + d*x^n], x] /; FreeQ[{c, d, m, n}, x]
+Int[x_^m_.*Cot[c_. + d_.*x_^n_]^2, x_Symbol] := -x^(m - n + 1)*Cot[c + d*x^n]/(d*n) - Int[x^m, x] + (m - n + 1)/(d*n)*Int[x^(m - n)*Cot[c + d*x^n], x] /; FreeQ[{c, d, m, n}, x]
+(* Int[x_^m_.*Tan[a_.+b_.*x_^n_]^p_,x_Symbol] := x^(m-n+1)*Tan[a+b*x^n]^(p-1)/(b*n*(p-1)) - (m-n+1)/(b*n*(p-1))*Int[x^(m-n)*Tan[a+b*x^n]^(p-1),x] - Int[x^m*Tan[a+b*x^n]^(p-2),x] /; FreeQ[{a,b},x] && LtQ[0,n,m+1] && GtQ[p,1] *)
+(* Int[x_^m_.*Cot[a_.+b_.*x_^n_]^p_,x_Symbol] := -x^(m-n+1)*Cot[a+b*x^n]^(p-1)/(b*n*(p-1)) + (m-n+1)/(b*n*(p-1))*Int[x^(m-n)*Cot[a+b*x^n]^(p-1),x] - Int[x^m*Cot[a+b*x^n]^(p-2),x] /; FreeQ[{a,b},x] && LtQ[0,n,m+1] && GtQ[p,1] *)
+(* Int[x_^m_.*Tan[a_.+b_.*x_^n_]^p_,x_Symbol] := x^(m-n+1)*Tan[a+b*x^n]^(p+1)/(b*n*(p+1)) - (m-n+1)/(b*n*(p+1))*Int[x^(m-n)*Tan[a+b*x^n]^(p+1),x] - Int[x^m*Tan[a+b*x^n]^(p+2),x] /; FreeQ[{a,b},x] && LtQ[0,n,m+1] && LtQ[p,-1] *)
+(* Int[x_^m_.*Cot[a_.+b_.*x_^n_]^p_,x_Symbol] := -x^(m-n+1)*Cot[a+b*x^n]^(p+1)/(b*n*(p+1)) + (m-n+1)/(b*n*(p+1))*Int[x^(m-n)*Cot[a+b*x^n]^(p+1),x] - Int[x^m*Cot[a+b*x^n]^(p+2),x] /; FreeQ[{a,b},x] && LtQ[0,n,m+1] && LtQ[p,-1] *)
+Int[x_^m_.*(a_. + b_.*Tan[c_. + d_.*x_^n_])^p_., x_Symbol] := Unintegrable[x^m*(a + b*Tan[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
+Int[x_^m_.*(a_. + b_.*Cot[c_. + d_.*x_^n_])^p_., x_Symbol] := Unintegrable[x^m*(a + b*Cot[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
+Int[(e_*x_)^m_.*(a_. + b_.*Tan[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Tan[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
+Int[(e_*x_)^m_.*(a_. + b_.*Cot[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Cot[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
+Int[(e_*x_)^m_.*(a_. + b_.*Tan[u_])^p_., x_Symbol] := Int[(e*x)^m*(a + b*Tan[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[(e_*x_)^m_.*(a_. + b_.*Cot[u_])^p_., x_Symbol] := Int[(e*x)^m*(a + b*Cot[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[x_^m_.*Sec[a_. + b_.*x_^n_.]^p_.*Tan[a_. + b_.*x_^n_.]^q_., x_Symbol] := x^(m - n + 1)*Sec[a + b*x^n]^p/(b*n*p) - (m - n + 1)/(b*n*p)*Int[x^(m - n)*Sec[a + b*x^n]^p, x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m, n] && EqQ[q, 1]
+Int[x_^m_.*Csc[a_. + b_.*x_^n_.]^p_.*Cot[a_. + b_.*x_^n_.]^q_., x_Symbol] := -x^(m - n + 1)*Csc[a + b*x^n]^p/(b*n*p) + (m - n + 1)/(b*n*p)*Int[x^(m - n)*Csc[a + b*x^n]^p, x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m, n] && EqQ[q, 1]
diff --git a/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.12 (d+e x)^m tan(a+b x+c x^2)^n.m b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.12 (d+e x)^m tan(a+b x+c x^2)^n.m
new file mode 100755
index 0000000..4ea104d
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.12 (d+e x)^m tan(a+b x+c x^2)^n.m	
@@ -0,0 +1,13 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.3.12 (d+e x)^m tan(a+b x+c x^2)^n *)
+Int[Tan[a_. + b_.*x_ + c_.*x_^2]^n_., x_Symbol] := Unintegrable[Tan[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, n}, x]
+Int[Cot[a_. + b_.*x_ + c_.*x_^2]^n_., x_Symbol] := Unintegrable[Cot[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, n}, x]
+Int[(d_ + e_.*x_)*Tan[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := -e*Log[Cos[a + b*x + c*x^2]]/(2*c) /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
+Int[(d_ + e_.*x_)*Cot[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*Log[Sin[a + b*x + c*x^2]]/(2*c) /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
+Int[(d_. + e_.*x_)*Tan[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := -e*Log[Cos[a + b*x + c*x^2]]/(2*c) + (2*c*d - b*e)/(2*c)*Int[Tan[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0]
+Int[(d_. + e_.*x_)*Cot[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*Log[Sin[a + b*x + c*x^2]]/(2*c) + (2*c*d - b*e)/(2*c)*Int[Cot[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0]
+(* Int[x_^m_*Tan[a_.+b_.*x_+c_.*x_^2],x_Symbol] := -x^(m-1)*Log[Cos[a+b*x+c*x^2]]/(2*c) - b/(2*c)*Int[x^(m-1)*Tan[a+b*x+c*x^2],x] + (m-1)/(2*c)*Int[x^(m-2)*Log[Cos[a+b*x+c*x^2]],x] /; FreeQ[{a,b,c},x] && GtQ[m,1] *)
+(* Int[x_^m_*Cot[a_.+b_.*x_+c_.*x_^2],x_Symbol] := x^(m-1)*Log[Sin[a+b*x+c*x^2]]/(2*c) - b/(2*c)*Int[x^(m-1)*Cot[a+b*x+c*x^2],x] - (m-1)/(2*c)*Int[x^(m-2)*Log[Sin[a+b*x+c*x^2]],x] /; FreeQ[{a,b,c},x] && GtQ[m,1]*)
+Int[(d_. + e_.*x_)^m_.*Tan[a_. + b_.*x_ + c_.*x_^2]^n_., x_Symbol] := Unintegrable[(d + e*x)^m*Tan[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[(d_. + e_.*x_)^m_.*Cot[a_. + b_.*x_ + c_.*x_^2]^n_., x_Symbol] := Unintegrable[(d + e*x)^m*Cot[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
diff --git a/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m
new file mode 100755
index 0000000..2c81aae
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m	
@@ -0,0 +1,63 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.3.2.1 (a+b tan)^m (c+d tan)^n *)
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_.*(c_ + d_.*tan[e_. + f_.*x_])^n_., x_Symbol] := a^m*c^m*Int[Sec[e + f*x]^(2*m)*(c + d*Tan[e + f*x])^(n - m), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0] && IntegerQ[m] && Not[IGtQ[n, 0] && (LtQ[m, 0] || GtQ[m, n])]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_ + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := a*c/f* Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1), x], x, Tan[e + f*x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
+Int[(a_ + b_.*tan[e_. + f_.*x_])*(c_ + d_.*tan[e_. + f_.*x_]), x_Symbol] := (a*c - b*d)*x + b*d*Tan[e + f*x]/f /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + a*d, 0]
+Int[(a_ + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol] := (a*c - b*d)*x + b*d*Tan[e + f*x]/f + (b*c + a*d)*Int[Tan[e + f*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol] := -(b*c - a*d)*(a + b*Tan[e + f*x])^m/(2*a*f*m) + (b*c + a*d)/(2*a*b)*Int[(a + b*Tan[e + f*x])^(m + 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol] := d*(a + b*Tan[e + f*x])^m/(f*m) + (b*c + a*d)/b* Int[(a + b*Tan[e + f*x])^m, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && Not[LtQ[m, 0]]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol] := d*(a + b*Tan[e + f*x])^m/(f*m) + Int[(a + b*Tan[e + f*x])^(m - 1)* Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol] := (b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 + b^2)) + 1/(a^2 + b^2)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]
+Int[(c_ + d_.*tan[e_. + f_.*x_])/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol] := c/(b*f)*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
+Int[(c_. + d_.*tan[e_. + f_.*x_])/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol] := (a*c + b*d)*x/(a^2 + b^2) + (b*c - a*d)/(a^2 + b^2)* Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]
+Int[(c_ + d_.*tan[e_. + f_.*x_])/Sqrt[b_.*tan[e_. + f_.*x_]], x_Symbol] := -2*d^2/f* Subst[Int[1/(2*c*d + b*x^2), x], x, (c - d*Tan[e + f*x])/Sqrt[b*Tan[e + f*x]]] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 - d^2, 0]
+(* Int[(c_+d_.*tan[e_.+f_.*x_])/Sqrt[b_.*tan[e_.+f_.*x_]],x_Symbol] := (c+d)/2*Int[(1+Tan[e+f*x])/Sqrt[b*Tan[e+f*x]],x] + (c-d)/2*Int[(1-Tan[e+f*x])/Sqrt[b*Tan[e+f*x]],x] /; FreeQ[{b,c,d,e,f},x] && NeQ[c^2+d^2,0] && NeQ[c^2-d^2,0] *)
+Int[(c_ + d_.*tan[e_. + f_.*x_])/Sqrt[b_.*tan[e_. + f_.*x_]], x_Symbol] := 2*c^2/f*Subst[Int[1/(b*c - d*x^2), x], x, Sqrt[b*Tan[e + f*x]]] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 + d^2, 0]
+(* Int[(c_+d_.*tan[e_.+f_.*x_])/Sqrt[b_.*tan[e_.+f_.*x_]],x_Symbol] := (c+I*d)/2*Int[(1-I*Tan[e+f*x])/Sqrt[b*Tan[e+f*x]],x] +  (c-I*d)/2*Int[(1+I*Tan[e+f*x])/Sqrt[b*Tan[e+f*x]],x] /; FreeQ[{b,c,d,e,f},x] && NeQ[c^2-d^2,0] && NeQ[c^2+d^2,0] *)
+Int[(c_ + d_.*tan[e_. + f_.*x_])/Sqrt[b_.*tan[e_. + f_.*x_]], x_Symbol] := 2/f*Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && NeQ[c^2 + d^2, 0]
+Int[(c_. + d_.*tan[e_. + f_.*x_])/Sqrt[a_ + b_.*tan[e_. + f_.*x_]], x_Symbol] := -2*d^2/f* Subst[Int[1/(2*b*c*d - 4*a*d^2 + x^2), x], x, (b*c - 2*a*d - b*d*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[2*a*c*d - b*(c^2 - d^2), 0]
+Int[(c_. + d_.*tan[e_. + f_.*x_])/Sqrt[a_ + b_.*tan[e_. + f_.*x_]], x_Symbol] := With[{q = Rt[a^2 + b^2, 2]}, 1/(2*q)* Int[(a*c + b*d + c*q + (b*c - a*d + d*q)*Tan[e + f*x])/ Sqrt[a + b*Tan[e + f*x]], x] - 1/(2*q)* Int[(a*c + b*d - c*q + (b*c - a*d - d*q)*Tan[e + f*x])/ Sqrt[a + b*Tan[e + f*x]], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && NeQ[2*a*c*d - b*(c^2 - d^2), 0] && (PerfectSquareQ[a^2 + b^2] || RationalQ[a, b, c, d])
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_ + d_.*tan[e_. + f_.*x_]), x_Symbol] := c*d/f*Subst[Int[(a + b/d*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
+Int[(b_.*tan[e_. + f_.*x_])^m_*(c_ + d_.*tan[e_. + f_.*x_]), x_Symbol] := c*Int[(b*Tan[e + f*x])^m, x] + d/b*Int[(b*Tan[e + f*x])^(m + 1), x] /; FreeQ[{b, c, d, e, f, m}, x] && NeQ[c^2 + d^2, 0] && Not[IntegerQ[2*m]]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol] := (c + I*d)/2*Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x] + (c - I*d)/2*Int[(a + b*Tan[e + f*x])^m*(1 + I*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Not[IntegerQ[m]]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^2, x_Symbol] := -b*(a*c + b*d)^2*(a + b*Tan[e + f*x])^m/(2*a^3*f*m) + 1/(2*a^2)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[a*c^2 - 2*b*c*d + a*d^2 - 2*b*d^2*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LeQ[m, -1] && EqQ[a^2 + b^2, 0]
+Int[(c_. + d_.*tan[e_. + f_.*x_])^2/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol] := d*(2*b*c - a*d)*x/b^2 + d^2/b*Int[Tan[e + f*x], x] + (b*c - a*d)^2/b^2* Int[1/(a + b*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^2, x_Symbol] := (b*c - a*d)^2*(a + b*Tan[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 + b^2)) + 1/(a^2 + b^2)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^2, x_Symbol] := d^2*(a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)) + Int[(a + b*Tan[e + f*x])^m* Simp[c^2 - d^2 + 2*c*d*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && Not[LeQ[m, -1]] && Not[EqQ[m, 2] && EqQ[a, 0]]
+Int[Sqrt[a_ + b_.*tan[e_. + f_.*x_]]/ Sqrt[c_. + d_.*tan[e_. + f_.*x_]], x_Symbol] := -2*a*b/f* Subst[Int[1/(a*c - b*d - 2*a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := a*b*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)/(f*(m - 1)*(a*c - b*d)) + 2*a^2/(a*c - b*d)* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[m + n, 0] && GtQ[m, 1/2]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := a*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n/(2*b*f*m) - (a*c - b*d)/(2*b^2)* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[m + n, 0] && LeQ[m, -1/2]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := a*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d)) + 1/(2*a)* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[m + n + 1, 0] && LtQ[m, -1]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := -d*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(f*m*(c^2 + d^2)) + a/(a*c - b*d)* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[m + n + 1, 0] && Not[LtQ[m, -1]]
+Int[(c_. + d_.*tan[e_. + f_.*x_])^n_/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol] := -(a*c + b*d)*(c + d*Tan[e + f*x])^ n/(2*(b*c - a*d)*f*(a + b*Tan[e + f*x])) + 1/(2*a*(b*c - a*d))* Int[(c + d*Tan[e + f*x])^(n - 1)* Simp[a*c*d*(n - 1) + b*c^2 + b*d^2*n - d*(b*c - a*d)*(n - 1)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[0, n, 1]
+Int[(c_. + d_.*tan[e_. + f_.*x_])^n_/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol] := (b*c - a*d)*(c + d*Tan[e + f*x])^(n - 1)/(2*a*f*(a + b*Tan[e + f*x])) + 1/(2*a^2)* Int[(c + d*Tan[e + f*x])^(n - 2)* Simp[a*c^2 + a*d^2*(n - 1) - b*c*d*n - d*(a*c*(n - 2) + b*d*n)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[n, 1]
+Int[1/((a_. + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])), x_Symbol] := b/(b*c - a*d)*Int[1/(a + b*Tan[e + f*x]), x] - d/(b*c - a*d)*Int[1/(c + d*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[(c_. + d_.*tan[e_. + f_.*x_])^n_/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol] := -a*(c + d*Tan[e + f*x])^(n + 1)/(2* f*(b*c - a*d)*(a + b*Tan[e + f*x])) + 1/(2*a*(b*c - a*d))* Int[(c + d*Tan[e + f*x])^n* Simp[b*c + a*d*(n - 1) - b*d*n*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Not[GtQ[n, 0]]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := -a^2*(b*c - a*d)*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(b*c + a*d)*(n + 1)) + a/(d*(b*c + a*d)*(n + 1))* Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)* Simp[b*(b*c*(m - 2) - a*d*(m - 2*n - 4)) + (a*b*c*(m - 2) + b^2*d*(n + 1) - a^2*d*(m + n - 1))*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
+Int[(a_ + b_.*tan[e_. + f_.*x_])^(3/2)/(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol] := 2*a^2/(a*c - b*d)*Int[Sqrt[a + b*Tan[e + f*x]], x] - (2*b*c*d + a*(c^2 - d^2))/(a*(c^2 + d^2))* Int[(a - b*Tan[e + f*x])* Sqrt[a + b*Tan[e + f*x]]/(c + d*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^(3/2)/ Sqrt[c_. + d_.*tan[e_. + f_.*x_]], x_Symbol] := 2*a*Int[Sqrt[a + b*Tan[e + f*x]]/Sqrt[c + d*Tan[e + f*x]], x] + b/a* Int[(b + a*Tan[e + f*x])* Sqrt[a + b*Tan[e + f*x]]/Sqrt[c + d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)/(d* f*(m + n - 1)) + a/(d*(m + n - 1))* Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n* Simp[b*c*(m - 2) + a*d*(m + 2*n) + (a*c*(m - 2) + b*d*(3*m + 2*n - 4))* Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 1] && NeQ[m + n - 1, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*Sqrt[c_. + d_.*tan[e_. + f_.*x_]], x_Symbol] := -b*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]]/(2*a*f*m) + 1/(4*a^2*m)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[2*a*c*m + b*d + a*d*(2*m + 1)*Tan[e + f*x], x]/ Sqrt[c + d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && IntegersQ[2*m]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := -(b*c - a*d)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n - 1)/(2*a*f*m) + 1/(2*a^2*m)* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)* Simp[c*(a*c*m + b*d*(n - 1)) - d*(b*c*m + a*d*(n - 1)) - d*(b*d*(m - n + 1) - a*c*(m + n - 1))*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && GtQ[n, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := a*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d)) + 1/(2*a*m*(b*c - a*d))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := d*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n - 1)/(f*(m + n - 1)) - 1/(a*(m + n - 1))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 2)* Simp[d*(b*c*m + a*d*(-1 + n)) - a*c^2*(m + n - 1) + d*(b*d*m - a*c*(m + 2*n - 2))*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[n, 1] && NeQ[m + n - 1, 0] && (IntegerQ[n] || IntegersQ[2*m, 2*n])
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := d*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 + d^2)) - 1/(a*(c^2 + d^2)*(n + 1))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)* Simp[b*d*m - a*c*(n + 1) + a*d*(m + n + 1)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || IntegersQ[2*m, 2*n])
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_/(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol] := a/(a*c - b*d)*Int[(a + b*Tan[e + f*x])^m, x] - d/(a*c - b*d)* Int[(a + b*Tan[e + f*x])^ m*(b + a*Tan[e + f*x])/(c + d*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[Sqrt[a_ + b_.*tan[e_. + f_.*x_]]* Sqrt[c_. + d_.*tan[e_. + f_.*x_]], x_Symbol] := (a*c - b*d)/a* Int[Sqrt[a + b*Tan[e + f*x]]/Sqrt[c + d*Tan[e + f*x]], x] + d/a* Int[Sqrt[a + b*Tan[e + f*x]]*(b + a*Tan[e + f*x])/ Sqrt[c + d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := a*b/f* Subst[Int[(a + x)^(m - 1)*(c + d/b*x)^n/(b^2 + a*x), x], x, b*Tan[e + f*x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := (b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2)) - 1/(d*(n + 1)*(c^2 + d^2))* Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)* Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d)* Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && IntegerQ[2*m]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)/(d* f*(m + n - 1)) + 1/(d*(m + n - 1))* Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n* Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) && Not[IGtQ[n, 2] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := (b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 + b^2)) + 1/((m + 1)*(a^2 + b^2))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)* Simp[a*c^2*(m + 1) + a*d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2*a*c*d - b*d^2)*(m + 1)* Tan[e + f*x] - d*(b*c - a*d)*(m + n)*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegerQ[2*m]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^ n/(f*(m + 1)*(a^2 + b^2)) + 1/((m + 1)*(a^2 + b^2))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)* Simp[a*c*(m + 1) - b*d*n - (b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[2*m]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)) + 1/((m + 1)*(a^2 + b^2)*(b*c - a*d))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) && Not[ILtQ[n, -1] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := b*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^ n/(f*(m + n - 1)) + 1/(m + n - 1)* Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n - 1)* Simp[a^2*c*(m + n - 1) - b*(b*c*(m - 1) + a*d*n) + (2*a*b*c + a^2*d - b^2*d)*(m + n - 1)*Tan[e + f*x] + b*(b*c*n + a*d*(2*m + n - 2))*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && GtQ[n, 0] && IntegerQ[2*n]
+Int[1/((a_ + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])), x_Symbol] := (a*c - b*d)*x/((a^2 + b^2)*(c^2 + d^2)) + b^2/((b*c - a*d)*(a^2 + b^2))* Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x] - d^2/((b*c - a*d)*(c^2 + d^2))* Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[Sqrt[a_. + b_.*tan[e_. + f_.*x_]]/(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol] := 1/(c^2 + d^2)* Int[Simp[a*c + b*d + (b*c - a*d)*Tan[e + f*x], x]/ Sqrt[a + b*Tan[e + f*x]], x] - d*(b*c - a*d)/(c^2 + d^2)* Int[(1 + Tan[e + f*x]^2)/(Sqrt[ a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^(3/2)/(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol] := 1/(c^2 + d^2)* Int[Simp[ a^2*c - b^2*c + 2*a*b*d + (2*a*b*c - a^2*d + b^2*d)*Tan[e + f*x], x]/ Sqrt[a + b*Tan[e + f*x]], x] + (b*c - a*d)^2/(c^2 + d^2)* Int[(1 + Tan[e + f*x]^2)/(Sqrt[ a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_/(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol] := 1/(c^2 + d^2)*Int[(a + b*Tan[e + f*x])^m*(c - d*Tan[e + f*x]), x] + d^2/(c^2 + d^2)* Int[(a + b*Tan[e + f*x])^ m*(1 + Tan[e + f*x]^2)/(c + d*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Not[IntegerQ[m]]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_ + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^n/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(d_./tan[e_. + f_.*x_])^n_, x_Symbol] := d^m*Int[(b + a*Cot[e + f*x])^m*(d*Cot[e + f*x])^(n - m), x] /; FreeQ[{a, b, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]
+Int[(a_. + b_.*cot[e_. + f_.*x_])^m_.*(d_./cot[e_. + f_.*x_])^n_, x_Symbol] := d^m*Int[(b + a*Tan[e + f*x])^m*(d*Tan[e + f*x])^(n - m), x] /; FreeQ[{a, b, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^ m_.*(c_.*(d_.*tan[e_. + f_.*x_])^p_)^n_, x_Symbol] := c^IntPart[n]*(c*(d*Tan[e + f*x])^p)^ FracPart[n]/(d*Tan[e + f*x])^(p*FracPart[n])* Int[(a + b*Tan[e + f*x])^m*(d*Tan[e + f*x])^(n*p), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[n]] && Not[IntegerQ[m]]
+Int[(a_. + b_.*cot[e_. + f_.*x_])^ m_.*(c_.*(d_.*cot[e_. + f_.*x_])^p_)^n_, x_Symbol] := c^IntPart[n]*(c*(d*Cot[e + f*x])^p)^ FracPart[n]/(d*Cot[e + f*x])^(p*FracPart[n])* Int[(a + b*Cot[e + f*x])^m*(d*Cot[e + f*x])^(n*p), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[n]] && Not[IntegerQ[m]]
diff --git a/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m
new file mode 100755
index 0000000..4411153
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m	
@@ -0,0 +1,11 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n *)
+Int[(g_.*tan[e_. + f_.*x_])^p_.*(a_ + b_.*tan[e_. + f_.*x_])^ m_*(c_ + d_.*tan[e_. + f_.*x_])^n_, x_Symbol] := Unintegrable[(g*Tan[e + f*x])^p*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
+Int[(g_.*cot[e_. + f_.*x_])^p_*(a_. + b_.*tan[e_. + f_.*x_])^ m_.*(c_ + d_.*tan[e_. + f_.*x_])^n_., x_Symbol] := g^(m + n)* Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^ m*(d + c*Cot[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && Not[IntegerQ[p]] && IntegerQ[m] && IntegerQ[n]
+Int[(g_.*tan[e_. + f_.*x_])^p_*(a_. + b_.*cot[e_. + f_.*x_])^ m_.*(c_ + d_.*cot[e_. + f_.*x_])^n_., x_Symbol] := g^(m + n)* Int[(g*Tan[e + f*x])^(p - m - n)*(b + a*Tan[e + f*x])^ m*(d + c*Tan[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && Not[IntegerQ[p]] && IntegerQ[m] && IntegerQ[n]
+Int[(g_.*tan[e_. + f_.*x_]^q_)^p_*(a_. + b_.*tan[e_. + f_.*x_])^ m_.*(c_ + d_.*tan[e_. + f_.*x_])^n_., x_Symbol] := (g*Tan[e + f*x]^q)^p/(g*Tan[e + f*x])^(p*q)* Int[(g*Tan[e + f*x])^(p*q)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q}, x] && Not[IntegerQ[p]] && Not[IntegerQ[m] && IntegerQ[n]]
+Int[(g_.*tan[e_. + f_.*x_])^p_.*(a_ + b_.*tan[e_. + f_.*x_])^ m_.*(c_ + d_.*cot[e_. + f_.*x_])^n_., x_Symbol] := g^n*Int[(g*Tan[e + f*x])^(p - n)*(a + b*Tan[e + f*x])^ m*(d + c*Tan[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IntegerQ[n]
+Int[tan[e_. + f_.*x_]^p_.*(a_ + b_.*tan[e_. + f_.*x_])^ m_.*(c_ + d_.*cot[e_. + f_.*x_])^n_, x_Symbol] := Int[(b + a*Cot[e + f*x])^m*(c + d*Cot[e + f*x])^n/ Cot[e + f*x]^(m + p), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m] && IntegerQ[p]
+Int[(g_.*tan[e_. + f_.*x_])^p_*(a_ + b_.*tan[e_. + f_.*x_])^ m_.*(c_ + d_.*cot[e_. + f_.*x_])^n_, x_Symbol] := Cot[e + f*x]^p*(g*Tan[e + f*x])^p* Int[(b + a*Cot[e + f*x])^m*(c + d*Cot[e + f*x])^n/ Cot[e + f*x]^(m + p), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && Not[IntegerQ[n]] && IntegerQ[m] && Not[IntegerQ[p]]
+Int[(g_.*tan[e_. + f_.*x_])^p_.*(a_ + b_.*tan[e_. + f_.*x_])^ m_*(c_ + d_.*cot[e_. + f_.*x_])^n_, x_Symbol] := (g*Tan[e + f*x])^n*(c + d*Cot[e + f*x])^n/(d + c*Tan[e + f*x])^n* Int[(g*Tan[e + f*x])^(p - n)*(a + b*Tan[e + f*x])^ m*(d + c*Tan[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && Not[IntegerQ[n]] && Not[IntegerQ[m]]
diff --git a/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m
new file mode 100755
index 0000000..7a80586
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m	
@@ -0,0 +1,35 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan) *)
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_.*(c_ + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := a*c/f* Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x], x, Tan[e + f*x]] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
+Int[(c_. + d_.*tan[e_. + f_.*x_])*(A_. + B_.*tan[e_. + f_.*x_])/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol] := B*d/b*Int[Tan[e + f*x], x] + 1/b*Int[Simp[A*b*c + (A*b*d + B*(b*c - a*d))*Tan[e + f*x], x]/(a + b*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^ m_*(c_. + d_.*tan[e_. + f_.*x_])*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := -(A*b - a*B)*(a*c + b*d)*(a + b*Tan[e + f*x])^m/(2*a^2*f*m) + 1/(2*a*b)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[A*b*c + a*B*c + a*A*d + b*B*d + 2*a*B*d*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && EqQ[a^2 + b^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^ m_*(c_. + d_.*tan[e_. + f_.*x_])*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := (b*c - a*d)*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2)) + 1/(a^2 + b^2)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[a*A*c + b*B*c + A*b*d - a*B*d - (A*b*c - a*B*c - a*A*d - b*B*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^ m_.*(c_. + d_.*tan[e_. + f_.*x_])*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := B*d*(a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)) + Int[(a + b*Tan[e + f*x])^m* Simp[A*c - B*d + (B*c + A*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && Not[LeQ[m, -1]]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := -a^2*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(b*c + a*d)*(n + 1)) - a/(d*(b*c + a*d)*(n + 1))* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)* Simp[A*b*d*(m - n - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m - 1) + b*d*(n + 1)))*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := b*B*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)/(d* f*(m + n)) + 1/(d*(m + n))* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n* Simp[a*A*d*(m + n) + B*(a*c*(m - 1) - b*d*(n + 1)) - (B*(b*c - a*d)*(m - 1) - d*(A*b + a*B)*(m + n))*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] && Not[LtQ[n, -1]]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := -(A*b - a*B)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^n/(2*a*f*m) + 1/(2*a^2*m)* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)* Simp[A*(a*c*m + b*d*n) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a*A*(m + n))*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && GtQ[n, 0]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := (a*A + b*B)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d)) + 1/(2*a*m*(b*c - a*d))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && Not[GtQ[n, 0]]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := B*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n/(f*(m + n)) + 1/(a*(m + n))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 1)* Simp[a*A*c*(m + n) - B*(b*c*m + a*d*n) + (a*A*d*(m + n) - B*(b*d*m - a*c*n))* Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[n, 0]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := (A*d - B*c)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 + d^2)) - 1/(a*(n + 1)*(c^2 + d^2))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)* Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c*m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := b*B/f* Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && EqQ[A*b + a*B, 0]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^ m_*(A_. + B_.*tan[e_. + f_.*x_])/(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol] := (A*b + a*B)/(b*c + a*d)*Int[(a + b*Tan[e + f*x])^m, x] - (B*c - A*d)/(b*c + a*d)* Int[(a + b*Tan[e + f*x])^ m*(a - b*Tan[e + f*x])/(c + d*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
+(* Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+ B_.*tan[e_.+f_.*x_]),x_Symbol] := (A*b-a*B)/b*Int[(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^n,x] + B/b*Int[(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^n,x] /; FreeQ[{a,b,c,d,e,f,A,B,m},x] && NeQ[b*c-a*d,0] && EqQ[a^2+b^2,0] &&  NeQ[c^2+d^2,0] *)
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := (A*b + a*B)/b* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x] - B/b* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^ n*(a - b*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_ + B_.*tan[e_. + f_.*x_]), x_Symbol] := A^2/f* Subst[Int[(a + b*x)^m*(c + d*x)^n/(A - B*x), x], x, Tan[e + f*x]] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[IntegersQ[2*m, 2*n]] && EqQ[A^2 + B^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := (A + I*B)/2* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^ n*(1 - I*Tan[e + f*x]), x] + (A - I*B)/2* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^ n*(1 + I*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[IntegersQ[2*m, 2*n]] && NeQ[A^2 + B^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^2*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := -(B*c - A*d)*(b*c - a*d)^2*(c + d*Tan[e + f*x])^(n + 1)/(f* d^2*(n + 1)*(c^2 + d^2)) + 1/(d*(c^2 + d^2))*Int[(c + d*Tan[e + f*x])^(n + 1)* Simp[B*(b*c - a*d)^2 + A*d*(a^2*c - b^2*c + 2*a*b*d) + d*(B*(a^2*c - b^2*c + 2*a*b*d) + A*(2*a*b*c - a^2*d + b^2*d))* Tan[e + f*x] + b^2*B*(c^2 + d^2)*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := (b*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2)) - 1/(d*(n + 1)*(c^2 + d^2))* Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)* Simp[a*A* d*(b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1) + a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)* Tan[e + f*x] - b*(d*(A*b*c + a*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
+Int[(a_. + b_.*tan[e_. + f_.*x_])^2*(A_. + B_.*tan[e_. + f_.*x_])/(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol] := b^2*B*Tan[e + f*x]/(d*f) + 1/d*Int[(a^2*A*d - b^2*B*c + (2*a*A*b + B*(a^2 - b^2))*d* Tan[e + f*x] + (A*b^2*d - b*B*(b*c - 2*a*d))* Tan[e + f*x]^2)/(c + d*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := b*B*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)/(d* f*(m + n)) + 1/(d*(m + n))* Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n* Simp[a^2*A*d*(m + n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m - 1) - b*(A*b + a*B)*d*(m + n))* Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && Not[IGtQ[n, 1] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := (A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^ n/(f*(m + 1)*(a^2 + b^2)) + 1/(b*(m + 1)*(a^2 + b^2))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)* Simp[b*B*(b*c*(m + 1) + a*d*n) + A*b*(a*c*(m + 1) - b*d*n) - b*(A*(b*c - a*d) - B*(a*c + b*d))*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := b*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)) + 1/((m + 1)*(b*c - a*d)*(a^2 + b^2))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && Not[ILtQ[n, -1] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := B*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n/(f*(m + n)) + 1/(m + n)* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n - 1)* Simp[a*A*c*(m + n) - B*(b*c*m + a*d*n) + (A*b*c + a*B*c + a*A*d - b*B*d)*(m + n)* Tan[e + f*x] + (A*b*d*(m + n) + B*(a*d*m + b*c*n))* Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[0, m, 1] && LtQ[0, n, 1]
+Int[(A_. + B_.*tan[e_. + f_.*x_])/((a_ + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])), x_Symbol] := (B*(b*c + a*d) + A*(a*c - b*d))*x/((a^2 + b^2)*(c^2 + d^2)) + b*(A*b - a*B)/((b*c - a*d)*(a^2 + b^2))* Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x] + d*(B*c - A*d)/((b*c - a*d)*(c^2 + d^2))* Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[Sqrt[ c_. + d_.*tan[e_. + f_.*x_]]*(A_. + B_.*tan[e_. + f_.*x_])/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol] := 1/(a^2 + b^2)* Int[Simp[ A*(a*c + b*d) + B*(b*c - a*d) - (A*(b*c - a*d) - B*(a*c + b*d))*Tan[e + f*x], x]/Sqrt[c + d*Tan[e + f*x]], x] - (b*c - a*d)*(B*a - A*b)/(a^2 + b^2)* Int[(1 + Tan[e + f*x]^2)/((a + b*Tan[e + f*x])* Sqrt[c + d*Tan[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_])/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol] := 1/(a^2 + b^2)* Int[(c + d*Tan[e + f*x])^n* Simp[a*A + b*B - (A*b - a*B)*Tan[e + f*x], x], x] + b*(A*b - a*B)/(a^2 + b^2)* Int[(c + d*Tan[e + f*x])^ n*(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[Sqrt[a_. + b_.*tan[e_. + f_.*x_]]*(A_. + B_.*tan[e_. + f_.*x_])/ Sqrt[c_. + d_.*tan[e_. + f_.*x_]], x_Symbol] := Int[Simp[a*A - b*B + (A*b + a*B)*Tan[e + f*x], x]/(Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]), x] + b*B* Int[(1 + Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]* Sqrt[c + d*Tan[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+(* Int[(A_.+B_.*tan[e_.+f_.*x_])/(Sqrt[a_.+b_.*tan[e_.+f_.*x_]]*Sqrt[ c_.+d_.*tan[e_.+f_.*x_]]),x_Symbol] := A^2/f*Subst[Int[1/((A-B*x)*Sqrt[a+b*x]*Sqrt[c+d*x]),x],x,Tan[e+f*x]]  /; FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d,0] && NeQ[a^2+b^2,0] &&  NeQ[c^2+d^2,0] && EqQ[A^2+B^2,0] *)
+(* Int[(A_.+B_.*tan[e_.+f_.*x_])/(Sqrt[a_.+b_.*tan[e_.+f_.*x_]]*Sqrt[ c_.+d_.*tan[e_.+f_.*x_]]),x_Symbol] := (A+I*B)/2*Int[(1-I*Tan[e+f*x])/(Sqrt[a+b*Tan[e+f*x]]*Sqrt[c+d*Tan[e+ f*x]]),x] + (A-I*B)/2*Int[(1+I*Tan[e+f*x])/(Sqrt[a+b*Tan[e+f*x]]*Sqrt[c+d*Tan[e+ f*x]]),x] /; FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d,0] && NeQ[a^2+b^2,0] &&  NeQ[c^2+d^2,0] && NeQ[A^2+B^2,0] *)
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := A^2/f* Subst[Int[(a + b*x)^m*(c + d*x)^n/(A - B*x), x], x, Tan[e + f*x]] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[A^2 + B^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol] := (A + I*B)/2* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^ n*(1 - I*Tan[e + f*x]), x] + (A - I*B)/2* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^ n*(1 + I*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2 + B^2, 0]
diff --git a/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m
new file mode 100755
index 0000000..dad493e
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m	
@@ -0,0 +1,18 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.3.4.1 (a+b tan)^m (A+B tan+C tan^2) *)
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(A_ + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := A/(b*f)*Subst[Int[(a + x)^m, x], x, b*Tan[e + f*x]] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]
+Int[(a_. + b_.*cot[e_. + f_.*x_])^m_.*(A_ + C_.*cot[e_. + f_.*x_]^2), x_Symbol] := -A/(b*f)*Subst[Int[(a + x)^m, x], x, b*Cot[e + f*x]] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^ m_.*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := 1/b^2* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[b*B - a*C + b*C*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := -C/b^2*Int[(a + b*Tan[e + f*x])^(m + 1)*(a - b*Tan[e + f*x]), x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A*b^2 + a^2*C, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^ m_.*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := -(a*A + b*B - a*C)* Tan[e + f*x]*(a + b*Tan[e + f*x])^m/(2*a*f*m) + 1/(2*a^2*m)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[(b*B - a*C) + a*A*(2*m + 1) - (b*C*(m - 1) + (A*b - a*B)*(m + 1))* Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LeQ[m, -1] && EqQ[a^2 + b^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := -(a*A - a*C)*Tan[e + f*x]*(a + b*Tan[e + f*x])^m/(2*a*f*m) + 1/(2*a^2*m)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[-a*C + a*A*(2*m + 1) - (b*C*(m - 1) + A*b*(m + 1))*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[A*b^2 + a^2*C, 0] && LeQ[m, -1] && EqQ[a^2 + b^2, 0]
+Int[(A_ + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2)/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol] := (a*A + b*B - a*C)*x/(a^2 + b^2) + (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2)* Int[(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 + b^2, 0] && EqQ[A*b - a*B - b*C, 0]
+Int[(A_ + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2)/ tan[e_. + f_.*x_], x_Symbol] := B*x + A*Int[1/Tan[e + f*x], x] + C*Int[Tan[e + f*x], x] /; FreeQ[{e, f, A, B, C}, x] && NeQ[A, C]
+Int[(A_ + C_.*tan[e_. + f_.*x_]^2)/tan[e_. + f_.*x_], x_Symbol] := A*Int[1/Tan[e + f*x], x] + C*Int[Tan[e + f*x], x] /; FreeQ[{e, f, A, C}, x] && NeQ[A, C]
+Int[(A_ + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2)/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol] := (a*A + b*B - a*C)*x/(a^2 + b^2) - (A*b - a*B - b*C)/(a^2 + b^2)*Int[Tan[e + f*x], x] + (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2)* Int[(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && NeQ[a^2 + b^2, 0] && NeQ[A*b - a*B - b*C, 0]
+Int[(A_ + C_.*tan[e_. + f_.*x_]^2)/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol] := a*(A - C)*x/(a^2 + b^2) - b*(A - C)/(a^2 + b^2)*Int[Tan[e + f*x], x] + (a^2*C + A*b^2)/(a^2 + b^2)* Int[(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2*C + A*b^2, 0] && NeQ[a^2 + b^2, 0] && NeQ[A, C]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^ m_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := (A*b^2 - a*b*B + a^2*C)*(a + b*Tan[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 + b^2)) + 1/(a^2 + b^2)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := (A*b^2 + a^2*C)*(a + b*Tan[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 + b^2)) + 1/(a^2 + b^2)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[a*(A - C) - (A*b - b*C)*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[A*b^2 + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^ m_.*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := C*(a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)) + Int[(a + b*Tan[e + f*x])^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && Not[LeQ[m, -1]]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := C*(a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)) + (A - C)* Int[(a + b*Tan[e + f*x])^m, x] /; FreeQ[{a, b, e, f, A, C, m}, x] && NeQ[A*b^2 + a^2*C, 0] && Not[LeQ[m, -1]]
diff --git a/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m
new file mode 100755
index 0000000..97b675f
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m	
@@ -0,0 +1,28 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2) *)
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := 1/b^2* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^ n*(b*B - a*C + b*C*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := -C/b^2* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^ n*(a - b*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 + a^2*C, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_ + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := A/f*Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
+Int[(a_. + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := -(b*c - a*d)*(c^2*C - B*c*d + A*d^2)*(c + d*Tan[e + f*x])^(n + 1)/(d^2* f*(n + 1)*(c^2 + d^2)) + 1/(d*(c^2 + d^2))*Int[(c + d*Tan[e + f*x])^(n + 1)* Simp[a*d*(A*c - c*C + B*d) + b*(c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d + a*C*d)* Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]
+Int[(a_. + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := -(b*c - a*d)*(c^2*C + A*d^2)*(c + d*Tan[e + f*x])^(n + 1)/(d^2* f*(n + 1)*(c^2 + d^2)) + 1/(d*(c^2 + d^2))*Int[(c + d*Tan[e + f*x])^(n + 1)* Simp[a*d*(A*c - c*C) + b*(c^2*C + A*d^2) + d*(A*b*c - b*c*C - a*A*d + a*C*d)*Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]
+Int[(a_ + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 2)) - 1/(d*(n + 2))*Int[(c + d*Tan[e + f*x])^n* Simp[b*c*C - a*A*d*(n + 2) - (A*b + a*B - b*C)*d*(n + 2)* Tan[e + f*x] - (a*C*d*(n + 2) - b*(c*C - B*d*(n + 2)))* Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && Not[LtQ[n, -1]]
+Int[(a_ + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 2)) - 1/(d*(n + 2))*Int[(c + d*Tan[e + f*x])^n* Simp[b*c*C - a*A*d*(n + 2) - (A*b - b*C)*d*(n + 2)* Tan[e + f*x] - (a*C*d*(n + 2) - b*c*C)*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && Not[LtQ[n, -1]]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := (a*A + b*B - a*C)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d)) + 1/(2*a*m*(b*c - a*d))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[b*(c*(A + C)*m - B*d*(n + 1)) + a*(B*c*m + C*d*(n + 1) - A*d*(2*m + n + 1)) + (b*C*d*(m - n - 1) + A*b*d*(m + n + 1) + a*(2*c*C*m - B*d*(m + n + 1)))*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && (LtQ[m, 0] || EqQ[m + n + 1, 0])
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := a*(A - C)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d)) + 1/(2*a*m*(b*c - a*d))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[b*c*(A + C)*m + a*(C*d*(n + 1) - A*d*(2*m + n + 1)) + (b*C*d*(m - n - 1) + A*b*d*(m + n + 1) + 2*a*c*C*m)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && (LtQ[m, 0] || EqQ[m + n + 1, 0])
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := (c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2)) - 1/(a*d*(n + 1)*(c^2 + d^2))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)* Simp[b*(c^2*C - B*c*d + A*d^2)*m - a*d*(n + 1)*(A*c - c*C + B*d) - a*(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))* Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && Not[LtQ[m, 0]] && LtQ[n, -1] && NeQ[c^2 + d^2, 0]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := (c^2*C + A*d^2)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2)) - 1/(a*d*(n + 1)*(c^2 + d^2))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)* Simp[b*(c^2*C + A*d^2)*m - a*d*(n + 1)*(A*c - c*C) - a*(-A*d^2*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && Not[LtQ[m, 0]] && LtQ[n, -1] && NeQ[c^2 + d^2, 0]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := C*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1)) + 1/(b*d*(m + n + 1))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n* Simp[A*b*d*(m + n + 1) + C*(a*c*m - b*d*(n + 1)) - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && Not[LtQ[m, 0]] && NeQ[m + n + 1, 0]
+Int[(a_ + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := C*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1)) + 1/(b*d*(m + n + 1))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n* Simp[A*b*d*(m + n + 1) + C*(a*c*m - b*d*(n + 1)) - C*m*(b*c - a*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && Not[LtQ[m, 0]] && NeQ[m + n + 1, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := (A*d^2 + c*(c*C - B*d))*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2)) - 1/(d*(n + 1)*(c^2 + d^2))* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)* Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b*(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))* Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := (A*d^2 + c^2*C)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2)) - 1/(d*(n + 1)*(c^2 + d^2))* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)* Simp[A*d*(b*d*m - a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d))*Tan[e + f*x] + b*(A*d^2*(m + n + 1) + C*(c^2*m - d^2*(n + 1)))* Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := C*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1)) + 1/(d*(m + n + 1))* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n* Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)* Tan[e + f*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))* Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && Not[IGtQ[n, 0] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := C*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1)) + 1/(d*(m + n + 1))* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n* Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b - b*C)*(m + n + 1)*Tan[e + f*x] - C*m*(b*c - a*d)*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && Not[IGtQ[n, 0] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := (A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)) + 1/((m + 1)*(b*c - a*d)*(a^2 + b^2))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && Not[ILtQ[n, -1] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := (A*b^2 + a^2*C)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)) + 1/((m + 1)*(b*c - a*d)*(a^2 + b^2))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - a*C*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - b*C)*Tan[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 2)*Tan[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && Not[ILtQ[n, -1] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]
+Int[(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2)/((a_ + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])), x_Symbol] := (a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d))* x/((a^2 + b^2)*(c^2 + d^2)) + (A*b^2 - a*b*B + a^2*C)/((b*c - a*d)*(a^2 + b^2))* Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x] - (c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2))* Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[(A_. + C_.*tan[e_. + f_.*x_]^2)/((a_ + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])), x_Symbol] := (a*(A*c - c*C) - b*(A*d - C*d))*x/((a^2 + b^2)*(c^2 + d^2)) + (A*b^2 + a^2*C)/((b*c - a*d)*(a^2 + b^2))* Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x] - (c^2*C + A*d^2)/((b*c - a*d)*(c^2 + d^2))* Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2)/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol] := 1/(a^2 + b^2)* Int[(c + d*Tan[e + f*x])^n* Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x] + (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2)* Int[(c + d*Tan[e + f*x])^ n*(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Not[GtQ[n, 0]] && Not[LeQ[n, -1]]
+Int[(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + C_.*tan[e_. + f_.*x_]^2)/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol] := 1/(a^2 + b^2)* Int[(c + d*Tan[e + f*x])^n* Simp[a*(A - C) - (A*b - b*C)*Tan[e + f*x], x], x] + (A*b^2 + a^2*C)/(a^2 + b^2)* Int[(c + d*Tan[e + f*x])^ n*(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Not[GtQ[n, 0]] && Not[LeQ[n, -1]]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^ n*(A + B*ff*x + C*ff^2*x^2)/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
+Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^ n*(A + C*ff^2*x^2)/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
diff --git a/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m
new file mode 100755
index 0000000..9ff7d52
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m	
@@ -0,0 +1,28 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.3.7 (d trig)^m (a+b (c tan)^n)^p *)
+Int[u_.*(a_ + b_.*tan[e_. + f_.*x_]^2)^p_, x_Symbol] := Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]
+Int[u_.*(b_.*tan[e_. + f_.*x_]^n_)^p_, x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, (b*ff^n)^ IntPart[p]*(b*Tan[e + f*x]^n)^ FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])* Int[ActivateTrig[u]*(Tan[e + f*x]/ff)^(n*p), x]] /; FreeQ[{b, e, f, n, p}, x] && Not[IntegerQ[p]] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, (d_.*trig_[e + f*x])^m_. /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
+Int[u_.*(b_.*(c_.*tan[e_. + f_.*x_])^n_)^p_, x_Symbol] := b^IntPart[p]*(b*(c*Tan[e + f*x])^n)^ FracPart[p]/(c*Tan[e + f*x])^(n*FracPart[p])* Int[ActivateTrig[u]*(c*Tan[e + f*x])^(n*p), x] /; FreeQ[{b, c, e, f, n, p}, x] && Not[IntegerQ[p]] && Not[IntegerQ[n]] && (EqQ[u, 1] || MatchQ[u, (d_.*trig_[e + f*x])^m_. /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
+Int[1/(a_ + b_.*tan[e_. + f_.*x_]^2), x_Symbol] := x/(a - b) - b/(a - b)*Int[Sec[e + f*x]^2/(a + b*Tan[e + f*x]^2), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a, b]
+Int[(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^p_, x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, c*ff/f* Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*Tan[e + f*x]/ff]] /; FreeQ[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
+Int[(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^p_., x_Symbol] := Unintegrable[(a + b*(c*Tan[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, e, f, n, p}, x]
+Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, c*ff^(m + 1)/f* Subst[Int[x^m*(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/2 + 1), x], x, c*Tan[e + f*x]/ff]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2]
+Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*tan[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Sec[e + f*x], x]}, 1/(f*ff^m)* Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*(a - b + b*ff^2*x^2)^p/ x^(m + 1), x], x, Sec[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
+Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*tan[e_. + f_.*x_]^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Sec[e + f*x], x]}, 1/(f*ff^m)* Subst[Int[(-1 + ff^2*x^2)^((m - 1)/ 2)*(a + b*(-1 + ff^2*x^2)^(n/2))^p/x^(m + 1), x], x, Sec[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2]
+Int[(d_.*sin[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_., x_Symbol] := Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
+Int[(d_.*sin[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_]^2)^p_, x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff*(d*Sin[e + f*x])^m*(Sec[e + f*x]^2)^(m/2)/(f*Tan[e + f*x]^m)* Subst[ Int[(ff*x)^m*(a + b*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, d, e, f, m, p}, x] && Not[IntegerQ[m]]
+Int[(d_.*sin[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_., x_Symbol] := Unintegrable[(d*Sin[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
+Int[(d_.*cos[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_, x_Symbol] := (d*Cos[e + f*x])^FracPart[m]*(Sec[e + f*x]/d)^FracPart[m]* Int[(Sec[e + f*x]/d)^(-m)*(a + b*(c*Tan[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]
+Int[(d_.*tan[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, c*ff/f* Subst[Int[(d*ff*x/c)^m*(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*Tan[e + f*x]/ff]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || IntegerQ[p] && RationalQ[n])
+Int[(d_.*tan[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_., x_Symbol] := Int[ExpandTrig[(d*tan[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
+Int[(d_.*tan[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_., x_Symbol] := Unintegrable[(d*Tan[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
+Int[(d_.*cot[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_]^n_.)^p_., x_Symbol] := d^(n*p)* Int[(d*Cot[e + f*x])^(m - n*p)*(b + a*Cot[e + f*x]^n)^p, x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && Not[IntegerQ[m]] && IntegersQ[n, p]
+Int[(d_.*cot[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_, x_Symbol] := (d*Cot[e + f*x])^FracPart[m]*(Tan[e + f*x]/d)^FracPart[m]* Int[(Tan[e + f*x]/d)^(-m)*(a + b*(c*Tan[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]
+Int[sec[e_. + f_.*x_]^m_*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/(c^(m - 1)*f)* Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)^n)^p, x], x, c*Tan[e + f*x]/ff]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[ m/2] && (IntegersQ[n, p] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
+Int[sec[e_. + f_.*x_]^m_.*(a_ + b_.*tan[e_. + f_.*x_]^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[ ExpandToSum[b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2), x]^ p/(1 - ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
+Int[sec[e_. + f_.*x_]^m_.*(a_ + b_.*tan[e_. + f_.*x_]^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[ 1/(1 - ff^2*x^2)^((m + 1)/ 2)*((b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2))/(1 - ff^2*x^2)^(n/2))^p, x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && Not[IntegerQ[p]]
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_., x_Symbol] := Int[ExpandTrig[(d*sec[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
+Int[(d_.*sec[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_]^2)^p_, x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff*(d*Sec[e + f*x])^m/(f*(Sec[e + f*x]^2)^(m/2))* Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*(a + b*ff^2*x^2)^p, x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, d, e, f, m, p}, x] && Not[IntegerQ[m]]
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_., x_Symbol] := Unintegrable[(d*Sec[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
+Int[(d_.*csc[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_, x_Symbol] := (d*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/d)^FracPart[m]* Int[(Sin[e + f*x]/d)^(-m)*(a + b*(c*Tan[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]
diff --git a/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m
new file mode 100755
index 0000000..0e2057a
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m	
@@ -0,0 +1,37 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p *)
+Int[(a_ + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_., x_Symbol] := 1/(4^p*c^p)*Int[(b + 2*c*Tan[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[(a_ + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.)^p_., x_Symbol] := 1/(4^p*c^p)*Int[(b + 2*c*Cot[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[(a_ + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := (a + b*Tan[d + e*x]^n + c*Tan[d + e*x]^(2*n))^ p/(b + 2*c*Tan[d + e*x]^n)^(2*p)* Int[(b + 2*c*Tan[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[(a_ + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := (a + b*Cot[d + e*x]^n + c*Cot[d + e*x]^(2*n))^ p/(b + 2*c*Cot[d + e*x]^n)^(2*p)* Int[(b + 2*c*Cot[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[1/(a_. + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.), x_Symbol] := Module[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[1/(b - q + 2*c*Tan[d + e*x]^n), x] - 2*c/q*Int[1/(b + q + 2*c*Tan[d + e*x]^n), x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
+Int[1/(a_. + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.), x_Symbol] := Module[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[1/(b - q + 2*c*Cot[d + e*x]^n), x] - 2*c/q*Int[1/(b + q + 2*c*Cot[d + e*x]^n), x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
+Int[sin[d_. + e_.*x_]^ m_*(a_. + b_.*(f_.*tan[d_. + e_.*x_])^n_. + c_.*(f_.*tan[d_. + e_.*x_])^n2_.)^p_, x_Symbol] := f/e*Subst[ Int[x^m*(a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)^(m/2 + 1), x], x, f*Tan[d + e*x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2]
+Int[cos[d_. + e_.*x_]^ m_*(a_. + b_.*(f_.*cot[d_. + e_.*x_])^n_. + c_.*(f_.*cot[d_. + e_.*x_])^n2_.)^p_, x_Symbol] := -f/e*Subst[ Int[x^m*(a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)^(m/2 + 1), x], x, f*Cot[d + e*x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2]
+Int[sin[d_. + e_.*x_]^ m_.*(a_. + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_., x_Symbol] := Module[{g = FreeFactors[Cos[d + e*x], x]}, -g/e* Subst[Int[(1 - g^2*x^2)^((m - 1)/2)* ExpandToSum[ a*(g*x)^(2*n) + b*(g*x)^n*(1 - g^2*x^2)^(n/2) + c*(1 - g^2*x^2)^n, x]^p/(g*x)^(2*n*p), x], x, Cos[d + e*x]/g]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
+Int[cos[d_. + e_.*x_]^ m_.*(a_. + b_.*cot[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_., x_Symbol] := Module[{g = FreeFactors[Sin[d + e*x], x]}, g/e* Subst[Int[(1 - g^2*x^2)^((m - 1)/2)* ExpandToSum[ a*(g*x)^(2*n) + b*(g*x)^n*(1 - g^2*x^2)^(n/2) + c*(1 - g^2*x^2)^n, x]^p/(g*x)^(2*n*p), x], x, Sin[d + e*x]/g]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
+Int[cos[d_. + e_.*x_]^ m_*(a_. + b_.*(f_.*tan[d_. + e_.*x_])^n_. + c_.*(f_.*tan[d_. + e_.*x_])^n2_.)^p_., x_Symbol] := f^(m + 1)/e* Subst[Int[(a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)^(m/2 + 1), x], x, f*Tan[d + e*x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2]
+Int[sin[d_. + e_.*x_]^ m_*(a_. + b_.*(f_.*cot[d_. + e_.*x_])^n_. + c_.*(f_.*cot[d_. + e_.*x_])^n2_.)^p_., x_Symbol] := -f^(m + 1)/e* Subst[Int[(a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)^(m/2 + 1), x], x, f*Cot[d + e*x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2]
+Int[cos[d_. + e_.*x_]^ m_*(a_. + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := Module[{g = FreeFactors[Sin[d + e*x], x]}, g/e* Subst[Int[(1 - g^2*x^2)^((m - 2*n*p - 1)/2)* ExpandToSum[c*x^(2*n) + b*x^n*(1 - x^2)^(n/2) + a*(1 - x^2)^n, x]^p, x], x, Sin[d + e*x]/g]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
+Int[sin[d_. + e_.*x_]^ m_*(a_. + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := Module[{g = FreeFactors[Cos[d + e*x], x]}, -g/e* Subst[Int[(1 - g^2*x^2)^((m - 2*n*p - 1)/2)* ExpandToSum[c*x^(2*n) + b*x^n*(1 - x^2)^(n/2) + a*(1 - x^2)^n, x]^p, x], x, Cos[d + e*x]/g]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
+Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := 1/(4^p*c^p)*Int[Tan[d + e*x]^m*(b + 2*c*Tan[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := 1/(4^p*c^p)*Int[Cot[d + e*x]^m*(b + 2*c*Cot[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := (a + b*Tan[d + e*x]^n + c*Tan[d + e*x]^(2*n))^ p/(b + 2*c*Tan[d + e*x]^n)^(2*p)* Int[Tan[d + e*x]^m*(b + 2*c*Tan[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := (a + b*Cot[d + e*x]^n + c*Cot[d + e*x]^(2*n))^ p/(b + 2*c*Cot[d + e*x]^n)^(2*p)* Int[Cot[d + e*x]^m*(b + 2*c*Cot[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*(f_.*tan[d_. + e_.*x_])^n_. + c_.*(f_.*tan[d_. + e_.*x_])^n2_.)^p_, x_Symbol] := f/e*Subst[Int[(x/f)^m*(a + b*x^n + c*x^(2*n))^p/(f^2 + x^2), x], x, f*Tan[d + e*x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
+Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*(f_.*cot[d_. + e_.*x_])^n_. + c_.*(f_.*cot[d_. + e_.*x_])^n2_.)^p_, x_Symbol] := -f/e*Subst[Int[(x/f)^m*(a + b*x^n + c*x^(2*n))^p/(f^2 + x^2), x], x, f*Cot[d + e*x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
+Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_., x_Symbol] := 1/(4^p*c^p)*Int[Cot[d + e*x]^m*(b + 2*c*Tan[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.)^p_., x_Symbol] := 1/(4^p*c^p)*Int[Tan[d + e*x]^m*(b + 2*c*Cot[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := (a + b*Tan[d + e*x]^n + c*Tan[d + e*x]^(2*n))^ p/(b + 2*c*Tan[d + e*x]^n)^(2*p)* Int[Cot[d + e*x]^m*(b + 2*c*Tan[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := (a + b*Cot[d + e*x]^n + c*Cot[d + e*x]^(2*n))^ p/(b + 2*c*Cot[d + e*x]^n)^(2*p)* Int[Tan[d + e*x]^m*(b + 2*c*Cot[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*tan[d_. + e_.*x_]^n_ + c_.*tan[d_. + e_.*x_]^n2_)^ p_., x_Symbol] := Module[{g = FreeFactors[Cot[d + e*x], x]}, g/e* Subst[Int[(g*x)^(m - 2*n*p)*(c + b*(g*x)^n + a*(g*x)^(2*n))^ p/(1 + g^2*x^2), x], x, Cot[d + e*x]/g]] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2]
+Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*cot[d_. + e_.*x_]^n_ + c_.*cot[d_. + e_.*x_]^n2_)^ p_., x_Symbol] := Module[{g = FreeFactors[Tan[d + e*x], x]}, -g/e* Subst[Int[(g*x)^(m - 2*n*p)*(c + b*(g*x)^n + a*(g*x)^(2*n))^ p/(1 + g^2*x^2), x], x, Tan[d + e*x]/g]] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2]
+Int[(A_ + B_.*tan[d_. + e_.*x_])*(a_ + b_.*tan[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_]^2)^n_, x_Symbol] := 1/(4^n*c^n)* Int[(A + B*Tan[d + e*x])*(b + 2*c*Tan[d + e*x])^(2*n), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]
+Int[(A_ + B_.*cot[d_. + e_.*x_])*(a_ + b_.*cot[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_]^2)^n_, x_Symbol] := 1/(4^n*c^n)* Int[(A + B*Cot[d + e*x])*(b + 2*c*Cot[d + e*x])^(2*n), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]
+Int[(A_ + B_.*tan[d_. + e_.*x_])*(a_ + b_.*tan[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_]^2)^n_, x_Symbol] := (a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^ n/(b + 2*c*Tan[d + e*x])^(2*n)* Int[(A + B*Tan[d + e*x])*(b + 2*c*Tan[d + e*x])^(2*n), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[n]]
+Int[(A_ + B_.*cot[d_. + e_.*x_])*(a_ + b_.*cot[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_]^2)^n_, x_Symbol] := (a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^ n/(b + 2*c*Cot[d + e*x])^(2*n)* Int[(A + B*Cot[d + e*x])*(b + 2*c*Cot[d + e*x])^(2*n), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[n]]
+Int[(A_ + B_.*tan[d_. + e_.*x_])/(a_. + b_.*tan[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_]^2), x_Symbol] := Module[{q = Rt[b^2 - 4*a*c, 2]}, (B + (b*B - 2*A*c)/q)* Int[1/Simp[b + q + 2*c*Tan[d + e*x], x], x] + (B - (b*B - 2*A*c)/q)* Int[1/Simp[b - q + 2*c*Tan[d + e*x], x], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]
+Int[(A_ + B_.*cot[d_. + e_.*x_])/(a_. + b_.*cot[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_]^2), x_Symbol] := Module[{q = Rt[b^2 - 4*a*c, 2]}, (B + (b*B - 2*A*c)/q)* Int[1/Simp[b + q + 2*c*Cot[d + e*x], x], x] + (B - (b*B - 2*A*c)/q)* Int[1/Simp[b - q + 2*c*Cot[d + e*x], x], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]
+Int[(A_ + B_.*tan[d_. + e_.*x_])*(a_. + b_.*tan[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_]^2)^n_, x_Symbol] := Int[ExpandTrig[(A + B*tan[d + e*x])*(a + b*tan[d + e*x] + c*tan[d + e*x]^2)^n, x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]
+Int[(A_ + B_.*cot[d_. + e_.*x_])*(a_. + b_.*cot[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_]^2)^n_, x_Symbol] := Int[ExpandTrig[(A + B*cot[d + e*x])*(a + b*cot[d + e*x] + c*cot[d + e*x]^2)^n, x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]
diff --git a/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m
new file mode 100755
index 0000000..b71ea2b
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m	
@@ -0,0 +1,24 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.5.1.1 (a+b sec)^n *)
+Int[csc[c_. + d_.*x_]^n_, x_Symbol] := -1/d*Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
+Int[(b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := -b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1)/(d*(n - 1)) + b^2*(n - 2)/(n - 1)*Int[(b*Csc[c + d*x])^(n - 2), x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
+Int[(b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1)/(b*d*n) + (n + 1)/(b^2*n)*Int[(b*Csc[c + d*x])^(n + 2), x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
+Int[csc[c_. + d_.*x_], x_Symbol] := (* -ArcCoth[Cos[c+d*x]]/d /; *) -ArcTanh[Cos[c + d*x]]/d /; FreeQ[{c, d}, x]
+(* Int[1/csc[c_.+d_.*x_],x_Symbol] := -Cos[c+d*x]/d /; FreeQ[{c,d},x] *)
+Int[(b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := (b*Csc[c + d*x])^n*Sin[c + d*x]^n*Int[1/Sin[c + d*x]^n, x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
+Int[(b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := (b*Csc[c + d*x])^(n - 1)*((Sin[c + d*x]/b)^(n - 1)* Int[1/(Sin[c + d*x]/b)^n, x]) /; FreeQ[{b, c, d, n}, x] && Not[IntegerQ[n]]
+Int[(a_ + b_.*csc[c_. + d_.*x_])^2, x_Symbol] := a^2*x + 2*a*b*Int[Csc[c + d*x], x] + b^2*Int[Csc[c + d*x]^2, x] /; FreeQ[{a, b, c, d}, x]
+Int[Sqrt[a_ + b_.*csc[c_. + d_.*x_]], x_Symbol] := -2*b/d* Subst[Int[1/(a + x^2), x], x, b*Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]]] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
+Int[(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := -b^2*Cot[c + d*x]*(a + b*Csc[c + d*x])^(n - 2)/(d*(n - 1)) + a/(n - 1)* Int[(a + b*Csc[c + d*x])^(n - 2)*(a*(n - 1) + b*(3*n - 4)*Csc[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
+Int[1/Sqrt[a_ + b_.*csc[c_. + d_.*x_]], x_Symbol] := 1/a*Int[Sqrt[a + b*Csc[c + d*x]], x] - b/a*Int[Csc[c + d*x]/Sqrt[a + b*Csc[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
+Int[(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := -Cot[c + d*x]*(a + b*Csc[c + d*x])^n/(d*(2*n + 1)) + 1/(a^2*(2*n + 1))* Int[(a + b*Csc[c + d*x])^(n + 1)*(a*(2*n + 1) - b*(n + 1)*Csc[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]
+Int[(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := a^n*Cot[ c + d*x]/(d*Sqrt[1 + Csc[c + d*x]]*Sqrt[1 - Csc[c + d*x]])* Subst[Int[(1 + b*x/a)^(n - 1/2)/(x*Sqrt[1 - b*x/a]), x], x, Csc[c + d*x]] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[2*n]] && GtQ[a, 0]
+Int[(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := a^IntPart[n]*(a + b*Csc[c + d*x])^ FracPart[n]/(1 + b/a*Csc[c + d*x])^FracPart[n]* Int[(1 + b/a*Csc[c + d*x])^n, x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[2*n]] && Not[GtQ[a, 0]]
+Int[Sqrt[a_ + b_.*csc[c_. + d_.*x_]], x_Symbol] := 2*(a + b*Csc[c + d*x])/(d*Rt[a + b, 2]*Cot[c + d*x])* Sqrt[b*(1 + Csc[c + d*x])/(a + b*Csc[c + d*x])]* Sqrt[-b*(1 - Csc[c + d*x])/(a + b*Csc[c + d*x])]* EllipticPi[a/(a + b), ArcSin[Rt[a + b, 2]/Sqrt[a + b*Csc[c + d*x]]], (a - b)/(a + b)] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
+Int[(a_ + b_.*csc[c_. + d_.*x_])^(3/2), x_Symbol] := Int[(a^2 + b*(2*a - b)*Csc[c + d*x])/Sqrt[a + b*Csc[c + d*x]], x] + b^2* Int[Csc[c + d*x]*(1 + Csc[c + d*x])/Sqrt[a + b*Csc[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
+Int[(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := -b^2*Cot[c + d*x]*(a + b*Csc[c + d*x])^(n - 2)/(d*(n - 1)) + 1/(n - 1)*Int[(a + b*Csc[c + d*x])^(n - 3)* Simp[a^3*(n - 1) + (b*(b^2*(n - 2) + 3*a^2*(n - 1)))* Csc[c + d*x] + (a*b^2*(3*n - 4))*Csc[c + d*x]^2, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]
+Int[1/(a_ + b_.*csc[c_. + d_.*x_]), x_Symbol] := x/a - 1/a*Int[1/(1 + a/b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
+Int[1/Sqrt[a_ + b_.*csc[c_. + d_.*x_]], x_Symbol] := 2*Rt[a + b, 2]/(a*d*Cot[c + d*x])* Sqrt[b*(1 - Csc[c + d*x])/(a + b)]* Sqrt[-b*(1 + Csc[c + d*x])/(a - b)]* EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
+Int[(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := b^2*Cot[ c + d*x]*(a + b*Csc[c + d*x])^(n + 1)/(a*d*(n + 1)*(a^2 - b^2)) + 1/(a*(n + 1)*(a^2 - b^2))* Int[(a + b*Csc[c + d*x])^(n + 1)* Simp[(a^2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
+Int[(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := Unintegrable[(a + b*Csc[c + d*x])^n, x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && Not[IntegerQ[2*n]]
diff --git a/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m
new file mode 100755
index 0000000..eaf1078
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m	
@@ -0,0 +1,89 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.5.1.2 (d sec)^n (a+b sec)^m *)
+Int[(a_ + b_.*csc[e_. + f_.*x_])*(d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := a*Int[(d*Csc[e + f*x])^n, x] + b/d*Int[(d*Csc[e + f*x])^(n + 1), x] /; FreeQ[{a, b, d, e, f, n}, x]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^2*(d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := 2*a*b/d*Int[(d*Csc[e + f*x])^(n + 1), x] + Int[(d*Csc[e + f*x])^n*(a^2 + b^2*Csc[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, n}, x]
+Int[csc[e_. + f_.*x_]^2/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := 1/b*Int[Csc[e + f*x], x] - a/b*Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x] /; FreeQ[{a, b, e, f}, x]
+Int[csc[e_. + f_.*x_]^3/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := -Cot[e + f*x]/(b*f) - a/b*Int[Csc[e + f*x]^2/(a + b*Csc[e + f*x]), x] /; FreeQ[{a, b, e, f}, x]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := Int[ExpandTrig[(a + b*csc[e + f*x])^m*(d*csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && RationalQ[n]
+Int[csc[e_. + f_.*x_]*Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := -2*b*Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]) /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := -b*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)/(f*m) + a*(2*m - 1)/m*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && IntegerQ[2*m]
+Int[csc[e_. + f_.*x_]/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := -Cot[e + f*x]/(f*(b + a*Csc[e + f*x])) /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := -2/f*Subst[Int[1/(2*a + x^2), x], x, b*Cot[e + f*x]/Sqrt[a + b*Csc[e + f*x]]] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := b*Cot[ e + f*x]*(a + b*Csc[e + f*x])^ m/(a*f*(2*m + 1)) + (m + 1)/(a*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2] && IntegerQ[2*m]
+Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := -Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(2*m + 1)) + m/(b*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]
+Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := -Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + a*m/(b*(m + 1))*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]
+Int[csc[e_. + f_.*x_]^3*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := b*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(a*f*(2*m + 1)) - 1/(a^2*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(a*m - b*(2*m + 1)*Csc[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]
+Int[csc[e_. + f_.*x_]^3*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := -Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^ m*(b*(m + 1) - a*Csc[e + f*x]), x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*Sqrt[d_.*csc[e_. + f_.*x_]], x_Symbol] := -2*a/(b*f)*Sqrt[a*d/b]* Subst[Int[1/Sqrt[1 + x^2/a], x], x, b*Cot[e + f*x]/Sqrt[a + b*Csc[e + f*x]]] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[a*d/b, 0]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*Sqrt[d_.*csc[e_. + f_.*x_]], x_Symbol] := -2*b*d/f* Subst[Int[1/(b - d*x^2), x], x, b*Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]])] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && Not[GtQ[a*d/b, 0]]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -2*b*d* Cot[e + f*x]*(d*Csc[e + f*x])^(n - 1)/(f*(2*n - 1)* Sqrt[a + b*Csc[e + f*x]]) + 2*a*d*(n - 1)/(b*(2*n - 1))* Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n - 1), x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/Sqrt[d_.*csc[e_. + f_.*x_]], x_Symbol] := -2*a*Cot[ e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]]) /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := a*Cot[ e + f*x]*(d*Csc[e + f*x])^n/(f*n*Sqrt[a + b*Csc[e + f*x]]) + a*(2*n + 1)/(2*b*d*n)* Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n + 1), x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1/2] && IntegerQ[2*n]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := a^2*d* Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]])* Subst[Int[(d*x)^(n - 1)/Sqrt[a - b*x], x], x, Csc[e + f*x]] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0]
+Int[Sqrt[d_.*csc[e_. + f_.*x_]]/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := -Sqrt[2]*Sqrt[a]/(b*f)* Subst[Int[1/Sqrt[1 + x^2], x], x, b*Cot[e + f*x]/(a + b*Csc[e + f*x])] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[d - a/b, 0] && GtQ[a, 0]
+Int[Sqrt[d_.*csc[e_. + f_.*x_]]/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := -2*b*d/(a*f)* Subst[Int[1/(2*b - d*x^2), x], x, b*Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]])] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -a*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^ n/(f*m) + b*(2*m - 1)/(d*m)* Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + n, 0] && GtQ[m, 1/2] && IntegerQ[2*m]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := b*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 1)/(a*f*(2*m + 1)) + d*(m + 1)/(b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + n, 0] && LtQ[m, -1/2] && IntegerQ[2*m]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(f*(2*m + 1)) + m/(a*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n, x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && LtQ[m, -1/2]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(f*(m + 1)) + a*m/(b*d*(m + 1))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && Not[LtQ[m, -1/2]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := b^2*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^ n/(f*n) - a/(d*n)* Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*(b*(m - 2*n - 2) - a*(m + 2*n - 1)*Csc[e + f*x]), x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && (LtQ[n, -1] || EqQ[m, 3/2] && EqQ[n, -1/2]) && IntegerQ[2*m]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -b^2*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^ n/(f*(m + n - 1)) + b/(m + n - 1)* Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^ n*(b*(m + 2*n - 1) + a*(3*m + 2*n - 4)*Csc[e + f*x]), x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + n - 1, 0] && IntegerQ[2*m]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := b*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 1)/(a*f*(2*m + 1)) - d/(a*b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)*(a*(n - 1) - b*(m + n)*Csc[e + f*x]), x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && (IntegersQ[2*m, 2*n] || IntegerQ[m])
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -d^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 2)/(f*(2*m + 1)) + d^2/(a*b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)*(b*(n - 2) + a*(m - n + 2)*Csc[e + f*x]), x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 2] && (IntegersQ[2*m, 2*n] || IntegerQ[m])
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(f*(2*m + 1)) + 1/(a^2*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n*(a*(2*m + n + 1) - b*(m + n + 1)*Csc[e + f*x]), x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m])
+Int[(d_.*csc[e_. + f_.*x_])^n_/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := d^2*Cot[ e + f*x]*(d*Csc[e + f*x])^(n - 2)/(f*(a + b*Csc[e + f*x])) - d^2/(a*b)* Int[(d*Csc[e + f*x])^(n - 2)*(b*(n - 2) - a*(n - 1)*Csc[e + f*x]), x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1]
+Int[(d_.*csc[e_. + f_.*x_])^n_/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*(a + b*Csc[e + f*x])) - 1/a^2*Int[(d*Csc[e + f*x])^n*(a*(n - 1) - b*n*Csc[e + f*x]), x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]
+Int[(d_.*csc[e_. + f_.*x_])^n_/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := -b*d*Cot[ e + f*x]*(d*Csc[e + f*x])^(n - 1)/(a*f*(a + b*Csc[e + f*x])) + d*(n - 1)/(a*b)* Int[(d*Csc[e + f*x])^(n - 1)*(a - b*Csc[e + f*x]), x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0]
+Int[(d_.*csc[e_. + f_.*x_])^(3/2)/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := d/b*Int[Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]], x] - a*d/b*Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0]
+Int[(d_.*csc[e_. + f_.*x_])^n_/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := -2*d^2* Cot[e + f*x]*(d*Csc[e + f*x])^(n - 2)/(f*(2*n - 3)* Sqrt[a + b*Csc[e + f*x]]) + d^2/(b*(2*n - 3))* Int[(d*Csc[e + f*x])^(n - 2)*(2*b*(n - 2) - a*Csc[e + f*x])/ Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]
+Int[(d_.*csc[e_. + f_.*x_])^n_/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*n*Sqrt[a + b*Csc[e + f*x]]) + 1/(2*b*d*n)* Int[(d*Csc[e + f*x])^(n + 1)*(a + b*(2*n + 1)*Csc[e + f*x])/ Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0] && IntegerQ[2*n]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -d^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 2)/(f*(m + n - 1)) + d^2/(b*(m + n - 1))* Int[(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 2)*(b*(n - 2) + a*m*Csc[e + f*x]), x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 2] && NeQ[m + n - 1, 0] && IntegerQ[n]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -(a*d/b)^n* Cot[e + f*x]/(a^(n - 2)*f*Sqrt[a + b*Csc[e + f*x]]* Sqrt[a - b*Csc[e + f*x]])* Subst[Int[(a - x)^(n - 1)*(2*a - x)^(m - 1/2)/Sqrt[x], x], x, a - b*Csc[e + f*x]] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && GtQ[a, 0] && Not[IntegerQ[n]] && GtQ[a*d/b, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -(-a*d/b)^n* Cot[e + f*x]/(a^(n - 1)*f*Sqrt[a + b*Csc[e + f*x]]* Sqrt[a - b*Csc[e + f*x]])* Subst[Int[x^(m - 1/2)*(a - x)^(n - 1)/Sqrt[2*a - x], x], x, a + b*Csc[e + f*x]] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && GtQ[a, 0] && Not[IntegerQ[n]] && LtQ[a*d/b, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := a^2*d* Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]* Sqrt[a - b*Csc[e + f*x]])* Subst[Int[(d*x)^(n - 1)*(a + b*x)^(m - 1/2)/Sqrt[a - b*x], x], x, Csc[e + f*x]] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && GtQ[a, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := a^IntPart[m]*(a + b*Csc[e + f*x])^ FracPart[m]/(1 + b/a*Csc[e + f*x])^FracPart[m]* Int[(1 + b/a*Csc[e + f*x])^m*(d*Csc[e + f*x])^n, x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && Not[GtQ[a, 0]]
+Int[csc[e_. + f_.*x_]*Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := (a - b)*Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] + b*Int[Csc[e + f*x]*(1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := -b*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)/(f*m) + 1/m* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*m + a*b*(2*m - 1)*Csc[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && IntegerQ[2*m]
+(* Int[csc[e_.+f_.*x_]/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol] := -2/f*Subst[Int[1/(a+b-(a-b)*x^2),x],x,Cot[e+f*x]/(1+Csc[e+f*x])] /; FreeQ[{a,b,e,f},x] && NeQ[a^2-b^2,0] *)
+Int[csc[e_. + f_.*x_]/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := 1/b*Int[1/(1 + a/b*Sin[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := -2*Rt[a + b, 2]/(b*f*Cot[e + f*x])* Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]* Sqrt[-b*(1 + Csc[e + f*x])/(a - b)]* EllipticF[ ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := -b*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + 2)*Csc[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := Cot[e + f*x]/(f*Sqrt[1 + Csc[e + f*x]]*Sqrt[1 - Csc[e + f*x]])* Subst[Int[(a + b*x)^m/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Csc[e + f*x]] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] && Not[IntegerQ[2*m]]
+Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := -Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + m/(m + 1)* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(b + a*Csc[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]
+Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := a*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2)) - 1/((m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(b*(m + 1) - a*(m + 2)*Csc[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
+Int[csc[e_. + f_.*x_]^2/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := -Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] + Int[Csc[e + f*x]*(1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := -a/b*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x] + 1/b*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]^3*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := -a^2*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 - b^2)) + 1/(b*(m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[a*b*(m + 1) - (a^2 + b^2*(m + 1))*Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
+Int[csc[e_. + f_.*x_]^3*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := -Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^ m*(b*(m + 1) - a*Csc[e + f*x]), x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] && Not[LtQ[m, -1]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := a^2*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^ n/(f*n) - 1/(d*n)*Int[(a + b*Csc[e + f*x])^(m - 3)*(d*Csc[e + f*x])^(n + 1)* Simp[a^2*b*(m - 2*n - 2) - a*(3*b^2*n + a^2*(n + 1))*Csc[e + f*x] - b*(b^2*n + a^2*(m + n - 1))*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 2] && (IntegerQ[m] && LtQ[n, -1] || IntegersQ[m + 1/2, 2*n] && LeQ[n, -1])
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -b^2*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^ n/(f*(m + n - 1)) + 1/(d*(m + n - 1))* Int[(a + b*Csc[e + f*x])^(m - 3)*(d*Csc[e + f*x])^n* Simp[a^3*d*(m + n - 1) + a*b^2*d*n + b*(b^2*d*(m + n - 2) + 3*a^2*d*(m + n - 1))*Csc[e + f*x] + a*b^2*d*(3*m + 2*n - 4)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && Not[IGtQ[n, 2] && Not[IntegerQ[m]]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -b*d*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)* Simp[b*d*(n - 1) + a*d*(m + 1)*Csc[e + f*x] - b*d*(m + n + 1)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && IntegersQ[2*m, 2*n]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := a*d^2* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)/(f*(m + 1)*(a^2 - b^2)) - d^2/((m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)*(a*(n - 2) + b*(m + 1)*Csc[e + f*x] - a*(m + n)*Csc[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegersQ[2*m, 2*n]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -a^2*d^3* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 3)/(b*f*(m + 1)*(a^2 - b^2)) + d^3/(b*(m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 3)* Simp[a^2*(n - 3) + a*b*(m + 1)*Csc[e + f*x] - (a^2*(n - 2) + b^2*(m + 1))* Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && (IGtQ[n, 3] || IntegersQ[n + 1/2, 2*m] && GtQ[n, 2])
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*n) - 1/(a*d*n)*Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)* Simp[b*(m + n + 1) - a*(n + 1)*Csc[e + f*x] - b*(m + n + 2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m + 1/2, 0] && ILtQ[n, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := b^2*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*(m + 1)*(a^2 - b^2)) + 1/(a*(m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n* (a^2*(m + 1) - b^2*(m + n + 1) - a*b*(m + 1)*Csc[e + f*x] + b^2*(m + n + 2)*Csc[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
+Int[Sqrt[d_.*csc[e_. + f_.*x_]]/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]]/d* Int[Sqrt[d*Sin[e + f*x]]/(b + a*Sin[e + f*x]), x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[(d_.*csc[e_. + f_.*x_])^(3/2)/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := d*Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]]* Int[1/(Sqrt[d*Sin[e + f*x]]*(b + a*Sin[e + f*x])), x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[(d_.*csc[e_. + f_.*x_])^(5/2)/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := d/b*Int[(d*Csc[e + f*x])^(3/2), x] - a*d/b*Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[(d_.*csc[e_. + f_.*x_])^n_/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := -d^3*Cot[e + f*x]*(d*Csc[e + f*x])^(n - 3)/(b*f*(n - 2)) + d^3/(b*(n - 2))* Int[(d*Csc[e + f*x])^(n - 3)* Simp[a*(n - 3) + b*(n - 3)*Csc[e + f*x] - a*(n - 2)*Csc[e + f*x]^2, x]/(a + b*Csc[e + f*x]), x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 3]
+Int[1/(Sqrt[d_.*csc[e_. + f_.*x_]]*(a_ + b_.*csc[e_. + f_.*x_])), x_Symbol] := b^2/(a^2*d^2)*Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x] + 1/a^2*Int[(a - b*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[(d_.*csc[e_. + f_.*x_])^n_/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := Cot[e + f*x]*(d*Csc[e + f*x])^n/(a*f*n) - 1/(a*d*n)*Int[(d*Csc[e + f*x])^(n + 1)/(a + b*Csc[e + f*x])* Simp[b*n - a*(n + 1)*Csc[e + f*x] - b*(n + 1)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*Sqrt[d_.*csc[e_. + f_.*x_]], x_Symbol] := a*Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x] + b/d*Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -2*d*Cos[e + f*x]* Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n - 1)/(f*(2*n - 1)) + d^2/(2*n - 1)* Int[(d*Csc[e + f*x])^(n - 2)* Simp[2*a*(n - 2) + b*(2*n - 3)*Csc[e + f*x] + a*Csc[e + f*x]^2, x]/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/Sqrt[d_.*csc[e_. + f_.*x_]], x_Symbol] := Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]* Sqrt[b + a*Sin[e + f*x]])*Int[Sqrt[b + a*Sin[e + f*x]], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := Cot[e + f*x]*Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^n/(f*n) - 1/(2*d*n)* Int[(d*Csc[e + f*x])^(n + 1)* Simp[b - 2*a*(n + 1)*Csc[e + f*x] - b*(2*n + 3)*Csc[e + f*x]^2, x]/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]
+Int[Sqrt[d_.*csc[e_. + f_.*x_]]/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := Sqrt[d*Csc[e + f*x]]* Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]* Int[1/Sqrt[b + a*Sin[e + f*x]], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[(d_.*csc[e_. + f_.*x_])^(3/2)/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := d*Sqrt[d*Csc[e + f*x]]* Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]* Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f*x]]), x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[(d_.*csc[e_. + f_.*x_])^n_/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := -2*d^2*Cos[e + f*x]*(d*Csc[e + f*x])^(n - 2)* Sqrt[a + b*Csc[e + f*x]]/(b*f*(2*n - 3)) + d^3/(b*(2*n - 3))* Int[(d*Csc[e + f*x])^(n - 3)/Sqrt[a + b*Csc[e + f*x]]* Simp[2*a*(n - 3) + b*(2*n - 5)*Csc[e + f*x] - 2*a*(n - 2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]
+Int[1/(csc[e_. + f_.*x_]*Sqrt[a_ + b_.*csc[e_. + f_.*x_]]), x_Symbol] := -Cos[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/(a*f) - b/(2*a)*Int[(1 + Csc[e + f*x]^2)/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[1/(Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*Sqrt[d_.*csc[e_. + f_.*x_]]), x_Symbol] := 1/a*Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x] - b/(a*d)*Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[(d_.*csc[e_. + f_.*x_])^n_/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := Cos[e + f*x]*(d*Csc[e + f*x])^(n + 1)* Sqrt[a + b*Csc[e + f*x]]/(a*d*f*n) + 1/(2*a*d*n)*Int[(d*Csc[e + f*x])^(n + 1)/Sqrt[a + b*Csc[e + f*x]]* Simp[-b*(2*n + 1) + 2*a*(n + 1)*Csc[e + f*x] + b*(2*n + 3)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^(3/2)*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := a*Cot[e + f*x]* Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^n/(f*n) + 1/(2*d*n)*Int[(d*Csc[e + f*x])^(n + 1)/Sqrt[a + b*Csc[e + f*x]]* Simp[a*b*(2*n - 1) + 2*(b^2*n + a^2*(n + 1))*Csc[e + f*x] + a*b*(2*n + 3)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegersQ[2*n]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -d^3*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 3)/(b*f*(m + n - 1)) + d^3/(b*(m + n - 1))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 3)* Simp[a*(n - 3) + b*(m + n - 2)*Csc[e + f*x] - a*(n - 2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 3] && (IntegerQ[n] || IntegersQ[2*m, 2*n]) && Not[IGtQ[m, 2]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -b*d*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n - 1)/(f*(m + n - 1)) + d/(m + n - 1)* Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n - 1)* Simp[a*b*(n - 1) + (b^2*(m + n - 2) + a^2*(m + n - 1))* Csc[e + f*x] + a*b*(2*m + n - 2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[0, m, 2] && LtQ[0, n, 3] && NeQ[m + n - 1, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -d^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 2)/(f*(m + n - 1)) + d^2/(b*(m + n - 1))* Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n - 2)* Simp[a*b*(n - 2) + b^2*(m + n - 2)*Csc[e + f*x] + a*b*m*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[-1, m, 2] && LtQ[1, n, 3] && NeQ[m + n - 1, 0] && (IntegerQ[n] || IntegersQ[2*m, 2*n])
+Int[(a_ + b_.*csc[e_. + f_.*x_])^(3/2)/Sqrt[d_.*csc[e_. + f_.*x_]], x_Symbol] := a*Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x] + b/d*Int[Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
+Int[(d_.*csc[e_. + f_.*x_])^n_.*(a_ + b_.*csc[e_. + f_.*x_])^m_., x_Symbol] := Sin[e + f*x]^n*(d*Csc[e + f*x])^n* Int[(b + a*Sin[e + f*x])^m/Sin[e + f*x]^(m + n), x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[m]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := Unintegrable[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n, x] /; FreeQ[{a, b, d, e, f, m, n}, x]
+Int[(d_.*cos[e_. + f_.*x_])^m_*(a_. + b_.*sec[e_. + f_.*x_])^p_, x_Symbol] := (d*Cos[e + f*x])^FracPart[m]*(Sec[e + f*x]/d)^FracPart[m]* Int[(Sec[e + f*x]/d)^(-m)*(a + b*Sec[e + f*x])^p, x] /; FreeQ[{a, b, d, e, f, m, p}, x] && Not[IntegerQ[m]] && Not[IntegerQ[p]]
diff --git a/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.3 (d sin)^n (a+b sec)^m.m b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.3 (d sin)^n (a+b sec)^m.m
new file mode 100755
index 0000000..ff18973
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.3 (d sin)^n (a+b sec)^m.m	
@@ -0,0 +1,11 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.5.1.3 (d sin)^n (a+b sec)^m *)
+Int[(g_.*cos[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^m_., x_Symbol] := Int[(g*Cos[e + f*x])^p*(b + a*Sin[e + f*x])^m/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
+Int[cos[e_. + f_.*x_]^p_.*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := -1/(f*b^(p - 1))* Subst[Int[(-a + b*x)^((p - 1)/2)*(a + b*x)^(m + (p - 1)/2)/ x^(p + 1), x], x, Csc[e + f*x]] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
+Int[cos[e_. + f_.*x_]^p_.*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := -1/f*Subst[ Int[(-1 + x)^((p - 1)/2)*(1 + x)^((p - 1)/2)*(a + b*x)^m/ x^(p + 1), x], x, Csc[e + f*x]] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_/cos[e_. + f_.*x_]^2, x_Symbol] := Tan[e + f*x]*(a + b*Csc[e + f*x])^m/f + b*m*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x] /; FreeQ[{a, b, e, f, m}, x]
+Int[(g_.*cos[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol] := Sin[e + f*x]^ FracPart[m]*(a + b*Csc[e + f*x])^FracPart[m]/(b + a*Sin[e + f*x])^ FracPart[m]* Int[(g*Cos[e + f*x])^p*(b + a*Sin[e + f*x])^m/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, m, p}, x] && (EqQ[a^2 - b^2, 0] || IntegersQ[2*m, p])
+Int[(g_.*cos[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^m_., x_Symbol] := Unintegrable[(g*Cos[e + f*x])^p*(a + b*Csc[e + f*x])^m, x] /; FreeQ[{a, b, e, f, g, m, p}, x]
+(* Int[(g_.*sec[e_.+f_.*x_])^p_*(a_+b_.*csc[e_.+f_.*x_])^m_.,x_Symbol]  := Int[(g*Sec[e+f*x])^p*(b+a*Sin[e+f*x])^m/Sin[e+f*x]^m,x] /; FreeQ[{a,b,e,f,g,p},x] && Not[IntegerQ[p]] && IntegerQ[m] *)
+Int[(g_.*sec[e_. + f_.*x_])^p_*(a_ + b_.*csc[e_. + f_.*x_])^m_., x_Symbol] := g^IntPart[p]*(g*Sec[e + f*x])^FracPart[p]*Cos[e + f*x]^FracPart[p]* Int[(a + b*Csc[e + f*x])^m/Cos[e + f*x]^p, x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Not[IntegerQ[p]]
diff --git a/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m
new file mode 100755
index 0000000..b3687e9
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m	
@@ -0,0 +1,27 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.5.1.4 (d tan)^n (a+b sec)^m *)
+Int[cot[c_. + d_.*x_]^m_.*(a_ + b_.*csc[c_. + d_.*x_])^n_., x_Symbol] := 1/(a^(m - n - 1)*b^n*d)* Subst[Int[(a - b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n)/ x^(m + n), x], x, Sin[c + d*x]] /; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]
+Int[cot[c_. + d_.*x_]^m_.*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := -1/(d*b^(m - 1))* Subst[Int[(-a + b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n)/x, x], x, Csc[c + d*x]] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[n]]
+Int[(e_.*cot[c_. + d_.*x_])^m_*(a_ + b_.*csc[c_. + d_.*x_]), x_Symbol] := -e*(e*Cot[c + d*x])^(m - 1)*(a*m + b*(m - 1)*Csc[c + d*x])/(d* m*(m - 1)) - e^2/m* Int[(e*Cot[c + d*x])^(m - 2)*(a*m + b*(m - 1)*Csc[c + d*x]), x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1]
+Int[(e_.*cot[c_. + d_.*x_])^m_*(a_ + b_.*csc[c_. + d_.*x_]), x_Symbol] := -(e*Cot[c + d*x])^(m + 1)*(a + b*Csc[c + d*x])/(d*e*(m + 1)) - 1/(e^2*(m + 1))* Int[(e*Cot[c + d*x])^(m + 2)*(a*(m + 1) + b*(m + 2)*Csc[c + d*x]), x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1]
+Int[(a_ + b_.*csc[c_. + d_.*x_])/cot[c_. + d_.*x_], x_Symbol] := Int[(b + a*Sin[c + d*x])/Cos[c + d*x], x] /; FreeQ[{a, b, c, d}, x]
+Int[(e_.*cot[c_. + d_.*x_])^m_.*(a_ + b_.*csc[c_. + d_.*x_]), x_Symbol] := a*Int[(e*Cot[c + d*x])^m, x] + b*Int[(e*Cot[c + d*x])^m*Csc[c + d*x], x] /; FreeQ[{a, b, c, d, e, m}, x]
+Int[cot[c_. + d_.*x_]^m_.*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := -(-1)^((m - 1)/2)/(d*b^(m - 1))* Subst[Int[(b^2 - x^2)^((m - 1)/2)*(a + x)^n/x, x], x, b*Csc[c + d*x]] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
+Int[(e_.*cot[c_. + d_.*x_])^m_*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]
+Int[cot[c_. + d_.*x_]^m_.*(a_ + b_.*csc[c_. + d_.*x_])^n_., x_Symbol] := -2*a^(m/2 + n + 1/2)/d* Subst[Int[x^m*(2 + a*x^2)^(m/2 + n - 1/2)/(1 + a*x^2), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]]] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && IntegerQ[n - 1/2]
+Int[(e_.*cot[c_. + d_.*x_])^m_*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := a^(2*n)*e^(-2*n)* Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[a^2 - b^2, 0] && ILtQ[n, 0]
+Int[(e_.*cot[c_. + d_.*x_])^m_*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := -2^(m + n + 1)*(e*Cot[c + d*x])^(m + 1)*(a + b*Csc[c + d*x])^ n/(d*e*(m + 1))*(a/(a + b*Csc[c + d*x]))^(m + n + 1)* AppellF1[(m + 1)/2, m + n, 1, (m + 3)/ 2, -(a - b*Csc[c + d*x])/(a + b*Csc[c + d*x]), (a - b*Csc[c + d*x])/(a + b*Csc[c + d*x])] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[n]]
+Int[Sqrt[e_.*cot[c_. + d_.*x_]]/(a_ + b_.*csc[c_. + d_.*x_]), x_Symbol] := 1/a*Int[Sqrt[e*Cot[c + d*x]], x] - b/a*Int[Sqrt[e*Cot[c + d*x]]/(b + a*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2, 0]
+Int[(e_.*cot[c_. + d_.*x_])^m_/(a_ + b_.*csc[c_. + d_.*x_]), x_Symbol] := -e^2/b^2*Int[(e*Cot[c + d*x])^(m - 2)*(a - b*Csc[c + d*x]), x] + e^2*(a^2 - b^2)/b^2* Int[(e*Cot[c + d*x])^(m - 2)/(a + b*Csc[c + d*x]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0]
+Int[1/(Sqrt[e_.*cot[c_. + d_.*x_]]*(a_ + b_.*csc[c_. + d_.*x_])), x_Symbol] := 1/a*Int[1/Sqrt[e*Cot[c + d*x]], x] - b/a*Int[1/(Sqrt[e*Cot[c + d*x]]*(b + a*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2, 0]
+Int[(e_.*cot[c_. + d_.*x_])^m_/(a_ + b_.*csc[c_. + d_.*x_]), x_Symbol] := 1/(a^2 - b^2)*Int[(e*Cot[c + d*x])^m*(a - b*Csc[c + d*x]), x] + b^2/(e^2*(a^2 - b^2))* Int[(e*Cot[c + d*x])^(m + 2)/(a + b*Csc[c + d*x]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m + 1/2, 0]
+Int[cot[c_. + d_.*x_]^2*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := Int[(-1 + Csc[c + d*x]^2)*(a + b*Csc[c + d*x])^n, x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0]
+Int[cot[c_. + d_.*x_]^m_*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[(a + b*Csc[c + d*x])^ n, (-1 + Csc[c + d*x]^2)^(m/2), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m/2, 0] && IntegerQ[n - 1/2]
+Int[cot[c_. + d_.*x_]^m_*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[(a + b*Csc[c + d*x])^ n, (-1 + Sec[c + d*x]^2)^(-m/2), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m/2, 0] && IntegerQ[n - 1/2] && EqQ[m, -2]
+Int[(e_.*cot[c_. + d_.*x_])^m_*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[a^2 - b^2, 0] && IGtQ[n, 0]
+Int[cot[c_. + d_.*x_]^m_.*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol] := Int[Cos[c + d*x]^m*(b + a*Sin[c + d*x])^n/Sin[c + d*x]^(m + n), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[n] && IntegerQ[m] && (IntegerQ[m/2] || LeQ[m, 1])
+Int[(e_.*cot[c_. + d_.*x_])^m_.*(a_. + b_.*csc[c_. + d_.*x_])^n_., x_Symbol] := Unintegrable[(e*Cot[c + d*x])^m*(a + b*Csc[c + d*x])^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[(e_.*cot[c_. + d_.*x_])^m_*(a_ + b_.*sec[c_. + d_.*x_])^n_., x_Symbol] := (e*Cot[c + d*x])^m*Tan[c + d*x]^m* Int[(a + b*Sec[c + d*x])^n/Tan[c + d*x]^m, x] /; FreeQ[{a, b, c, d, e, m, n}, x] && Not[IntegerQ[m]]
+Int[(e_.*tan[c_. + d_.*x_]^p_)^m_*(a_ + b_.*sec[c_. + d_.*x_])^n_., x_Symbol] := (e*Tan[c + d*x]^p)^m/(e*Tan[c + d*x])^(m*p)* Int[(e*Tan[c + d*x])^(m*p)*(a + b*Sec[c + d*x])^n, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && Not[IntegerQ[m]]
+Int[(e_.*cot[c_. + d_.*x_]^p_)^m_*(a_ + b_.*csc[c_. + d_.*x_])^n_., x_Symbol] := (e*Cot[c + d*x]^p)^m/(e*Cot[c + d*x])^(m*p)* Int[(e*Cot[c + d*x])^(m*p)*(a + b*Csc[c + d*x])^n, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && Not[IntegerQ[m]]
diff --git a/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m
new file mode 100755
index 0000000..90a9fbd
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m	
@@ -0,0 +1,19 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.5.10 (c+d x)^m (a+b sec)^n *)
+Int[(c_. + d_.*x_)^m_.*csc[e_. + k_.*Pi + f_.*Complex[0, fz_]*x_], x_Symbol] := -2*(c + d*x)^m*ArcTanh[E^(-I*k*Pi)*E^(-I*e + f*fz*x)]/(f*fz*I) - d*m/(f*fz*I)* Int[(c + d*x)^(m - 1)*Log[1 - E^(-I*k*Pi)*E^(-I*e + f*fz*x)], x] + d*m/(f*fz*I)* Int[(c + d*x)^(m - 1)*Log[1 + E^(-I*k*Pi)*E^(-I*e + f*fz*x)], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*csc[e_. + k_.*Pi + f_.*x_], x_Symbol] := -2*(c + d*x)^m*ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f - d*m/f* Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x] + d*m/f* Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*csc[e_. + f_.*Complex[0, fz_]*x_], x_Symbol] := -2*(c + d*x)^m*ArcTanh[E^(-I*e + f*fz*x)]/(f*fz*I) - d*m/(f*fz*I)* Int[(c + d*x)^(m - 1)*Log[1 - E^(-I*e + f*fz*x)], x] + d*m/(f*fz*I)* Int[(c + d*x)^(m - 1)*Log[1 + E^(-I*e + f*fz*x)], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*csc[e_. + f_.*x_], x_Symbol] := -2*(c + d*x)^m*ArcTanh[E^(I*(e + f*x))]/f - d*m/f*Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x] + d*m/f*Int[(c + d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x] /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*csc[e_. + f_.*x_]^2, x_Symbol] := -(c + d*x)^m*Cot[e + f*x]/f + d*m/f*Int[(c + d*x)^(m - 1)*Cot[e + f*x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
+Int[(c_. + d_.*x_)*(b_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -b^2*(c + d*x)* Cot[e + f*x]*(b*Csc[e + f*x])^(n - 2)/(f*(n - 1)) - b^2*d*(b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2)) + b^2*(n - 2)/(n - 1)*Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
+Int[(c_. + d_.*x_)^m_*(b_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -b^2*(c + d*x)^m* Cot[e + f*x]*(b*Csc[e + f*x])^(n - 2)/(f*(n - 1)) - b^2*d* m*(c + d*x)^(m - 1)*(b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2)) + b^2*(n - 2)/(n - 1)* Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x] + b^2*d^2*m*(m - 1)/(f^2*(n - 1)*(n - 2))* Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
+Int[(c_. + d_.*x_)*(b_.*csc[e_. + f_.*x_])^n_, x_Symbol] := d*(b*Csc[e + f*x])^n/(f^2*n^2) + (c + d*x)*Cos[e + f*x]*(b*Csc[e + f*x])^(n + 1)/(b*f*n) + (n + 1)/(b^2*n)*Int[(c + d*x)*(b*Csc[e + f*x])^(n + 2), x] /; FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1]
+Int[(c_. + d_.*x_)^m_*(b_.*csc[e_. + f_.*x_])^n_, x_Symbol] := d*m*(c + d*x)^(m - 1)*(b*Csc[e + f*x])^n/(f^2*n^2) + (c + d*x)^m*Cos[e + f*x]*(b*Csc[e + f*x])^(n + 1)/(b*f*n) + (n + 1)/(b^2*n)*Int[(c + d*x)^m*(b*Csc[e + f*x])^(n + 2), x] - d^2*m*(m - 1)/(f^2*n^2)* Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^n, x] /; FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1] && GtQ[m, 1]
+Int[(c_. + d_.*x_)^m_.*(b_.*csc[e_. + f_.*x_])^n_, x_Symbol] := (b*Sin[e + f*x])^n*(b*Csc[e + f*x])^n* Int[(c + d*x)^m/(b*Sin[e + f*x])^n, x] /; FreeQ[{b, c, d, e, f, m, n}, x] && Not[IntegerQ[n]]
+Int[(c_. + d_.*x_)^m_.*(a_ + b_.*csc[e_. + f_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(c_. + d_.*x_)^m_.*(a_ + b_.*csc[e_. + f_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(c + d*x)^m, Sin[e + f*x]^(-n)/(b + a*Sin[e + f*x])^(-n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*csc[e_. + f_.*x_]^n_., x_Symbol] := If[MatchQ[f, f1_.*Complex[0, j_]], If[MatchQ[e, e1_. + Pi/2], Unintegrable[(c + d*x)^m*Sech[I*(e - Pi/2) + I*f*x]^n, x], (-I)^n*Unintegrable[(c + d*x)^m*Csch[-I*e - I*f*x]^n, x]], If[MatchQ[e, e1_. + Pi/2], Unintegrable[(c + d*x)^m*Sec[e - Pi/2 + f*x]^n, x], Unintegrable[(c + d*x)^m*Csc[e + f*x]^n, x]]] /; FreeQ[{c, d, e, f, m, n}, x] && IntegerQ[n]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*csc[e_. + f_.*x_])^n_., x_Symbol] := Unintegrable[(c + d*x)^m*(a + b*Csc[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
+Int[u_^m_.*(a_. + b_.*Sec[v_])^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*(a + b*Sec[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
+Int[u_^m_.*(a_. + b_.*Csc[v_])^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*(a + b*Csc[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
diff --git a/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m
new file mode 100755
index 0000000..1ace2ee
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m	
@@ -0,0 +1,21 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.5.11 (e x)^m (a+b sec(c+d x^n))^p *)
+Int[(a_. + b_.*Sec[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[Int[x^(1/n - 1)*(a + b*Sec[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
+Int[(a_. + b_.*Csc[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[Int[x^(1/n - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
+Int[(a_. + b_.*Sec[c_. + d_.*x_^n_])^p_., x_Symbol] := Unintegrable[(a + b*Sec[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
+Int[(a_. + b_.*Csc[c_. + d_.*x_^n_])^p_., x_Symbol] := Unintegrable[(a + b*Csc[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
+Int[(a_. + b_.*Sec[c_. + d_.*u_^n_])^p_., x_Symbol] := 1/Coefficient[u, x, 1]* Subst[Int[(a + b*Sec[c + d*x^n])^p, x], x, u] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
+Int[(a_. + b_.*Csc[c_. + d_.*u_^n_])^p_., x_Symbol] := 1/Coefficient[u, x, 1]* Subst[Int[(a + b*Csc[c + d*x^n])^p, x], x, u] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
+Int[(a_. + b_.*Sec[u_])^p_., x_Symbol] := Int[(a + b*Sec[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[(a_. + b_.*Csc[u_])^p_., x_Symbol] := Int[(a + b*Csc[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[x_^m_.*(a_. + b_.*Sec[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sec[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
+Int[x_^m_.*(a_. + b_.*Csc[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
+Int[x_^m_.*(a_. + b_.*Sec[c_. + d_.*x_^n_])^p_., x_Symbol] := Unintegrable[x^m*(a + b*Sec[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
+Int[x_^m_.*(a_. + b_.*Csc[c_. + d_.*x_^n_])^p_., x_Symbol] := Unintegrable[x^m*(a + b*Csc[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
+Int[(e_*x_)^m_.*(a_. + b_.*Sec[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Sec[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
+Int[(e_*x_)^m_.*(a_. + b_.*Csc[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Csc[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
+Int[(e_*x_)^m_.*(a_. + b_.*Sec[u_])^p_., x_Symbol] := Int[(e*x)^m*(a + b*Sec[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[(e_*x_)^m_.*(a_. + b_.*Csc[u_])^p_., x_Symbol] := Int[(e*x)^m*(a + b*Csc[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[x_^m_.*Sec[a_. + b_.*x_^n_.]^p_*Sin[a_. + b_.*x_^n_.], x_Symbol] := x^(m - n + 1)*Sec[a + b*x^n]^(p - 1)/(b*n*(p - 1)) - (m - n + 1)/(b*n*(p - 1))* Int[x^(m - n)*Sec[a + b*x^n]^(p - 1), x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]
+Int[x_^m_.*Csc[a_. + b_.*x_^n_.]^p_*Cos[a_. + b_.*x_^n_.], x_Symbol] := -x^(m - n + 1)*Csc[a + b*x^n]^(p - 1)/(b*n*(p - 1)) + (m - n + 1)/(b*n*(p - 1))* Int[x^(m - n)*Csc[a + b*x^n]^(p - 1), x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]
diff --git a/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m
new file mode 100755
index 0000000..adf73cb
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m	
@@ -0,0 +1,51 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.5.2.1 (a+b sec)^m (c+d sec)^n *)
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := c^n*Int[ ExpandTrig[(1 + d/c*csc[e + f*x])^n, (a + b*csc[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && ILtQ[n, 0] && LtQ[m + n, 2]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := (-a*c)^m*Int[Cot[e + f*x]^(2*m)*(c + d*Csc[e + f*x])^(n - m), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[n] && Not[IntegerQ[n] && GtQ[m - n, 0]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_])^m_, x_Symbol] := (-a*c)^(m + 1/2)* Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])* Int[Cot[e + f*x]^(2*m), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m + 1/2]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := 2*a*c* Cot[e + f*x]*(c + d*Csc[e + f*x])^(n - 1)/(f*(2*n - 1)* Sqrt[a + b*Csc[e + f*x]]) + c*Int[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && GtQ[n, 1/2]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -2*a*Cot[ e + f*x]*(c + d*Csc[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]) + 1/c* Int[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, -1/2]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^(3/2)*(c_ + d_.*csc[e_. + f_.*x_])^ n_., x_Symbol] := -4*a^2* Cot[e + f*x]*(c + d*Csc[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]) + a/c* Int[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, -1/2]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^(3/2)*(c_ + d_.*csc[e_. + f_.*x_])^ n_., x_Symbol] := -2*a^2* Cot[e + f*x]*(c + d*Csc[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]) + a*Int[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LeQ[n, -1/2]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^(5/2)*(c_ + d_.*csc[e_. + f_.*x_])^ n_., x_Symbol] := -8*a^3* Cot[e + f*x]*(c + d*Csc[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]) + a^2/c^2* Int[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n + 2), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, -1/2]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -a*c*Cot[ e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])* Subst[ Int[(b + a*x)^(m - 1/2)*(d + c*x)^(n - 1/2)/x^(m + n), x], x, Sin[e + f*x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && EqQ[m + n, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := a*c*Cot[ e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])* Subst[Int[(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2)/x, x], x, Csc[e + f*x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := a*c*x + b*d*Int[Csc[e + f*x]^2, x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := a*c*x + (b*c + a*d)*Int[Csc[e + f*x], x] + b*d*Int[Csc[e + f*x]^2, x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := c*Int[Sqrt[a + b*Csc[e + f*x]], x] + d*Int[Sqrt[a + b*Csc[e + f*x]]*Csc[e + f*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := a*c*Int[1/Sqrt[a + b*Csc[e + f*x]], x] + Int[Csc[e + f*x]*(b*c + a*d + b*d*Csc[e + f*x])/ Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := -b*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)/(f*m) + 1/m* Int[(a + b*Csc[e + f*x])^(m - 1)* Simp[a*c*m + (b*c*m + a*d*(2*m - 1))*Csc[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && GtQ[m, 1] && EqQ[a^2 - b^2, 0] && IntegerQ[2*m]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := -b*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)/(f*m) + 1/m*Int[(a + b*Csc[e + f*x])^(m - 2)* Simp[a^2*c*m + (b^2*d*(m - 1) + 2*a*b*c*m + a^2*d*m)* Csc[e + f*x] + b*(b*c*m + a*d*(2*m - 1))*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && GtQ[m, 1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]
+Int[(c_ + d_.*csc[e_. + f_.*x_])/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := c*x/a - (b*c - a*d)/a*Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
+Int[(c_ + d_.*csc[e_. + f_.*x_])/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := c/a*Int[Sqrt[a + b*Csc[e + f*x]], x] - (b*c - a*d)/a* Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[(c_ + d_.*csc[e_. + f_.*x_])/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := c*Int[1/Sqrt[a + b*Csc[e + f*x]], x] + d*Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := -(b*c - a*d)* Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(b*f*(2*m + 1)) + 1/(a^2*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)* Simp[a*c*(2*m + 1) - (b*c - a*d)*(m + 1)*Csc[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && EqQ[a^2 - b^2, 0] && IntegerQ[2*m]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := b*(b*c - a*d)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(a* f*(m + 1)*(a^2 - b^2)) + 1/(a*(m + 1)*(a^2 - b^2))*Int[(a + b*Csc[e + f*x])^(m + 1)* Simp[c*(a^2 - b^2)*(m + 1) - (a*(b*c - a*d)*(m + 1))* Csc[e + f*x] + b*(b*c - a*d)*(m + 2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := c*Int[(a + b*Csc[e + f*x])^m, x] + d*Int[(a + b*Csc[e + f*x])^m*Csc[e + f*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && Not[IntegerQ[2*m]]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := 1/c*Int[Sqrt[a + b*Csc[e + f*x]], x] - d/c*Int[Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := a/c*Int[1/Sqrt[a + b*Csc[e + f*x]], x] + (b*c - a*d)/c* Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^(3/2)/(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := a/c*Int[Sqrt[a + b*Csc[e + f*x]], x] + (b*c - a*d)/c* Int[Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
+(* Int[(a_+b_.*csc[e_.+f_.*x_])^(3/2)/(c_+d_.*csc[e_.+f_.*x_]),x_ Symbol] := b/d*Int[Sqrt[a+b*Csc[e+f*x]],x] -  (b*c-a*d)/d*Int[Sqrt[a+b*Csc[e+f*x]]/(c+d*Csc[e+f*x]),x] /; FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d,0] && NeQ[a^2-b^2,0] &&  NeQ[c^2-d^2,0] *)
+Int[(a_ + b_.*csc[e_. + f_.*x_])^(3/2)/(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := 1/(c*d)* Int[(a^2*d + b^2*c*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] - (b*c - a*d)^2/(c*d)* Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[1/(Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])), x_Symbol] := 1/(c*(b*c - a*d))* Int[(b*c - a*d - b*d*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + d^2/(c*(b*c - a*d))* Int[Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
+Int[1/(Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])), x_Symbol] := 1/c*Int[1/Sqrt[a + b*Csc[e + f*x]], x] - d/c*Int[Csc[ e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol] := Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]/Cot[e + f*x]* Int[Cot[e + f*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol] := c*Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x] + d*Int[ Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol] := 1/c*Int[Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]], x] - d/c* Int[Csc[e + f*x]* Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol] := -2*a/f* Subst[Int[1/(1 + a*c*x^2), x], x, Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]* Sqrt[c + d*Csc[e + f*x]])] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol] := a/c*Int[Sqrt[c + d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x] + (b*c - a*d)/c* Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]* Sqrt[c + d*Csc[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol] := 2*(a + b*Csc[e + f*x])/(c*f*Rt[(a + b)/(c + d), 2]*Cot[e + f*x])* Sqrt[(b*c - a*d)*(1 + Csc[e + f*x])/((c - d)*(a + b*Csc[e + f*x]))]* Sqrt[-(b*c - a*d)*(1 - Csc[e + f*x])/((c + d)*(a + b*Csc[e + f*x]))]* EllipticPi[a*(c + d)/(c*(a + b)), ArcSin[Rt[(a + b)/(c + d), 2]* Sqrt[c + d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]], (a - b)*(c + d)/((a + b)*(c - d))] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[1/(Sqrt[a_ + b_.*csc[e_. + f_.*x_]]* Sqrt[c_ + d_.*csc[e_. + f_.*x_]]), x_Symbol] := Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])* Int[1/Cot[e + f*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
+Int[1/(Sqrt[a_ + b_.*csc[e_. + f_.*x_]]* Sqrt[c_ + d_.*csc[e_. + f_.*x_]]), x_Symbol] := 1/a*Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x] - b/a* Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]* Sqrt[c + d*Csc[e + f*x]]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
+Int[Sqrt[ a_ + b_.*csc[e_. + f_.*x_]]/(c_ + d_.*csc[e_. + f_.*x_])^(3/2), x_Symbol] := 1/c*Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x] - d/c* Int[Csc[e + f*x]* Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x])^(3/2), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 - d^2, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := a^2*Cot[ e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]])* Subst[ Int[(a + b*x)^(m - 1/2)*(c + d*x)^n/(x*Sqrt[a - b*x]), x], x, Csc[e + f*x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && IntegerQ[m - 1/2]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := Int[(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/ Sin[e + f*x]^(m + n), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IntegerQ[n] && LeQ[-2, m + n, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := Sqrt[d + c*Sin[e + f*x]]* Sqrt[a + b*Csc[e + f*x]]/(Sqrt[b + a*Sin[e + f*x]]* Sqrt[c + d*Csc[e + f*x]])* Int[(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/ Sin[e + f*x]^(m + n), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m + 1/2] && IntegerQ[n + 1/2] && LeQ[-2, m + n, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := Sin[e + f*x]^(m + n)*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^ n/((b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)* Int[(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/ Sin[e + f*x]^Simplify[m + n], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n, 0] && Not[IntegerQ[2*m]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := Int[ExpandTrig[(a + b*csc[e + f*x])^m, (c + d*csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[n, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := Unintegrable[(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
+Int[(a_. + b_.*sec[e_. + f_.*x_])^m_.*(d_./sec[e_. + f_.*x_])^n_, x_Symbol] := d^m*Int[(b + a*Cos[e + f*x])^m*(d*Cos[e + f*x])^(n - m), x] /; FreeQ[{a, b, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]
+Int[(a_. + b_.*csc[e_. + f_.*x_])^m_.*(d_./csc[e_. + f_.*x_])^n_, x_Symbol] := d^m*Int[(b + a*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n - m), x] /; FreeQ[{a, b, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]
+Int[(a_. + b_.*sec[e_. + f_.*x_])^ m_.*(c_.*(d_.*sec[e_. + f_.*x_])^p_)^n_, x_Symbol] := c^IntPart[n]*(c*(d*Sec[e + f*x])^p)^ FracPart[n]/(d*Sec[e + f*x])^(p*FracPart[n])* Int[(a + b*Sec[e + f*x])^m*(d*Sec[e + f*x])^(n*p), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[n]]
+Int[(a_. + b_.*csc[e_. + f_.*x_])^ m_.*(c_.*(d_.*csc[e_. + f_.*x_])^p_)^n_, x_Symbol] := c^IntPart[n]*(c*(d*Csc[e + f*x])^p)^ FracPart[n]/(d*Csc[e + f*x])^(p*FracPart[n])* Int[(a + b*Cos[e + f*x])^m*(d*Cos[e + f*x])^(n*p), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[n]]
diff --git a/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m
new file mode 100755
index 0000000..95d07fc
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m	
@@ -0,0 +1,49 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n *)
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := b*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^n/(a*f*(2*m + 1)) /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && NeQ[2*m + 1, 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := b*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^n/(a*f*(2*m + 1)) + (m + n + 1)/(a*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[m + n + 1, 0] && NeQ[2*m + 1, 0] && Not[LtQ[n, 0]] && Not[IGtQ[n + 1/2, 0] && LtQ[n + 1/2, -(m + n)]]
+Int[csc[e_. + f_.*x_]* Sqrt[c_ + d_.*csc[e_. + f_.*x_]]/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := a*c*Log[1 + b/a*Csc[e + f*x]]* Cot[e + f*x]/(b*f*Sqrt[a + b*Csc[e + f*x]]* Sqrt[c + d*Csc[e + f*x]]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^m_.* Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol] := 2*a*c* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m/(b*f*(2*m + 1)*Sqrt[c + d*Csc[e + f*x]]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[m, -1/2]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := 2*a*c* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^(n - 1)/(b*f*(2*m + 1)) - d*(2*n - 1)/(b*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 0] && LtQ[m, -1/2]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := -d*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^(n - 1)/(f*(m + n)) + c*(2*n - 1)/(m + n)* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 0] && Not[LtQ[m, -1/2]] && Not[IGtQ[m - 1/2, 0] && LtQ[m, n]]
+Int[csc[e_. + f_.*x_]*(c_ + d_.*csc[e_. + f_.*x_])^n_./ Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := -2*d*Cot[ e + f*x]*(c + d*Csc[e + f*x])^(n - 1)/(f*(2*n - 1)* Sqrt[a + b*Csc[e + f*x]]) + 2*c*(2*n - 1)/(2*n - 1)* Int[Csc[e + f*x]*(c + d*Csc[e + f*x])^(n - 1)/ Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := 2*a*c* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^(n - 1)/(b*f*(2*m + 1)) - d*(2*n - 1)/(b*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0] && LtQ[m, -1/2] && IntegerQ[2*m]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := (-a*c)^m* Int[ExpandTrig[ csc[e + f*x]*cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n] && GeQ[n - m, 0] && GtQ[m*n, 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^m_, x_Symbol] := (-a*c)^(m + 1/2)* Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])* Int[Csc[e + f*x]*Cot[e + f*x]^(2*m), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m + 1/2]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := b*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^n/(a*f*(2*m + 1)) + (m + n + 1)/(a*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (ILtQ[m, 0] && ILtQ[n - 1/2, 0] || ILtQ[m - 1/2, 0] && ILtQ[n - 1/2, 0] && LtQ[m, n])
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := a*c*Cot[ e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])* Subst[Int[(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2), x], x, Csc[e + f*x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol] := (-a*c)^m* Int[ExpandTrig[(g*csc[e + f*x])^p* cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n] && GeQ[n - m, 0] && GtQ[m*n, 0]
+Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^m_, x_Symbol] := (-a*c)^(m + 1/2)* Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])* Int[(g*Csc[e + f*x])^p*Cot[e + f*x]^(2*m), x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m + 1/2]
+Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := a*c*g* Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]* Sqrt[c + d*Csc[e + f*x]])* Subst[ Int[(g*x)^(p - 1)*(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2), x], x, Csc[e + f*x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[Sqrt[g_.*csc[e_. + f_.*x_]]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := -2*b*g/f* Subst[Int[1/(b*c + a*d - c*g*x^2), x], x, b*Cot[e + f*x]/(Sqrt[g*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]])] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[Sqrt[g_.*csc[e_. + f_.*x_]]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := a/c*Int[Sqrt[g*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x] + (b*c - a*d)/(c*g)* Int[(g*Csc[e + f*x])^(3/2)/(Sqrt[ a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := -2*b/f* Subst[Int[1/(b*c + a*d + d*x^2), x], x, b*Cot[e + f*x]/Sqrt[a + b*Csc[e + f*x]]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := -Sqrt[a + b*Csc[e + f*x]]* Sqrt[c/(c + d*Csc[e + f*x])]/(d*f* Sqrt[c*d*(a + b*Csc[e + f*x])/((b*c + a*d)*(c + d*Csc[e + f*x]))])* EllipticE[ ArcSin[c* Cot[e + f*x]/(c + d*Csc[e + f*x])], -(b*c - a*d)/(b*c + a*d)] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
+Int[csc[e_. + f_.*x_]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := b/d*Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] - (b*c - a*d)/d* Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(g_.*csc[e_. + f_.*x_])^(3/2)* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := g/d*Int[Sqrt[g*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]], x] - c*g/d* Int[Sqrt[g*Csc[e + f*x]]* Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[(g_.*csc[e_. + f_.*x_])^(3/2)* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol] := b/d*Int[(g*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x] - (b*c - a*d)/d* Int[(g*Csc[e + f*x])^(3/2)/(Sqrt[ a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]/(Sqrt[ a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])), x_Symbol] := b/(b*c - a*d)*Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] - d/(b*c - a*d)* Int[Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
+Int[csc[e_. + f_.*x_]/(Sqrt[ a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])), x_Symbol] := -2*Cot[ e + f*x]/(f*(c + d)*Sqrt[a + b*Csc[e + f*x]]* Sqrt[-Cot[e + f*x]^2])*Sqrt[(a + b*Csc[e + f*x])/(a + b)]* EllipticPi[2*d/(c + d), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], 2*b/(a + b)] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(g_.*csc[e_. + f_.*x_])^(3/2)/(Sqrt[ a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])), x_Symbol] := -a*g/(b*c - a*d)* Int[Sqrt[g*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x] + c*g/(b*c - a*d)* Int[Sqrt[g*Csc[e + f*x]]* Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[(g_.*csc[e_. + f_.*x_])^(3/2)/(Sqrt[ a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])), x_Symbol] := g*Sqrt[g*Csc[e + f*x]]* Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]* Int[1/(Sqrt[b + a*Sin[e + f*x]]*(d + c*Sin[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]^2/(Sqrt[ a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])), x_Symbol] := -a/(b*c - a*d)*Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] + c/(b*c - a*d)* Int[Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
+Int[csc[e_. + f_.*x_]^2/(Sqrt[ a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])), x_Symbol] := 1/d*Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] - c/d* Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(g_.*csc[e_. + f_.*x_])^(5/2)/(Sqrt[ a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])), x_Symbol] := -c^2*g^2/(d*(b*c - a*d))* Int[Sqrt[g*Csc[e + f*x]]* Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x]), x] + g^2/(d*(b*c - a*d))* Int[Sqrt[g*Csc[e + f*x]]*(a*c + (b*c - a*d)*Csc[e + f*x])/ Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
+Int[(g_.*csc[e_. + f_.*x_])^(5/2)/(Sqrt[ a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])), x_Symbol] := g/d*Int[(g*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x] - c*g/d* Int[(g*Csc[e + f*x])^(3/2)/(Sqrt[ a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol] := -2*b/f* Subst[Int[1/(1 - b*d*x^2), x], x, Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]* Sqrt[c + d*Csc[e + f*x]])] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[csc[e_. + f_.*x_]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol] := -(b*c - a*d)/d* Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]* Sqrt[c + d*Csc[e + f*x]]), x] + b/d* Int[Csc[e + f*x]* Sqrt[c + d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]
+Int[csc[e_. + f_.*x_]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol] := -2*(a + b*Csc[e + f*x])/(d*f*Sqrt[(a + b)/(c + d)]*Cot[e + f*x])* Sqrt[-(b*c - a*d)*(1 - Csc[e + f*x])/((c + d)*(a + b*Csc[e + f*x]))]* Sqrt[(b*c - a*d)*(1 + Csc[e + f*x])/((c - d)*(a + b*Csc[e + f*x]))]* EllipticPi[b*(c + d)/(d*(a + b)), ArcSin[Sqrt[(a + b)/(c + d)]* Sqrt[c + d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]], (a - b)*(c + d)/((a + b)*(c - d))] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[csc[e_. + f_.*x_]/(Sqrt[a_ + b_.*csc[e_. + f_.*x_]]* Sqrt[c_ + d_.*csc[e_. + f_.*x_]]), x_Symbol] := -2*a/(b*f)* Subst[Int[1/(2 + (a*c - b*d)*x^2), x], x, Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]* Sqrt[c + d*Csc[e + f*x]])] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[csc[e_. + f_.*x_]/(Sqrt[a_ + b_.*csc[e_. + f_.*x_]]* Sqrt[c_ + d_.*csc[e_. + f_.*x_]]), x_Symbol] := -2*(c + d*Csc[e + f*x])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]* Cot[e + f*x])* Sqrt[(b*c - a*d)*(1 - Csc[e + f*x])/((a + b)*(c + d*Csc[e + f*x]))]* Sqrt[-(b*c - a*d)*(1 + Csc[e + f*x])/((a - b)*(c + d*Csc[e + f*x]))]* EllipticF[ ArcSin[Rt[(c + d)/(a + b), 2]*(Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]])], (a + b)*(c - d)/((a - b)*(c + d))] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[csc[e_. + f_.*x_]^2/(Sqrt[a_ + b_.*csc[e_. + f_.*x_]]* Sqrt[c_ + d_.*csc[e_. + f_.*x_]]), x_Symbol] := -a/b*Int[ Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), x] + 1/b* Int[Csc[e + f*x]* Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
+Int[csc[e_. + f_.*x_]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/(c_ + d_.*csc[e_. + f_.*x_])^(3/2), x_Symbol] := (a - b)/(c - d)* Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]* Sqrt[c + d*Csc[e + f*x]]), x] + (b*c - a*d)/(c - d)* Int[Csc[e + f*x]*(1 + Csc[e + f*x])/(Sqrt[ a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(3/2)), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
+Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := a^2*g* Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]* Sqrt[a - b*Csc[e + f*x]])* Subst[ Int[(g*x)^(p - 1)*(a + b*x)^(m - 1/2)*(c + d*x)^n/Sqrt[a - b*x], x], x, Csc[e + f*x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && (EqQ[p, 1] || IntegerQ[m - 1/2])
+Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := 1/g^(m + n)* Int[(g*Csc[e + f*x])^(m + n + p)*(b + a*Sin[e + f*x])^ m*(d + c*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IntegerQ[n]
+Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := (g*Csc[e + f*x])^(m + p)*(c + d*Csc[e + f*x])^ n/(g^m*(d + c*Sin[e + f*x])^n)* Int[(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + p, 0] && IntegerQ[m]
+Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := (g*Csc[e + f*x])^p*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^ n/((b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)* Int[(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + p, 0] && Not[IntegerQ[m]]
+Int[csc[e_. + f_.*x_]^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := Sqrt[d + c*Sin[e + f*x]]* Sqrt[a + b*Csc[e + f*x]]/(Sqrt[b + a*Sin[e + f*x]]* Sqrt[c + d*Csc[e + f*x]])* Int[(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/ Sin[e + f*x]^(m + n + p), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m - 1/2] && IntegerQ[n - 1/2] && IntegerQ[p] && LeQ[-2, m + n + p, -1]
+Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := Int[ExpandTrig[(g*csc[e + f*x])^p*(a + b*csc[e + f*x])^ m*(c + d*csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && (IntegersQ[m, n] || IntegersQ[m, p] || IntegersQ[n, p])
+Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_. + b_.*csc[e_. + f_.*x_])^ m_*(c_. + d_.*csc[e_. + f_.*x_])^n_, x_Symbol] := Unintegrable[(g*Csc[e + f*x])^p*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
+Int[sec[e_. + f_.*x_]*(A_ + B_.*sec[e_. + f_.*x_])/(Sqrt[ a_ + b_.*sec[e_. + f_.*x_]]*(c_ + d_.*sec[e_. + f_.*x_])^(3/ 2)), x_Symbol] := 2*A*(1 + Sec[e + f*x])* Sqrt[(b*c - a*d)*(1 - Sec[e + f*x])/((a + b)*(c + d*Sec[e + f*x]))]/ (f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Tan[e + f*x]* Sqrt[-(b*c - a*d)*(1 + Sec[e + f*x])/((a - b)*(c + d*Sec[e + f*x]))])* EllipticE[ ArcSin[Rt[(c + d)/(a + b), 2]* Sqrt[a + b*Sec[e + f*x]]/Sqrt[c + d*Sec[e + f*x]]], (a + b)*(c - d)/((a - b)*(c + d))] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B]
+Int[csc[e_. + f_.*x_]*(A_ + B_.*csc[e_. + f_.*x_])/(Sqrt[ a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])^(3/ 2)), x_Symbol] := -2*A*(1 + Csc[e + f*x])* Sqrt[(b*c - a*d)*(1 - Csc[e + f*x])/((a + b)*(c + d*Csc[e + f*x]))]/ (f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Cot[e + f*x]* Sqrt[-(b*c - a*d)*(1 + Csc[e + f*x])/((a - b)*(c + d*Csc[e + f*x]))])* EllipticE[ ArcSin[Rt[(c + d)/(a + b), 2]* Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]]], (a + b)*(c - d)/((a - b)*(c + d))] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B]
diff --git a/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m
new file mode 100755
index 0000000..73ba9ab
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m	
@@ -0,0 +1,49 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec) *)
+Int[(a_ + b_.*csc[e_. + f_.*x_])*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*n) + 1/(d*n)* Int[(d*Csc[e + f*x])^(n + 1)* Simp[n*(B*a + A*b) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]
+Int[(a_ + b_.*csc[e_. + f_.*x_])*(d_.*csc[e_. + f_.*x_])^ n_.*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -b*B*Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*(n + 1)) + 1/(n + 1)* Int[(d*Csc[e + f*x])^n* Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && Not[LeQ[n, -1]]
+Int[csc[e_. + f_.*x_]*(A_ + B_.*csc[e_. + f_.*x_])/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := B/b*Int[Csc[e + f*x], x] + (A*b - a*B)/b* Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -B*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) /; FreeQ[{a, b, A, B, e, f, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[a*B*m + A*b*(m + 1), 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := (A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(a*f*(2*m + 1)) + (a*B*m + A*b*(m + 1))/(a*b*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x] /; FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[a*B*m + A*b*(m + 1), 0] && LtQ[m, -1/2]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -B*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + (a*B*m + A*b*(m + 1))/(b*(m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x] /; FreeQ[{a, b, A, B, e, f, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[a*B*m + A*b*(m + 1), 0] && Not[LtQ[m, -1/2]]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -B*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + 1/(m + 1)* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)* Simp[b*B*m + a*A*(m + 1) + (a*B*m + A*b*(m + 1))*Csc[e + f*x], x], x] /; FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -(A*b - a*B)* Cot[e + f* x]*(a + b*Csc[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[(a*A - b*B)*(m + 1) - (A*b - a*B)*(m + 2)*Csc[e + f*x], x], x] /; FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
+Int[csc[e_. + f_.*x_]*(A_ + B_.*csc[e_. + f_.*x_])/ Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := -2*(A*b - a*B)*Rt[a + b*B/A, 2]*Sqrt[b*(1 - Csc[e + f*x])/(a + b)]* Sqrt[-b*(1 + Csc[e + f*x])/(a - b)]/(b^2*f*Cot[e + f*x])* EllipticE[ ArcSin[Sqrt[a + b*Csc[e + f*x]]/ Rt[a + b*B/A, 2]], (a*A + b*B)/(a*A - b*B)] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
+Int[csc[e_. + f_.*x_]*(A_ + B_.*csc[e_. + f_.*x_])/ Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := (A - B)*Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] + B*Int[Csc[e + f*x]*(1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && NeQ[A^2 - B^2, 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := 2*Sqrt[2]*A*(a + b*Csc[e + f*x])^m*(A - B*Csc[e + f*x])* Sqrt[(A + B*Csc[e + f*x])/A]/(B*f* Cot[e + f*x]*(A*(a + b*Csc[e + f*x])/(a*A + b*B))^m)* AppellF1[1/2, -(1/2), -m, 3/2, (A - B*Csc[e + f*x])/(2*A), (b*(A - B*Csc[e + f*x]))/(A*b + a*B)] /; FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0] && Not[IntegerQ[2*m]]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := (A*b - a*B)/b*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x] + B/b*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x] /; FreeQ[{a, b, A, B, e, f, m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(b*f*(2*m + 1)) + 1/(b^2*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[A*b*m - a*B*m + b*B*(2*m + 1)*Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]
+Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := a*(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 - b^2)) - 1/(b*(m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[b*(A*b - a*B)*(m + 1) - (a*A*b*(m + 2) - B*(a^2 + b^2*(m + 1)))*Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
+Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -B*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m* Simp[b*B*(m + 1) + (A*b*(m + 2) - a*B)*Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && Not[LtQ[m, -1]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n) /; FreeQ[{a, b, d, e, f, A, B, m, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && EqQ[a*A*m - b*B*n, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(b*f*(2*m + 1)) + (a*A*m + b*B*(m + 1))/(a^2*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n, x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && LeQ[m, -1]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n) - (a*A*m - b*B*n)/(b*d*n)* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1), x] /; FreeQ[{a, b, d, e, f, A, B, m, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && Not[LeQ[m, -1]]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -2*b*B* Cot[e + f*x]*(d*Csc[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]) /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[A*b*(2*n + 1) + 2*a*B*n, 0]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := A*b^2* Cot[e + f*x]*(d*Csc[e + f*x])^n/(a*f*n*Sqrt[a + b*Csc[e + f*x]]) + (A*b*(2*n + 1) + 2*a*B*n)/(2*a*d*n)* Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n + 1), x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n, 0] && LtQ[n, 0]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -2*b*B* Cot[e + f*x]*(d*Csc[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]) + (A*b*(2*n + 1) + 2*a*B*n)/(b*(2*n + 1))* Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^n, x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n, 0] && Not[LtQ[n, 0]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := a*A*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^ n/(f*n) - b/(a*d*n)* Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)* Simp[a*A*(m - n - 1) - b*B*n - (a*B*n + A*b*(m + n))*Csc[e + f*x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && LtQ[n, -1]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -b*B*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^ n/(f*(m + n)) + 1/(d*(m + n))* Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n* Simp[a*A*d*(m + n) + B*(b*d*n) + (A*b*d*(m + n) + a*B*d*(2*m + n - 1))* Csc[e + f*x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && Not[LtQ[n, -1]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := d*(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 1)/(a*f*(2*m + 1)) - 1/(a*b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)* Simp[A*(a*d*(n - 1)) - B*(b*d*(n - 1)) - d*(a*B*(m - n + 1) + A*b*(m + n))*Csc[e + f*x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2] && GtQ[n, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(b*f*(2*m + 1)) - 1/(a^2*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n* Simp[b*B*n - a*A*(2*m + n + 1) + (A*b - a*B)*(m + n + 1)*Csc[e + f*x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2] && Not[GtQ[n, 0]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -B*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 1)/(f*(m + n)) + d/(b*(m + n))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)* Simp[b*B*(n - 1) + (A*b*(m + n) + a*B*m)*Csc[e + f*x], x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[n, 1]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n) - 1/(b*d*n)* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)* Simp[a*A*m - b*B*n - A*b*(m + n + 1)*Csc[e + f*x], x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := (A*b - a*B)/b*Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n, x] + B/b*Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n, x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^2*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := a^2*A*Cos[e + f*x]*(d*Csc[e + f*x])^(n + 1)/(d*f*n) + 1/(d*n)* Int[(d*Csc[e + f*x])^(n + 1)*(a*(2*A*b + a*B)* n + (2*a*b*B*n + A*(b^2*n + a^2*(n + 1)))*Csc[e + f*x] + b^2*B*n*Csc[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := a*A*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^ n/(f*n) + 1/(d*n)*Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)* Simp[a*(a*B*n - A*b*(m - n - 1)) + (2*a*b*B*n + A*(b^2*n + a^2*(1 + n)))*Csc[e + f*x] + b*(b*B*n + a*A*(m + n))*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LeQ[n, -1]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -b*B*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^ n/(f*(m + n)) + 1/(m + n)*Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n* Simp[a^2*A*(m + n) + a*b*B*n + (a*(2*A*b + a*B)*(m + n) + b^2*B*(m + n - 1))* Csc[e + f*x] + b*(A*b*(m + n) + a*B*(2*m + n - 1))*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && Not[IGtQ[n, 1] && Not[IntegerQ[m]]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -d*(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)* Simp[d*(n - 1)*(A*b - a*B) + d*(a*A - b*B)*(m + 1)*Csc[e + f*x] - d*(A*b - a*B)*(m + n + 1)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && LtQ[0, n, 1]
+Int[csc[e_. + f_.*x_]^3*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -a^2*(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b^2* f*(m + 1)*(a^2 - b^2)) + 1/(b^2*(m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[a*b*(A*b - a*B)*(m + 1) - (A*b - a*B)*(a^2 + b^2*(m + 1))* Csc[e + f*x] + b*B*(m + 1)*(a^2 - b^2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := a*d^2*(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)/(b*f*(m + 1)*(a^2 - b^2)) - d/(b*(m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)* Simp[a*d*(A*b - a*B)*(n - 2) + b*d*(A*b - a*B)*(m + 1)* Csc[e + f*x] - (a*A*b*d*(m + n) - d*B*(a^2*(n - 1) + b^2*(m + 1)))*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 1]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := b*(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*(m + 1)*(a^2 - b^2)) + 1/(a*(m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n* Simp[A*(a^2*(m + 1) - b^2*(m + n + 1)) + a*b*B*n - a*(A*b - a*B)*(m + 1)*Csc[e + f*x] + b*(A*b - a*B)*(m + n + 2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && Not[ILtQ[m + 1/2, 0] && ILtQ[n, 0]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -B*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 1)/(f*(m + n)) + d/(m + n)* Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n - 1)* Simp[a*B*(n - 1) + (b*B*(m + n - 1) + a*A*(m + n))* Csc[e + f*x] + (a*B*m + A*b*(m + n))*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[0, m, 1] && GtQ[n, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n) - 1/(d*n)*Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)* Simp[A*b*m - a*B*n - (b*B*n + a*A*(n + 1))*Csc[e + f*x] - A*b*(m + n + 1)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[0, m, 1] && LeQ[n, -1]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := -B*d^2* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)/(b*f*(m + n)) + d^2/(b*(m + n))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 2)* Simp[a*B*(n - 2) + B*b*(m + n - 1)*Csc[e + f*x] + (A*b*(m + n) - a*B*(n - 1))* Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && NeQ[m + n, 0] && Not[IGtQ[m, 1]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := A*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*n) + 1/(a*d*n)*Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)* Simp[a*B*n - A*b*(m + n + 1) + A*a*(n + 1)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
+Int[(A_ + B_.*csc[e_. + f_.*x_])/(Sqrt[d_.*csc[e_. + f_.*x_]]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]), x_Symbol] := A/a*Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x] - (A*b - a*B)/(a*d)* Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
+Int[Sqrt[d_.*csc[e_. + f_.*x_]]*(A_ + B_.*csc[e_. + f_.*x_])/ Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := A*Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x] + B/d*Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
+Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(A_ + B_.*csc[e_. + f_.*x_])/ Sqrt[d_.*csc[e_. + f_.*x_]], x_Symbol] := B/d*Int[Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]], x] + A*Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
+Int[(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_])/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := A/a*Int[(d*Csc[e + f*x])^n, x] - (A*b - a*B)/(a*d)* Int[(d*Csc[e + f*x])^(n + 1)/(a + b*Csc[e + f*x]), x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_.*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol] := Unintegrable[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^ n*(A + B*Csc[e + f*x]), x] /; FreeQ[{a, b, d, e, f, A, B, m, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
+(* Int[(a_+b_.*csc[e_.+f_.*x_])^m_.*(c_+d_.*csc[e_.+f_.*x_])^n_.*(A_.+ B_.*csc[e_.+f_.*x_])^p_.,x_Symbol] := (-a*c)^m*Int[Cot[e+f*x]^(2*m)*(c+d*Csc[e+f*x])^(n-m)*(A+B*Csc[e+f*x] )^p,x] /; FreeQ[{a,b,c,d,e,f,A,B,n,p},x] && EqQ[b*c+a*d,0] && EqQ[a^2-b^2,0] &&  IntegerQ[m] && Not[IntegerQ[n] && (LtQ[m,0] && GtQ[n,0] || LtQ[0,n,m] ||  LtQ[m,n,0])] *)
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^ n_.*(A_. + B_.*csc[e_. + f_.*x_])^p_., x_Symbol] := (-a*c)^m* Int[Cos[e + f*x]^(2*m)*(d + c*Sin[e + f*x])^(n - m)*(B + A*Sin[e + f*x])^p/Sin[e + f*x]^(m + n + p), x] /; FreeQ[{a, b, c, d, e, f, A, B, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p]
diff --git a/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m
new file mode 100755
index 0000000..9396261
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m	
@@ -0,0 +1,34 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.5.4.1 (a+b sec)^m (A+B sec+C sec^2) *)
+Int[(a_ + b_.*csc[e_. + f_.*x_])^ m_.*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := 1/b^2* Int[(a + b*Csc[e + f*x])^(m + 1)* Simp[b*B - a*C + b*C*Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := C/b^2* Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[-a + b*Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A*b^2 + a^2*C, 0]
+Int[(b_.*csc[e_. + f_.*x_])^m_.*(A_ + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := A*Cot[e + f*x]*(b*Csc[e + f*x])^m/(f*m) /; FreeQ[{b, e, f, A, C, m}, x] && EqQ[C*m + A*(m + 1), 0]
+Int[csc[e_. + f_.*x_]^m_.*(A_ + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := Int[(C + A*Sin[e + f*x]^2)/Sin[e + f*x]^(m + 2), x] /; FreeQ[{e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && ILtQ[(m + 1)/2, 0]
+Int[(b_.*csc[e_. + f_.*x_])^m_.*(A_ + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := A*Cot[e + f*x]*(b*Csc[e + f*x])^m/(f*m) + (C*m + A*(m + 1))/(b^2*m)*Int[(b*Csc[e + f*x])^(m + 2), x] /; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]
+Int[(b_.*csc[e_. + f_.*x_])^m_.*(A_ + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C*Cot[e + f*x]*(b*Csc[e + f*x])^m/(f*(m + 1)) + (C*m + A*(m + 1))/(m + 1)*Int[(b*Csc[e + f*x])^m, x] /; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] && Not[LeQ[m, -1]]
+Int[(b_.*csc[e_. + f_.*x_])^ m_.*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := B/b*Int[(b*Csc[e + f*x])^(m + 1), x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]
+Int[(a_ + b_.*csc[e_. + f_.*x_])*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -b*C*Csc[e + f*x]*Cot[e + f*x]/(2*f) + 1/2* Int[Simp[ 2*A*a + (2*B*a + b*(2*A + C))*Csc[e + f*x] + 2*(a*C + B*b)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, A, B, C}, x]
+Int[(a_ + b_.*csc[e_. + f_.*x_])*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -b*C*Csc[e + f*x]*Cot[e + f*x]/(2*f) + 1/2* Int[Simp[2*A*a + b*(2*A + C)*Csc[e + f*x] + 2*a*C*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, A, C}, x]
+Int[(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2)/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := C/b*Int[Csc[e + f*x], x] + 1/b*Int[(A*b + (b*B - a*C)*Csc[e + f*x])/(a + b*Csc[e + f*x]), x] /; FreeQ[{a, b, e, f, A, B, C}, x]
+Int[(A_. + C_.*csc[e_. + f_.*x_]^2)/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol] := C/b*Int[Csc[e + f*x], x] + 1/b*Int[(A*b - a*C*Csc[e + f*x])/(a + b*Csc[e + f*x]), x] /; FreeQ[{a, b, e, f, A, C}, x]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -(a*A - b*B + a*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(a*f*(2*m + 1)) + 1/(a*b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)* Simp[A*b*(2*m + 1) + (b*B*(m + 1) - a*(A*(m + 1) - C*m))* Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -a*(A + C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(a*f*(2*m + 1)) + 1/(a*b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)* Simp[A*b*(2*m + 1) - a*(A*(m + 1) - C*m)*Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^ m_.*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + 1/(b*(m + 1))* Int[(a + b*Csc[e + f*x])^m* Simp[A*b*(m + 1) + (a*C*m + b*B*(m + 1))*Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + 1/(b*(m + 1))* Int[(a + b*Csc[e + f*x])^m* Simp[A*b*(m + 1) + a*C*m*Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^ m_.*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + 1/(m + 1)*Int[(a + b*Csc[e + f*x])^(m - 1)* Simp[a*A*(m + 1) + ((A*b + a*B)*(m + 1) + b*C*m)* Csc[e + f*x] + (b*B*(m + 1) + a*C*m)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && IGtQ[2*m, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + 1/(m + 1)* Int[(a + b*Csc[e + f*x])^(m - 1)* Simp[a*A*(m + 1) + (A*b*(m + 1) + b*C*m)*Csc[e + f*x] + a*C*m*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && IGtQ[2*m, 0]
+Int[(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2)/ Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := Int[(A + (B - C)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + C*Int[Csc[e + f*x]*(1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
+Int[(A_. + C_.*csc[e_. + f_.*x_]^2)/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol] := Int[(A - C*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + C*Int[Csc[e + f*x]*(1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := (A*b^2 - a*b*B + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(a* f*(m + 1)*(a^2 - b^2)) + 1/(a*(m + 1)*(a^2 - b^2))*Int[(a + b*Csc[e + f*x])^(m + 1)* Simp[A*(a^2 - b^2)*(m + 1) - a*(A*b - a*B + b*C)*(m + 1)* Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)* Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := (A*b^2 + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(a* f*(m + 1)*(a^2 - b^2)) + 1/(a*(m + 1)*(a^2 - b^2))*Int[(a + b*Csc[e + f*x])^(m + 1)* Simp[A*(a^2 - b^2)*(m + 1) - a*b*(A + C)*(m + 1)*Csc[e + f*x] + (A*b^2 + a^2*C)*(m + 2)* Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m] && LtQ[m, -1]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := 1/b*Int[(a + b*Csc[e + f*x])^m*(A*b + (b*B - a*C)*Csc[e + f*x]), x] + C/b*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && Not[IntegerQ[2*m]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := 1/b*Int[(a + b*Csc[e + f*x])^m*(A*b - a*C*Csc[e + f*x]), x] + C/b*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x] /; FreeQ[{a, b, e, f, A, C, m}, x] && NeQ[a^2 - b^2, 0] && Not[IntegerQ[2*m]]
+Int[(b_.*cos[e_. + f_.*x_])^ m_*(A_. + B_.*sec[e_. + f_.*x_] + C_.*sec[e_. + f_.*x_]^2), x_Symbol] := b^2*Int[(b*Cos[e + f*x])^(m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x] && Not[IntegerQ[m]]
+Int[(b_.*sin[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := b^2*Int[(b*Sin[e + f*x])^(m - 2)*(C + B*Sin[e + f*x] + A*Sin[e + f*x]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x] && Not[IntegerQ[m]]
+Int[(b_.*cos[e_. + f_.*x_])^m_*(A_. + C_.*sec[e_. + f_.*x_]^2), x_Symbol] := b^2*Int[(b*Cos[e + f*x])^(m - 2)*(C + A*Cos[e + f*x]^2), x] /; FreeQ[{b, e, f, A, C, m}, x] && Not[IntegerQ[m]]
+Int[(b_.*sin[e_. + f_.*x_])^m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := b^2*Int[(b*Sin[e + f*x])^(m - 2)*(C + A*Sin[e + f*x]^2), x] /; FreeQ[{b, e, f, A, C, m}, x] && Not[IntegerQ[m]]
+Int[(a_.*(b_.*sec[e_. + f_.*x_])^p_)^ m_*(A_. + B_.*sec[e_. + f_.*x_] + C_.*sec[e_. + f_.*x_]^2), x_Symbol] := a^IntPart[m]*(a*(b*Sec[e + f*x])^p)^ FracPart[m]/(b*Sec[e + f*x])^(p*FracPart[m])* Int[(b*Sec[e + f*x])^(m*p)*(A + B*Sec[e + f*x] + C*Sec[e + f*x]^2), x] /; FreeQ[{a, b, e, f, A, B, C, m, p}, x] && Not[IntegerQ[m]]
+Int[(a_.*(b_.*csc[e_. + f_.*x_])^p_)^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := a^IntPart[m]*(a*(b*Csc[e + f*x])^p)^ FracPart[m]/(b*Csc[e + f*x])^(p*FracPart[m])* Int[(b*Csc[e + f*x])^(m*p)*(A + B*Csc[e + f*x] + C*Csc[e + f*x]^2), x] /; FreeQ[{a, b, e, f, A, B, C, m, p}, x] && Not[IntegerQ[m]]
+Int[(a_.*(b_.*sec[e_. + f_.*x_])^p_)^ m_*(A_. + C_.*sec[e_. + f_.*x_]^2), x_Symbol] := a^IntPart[m]*(a*(b*Sec[e + f*x])^p)^ FracPart[m]/(b*Sec[e + f*x])^(p*FracPart[m])* Int[(b*Sec[e + f*x])^(m*p)*(A + C*Sec[e + f*x]^2), x] /; FreeQ[{a, b, e, f, A, C, m, p}, x] && Not[IntegerQ[m]]
+Int[(a_.*(b_.*csc[e_. + f_.*x_])^p_)^ m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := a^IntPart[m]*(a*(b*Csc[e + f*x])^p)^ FracPart[m]/(b*Csc[e + f*x])^(p*FracPart[m])* Int[(b*Csc[e + f*x])^(m*p)*(A + C*Csc[e + f*x]^2), x] /; FreeQ[{a, b, e, f, A, C, m, p}, x] && Not[IntegerQ[m]]
diff --git a/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m
new file mode 100755
index 0000000..bc44059
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m	
@@ -0,0 +1,51 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2) *)
+Int[(a_. + b_.*csc[e_. + f_.*x_])^m_.*(c_. + d_.*csc[e_. + f_.*x_])^ n_.*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := 1/b^2* Int[(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^ n*(b*B - a*C + b*C*Csc[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
+Int[(a_. + b_.*csc[e_. + f_.*x_])^m_.*(c_. + d_.*csc[e_. + f_.*x_])^ n_.*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C/b^2* Int[(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^ n*(a - b*Csc[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A*b^2 + a^2*C, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*n) + 1/(d*n)* Int[(d*Csc[e + f*x])^(n + 1)* Simp[n*(B*a + A*b) + (n*(a*C + B*b) + A*a*(n + 1))* Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]
+Int[(a_ + b_.*csc[e_. + f_.*x_])*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*n) + 1/(d*n)* Int[(d*Csc[e + f*x])^(n + 1)* Simp[A*b*n + a*(C*n + A*(n + 1))*Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && LtQ[n, -1]
+Int[(d_.*csc[e_. + f_.*x_])^ n_.*(a_ + b_.*csc[e_. + f_.*x_])*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -b*C*Csc[e + f*x]*Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*(n + 2)) + 1/(n + 2)* Int[(d*Csc[e + f*x])^n* Simp[A*a*(n + 2) + (B*a*(n + 2) + b*(C*(n + 1) + A*(n + 2)))* Csc[e + f*x] + (a*C + B*b)*(n + 2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && Not[LtQ[n, -1]]
+Int[(d_.*csc[e_. + f_.*x_])^ n_.*(a_ + b_.*csc[e_. + f_.*x_])*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -b*C*Csc[e + f*x]*Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*(n + 2)) + 1/(n + 2)* Int[(d*Csc[e + f*x])^n* Simp[A*a*(n + 2) + b*(C*(n + 1) + A*(n + 2))*Csc[e + f*x] + a*C*(n + 2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && Not[LtQ[n, -1]]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -(a*A - b*B + a*C)*Cot[e + f*x]* Csc[e + f*x]*(a + b*Csc[e + f*x])^m/(a*f*(2*m + 1)) - 1/(a*b*(2*m + 1))*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[a*B - b*C - 2*A*b*(m + 1) - (b*B*(m + 2) - a*(A*(m + 2) - C*(m - 1)))* Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -(A + C)*Cot[e + f*x]* Csc[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(2*m + 1)) - 1/(a*b*(2*m + 1))*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[-b*C - 2*A*b*(m + 1) + a*(A*(m + 2) - C*(m - 1))*Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -(A*b^2 - a*b*B + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 - b^2)) + 1/(b*(m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -(A*b^2 + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 - b^2)) + 1/(b*(m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[a*b*(A + C)*(m + 1) - (A*b^2 + a^2*C + b*(A*b + b*C)*(m + 1))*Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m* Simp[b*A*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && Not[LtQ[m, -1]]
+Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m* Simp[b*A*(m + 2) + b*C*(m + 1) - a*C*Csc[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && Not[LtQ[m, -1]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -(a*A - b*B + a*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(a*f*(2*m + 1)) - 1/(a*b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n* Simp[a*B*n - b*C*n - A*b*(2*m + n + 1) - (b*B*(m + n + 1) - a*(A*(m + n + 1) - C*(m - n)))*Csc[e + f*x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -a*(A + C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(a*f*(2*m + 1)) + 1/(a*b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n* Simp[b*C*n + A*b*(2*m + n + 1) - (a*(A*(m + n + 1) - C*(m - n)))* Csc[e + f*x], x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n) - 1/(b*d*n)* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)* Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]] && (LtQ[n, -1/2] || EqQ[m + n + 1, 0])
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n) - 1/(b*d*n)* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)* Simp[a*A*m - b*(A*(m + n + 1) + C*n)*Csc[e + f*x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]] && (LtQ[n, -1/2] || EqQ[m + n + 1, 0])
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(f*(m + n + 1)) + 1/(b*(m + n + 1))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n* Simp[A*b*(m + n + 1) + b*C*n + (a*C*m + b*B*(m + n + 1))*Csc[e + f*x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]] && Not[LtQ[n, -1/2]] && NeQ[m + n + 1, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(f*(m + n + 1)) + 1/(b*(m + n + 1))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n* Simp[A*b*(m + n + 1) + b*C*n + a*C*m*Csc[e + f*x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]] && Not[LtQ[n, -1/2]] && NeQ[m + n + 1, 0]
+Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := a*(A*b^2 - a*b*B + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b^2* f*(m + 1)*(a^2 - b^2)) - 1/(b^2*(m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[b*(m + 1)*(-a*(b*B - a*C) + A*b^2) + (b*B*(a^2 + b^2*(m + 1)) - a*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Csc[e + f*x] - b*C*(m + 1)*(a^2 - b^2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
+Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := a*(A*b^2 + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b^2* f*(m + 1)*(a^2 - b^2)) - 1/(b^2*(m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[b*(m + 1)*(a^2*C + A*b^2) - a*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)))*Csc[e + f*x] - b*C*(m + 1)*(a^2 - b^2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
+Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C*Csc[e + f*x]* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 3)) + 1/(b*(m + 3))*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m* Simp[a*C + b*(C*(m + 2) + A*(m + 3))* Csc[e + f*x] - (2*a*C - b*B*(m + 3))*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && Not[LtQ[m, -1]]
+Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C*Csc[e + f*x]* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 3)) + 1/(b*(m + 3))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m* Simp[a*C + b*(C*(m + 2) + A*(m + 3))*Csc[e + f*x] - 2*a*C*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && NeQ[a^2 - b^2, 0] && Not[LtQ[m, -1]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n) - 1/(d*n)*Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)* Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Csc[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n) - 1/(d*n)*Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)* Simp[A*b*m - a*(C*n + A*(n + 1))*Csc[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(f*(m + n + 1)) + 1/(m + n + 1)*Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n* Simp[a*A*(m + n + 1) + a*C*n + ((A*b + a*B)*(m + n + 1) + b*C*(m + n))* Csc[e + f*x] + (b*B*(m + n + 1) + a*C*m)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && Not[LeQ[n, -1]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(f*(m + n + 1)) + 1/(m + n + 1)*Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n* Simp[a*A*(m + n + 1) + a*C*n + b*(A*(m + n + 1) + C*(m + n))*Csc[e + f*x] + a*C*m*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && Not[LeQ[n, -1]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -d*(A*b^2 - a*b*B + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)/(b*f*(a^2 - b^2)*(m + 1)) + d/(b*(a^2 - b^2)*(m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)* Simp[A*b^2*(n - 1) - a*(b*B - a*C)*(n - 1) + b*(a*A - b*B + a*C)*(m + 1)*Csc[e + f*x] - (b*(A*b - a*B)*(m + n + 1) + C*(a^2*n + b^2*(m + 1)))* Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -d*(A*b^2 + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)/(b*f*(a^2 - b^2)*(m + 1)) + d/(b*(a^2 - b^2)*(m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)* Simp[A*b^2*(n - 1) + a^2*C*(n - 1) + a*b*(A + C)*(m + 1)* Csc[e + f*x] - (A*b^2*(m + n + 1) + C*(a^2*n + b^2*(m + 1)))* Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := (A*b^2 - a*b*B + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*(m + 1)*(a^2 - b^2)) + 1/(a*(m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n* Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && Not[ILtQ[m + 1/2, 0] && ILtQ[n, 0]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := (A*b^2 + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*(m + 1)*(a^2 - b^2)) + 1/(a*(m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n* Simp[a^2*(A + C)*(m + 1) - (A*b^2 + a^2*C)*(m + n + 1) - a*b*(A + C)*(m + 1)* Csc[e + f*x] + (A*b^2 + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && Not[ILtQ[m + 1/2, 0] && ILtQ[n, 0]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C*d*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1)) + d/(b*(m + n + 1))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)* Simp[a*C*(n - 1) + (A*b*(m + n + 1) + b*C*(m + n))* Csc[e + f*x] + (b*B*(m + n + 1) - a*C*n)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 0] (* && Not[IGtQ[m,0] && Not[IntegerQ[n]]] *)
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := -C*d*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1)) + d/(b*(m + n + 1))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)* Simp[a*C*(n - 1) + (A*b*(m + n + 1) + b*C*(m + n))* Csc[e + f*x] - a*C*n*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, C, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 0] (* && Not[IGtQ[m,0] && Not[IntegerQ[n]]] *)
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := A*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*n) + 1/(a*d*n)*Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)* Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := A*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*n) + 1/(a*d*n)*Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)* Simp[-A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x] /; FreeQ[{a, b, d, e, f, A, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
+Int[(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2)/(Sqrt[ d_.*csc[e_. + f_.*x_]]*(a_ + b_.*csc[e_. + f_.*x_])), x_Symbol] := (A*b^2 - a*b*B + a^2*C)/(a^2*d^2)* Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x] + 1/a^2* Int[(a*A - (A*b - a*B)*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
+Int[(A_. + C_.*csc[e_. + f_.*x_]^2)/(Sqrt[ d_.*csc[e_. + f_.*x_]]*(a_ + b_.*csc[e_. + f_.*x_])), x_Symbol] := (A*b^2 + a^2*C)/(a^2*d^2)* Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x] + 1/a^2*Int[(a*A - A*b*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0]
+Int[(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2)/(Sqrt[d_.*csc[e_. + f_.*x_]]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]), x_Symbol] := C/d^2*Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x] + Int[(A + B*Csc[e + f*x])/(Sqrt[d*Csc[e + f*x]]* Sqrt[a + b*Csc[e + f*x]]), x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
+Int[(A_. + C_.*csc[e_. + f_.*x_]^2)/(Sqrt[d_.*csc[e_. + f_.*x_]]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]), x_Symbol] := C/d^2*Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x] + A*Int[1/(Sqrt[d*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]]), x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(d_.*csc[e_. + f_.*x_])^ n_.*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := Unintegrable[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^ n*(A + B*Csc[e + f*x] + C*Csc[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, A, B, C, m, n}, x]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(d_.*csc[e_. + f_.*x_])^ n_.*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := Unintegrable[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^ n*(A + C*Csc[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, A, C, m, n}, x]
+Int[(a_ + b_.*sec[e_. + f_.*x_])^m_.*(d_.*cos[e_. + f_.*x_])^ n_*(A_. + B_.*sec[e_. + f_.*x_] + C_.*sec[e_. + f_.*x_]^2), x_Symbol] := d^(m + 2)* Int[(b + a*Cos[e + f*x])^ m*(d*Cos[e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := d^(m + 2)* Int[(b + a*Sin[e + f*x])^ m*(d*Sin[e + f*x])^(n - m - 2)*(C + B*Sin[e + f*x] + A*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]
+Int[(a_ + b_.*sec[e_. + f_.*x_])^m_.*(d_.*cos[e_. + f_.*x_])^ n_*(A_. + C_.*sec[e_. + f_.*x_]^2), x_Symbol] := d^(m + 2)* Int[(b + a*Cos[e + f*x])^ m*(d*Cos[e + f*x])^(n - m - 2)*(C + A*Cos[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(d_.*sin[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := d^(m + 2)* Int[(b + a*Sin[e + f*x])^ m*(d*Sin[e + f*x])^(n - m - 2)*(C + A*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]
+Int[(a_ + b_.*sec[e_. + f_.*x_])^m_.*(c_.*(d_.*sec[e_. + f_.*x_])^p_)^ n_*(A_. + B_.*sec[e_. + f_.*x_] + C_.*sec[e_. + f_.*x_]^2), x_Symbol] := c^IntPart[n]*(c*(d*Sec[e + f*x])^p)^ FracPart[n]/(d*Sec[e + f*x])^(p*FracPart[n])* Int[(a + b*Sec[e + f*x])^ m*(d*Sec[e + f*x])^(n*p)*(A + B*Sec[e + f*x] + C*Sec[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && Not[IntegerQ[n]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_.*(d_.*csc[e_. + f_.*x_])^p_)^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := c^IntPart[n]*(c*(d*Csc[e + f*x])^p)^ FracPart[n]/(d*Csc[e + f*x])^(p*FracPart[n])* Int[(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n*p)*(A + B*Csc[e + f*x] + C*Csc[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && Not[IntegerQ[n]]
+Int[(a_ + b_.*sec[e_. + f_.*x_])^m_.*(c_.*(d_.*sec[e_. + f_.*x_])^p_)^ n_*(A_. + C_.*sec[e_. + f_.*x_]^2), x_Symbol] := c^IntPart[n]*(c*(d*Sec[e + f*x])^p)^ FracPart[n]/(d*Sec[e + f*x])^(p*FracPart[n])* Int[(a + b*Sec[e + f*x])^ m*(d*Sec[e + f*x])^(n*p)*(A + C*Sec[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n, p}, x] && Not[IntegerQ[n]]
+Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_.*(d_.*csc[e_. + f_.*x_])^p_)^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol] := c^IntPart[n]*(c*(d*Csc[e + f*x])^p)^ FracPart[n]/(d*Csc[e + f*x])^(p*FracPart[n])* Int[(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n*p)*(A + C*Csc[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n, p}, x] && Not[IntegerQ[n]]
diff --git a/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m
new file mode 100755
index 0000000..93472d4
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m	
@@ -0,0 +1,35 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.5.7 (d trig)^m (a+b (c sec)^n)^p *)
+Int[u_.*(a_ + b_.*sec[e_. + f_.*x_]^2)^p_, x_Symbol] := b^p*Int[ActivateTrig[u*tan[e + f*x]^(2*p)], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]
+Int[u_.*(a_ + b_.*sec[e_. + f_.*x_]^2)^p_, x_Symbol] := Int[ActivateTrig[u*(b*tan[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]
+Int[(b_.*sec[e_. + f_.*x_]^2)^p_, x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, b*ff/f* Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{b, e, f, p}, x] && Not[IntegerQ[p]]
+Int[(b_.*(c_.*sec[e_. + f_.*x_])^n_)^p_, x_Symbol] := b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^ FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])* Int[(c*Sec[e + f*x])^(n*p), x] /; FreeQ[{b, c, e, f, n, p}, x] && Not[IntegerQ[p]]
+Int[tan[e_. + f_.*x_]^m_.*(b_.*sec[e_. + f_.*x_]^2)^p_., x_Symbol] := b/(2*f)* Subst[Int[(-1 + x)^((m - 1)/2)*(b*x)^(p - 1), x], x, Sec[e + f*x]^2] /; FreeQ[{b, e, f, p}, x] && Not[IntegerQ[p]] && IntegerQ[(m - 1)/2]
+Int[u_.*(b_.*sec[e_. + f_.*x_]^n_)^p_, x_Symbol] := With[{ff = FreeFactors[Sec[e + f*x], x]}, (b*ff^n)^ IntPart[p]*(b*Sec[e + f*x]^n)^ FracPart[p]/(Sec[e + f*x]/ff)^(n*FracPart[p])* Int[ActivateTrig[u]*(Sec[e + f*x]/ff)^(n*p), x]] /; FreeQ[{b, e, f, n, p}, x] && Not[IntegerQ[p]] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, (d_.*trig_[e + f*x])^m_. /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
+Int[u_.*(b_.*(c_.*sec[e_. + f_.*x_])^n_)^p_, x_Symbol] := b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^ FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])* Int[ActivateTrig[u]*(c*Sec[e + f*x])^(n*p), x] /; FreeQ[{b, c, e, f, n, p}, x] && Not[IntegerQ[p]] && Not[IntegerQ[n]] && (EqQ[u, 1] || MatchQ[u, (d_.*trig_[e + f*x])^m_. /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
+Int[1/(a_ + b_.*sec[e_. + f_.*x_]^2), x_Symbol] := x/a - b/a*Int[1/(b + a*Cos[e + f*x]^2), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0]
+Int[(a_ + b_.*sec[e_. + f_.*x_]^2)^p_, x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && NeQ[a + b, 0] && NeQ[p, -1]
+Int[(a_ + b_.*sec[e_. + f_.*x_]^4)^p_, x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + b + 2*b*ff^2*x^2 + b*ff^4*x^4)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[2*p]
+Int[(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_, x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[n/2] && IGtQ[p, -2]
+Int[(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^p_, x_Symbol] := Unintegrable[(a + b*(c*Sec[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, e, f, n, p}, x]
+Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff^(m + 1)/f* Subst[Int[ x^m*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^ p/(1 + ff^2*x^2)^(m/2 + 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]
+Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Cos[e + f*x], x]}, -ff/f* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(b + a*(ff*x)^n)^ p/(ff*x)^(n*p), x], x, Cos[e + f*x]/ff]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
+Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Cos[e + f*x], x]}, 1/(f*ff^m)* Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p/ x^(m + 1), x], x, Sec[e + f*x]/ff]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (GtQ[m, 0] || EqQ[n, 2] || EqQ[n, 4])
+Int[(d_.*sin[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^ p_., x_Symbol] := Unintegrable[(d*Sin[e + f*x])^m*(a + b*(c*Sec[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
+Int[(d_.*cos[e_. + f_.*x_])^m_*(a_ + b_.*sec[e_. + f_.*x_]^n_.)^p_., x_Symbol] := d^(n*p)* Int[(d*Cos[e + f*x])^(m - n*p)*(b + a*Cos[e + f*x]^n)^p, x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && Not[IntegerQ[m]] && IntegersQ[n, p]
+Int[(d_.*cos[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^ p_, x_Symbol] := (d*Cos[e + f*x])^FracPart[m]*(Sec[e + f*x]/d)^FracPart[m]* Int[(Sec[e + f*x]/d)^(-m)*(a + b*(c*Sec[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]
+Int[tan[e_. + f_.*x_]^m_.*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_., x_Symbol] := Module[{ff = FreeFactors[Cos[e + f*x], x]}, -1/(f*ff^(m + n*p - 1))* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(b + a*(ff*x)^n)^p/ x^(m + n*p), x], x, Cos[e + f*x]/ff]] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
+Int[tan[e_. + f_.*x_]^m_.*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Sec[e + f*x], x]}, 1/f* Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p/x, x], x, Sec[e + f*x]/ff]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (GtQ[m, 0] || EqQ[n, 2] || EqQ[n, 4] || IGtQ[p, 0] || IntegersQ[2*n, p])
+Int[(d_.*tan[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_]^2)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, b*ff/f* Subst[Int[(d*ff*x)^m*(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff]] /; FreeQ[{b, d, e, f, m, p}, x]
+Int[(d_.*tan[e_. + f_.*x_])^m_*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_., x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(d*ff*x)^ m*(a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] || EqQ[n, 2])
+Int[(d_.*tan[e_. + f_.*x_])^m_*(b_.*(c_.*sec[e_. + f_.*x_])^n_)^p_., x_Symbol] := d*(d*Tan[e + f*x])^(m - 1)*(b*(c*Sec[e + f*x])^n)^ p/(f*(p*n + m - 1)) - d^2*(m - 1)/(p*n + m - 1)* Int[(d*Tan[e + f*x])^(m - 2)*(b*(c*Sec[e + f*x])^n)^p, x] /; FreeQ[{b, c, d, e, f, p, n}, x] && GtQ[m, 1] && NeQ[p*n + m - 1, 0] && IntegersQ[2*p*n, 2*m]
+Int[(d_.*tan[e_. + f_.*x_])^m_*(b_.*(c_.*sec[e_. + f_.*x_])^n_)^p_., x_Symbol] := (d*Tan[e + f*x])^(m + 1)*(b*(c*Sec[e + f*x])^n)^p/(d*f*(m + 1)) - (p*n + m + 1)/(d^2*(m + 1))* Int[(d*Tan[e + f*x])^(m + 2)*(b*(c*Sec[e + f*x])^n)^p, x] /; FreeQ[{b, c, d, e, f, p, n}, x] && LtQ[m, -1] && NeQ[p*n + m + 1, 0] && IntegersQ[2*p*n, 2*m]
+Int[(d_.*tan[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^ p_., x_Symbol] := Unintegrable[(d*Tan[e + f*x])^m*(a + b*(c*Sec[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
+Int[(d_.*cot[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^ p_, x_Symbol] := (d*Cot[e + f*x])^FracPart[m]*(Tan[e + f*x]/d)^FracPart[m]* Int[(Tan[e + f*x]/d)^(-m)*(a + b*(c*Sec[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]
+Int[sec[e_. + f_.*x_]^m_*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_, x_Symbol] := With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)* ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p, x], x, Tan[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]
+Int[sec[e_. + f_.*x_]^m_.*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_, x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[ ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^ p/(1 - ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
+Int[sec[e_. + f_.*x_]^m_.*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_, x_Symbol] := With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[(a + b/(1 - ff^2*x^2)^(n/2))^ p/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && Not[IntegerQ[p]]
+Int[sec[e_. + f_.*x_]^m_.*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_, x_Symbol] := Int[ExpandTrig[sec[e + f*x]^m*(a + b*sec[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, n, p]
+Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^ p_., x_Symbol] := Unintegrable[(d*Sec[e + f*x])^m*(a + b*(c*Sec[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
+Int[(d_.*csc[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^ p_, x_Symbol] := (d*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/d)^FracPart[m]* Int[(Sin[e + f*x]/d)^(-m)*(a + b*(c*Sec[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]
diff --git a/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m
new file mode 100755
index 0000000..3491eeb
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m	
@@ -0,0 +1,31 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p *)
+Int[(a_. + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := 1/(4^p*c^p)*Int[(b + 2*c*Sec[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[(a_. + b_.*csc[d_. + e_.*x_]^n_. + c_.*csc[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := 1/(4^p*c^p)*Int[(b + 2*c*Csc[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[(a_. + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := (a + b*Sec[d + e*x]^n + c*Sec[d + e*x]^(2*n))^ p/(b + 2*c*Sec[d + e*x]^n)^(2*p)* Int[u*(b + 2*c*Sec[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[(a_. + b_.*csc[d_. + e_.*x_]^n_. + c_.*csc[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := (a + b*Csc[d + e*x]^n + c*Csc[d + e*x]^(2*n))^ p/(b + 2*c*Csc[d + e*x]^n)^(2*p)* Int[u*(b + 2*c*Csc[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[1/(a_. + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.), x_Symbol] := Module[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[1/(b - q + 2*c*Sec[d + e*x]^n), x] - 2*c/q*Int[1/(b + q + 2*c*Sec[d + e*x]^n), x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
+Int[1/(a_. + b_.*csc[d_. + e_.*x_]^n_. + c_.*csc[d_. + e_.*x_]^n2_.), x_Symbol] := Module[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[1/(b - q + 2*c*Csc[d + e*x]^n), x] - 2*c/q*Int[1/(b + q + 2*c*Csc[d + e*x]^n), x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
+Int[sin[d_. + e_.*x_]^ m_.*(a_. + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_)^ p_., x_Symbol] := Module[{f = FreeFactors[Cos[d + e*x], x]}, -f/e* Subst[Int[(1 - f^2*x^2)^((m - 1)/2)*(b + a*(f*x)^n)^p/(f*x)^(n*p), x], x, Cos[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegersQ[n, p]
+Int[cos[d_. + e_.*x_]^ m_.*(a_. + b_.*csc[d_. + e_.*x_]^n_. + c_.*csc[d_. + e_.*x_]^n2_)^ p_., x_Symbol] := Module[{f = FreeFactors[Sin[d + e*x], x]}, f/e* Subst[Int[(1 - f^2*x^2)^((m - 1)/2)*(b + a*(f*x)^n)^p/(f*x)^(n*p), x], x, Sin[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegersQ[n, p]
+Int[sin[d_. + e_.*x_]^ m_*(a_. + b_.*sec[d_. + e_.*x_]^n_ + c_.*sec[d_. + e_.*x_]^n2_)^ p_., x_Symbol] := Module[{f = FreeFactors[Tan[d + e*x], x]}, f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[a + b*(1 + f^2*x^2)^(n/2) + c*(1 + f^2*x^2)^n, x]^p/(1 + f^2*x^2)^(m/2 + 1), x], x, Tan[d + e*x]/f]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && IntegerQ[n/2]
+Int[cos[d_. + e_.*x_]^ m_*(a_. + b_.*csc[d_. + e_.*x_]^n_ + c_.*csc[d_. + e_.*x_]^n2_)^ p_., x_Symbol] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[a + b*(1 + f^2*x^2)^(n/2) + c*(1 + f^2*x^2)^n, x]^p/(1 + f^2*x^2)^(m/2 + 1), x], x, Cot[d + e*x]/f]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && IntegerQ[n/2]
+Int[sec[d_. + e_.*x_]^ m_.*(a_. + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := 1/(4^p*c^p)*Int[Sec[d + e*x]^m*(b + 2*c*Sec[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[csc[d_. + e_.*x_]^ m_.*(a_. + b_.*csc[d_. + e_.*x_]^n_. + c_.*csc[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := 1/(4^p*c^p)*Int[Csc[d + e*x]^m*(b + 2*c*Csc[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
+Int[sec[d_. + e_.*x_]^ m_.*(a_. + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := (a + b*Sec[d + e*x]^n + c*Sec[d + e*x]^(2*n))^ p/(b + 2*c*Sec[d + e*x]^n)^(2*p)* Int[Sec[d + e*x]^m*(b + 2*c*Sec[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[csc[d_. + e_.*x_]^ m_.*(a_. + b_.*csc[d_. + e_.*x_]^n_. + c_.*csc[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := (a + b*Csc[d + e*x]^n + c*Csc[d + e*x]^(2*n))^ p/(b + 2*c*Csc[d + e*x]^n)^(2*p)* Int[Csc[d + e*x]^m*(b + 2*c*Csc[d + e*x]^n)^(2*p), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]
+Int[sec[d_. + e_.*x_]^ m_.*(a_. + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := Int[ExpandTrig[ sec[d + e*x]^m*(a + b*sec[d + e*x]^n + c*sec[d + e*x]^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegersQ[m, n, p]
+Int[csc[d_. + e_.*x_]^ m_.*(a_. + b_.*csc[d_. + e_.*x_]^n_. + c_.*csc[d_. + e_.*x_]^n2_.)^p_, x_Symbol] := Int[ExpandTrig[ csc[d + e*x]^m*(a + b*csc[d + e*x]^n + c*csc[d + e*x]^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegersQ[m, n, p]
+Int[tan[d_. + e_.*x_]^ m_.*(a_ + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := Module[{f = FreeFactors[Cos[d + e*x], x]}, -1/(e*f^(m + n*p - 1))* Subst[Int[(1 - f^2*x^2)^((m - 1)/ 2)*(c + b*(f*x)^n + c*(f*x)^(2*n))^p/x^(m + 2*n*p), x], x, Cos[d + e*x]/f]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
+Int[cot[d_. + e_.*x_]^ m_.*(a_ + b_.*csc[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := Module[{f = FreeFactors[Sin[d + e*x], x]}, 1/(e*f^(m + n*p - 1))* Subst[Int[(1 - f^2*x^2)^((m - 1)/ 2)*(c + b*(f*x)^n + c*(f*x)^(2*n))^p/x^(m + 2*n*p), x], x, Sin[d + e*x]/f]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
+Int[tan[d_. + e_.*x_]^ m_.*(a_ + b_.*sec[d_. + e_.*x_]^n_ + c_.*sec[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := Module[{f = FreeFactors[Tan[d + e*x], x]}, f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[a + b*(1 + f^2*x^2)^(n/2) + c*(1 + f^2*x^2)^n, x]^p/(1 + f^2*x^2), x], x, Tan[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && IntegerQ[n/2]
+Int[cot[d_. + e_.*x_]^ m_.*(a_ + b_.*csc[d_. + e_.*x_]^n_ + c_.*sec[d_. + e_.*x_]^n2_.)^ p_., x_Symbol] := Module[{f = FreeFactors[Cot[d + e*x], x]}, -f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[a + b*(1 + f^2*x^2)^(n/2) + c*(1 + f^2*x^2)^n, x]^p/(1 + f^2*x^2), x], x, Cot[d + e*x]/f]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && IntegerQ[n/2]
+Int[(A_ + B_.*sec[d_. + e_.*x_])*(a_ + b_.*sec[d_. + e_.*x_] + c_.*sec[d_. + e_.*x_]^2)^n_, x_Symbol] := 1/(4^n*c^n)* Int[(A + B*Sec[d + e*x])*(b + 2*c*Sec[d + e*x])^(2*n), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]
+Int[(A_ + B_.*csc[d_. + e_.*x_])*(a_ + b_.*csc[d_. + e_.*x_] + c_.*csc[d_. + e_.*x_]^2)^n_, x_Symbol] := 1/(4^n*c^n)* Int[(A + B*Csc[d + e*x])*(b + 2*c*Csc[d + e*x])^(2*n), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]
+Int[(A_ + B_.*sec[d_. + e_.*x_])*(a_ + b_.*sec[d_. + e_.*x_] + c_.*sec[d_. + e_.*x_]^2)^n_, x_Symbol] := (a + b*Sec[d + e*x] + c*Sec[d + e*x]^2)^ n/(b + 2*c*Sec[d + e*x])^(2*n)* Int[(A + B*Sec[d + e*x])*(b + 2*c*Sec[d + e*x])^(2*n), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[n]]
+Int[(A_ + B_.*csc[d_. + e_.*x_])*(a_ + b_.*csc[d_. + e_.*x_] + c_.*csc[d_. + e_.*x_]^2)^n_, x_Symbol] := (a + b*Csc[d + e*x] + c*Csc[d + e*x]^2)^ n/(b + 2*c*Csc[d + e*x])^(2*n)* Int[(A + B*Csc[d + e*x])*(b + 2*c*Csc[d + e*x])^(2*n), x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[n]]
+Int[(A_ + B_.*sec[d_. + e_.*x_])/(a_. + b_.*sec[d_. + e_.*x_] + c_.*sec[d_. + e_.*x_]^2), x_Symbol] := Module[{q = Rt[b^2 - 4*a*c, 2]}, (B + (b*B - 2*A*c)/q)*Int[1/(b + q + 2*c*Sec[d + e*x]), x] + (B - (b*B - 2*A*c)/q)*Int[1/(b - q + 2*c*Sec[d + e*x]), x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]
+Int[(A_ + B_.*csc[d_. + e_.*x_])/(a_. + b_.*csc[d_. + e_.*x_] + c_.*csc[d_. + e_.*x_]^2), x_Symbol] := Module[{q = Rt[b^2 - 4*a*c, 2]}, (B + (b*B - 2*A*c)/q)*Int[1/(b + q + 2*c*Csc[d + e*x]), x] + (B - (b*B - 2*A*c)/q)*Int[1/(b - q + 2*c*Csc[d + e*x]), x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]
+Int[(A_ + B_.*sec[d_. + e_.*x_])*(a_. + b_.*sec[d_. + e_.*x_] + c_.*sec[d_. + e_.*x_]^2)^n_, x_Symbol] := Int[ExpandTrig[(A + B*sec[d + e*x])*(a + b*sec[d + e*x] + c*sec[d + e*x]^2)^n, x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]
+Int[(A_ + B_.*csc[d_. + e_.*x_])*(a_. + b_.*csc[d_. + e_.*x_] + c_.*csc[d_. + e_.*x_]^2)^n_, x_Symbol] := Int[ExpandTrig[(A + B*csc[d + e*x])*(a + b*csc[d + e*x] + c*csc[d + e*x]^2)^n, x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]
diff --git a/IntegrationRules/4 Trig functions/4.7 Miscellaneous/(a sin(m x) + b cos(n x))^p.m b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/(a sin(m x) + b cos(n x))^p.m
new file mode 100755
index 0000000..acbf30b
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/(a sin(m x) + b cos(n x))^p.m	
@@ -0,0 +1,154 @@
+(* ::Package:: *)
+
+(************************************************************************)
+(* This file was generated automatically by the Mathematica front end.  *)
+(* It contains Initialization cells from a Notebook file, which         *)
+(* typically will have the same name as this file except ending in      *)
+(* ".nb" instead of ".m".                                               *)
+(*                                                                      *)
+(* This file is intended to be loaded into the Mathematica kernel using *)
+(* the package loading commands Get or Needs.  Doing so is equivalent   *)
+(* to using the Evaluate Initialization Cells menu command in the front *)
+(* end.                                                                 *)
+(*                                                                      *)
+(* DO NOT EDIT THIS FILE.  This entire file is regenerated              *)
+(* automatically each time the parent Notebook file is saved in the     *)
+(* Mathematica front end.  Any changes you make to this file will be    *)
+(* overwritten.                                                         *)
+(************************************************************************)
+
+
+
+(* ::Code:: *)
+Int[(a_.*sin[m_.*(c_.+d_.*x_)]+b_.*sin[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Sin[m*ArcTan[x]]+b*Sin[n*ArcTan[x]]]]^p/(1+x^2),x],x,Tan[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p,0] && IntegerQ[m/2] && IntegerQ[n/2]
+
+
+(* ::Code:: *)
+Int[(a_.*cos[m_.*(c_.+d_.*x_)]+b_.*cos[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  -1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Cos[m*ArcCot[x]]+b*Cos[n*ArcCot[x]]]]^p/(1+x^2),x],x,Cot[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p,0] && IntegerQ[m/2] && IntegerQ[n/2]
+
+
+(* ::Code:: *)
+Int[(a_.*sin[m_.*(c_.+d_.*x_)]+b_.*sin[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Sin[m*ArcTan[x]]+b*Sin[n*ArcTan[x]]]]^p/(1+x^2),x],x,Tan[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p/2,0] && IntegerQ[(m-1)/2] && IntegerQ[(n-1)/2]
+
+
+(* ::Code:: *)
+Int[(a_.*cos[m_.*(c_.+d_.*x_)]+b_.*cos[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Cos[m*ArcTan[x]]+b*Cos[n*ArcTan[x]]]]^p/(1+x^2),x],x,Tan[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p/2,0] && IntegerQ[(m-1)/2] && IntegerQ[(n-1)/2]
+
+
+(* ::Code:: *)
+Int[(a_.*sin[m_.*(c_.+d_.*x_)]+b_.*sin[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  -1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Sin[m*ArcCos[x]]+b*Sin[n*ArcCos[x]]]]^p/Sqrt[1-x^2],x],x,Cos[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[(p-1)/2,0] && IntegerQ[(m-1)/2] && IntegerQ[(n-1)/2]
+
+
+(* ::Code:: *)
+Int[(a_.*cos[m_.*(c_.+d_.*x_)]+b_.*cos[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Cos[m*ArcSin[x]]+b*Cos[n*ArcSin[x]]]]^p/Sqrt[1-x^2],x],x,Sin[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[(p-1)/2,0] && IntegerQ[(m-1)/2] && IntegerQ[(n-1)/2]
+
+
+(* ::Code:: *)
+Int[(a_.*sin[m_.*(c_.+d_.*x_)]+b_.*sin[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  2/d \[Star] Subst[Int[Simplify[TrigExpand[a*Sin[2*m*ArcTan[x]]+b*Sin[2*n*ArcTan[x]]]]^p/(1+x^2),x],x,Tan[1/2*(c+d*x)]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p,0] && IntegerQ[m/2] && IntegerQ[(n-1)/2]
+
+
+(* ::Code:: *)
+Int[(a_.*cos[m_.*(c_.+d_.*x_)]+b_.*cos[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  -2/d \[Star] Subst[Int[Simplify[TrigExpand[a*Cos[2*m*ArcCot[x]]+b*Cos[2*n*ArcCot[x]]]]^p/(1+x^2),x],x,Cot[1/2*(c+d*x)]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p,0] && IntegerQ[m/2] && IntegerQ[(n-1)/2]
+
+
+(* ::Code:: *)
+Int[(a_.*sin[m_.*(c_.+d_.*x_)]+b_.*cos[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Sin[m*ArcTan[x]]+b*Cos[n*ArcTan[x]]]]^p/(1+x^2),x],x,Tan[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p,0] && IntegerQ[m/2] && IntegerQ[n/2]
+
+
+(* ::Code:: *)
+Int[(a_.*sin[m_.*(c_.+d_.*x_)]+b_.*cos[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Sin[m*ArcSin[x]]+b*Cos[n*ArcSin[x]]]]^p/Sqrt[1-x^2],x],x,Sin[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[(p-1)/2,0] && IntegerQ[m/2] && IntegerQ[(n-1)/2]
+
+
+(* ::Code:: *)
+Int[(a_.*sin[m_.*(c_.+d_.*x_)]+b_.*cos[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  2/d \[Star] Subst[Int[Simplify[TrigExpand[a*Sin[2*m*ArcTan[x]]+b*Cos[2*n*ArcTan[x]]]]^p/(1+x^2),x],x,Tan[1/2*(c+d*x)]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p,0] && IntegerQ[m] && IntegerQ[n]
+
+
+(* ::Code:: *)
+Int[(a_.*sin[m_.*(c_.+d_.*x_)]+b_.*sin[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Sin[m*ArcTan[x]]+b*Sin[n*ArcTan[x]]]]^p/(1+x^2),x],x,Tan[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p,0] && IntegerQ[m/2] && IntegerQ[n/2]
+
+
+(* ::Code:: *)
+Int[(a_.*cos[m_.*(c_.+d_.*x_)]+b_.*cos[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  -1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Cos[m*ArcCot[x]]+b*Cos[n*ArcCot[x]]]]^p/(1+x^2),x],x,Cot[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p,0] && IntegerQ[m/2] && IntegerQ[n/2]
+
+
+(* ::Code:: *)
+Int[(a_.*sin[m_.*(c_.+d_.*x_)]+b_.*sin[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Sin[m*ArcTan[x]]+b*Sin[n*ArcTan[x]]]]^p/(1+x^2),x],x,Tan[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p/2,0] && IntegerQ[(m-1)/2] && IntegerQ[(n-1)/2]
+
+
+(* ::Code:: *)
+Int[(a_.*cos[m_.*(c_.+d_.*x_)]+b_.*cos[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Cos[m*ArcTan[x]]+b*Cos[n*ArcTan[x]]]]^p/(1+x^2),x],x,Tan[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p/2,0] && IntegerQ[(m-1)/2] && IntegerQ[(n-1)/2]
+
+
+(* ::Code:: *)
+Int[(a_.*sin[m_.*(c_.+d_.*x_)]+b_.*sin[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  -1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Sin[m*ArcCos[x]]+b*Sin[n*ArcCos[x]]]]^p/Sqrt[1-x^2],x],x,Cos[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[(p-1)/2,0] && IntegerQ[(m-1)/2] && IntegerQ[(n-1)/2]
+
+
+(* ::Code:: *)
+Int[(a_.*cos[m_.*(c_.+d_.*x_)]+b_.*cos[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Cos[m*ArcSin[x]]+b*Cos[n*ArcSin[x]]]]^p/Sqrt[1-x^2],x],x,Sin[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[(p-1)/2,0] && IntegerQ[(m-1)/2] && IntegerQ[(n-1)/2]
+
+
+(* ::Code:: *)
+Int[(a_.*sin[m_.*(c_.+d_.*x_)]+b_.*sin[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  2/d \[Star] Subst[Int[Simplify[TrigExpand[a*Sin[2*m*ArcTan[x]]+b*Sin[2*n*ArcTan[x]]]]^p/(1+x^2),x],x,Tan[1/2*(c+d*x)]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p,0] && IntegerQ[m/2] && IntegerQ[(n-1)/2]
+
+
+(* ::Code:: *)
+Int[(a_.*cos[m_.*(c_.+d_.*x_)]+b_.*cos[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  -2/d \[Star] Subst[Int[Simplify[TrigExpand[a*Cos[2*m*ArcCot[x]]+b*Cos[2*n*ArcCot[x]]]]^p/(1+x^2),x],x,Cot[1/2*(c+d*x)]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p,0] && IntegerQ[m/2] && IntegerQ[(n-1)/2]
+
+
+(* ::Code:: *)
+Int[(a_.*sin[m_.*(c_.+d_.*x_)]+b_.*cos[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Sin[m*ArcTan[x]]+b*Cos[n*ArcTan[x]]]]^p/(1+x^2),x],x,Tan[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p,0] && IntegerQ[m/2] && IntegerQ[n/2]
+
+
+(* ::Code:: *)
+Int[(a_.*sin[m_.*(c_.+d_.*x_)]+b_.*cos[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  1/d \[Star] Subst[Int[Simplify[TrigExpand[a*Sin[m*ArcSin[x]]+b*Cos[n*ArcSin[x]]]]^p/Sqrt[1-x^2],x],x,Sin[c+d*x]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[(p-1)/2,0] && IntegerQ[m/2] && IntegerQ[(n-1)/2]
+
+
+(* ::Code:: *)
+Int[(a_.*sin[m_.*(c_.+d_.*x_)]+b_.*cos[n_.*(c_.+d_.*x_)])^p_,x_Symbol] :=
+  2/d \[Star] Subst[Int[Simplify[TrigExpand[a*Sin[2*m*ArcTan[x]]+b*Cos[2*n*ArcTan[x]]]]^p/(1+x^2),x],x,Tan[1/2*(c+d*x)]] /;
+FreeQ[{a,b,c,d},x] && ILtQ[p,0] && IntegerQ[m] && IntegerQ[n]
+
+
+
diff --git a/IntegrationRules/4 Trig functions/4.7 Miscellaneous/(a sin(m x) + b cos(n x))^p.pdf b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/(a sin(m x) + b cos(n x))^p.pdf
new file mode 100755
index 0000000..3ecd18b
Binary files /dev/null and b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/(a sin(m x) + b cos(n x))^p.pdf differ
diff --git a/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m
new file mode 100755
index 0000000..ceb114e
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m	
@@ -0,0 +1,29 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.7.1 Sine normalization rules *)
+Int[u_*(c_.*tan[a_. + b_.*x_])^m_.*(d_.*sin[a_. + b_.*x_])^n_., x_Symbol] := (c*Tan[a + b*x])^m*(d*Cos[a + b*x])^m/(d*Sin[a + b*x])^m* Int[ActivateTrig[u]*(d*Sin[a + b*x])^(m + n)/(d*Cos[a + b*x])^m, x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x] && Not[IntegerQ[m]]
+Int[u_*(c_.*tan[a_. + b_.*x_])^m_.*(d_.*cos[a_. + b_.*x_])^n_., x_Symbol] := (c*Tan[a + b*x])^m*(d*Cos[a + b*x])^m/(d*Sin[a + b*x])^m* Int[ActivateTrig[u]*(d*Sin[a + b*x])^m/(d*Cos[a + b*x])^(m - n), x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x] && Not[IntegerQ[m]]
+Int[u_*(c_.*cot[a_. + b_.*x_])^m_.*(d_.*sin[a_. + b_.*x_])^n_., x_Symbol] := (c*Cot[a + b*x])^m*(d*Sin[a + b*x])^m/(d*Cos[a + b*x])^m* Int[ActivateTrig[u]*(d*Cos[a + b*x])^m/(d*Sin[a + b*x])^(m - n), x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x] && Not[IntegerQ[m]]
+Int[u_*(c_.*cot[a_. + b_.*x_])^m_.*(d_.*cos[a_. + b_.*x_])^n_., x_Symbol] := (c*Cot[a + b*x])^m*(d*Sin[a + b*x])^m/(d*Cos[a + b*x])^m* Int[ActivateTrig[u]*(d*Cos[a + b*x])^(m + n)/(d*Sin[a + b*x])^m, x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x] && Not[IntegerQ[m]]
+Int[u_*(c_.*sec[a_. + b_.*x_])^m_.*(d_.*cos[a_. + b_.*x_])^n_., x_Symbol] := (c*Sec[a + b*x])^m*(d*Cos[a + b*x])^m* Int[ActivateTrig[u]*(d*Cos[a + b*x])^(n - m), x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x]
+Int[u_*(c_.*sec[a_. + b_.*x_])^m_.*(d_.*cos[a_. + b_.*x_])^n_., x_Symbol] := (c*Csc[a + b*x])^m*(d*Sin[a + b*x])^m* Int[ActivateTrig[u]*(d*Sin[a + b*x])^(n - m), x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x]
+Int[u_*(c_.*tan[a_. + b_.*x_])^m_., x_Symbol] := (c*Tan[a + b*x])^m*(c*Cos[a + b*x])^m/(c*Sin[a + b*x])^m* Int[ActivateTrig[u]*(c*Sin[a + b*x])^m/(c*Cos[a + b*x])^m, x] /; FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSineIntegrandQ[u, x]
+Int[u_*(c_.*cot[a_. + b_.*x_])^m_., x_Symbol] := (c*Cot[a + b*x])^m*(c*Sin[a + b*x])^m/(c*Cos[a + b*x])^m* Int[ActivateTrig[u]*(c*Cos[a + b*x])^m/(c*Sin[a + b*x])^m, x] /; FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSineIntegrandQ[u, x]
+Int[u_*(c_.*sec[a_. + b_.*x_])^m_., x_Symbol] := (c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m* Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x] /; FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSineIntegrandQ[u, x]
+Int[u_*(c_.*csc[a_. + b_.*x_])^m_., x_Symbol] := (c*Csc[a + b*x])^m*(c*Sin[a + b*x])^m* Int[ActivateTrig[u]/(c*Sin[a + b*x])^m, x] /; FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSineIntegrandQ[u, x]
+Int[u_*(c_.*sin[a_. + b_.*x_])^n_.*(A_ + B_.*csc[a_. + b_.*x_]), x_Symbol] := c*Int[ ActivateTrig[u]*(c*Sin[a + b*x])^(n - 1)*(B + A*Sin[a + b*x]), x] /; FreeQ[{a, b, c, A, B, n}, x] && KnownSineIntegrandQ[u, x]
+Int[u_*(c_.*cos[a_. + b_.*x_])^n_.*(A_ + B_.*sec[a_. + b_.*x_]), x_Symbol] := c*Int[ ActivateTrig[u]*(c*Cos[a + b*x])^(n - 1)*(B + A*Cos[a + b*x]), x] /; FreeQ[{a, b, c, A, B, n}, x] && KnownSineIntegrandQ[u, x]
+Int[u_*(A_ + B_.*csc[a_. + b_.*x_]), x_Symbol] := Int[ActivateTrig[u]*(B + A*Sin[a + b*x])/Sin[a + b*x], x] /; FreeQ[{a, b, A, B}, x] && KnownSineIntegrandQ[u, x]
+Int[u_*(A_ + B_.*sec[a_. + b_.*x_]), x_Symbol] := Int[ActivateTrig[u]*(B + A*Cos[a + b*x])/Cos[a + b*x], x] /; FreeQ[{a, b, A, B}, x] && KnownSineIntegrandQ[u, x]
+Int[u_.*(c_.*sin[a_. + b_.*x_])^ n_.*(A_. + B_.*csc[a_. + b_.*x_] + C_.*csc[a_. + b_.*x_]^2), x_Symbol] := c^2*Int[ ActivateTrig[ u]*(c*Sin[a + b*x])^(n - 2)*(C + B*Sin[a + b*x] + A*Sin[a + b*x]^2), x] /; FreeQ[{a, b, c, A, B, C, n}, x] && KnownSineIntegrandQ[u, x]
+Int[u_.*(c_.*cos[a_. + b_.*x_])^ n_.*(A_. + B_.*sec[a_. + b_.*x_] + C_.*sec[a_. + b_.*x_]^2), x_Symbol] := c^2*Int[ ActivateTrig[ u]*(c*Cos[a + b*x])^(n - 2)*(C + B*Cos[a + b*x] + A*Cos[a + b*x]^2), x] /; FreeQ[{a, b, c, A, B, C, n}, x] && KnownSineIntegrandQ[u, x]
+Int[u_.*(c_.*sin[a_. + b_.*x_])^n_.*(A_ + C_.*csc[a_. + b_.*x_]^2), x_Symbol] := c^2*Int[ ActivateTrig[u]*(c*Sin[a + b*x])^(n - 2)*(C + A*Sin[a + b*x]^2), x] /; FreeQ[{a, b, c, A, C, n}, x] && KnownSineIntegrandQ[u, x]
+Int[u_.*(c_.*cos[a_. + b_.*x_])^n_.*(A_ + C_.*sec[a_. + b_.*x_]^2), x_Symbol] := c^2*Int[ ActivateTrig[u]*(c*Cos[a + b*x])^(n - 2)*(C + A*Cos[a + b*x]^2), x] /; FreeQ[{a, b, c, A, C, n}, x] && KnownSineIntegrandQ[u, x]
+Int[u_*(A_. + B_.*csc[a_. + b_.*x_] + C_.*csc[a_. + b_.*x_]^2), x_Symbol] := Int[ActivateTrig[u]*(C + B*Sin[a + b*x] + A*Sin[a + b*x]^2)/ Sin[a + b*x]^2, x] /; FreeQ[{a, b, A, B, C}, x] && KnownSineIntegrandQ[u, x]
+Int[u_*(A_. + B_.*sec[a_. + b_.*x_] + C_.*sec[a_. + b_.*x_]^2), x_Symbol] := Int[ActivateTrig[u]*(C + B*Cos[a + b*x] + A*Cos[a + b*x]^2)/ Cos[a + b*x]^2, x] /; FreeQ[{a, b, A, B, C}, x] && KnownSineIntegrandQ[u, x]
+Int[u_*(A_ + C_.*csc[a_. + b_.*x_]^2), x_Symbol] := Int[ActivateTrig[u]*(C + A*Sin[a + b*x]^2)/Sin[a + b*x]^2, x] /; FreeQ[{a, b, A, C}, x] && KnownSineIntegrandQ[u, x]
+Int[u_*(A_ + C_.*sec[a_. + b_.*x_]^2), x_Symbol] := Int[ActivateTrig[u]*(C + A*Cos[a + b*x]^2)/Cos[a + b*x]^2, x] /; FreeQ[{a, b, A, C}, x] && KnownSineIntegrandQ[u, x]
+Int[u_*(A_. + B_.*sin[a_. + b_.*x_] + C_.*csc[a_. + b_.*x_]), x_Symbol] := Int[ActivateTrig[u]*(C + A*Sin[a + b*x] + B*Sin[a + b*x]^2)/ Sin[a + b*x], x] /; FreeQ[{a, b, A, B, C}, x]
+Int[u_*(A_. + B_.*cos[a_. + b_.*x_] + C_.*sec[a_. + b_.*x_]), x_Symbol] := Int[ActivateTrig[u]*(C + A*Cos[a + b*x] + B*Cos[a + b*x]^2)/ Cos[a + b*x], x] /; FreeQ[{a, b, A, B, C}, x]
+Int[u_*(A_.*sin[a_. + b_.*x_]^n_. + B_.*sin[a_. + b_.*x_]^n1_ + C_.*sin[a_. + b_.*x_]^n2_), x_Symbol] := Int[ActivateTrig[u]* Sin[a + b*x]^n*(A + B*Sin[a + b*x] + C*Sin[a + b*x]^2), x] /; FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]
+Int[u_*(A_.*cos[a_. + b_.*x_]^n_. + B_.*cos[a_. + b_.*x_]^n1_ + C_.*cos[a_. + b_.*x_]^n2_), x_Symbol] := Int[ActivateTrig[u]* Cos[a + b*x]^n*(A + B*Cos[a + b*x] + C*Cos[a + b*x]^2), x] /; FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]
diff --git a/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m
new file mode 100755
index 0000000..05bdef3
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m	
@@ -0,0 +1,22 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.7.2 Tangent normalization rules *)
+Int[u_*(c_.*cot[a_. + b_.*x_])^m_.*(d_.*tan[a_. + b_.*x_])^n_., x_Symbol] := (c*Cot[a + b*x])^m*(d*Tan[a + b*x])^m* Int[ActivateTrig[u]*(d*Tan[a + b*x])^(n - m), x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownTangentIntegrandQ[u, x]
+Int[u_*(c_.*tan[a_. + b_.*x_])^m_.*(d_.*cos[a_. + b_.*x_])^n_., x_Symbol] := (c*Tan[a + b*x])^m*(d*Cos[a + b*x])^m/(d*Sin[a + b*x])^m* Int[ActivateTrig[u]*(d*Sin[a + b*x])^m/(d*Cos[a + b*x])^(m - n), x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownCotangentIntegrandQ[u, x]
+Int[u_*(c_.*cot[a_. + b_.*x_])^m_., x_Symbol] := (c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m* Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x] /; FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownTangentIntegrandQ[u, x]
+Int[u_*(c_.*tan[a_. + b_.*x_])^m_., x_Symbol] := (c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m* Int[ActivateTrig[u]/(c*Cot[a + b*x])^m, x] /; FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownCotangentIntegrandQ[u, x]
+Int[u_*(c_.*tan[a_. + b_.*x_])^n_.*(A_ + B_.*cot[a_. + b_.*x_]), x_Symbol] := c*Int[ ActivateTrig[u]*(c*Tan[a + b*x])^(n - 1)*(B + A*Tan[a + b*x]), x] /; FreeQ[{a, b, c, A, B, n}, x] && KnownTangentIntegrandQ[u, x]
+Int[u_*(c_.*cot[a_. + b_.*x_])^n_.*(A_ + B_.*tan[a_. + b_.*x_]), x_Symbol] := c*Int[ ActivateTrig[u]*(c*Cot[a + b*x])^(n - 1)*(B + A*Cot[a + b*x]), x] /; FreeQ[{a, b, c, A, B, n}, x] && KnownCotangentIntegrandQ[u, x]
+Int[u_*(A_ + B_.*cot[a_. + b_.*x_]), x_Symbol] := Int[ActivateTrig[u]*(B + A*Tan[a + b*x])/Tan[a + b*x], x] /; FreeQ[{a, b, A, B}, x] && KnownTangentIntegrandQ[u, x]
+Int[u_*(A_ + B_.*tan[a_. + b_.*x_]), x_Symbol] := Int[ActivateTrig[u]*(B + A*Cot[a + b*x])/Cot[a + b*x], x] /; FreeQ[{a, b, A, B}, x] && KnownCotangentIntegrandQ[u, x]
+Int[u_.*(c_.*tan[a_. + b_.*x_])^ n_.*(A_. + B_.*cot[a_. + b_.*x_] + C_.*cot[a_. + b_.*x_]^2), x_Symbol] := c^2*Int[ ActivateTrig[ u]*(c*Tan[a + b*x])^(n - 2)*(C + B*Tan[a + b*x] + A*Tan[a + b*x]^2), x] /; FreeQ[{a, b, c, A, B, C, n}, x] && KnownTangentIntegrandQ[u, x]
+Int[u_.*(c_.*cot[a_. + b_.*x_])^ n_.*(A_. + B_.*tan[a_. + b_.*x_] + C_.*tan[a_. + b_.*x_]^2), x_Symbol] := c^2*Int[ ActivateTrig[ u]*(c*Cot[a + b*x])^(n - 2)*(C + B*Cot[a + b*x] + A*Cot[a + b*x]^2), x] /; FreeQ[{a, b, c, A, B, C, n}, x] && KnownCotangentIntegrandQ[u, x]
+Int[u_.*(c_.*tan[a_. + b_.*x_])^n_.*(A_ + C_.*cot[a_. + b_.*x_]^2), x_Symbol] := c^2*Int[ ActivateTrig[u]*(c*Tan[a + b*x])^(n - 2)*(C + A*Tan[a + b*x]^2), x] /; FreeQ[{a, b, c, A, C, n}, x] && KnownTangentIntegrandQ[u, x]
+Int[u_.*(c_.*cot[a_. + b_.*x_])^n_.*(A_ + C_.*tan[a_. + b_.*x_]^2), x_Symbol] := c^2*Int[ ActivateTrig[u]*(c*Cot[a + b*x])^(n - 2)*(C + A*Cot[a + b*x]^2), x] /; FreeQ[{a, b, c, A, C, n}, x] && KnownCotangentIntegrandQ[u, x]
+Int[u_*(A_. + B_.*cot[a_. + b_.*x_] + C_.*cot[a_. + b_.*x_]^2), x_Symbol] := Int[ActivateTrig[u]*(C + B*Tan[a + b*x] + A*Tan[a + b*x]^2)/ Tan[a + b*x]^2, x] /; FreeQ[{a, b, A, B, C}, x] && KnownTangentIntegrandQ[u, x]
+Int[u_*(A_. + B_.*tan[a_. + b_.*x_] + C_.*tan[a_. + b_.*x_]^2), x_Symbol] := Int[ActivateTrig[u]*(C + B*Cot[a + b*x] + A*Cot[a + b*x]^2)/ Cot[a + b*x]^2, x] /; FreeQ[{a, b, A, B, C}, x] && KnownCotangentIntegrandQ[u, x]
+Int[u_*(A_ + C_.*cot[a_. + b_.*x_]^2), x_Symbol] := Int[ActivateTrig[u]*(C + A*Tan[a + b*x]^2)/Tan[a + b*x]^2, x] /; FreeQ[{a, b, A, C}, x] && KnownTangentIntegrandQ[u, x]
+Int[u_*(A_ + C_.*tan[a_. + b_.*x_]^2), x_Symbol] := Int[ActivateTrig[u]*(C + A*Cot[a + b*x]^2)/Cot[a + b*x]^2, x] /; FreeQ[{a, b, A, C}, x] && KnownCotangentIntegrandQ[u, x]
+Int[u_*(A_. + B_.*tan[a_. + b_.*x_] + C_.*cot[a_. + b_.*x_]), x_Symbol] := Int[ActivateTrig[u]*(C + A*Tan[a + b*x] + B*Tan[a + b*x]^2)/ Tan[a + b*x], x] /; FreeQ[{a, b, A, B, C}, x]
+Int[u_*(A_.*tan[a_. + b_.*x_]^n_. + B_.*tan[a_. + b_.*x_]^n1_ + C_.*tan[a_. + b_.*x_]^n2_), x_Symbol] := Int[ActivateTrig[u]* Tan[a + b*x]^n*(A + B*Tan[a + b*x] + C*Tan[a + b*x]^2), x] /; FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]
+Int[u_*(A_.*cot[a_. + b_.*x_]^n_. + B_.*cot[a_. + b_.*x_]^n1_ + C_.*cot[a_. + b_.*x_]^n2_), x_Symbol] := Int[ActivateTrig[u]* Cot[a + b*x]^n*(A + B*Cot[a + b*x] + C*Cot[a + b*x]^2), x] /; FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]
diff --git a/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m
new file mode 100755
index 0000000..deddefa
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m	
@@ -0,0 +1,27 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.7.3 Secant normalization rules *)
+Int[u_*(c_.*sin[a_. + b_.*x_])^m_.*(d_.*csc[a_. + b_.*x_])^n_., x_Symbol] := (c*Sin[a + b*x])^m*(d*Csc[a + b*x])^m* Int[ActivateTrig[u]*(d*Csc[a + b*x])^(n - m), x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x]
+Int[u_*(c_.*cos[a_. + b_.*x_])^m_.*(d_.*sec[a_. + b_.*x_])^n_., x_Symbol] := (c*Cos[a + b*x])^m*(d*Sec[a + b*x])^m* Int[ActivateTrig[u]*(d*Sec[a + b*x])^(n - m), x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x]
+Int[u_*(c_.*tan[a_. + b_.*x_])^m_.*(d_.*sec[a_. + b_.*x_])^n_., x_Symbol] := (c*Tan[a + b*x])^m*(d*Csc[a + b*x])^m/(d*Sec[a + b*x])^m* Int[ActivateTrig[u]*(d*Sec[a + b*x])^(m + n)/(d*Csc[a + b*x])^m, x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x] && Not[IntegerQ[m]]
+Int[u_*(c_.*tan[a_. + b_.*x_])^m_.*(d_.*csc[a_. + b_.*x_])^n_., x_Symbol] := (c*Tan[a + b*x])^m*(d*Csc[a + b*x])^m/(d*Sec[a + b*x])^m* Int[ActivateTrig[u]*(d*Sec[a + b*x])^m/(d*Csc[a + b*x])^(m - n), x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x] && Not[IntegerQ[m]]
+Int[u_*(c_.*cot[a_. + b_.*x_])^m_.*(d_.*sec[a_. + b_.*x_])^n_., x_Symbol] := (c*Cot[a + b*x])^m*(d*Sec[a + b*x])^m/(d*Csc[a + b*x])^m* Int[ActivateTrig[u]*(d*Csc[a + b*x])^m/(d*Sec[a + b*x])^(m - n), x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x] && Not[IntegerQ[m]]
+Int[u_*(c_.*cot[a_. + b_.*x_])^m_.*(d_.*csc[a_. + b_.*x_])^n_., x_Symbol] := (c*Cot[a + b*x])^m*(d*Sec[a + b*x])^m/(d*Csc[a + b*x])^m* Int[ActivateTrig[u]*(d*Csc[a + b*x])^(m + n)/(d*Sec[a + b*x])^m, x] /; FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x] && Not[IntegerQ[m]]
+Int[u_*(c_.*sin[a_. + b_.*x_])^m_., x_Symbol] := (c*Csc[a + b*x])^m*(c*Sin[a + b*x])^m* Int[ActivateTrig[u]/(c*Csc[a + b*x])^m, x] /; FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSecantIntegrandQ[u, x]
+Int[u_*(c_.*cos[a_. + b_.*x_])^m_., x_Symbol] := (c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m* Int[ActivateTrig[u]/(c*Sec[a + b*x])^m, x] /; FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSecantIntegrandQ[u, x]
+Int[u_*(c_.*tan[a_. + b_.*x_])^m_., x_Symbol] := (c*Tan[a + b*x])^m*(c*Csc[a + b*x])^m/(c*Sec[a + b*x])^m* Int[ActivateTrig[u]*(c*Sec[a + b*x])^m/(c*Csc[a + b*x])^m, x] /; FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSecantIntegrandQ[u, x]
+Int[u_*(c_.*cot[a_. + b_.*x_])^m_., x_Symbol] := (c*Cot[a + b*x])^m*(c*Sec[a + b*x])^m/(c*Csc[a + b*x])^m* Int[ActivateTrig[u]*(c*Csc[a + b*x])^m/(c*Sec[a + b*x])^m, x] /; FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSecantIntegrandQ[u, x]
+Int[u_*(c_.*sec[a_. + b_.*x_])^n_.*(A_ + B_.*cos[a_. + b_.*x_]), x_Symbol] := c*Int[ ActivateTrig[u]*(c*Sec[a + b*x])^(n - 1)*(B + A*Sec[a + b*x]), x] /; FreeQ[{a, b, c, A, B, n}, x] && KnownSecantIntegrandQ[u, x]
+Int[u_*(c_.*csc[a_. + b_.*x_])^n_.*(A_ + B_.*sin[a_. + b_.*x_]), x_Symbol] := c*Int[ ActivateTrig[u]*(c*Csc[a + b*x])^(n - 1)*(B + A*Csc[a + b*x]), x] /; FreeQ[{a, b, c, A, B, n}, x] && KnownSecantIntegrandQ[u, x]
+Int[u_*(A_ + B_.*cos[a_. + b_.*x_]), x_Symbol] := Int[ActivateTrig[u]*(B + A*Sec[a + b*x])/Sec[a + b*x], x] /; FreeQ[{a, b, A, B}, x] && KnownSecantIntegrandQ[u, x]
+Int[u_*(A_ + B_.*sin[a_. + b_.*x_]), x_Symbol] := Int[ActivateTrig[u]*(B + A*Csc[a + b*x])/Csc[a + b*x], x] /; FreeQ[{a, b, A, B}, x] && KnownSecantIntegrandQ[u, x]
+Int[u_.*(c_.*sec[a_. + b_.*x_])^ n_.*(A_. + B_.*cos[a_. + b_.*x_] + C_.*cos[a_. + b_.*x_]^2), x_Symbol] := c^2*Int[ ActivateTrig[ u]*(c*Sec[a + b*x])^(n - 2)*(C + B*Sec[a + b*x] + A*Sec[a + b*x]^2), x] /; FreeQ[{a, b, c, A, B, C, n}, x] && KnownSecantIntegrandQ[u, x]
+Int[u_.*(c_.*csc[a_. + b_.*x_])^ n_.*(A_. + B_.*sin[a_. + b_.*x_] + C_.*sin[a_. + b_.*x_]^2), x_Symbol] := c^2*Int[ ActivateTrig[ u]*(c*Csc[a + b*x])^(n - 2)*(C + B*Csc[a + b*x] + A*Csc[a + b*x]^2), x] /; FreeQ[{a, b, c, A, B, C, n}, x] && KnownSecantIntegrandQ[u, x]
+Int[u_.*(c_.*sec[a_. + b_.*x_])^n_.*(A_ + C_.*cos[a_. + b_.*x_]^2), x_Symbol] := c^2*Int[ ActivateTrig[u]*(c*Sec[a + b*x])^(n - 2)*(C + A*Sec[a + b*x]^2), x] /; FreeQ[{a, b, c, A, C, n}, x] && KnownSecantIntegrandQ[u, x]
+Int[u_.*(c_.*csc[a_. + b_.*x_])^n_.*(A_ + C_.*sin[a_. + b_.*x_]^2), x_Symbol] := c^2*Int[ ActivateTrig[u]*(c*Csc[a + b*x])^(n - 2)*(C + A*Csc[a + b*x]^2), x] /; FreeQ[{a, b, c, A, C, n}, x] && KnownSecantIntegrandQ[u, x]
+Int[u_*(A_. + B_.*cos[a_. + b_.*x_] + C_.*cos[a_. + b_.*x_]^2), x_Symbol] := Int[ActivateTrig[u]*(C + B*Sec[a + b*x] + A*Sec[a + b*x]^2)/ Sec[a + b*x]^2, x] /; FreeQ[{a, b, A, B, C}, x] && KnownSecantIntegrandQ[u, x]
+Int[u_*(A_. + B_.*sin[a_. + b_.*x_] + C_.*sin[a_. + b_.*x_]^2), x_Symbol] := Int[ActivateTrig[u]*(C + B*Csc[a + b*x] + A*Csc[a + b*x]^2)/ Csc[a + b*x]^2, x] /; FreeQ[{a, b, A, B, C}, x] && KnownSecantIntegrandQ[u, x]
+Int[u_*(A_ + C_.*cos[a_. + b_.*x_]^2), x_Symbol] := Int[ActivateTrig[u]*(C + A*Sec[a + b*x]^2)/Sec[a + b*x]^2, x] /; FreeQ[{a, b, A, C}, x] && KnownSecantIntegrandQ[u, x]
+Int[u_*(A_ + C_.*sin[a_. + b_.*x_]^2), x_Symbol] := Int[ActivateTrig[u]*(C + A*Csc[a + b*x]^2)/Csc[a + b*x]^2, x] /; FreeQ[{a, b, A, C}, x] && KnownSecantIntegrandQ[u, x]
+Int[u_*(A_.*sec[a_. + b_.*x_]^n_. + B_.*sec[a_. + b_.*x_]^n1_ + C_.*sec[a_. + b_.*x_]^n2_), x_Symbol] := Int[ActivateTrig[u]* Sec[a + b*x]^n*(A + B*Sec[a + b*x] + C*Sec[a + b*x]^2), x] /; FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]
+Int[u_*(A_.*csc[a_. + b_.*x_]^n_. + B_.*csc[a_. + b_.*x_]^n1_ + C_.*csc[a_. + b_.*x_]^n2_), x_Symbol] := Int[ActivateTrig[u]* Csc[a + b*x]^n*(A + B*Csc[a + b*x] + C*Csc[a + b*x]^2), x] /; FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]
diff --git a/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m
new file mode 100755
index 0000000..90ad8a7
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m	
@@ -0,0 +1,56 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.7.4 (c trig)^m (d trig)^n *)
+Int[sin[a_. + b_.*x_]*sin[c_. + d_.*x_], x_Symbol] := Sin[a - c + (b - d)*x]/(2*(b - d)) - Sin[a + c + (b + d)*x]/(2*(b + d)) /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
+Int[cos[a_. + b_.*x_]*cos[c_. + d_.*x_], x_Symbol] := Sin[a - c + (b - d)*x]/(2*(b - d)) + Sin[a + c + (b + d)*x]/(2*(b + d)) /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
+Int[sin[a_. + b_.*x_]*cos[c_. + d_.*x_], x_Symbol] := -Cos[a - c + (b - d)*x]/(2*(b - d)) - Cos[a + c + (b + d)*x]/(2*(b + d)) /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
+Int[cos[a_. + b_.*x_]^2*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := 1/2*Int[(g*Sin[c + d*x])^p, x] + 1/2*Int[Cos[c + d*x]*(g*Sin[c + d*x])^p, x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IGtQ[p/2, 0]
+Int[sin[a_. + b_.*x_]^2*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := 1/2*Int[(g*Sin[c + d*x])^p, x] - 1/2*Int[Cos[c + d*x]*(g*Sin[c + d*x])^p, x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IGtQ[p/2, 0]
+Int[(e_.*cos[a_. + b_.*x_])^m_.*sin[c_. + d_.*x_]^p_., x_Symbol] := 2^p/e^p*Int[(e*Cos[a + b*x])^(m + p)*Sin[a + b*x]^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IntegerQ[p]
+Int[(f_.*sin[a_. + b_.*x_])^n_.*sin[c_. + d_.*x_]^p_., x_Symbol] := 2^p/f^p*Int[Cos[a + b*x]^p*(f*Sin[a + b*x])^(n + p), x] /; FreeQ[{a, b, c, d, f, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IntegerQ[p]
+Int[(e_.*cos[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := e^2*(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1)/(2*b* g*(p + 1)) /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && EqQ[m + p - 1, 0]
+Int[(e_.*sin[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -e^2*(e*Sin[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1)/(2*b* g*(p + 1)) /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && EqQ[m + p - 1, 0]
+Int[(e_.*cos[a_. + b_.*x_])^m_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -(e*Cos[a + b*x])^m*(g*Sin[c + d*x])^(p + 1)/(b*g*m) /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && EqQ[m + 2*p + 2, 0]
+Int[(e_.*sin[a_. + b_.*x_])^m_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := (e*Sin[a + b*x])^m*(g*Sin[c + d*x])^(p + 1)/(b*g*m) /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && EqQ[m + 2*p + 2, 0]
+Int[(e_.*cos[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := e^2*(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1)/(2*b* g*(p + 1)) + e^4*(m + p - 1)/(4*g^2*(p + 1))* Int[(e*Cos[a + b*x])^(m - 4)*(g*Sin[c + d*x])^(p + 2), x] /; FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 2] && LtQ[p, -1] && (GtQ[m, 3] || EqQ[p, -3/2]) && IntegersQ[2*m, 2*p]
+Int[(e_.*sin[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -e^2*(e*Sin[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1)/(2*b* g*(p + 1)) + e^4*(m + p - 1)/(4*g^2*(p + 1))* Int[(e*Sin[a + b*x])^(m - 4)*(g*Sin[c + d*x])^(p + 2), x] /; FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 2] && LtQ[p, -1] && (GtQ[m, 3] || EqQ[p, -3/2]) && IntegersQ[2*m, 2*p]
+Int[(e_.*cos[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := (e*Cos[a + b*x])^m*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(p + 1)) + e^2*(m + 2*p + 2)/(4*g^2*(p + 1))* Int[(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 2), x] /; FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && LtQ[p, -1] && NeQ[m + 2*p + 2, 0] && (LtQ[p, -2] || EqQ[m, 2]) && IntegersQ[2*m, 2*p]
+Int[(e_.*sin[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -(e*Sin[a + b*x])^m*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(p + 1)) + e^2*(m + 2*p + 2)/(4*g^2*(p + 1))* Int[(e*Sin[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 2), x] /; FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && LtQ[p, -1] && NeQ[m + 2*p + 2, 0] && (LtQ[p, -2] || EqQ[m, 2]) && IntegersQ[2*m, 2*p]
+Int[(e_.*cos[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := e^2*(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1)/(2*b* g*(m + 2*p)) + e^2*(m + p - 1)/(m + 2*p)* Int[(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && NeQ[m + 2*p, 0] && IntegersQ[2*m, 2*p]
+Int[(e_.*sin[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -e^2*(e*Sin[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1)/(2*b* g*(m + 2*p)) + e^2*(m + p - 1)/(m + 2*p)* Int[(e*Sin[a + b*x])^(m - 2)*(g*Sin[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && NeQ[m + 2*p, 0] && IntegersQ[2*m, 2*p]
+Int[(e_.*cos[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -(e*Cos[a + b*x])^ m*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(m + p + 1)) + (m + 2*p + 2)/(e^2*(m + p + 1))* Int[(e*Cos[a + b*x])^(m + 2)*(g*Sin[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && NeQ[m + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
+Int[(e_.*sin[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := (e*Sin[a + b*x])^m*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(m + p + 1)) + (m + 2*p + 2)/(e^2*(m + p + 1))* Int[(e*Sin[a + b*x])^(m + 2)*(g*Sin[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && NeQ[m + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
+Int[cos[a_. + b_.*x_]*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := 2*Sin[a + b*x]*(g*Sin[c + d*x])^p/(d*(2*p + 1)) + 2*p*g/(2*p + 1)*Int[Sin[a + b*x]*(g*Sin[c + d*x])^(p - 1), x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[p, 0] && IntegerQ[2*p]
+Int[sin[a_. + b_.*x_]*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -2*Cos[a + b*x]*(g*Sin[c + d*x])^p/(d*(2*p + 1)) + 2*p*g/(2*p + 1)*Int[Cos[a + b*x]*(g*Sin[c + d*x])^(p - 1), x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[p, 0] && IntegerQ[2*p]
+Int[cos[a_. + b_.*x_]*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := Cos[a + b*x]*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(p + 1)) + (2*p + 3)/(2*g*(p + 1))* Int[Sin[a + b*x]*(g*Sin[c + d*x])^(p + 1), x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[p, -1] && IntegerQ[2*p]
+Int[sin[a_. + b_.*x_]*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -Sin[a + b*x]*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(p + 1)) + (2*p + 3)/(2*g*(p + 1))* Int[Cos[a + b*x]*(g*Sin[c + d*x])^(p + 1), x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[p, -1] && IntegerQ[2*p]
+Int[cos[a_. + b_.*x_]/Sqrt[sin[c_. + d_.*x_]], x_Symbol] := -ArcSin[Cos[a + b*x] - Sin[a + b*x]]/d + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[c + d*x]]]/d /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
+Int[sin[a_. + b_.*x_]/Sqrt[sin[c_. + d_.*x_]], x_Symbol] := -ArcSin[Cos[a + b*x] - Sin[a + b*x]]/d - Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[c + d*x]]]/d /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
+Int[(g_.*sin[c_. + d_.*x_])^p_/cos[a_. + b_.*x_], x_Symbol] := 2*g*Int[Sin[a + b*x]*(g*Sin[c + d*x])^(p - 1), x] /; FreeQ[{a, b, c, d, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && IntegerQ[2*p]
+Int[(g_.*sin[c_. + d_.*x_])^p_/sin[a_. + b_.*x_], x_Symbol] := 2*g*Int[Cos[a + b*x]*(g*Sin[c + d*x])^(p - 1), x] /; FreeQ[{a, b, c, d, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && IntegerQ[2*p]
+(* Int[(e_.*cos[a_.+b_.*x_])^m_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := -(e*Cos[a+b*x])^(m+1)*Sin[a+b*x]*(g*Sin[c+d*x])^p/(b*e*(m+p+1)*(Sin[ a+b*x]^2)^((p+1)/2))* Hypergeometric2F1[-(p-1)/2,(m+p+1)/2,(m+p+3)/2,Cos[a+b*x]^2] /; FreeQ[{a,b,c,d,e,g,m,p},x] && EqQ[b*c-a*d,0] && EqQ[d/b,2] &&  Not[IntegerQ[p]] && Not[IntegerQ[m+p]] *)
+(* Int[(f_.*sin[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol] := -Cos[a+b*x]*(f*Sin[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*f*(p+1)*(Sin[a+ b*x]^2)^((n+p+1)/2))* Hypergeometric2F1[-(n+p-1)/2,(p+1)/2,(p+3)/2,Cos[a+b*x]^2] /; FreeQ[{a,b,c,d,f,g,n,p},x] && EqQ[b*c-a*d,0] && EqQ[d/b,2] &&  Not[IntegerQ[p]] && Not[IntegerQ[n+p]] *)
+Int[(e_.*cos[a_. + b_.*x_])^m_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := (g*Sin[c + d*x])^p/((e*Cos[a + b*x])^p*Sin[a + b*x]^p)* Int[(e*Cos[a + b*x])^(m + p)*Sin[a + b*x]^p, x] /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]]
+Int[(f_.*sin[a_. + b_.*x_])^n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := (g*Sin[c + d*x])^p/(Cos[a + b*x]^p*(f*Sin[a + b*x])^p)* Int[Cos[a + b*x]^p*(f*Sin[a + b*x])^(n + p), x] /; FreeQ[{a, b, c, d, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]]
+Int[cos[a_. + b_.*x_]^2* sin[a_. + b_.*x_]^2*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := 1/4*Int[(g*Sin[c + d*x])^p, x] - 1/4*Int[Cos[c + d*x]^2*(g*Sin[c + d*x])^p, x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IGtQ[p/2, 0]
+Int[(e_.*cos[a_. + b_.*x_])^m_.*(f_.*sin[a_. + b_.*x_])^n_.* sin[c_. + d_.*x_]^p_., x_Symbol] := 2^p/(e^p*f^p)* Int[(e*Cos[a + b*x])^(m + p)*(f*Sin[a + b*x])^(n + p), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IntegerQ[p]
+Int[(e_.*cos[a_. + b_.*x_])^m_.*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := e*(e*Cos[a + b*x])^(m - 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*f*(n + p + 1)) /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && EqQ[m + p + 1, 0]
+Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -e*(e*Sin[a + b*x])^(m - 1)*(f*Cos[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*f*(n + p + 1)) /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && EqQ[m + p + 1, 0]
+Int[(e_.*cos[a_. + b_.*x_])^m_.*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -(e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*e*f*(m + p + 1)) /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && EqQ[m + n + 2*p + 2, 0] && NeQ[m + p + 1, 0]
+Int[(e_.*cos[a_. + b_.*x_])^m_*(f_.*sin[a_. + b_.*x_])^ n_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := e^2*(e*Cos[a + b*x])^(m - 2)*(f*Sin[a + b*x])^ n*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(n + p + 1)) + e^4*(m + p - 1)/(4*g^2*(n + p + 1))* Int[(e*Cos[a + b*x])^(m - 4)*(f*Sin[a + b*x])^ n*(g*Sin[c + d*x])^(p + 2), x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 3] && LtQ[p, -1] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
+Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -e^2*(e*Sin[a + b*x])^(m - 2)*(f*Cos[a + b*x])^ n*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(n + p + 1)) + e^4*(m + p - 1)/(4*g^2*(n + p + 1))* Int[(e*Sin[a + b*x])^(m - 4)*(f*Cos[a + b*x])^ n*(g*Sin[c + d*x])^(p + 2), x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 3] && LtQ[p, -1] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
+Int[(e_.*cos[a_. + b_.*x_])^m_*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := (e*Cos[a + b*x])^m*(f*Sin[a + b*x])^ n*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(n + p + 1)) + e^2*(m + n + 2*p + 2)/(4*g^2*(n + p + 1))* Int[(e*Cos[a + b*x])^(m - 2)*(f*Sin[a + b*x])^ n*(g*Sin[c + d*x])^(p + 2), x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && LtQ[p, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p] && (LtQ[p, -2] || EqQ[m, 2] || EqQ[m, 3])
+Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -(e*Sin[a + b*x])^m*(f*Cos[a + b*x])^ n*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(n + p + 1)) + e^2*(m + n + 2*p + 2)/(4*g^2*(n + p + 1))* Int[(e*Sin[a + b*x])^(m - 2)*(f*Cos[a + b*x])^ n*(g*Sin[c + d*x])^(p + 2), x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && LtQ[p, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p] && (LtQ[p, -2] || EqQ[m, 2] || EqQ[m, 3])
+Int[(e_.*cos[a_. + b_.*x_])^m_*(f_.*sin[a_. + b_.*x_])^ n_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := e*(e*Cos[a + b*x])^(m - 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*f*(n + p + 1)) + e^2*(m + p - 1)/(f^2*(n + p + 1))* Int[(e*Cos[a + b*x])^(m - 2)*(f*Sin[a + b*x])^(n + 2)*(g* Sin[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && LtQ[n, -1] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
+Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -e*(e*Sin[a + b*x])^(m - 1)*(f*Cos[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*f*(n + p + 1)) + e^2*(m + p - 1)/(f^2*(n + p + 1))* Int[(e*Sin[a + b*x])^(m - 2)*(f*Cos[a + b*x])^(n + 2)*(g* Sin[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && LtQ[n, -1] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
+Int[(e_.*cos[a_. + b_.*x_])^m_*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := e*(e*Cos[a + b*x])^(m - 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*f*(m + n + 2*p)) + e^2*(m + p - 1)/(m + n + 2*p)* Int[(e*Cos[a + b*x])^(m - 2)*(f*Sin[a + b*x])^n*(g*Sin[c + d*x])^ p, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && NeQ[m + n + 2*p, 0] && IntegersQ[2*m, 2*n, 2*p]
+Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -e*(e*Sin[a + b*x])^(m - 1)*(f*Cos[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*f*(m + n + 2*p)) + e^2*(m + p - 1)/(m + n + 2*p)* Int[(e*Sin[a + b*x])^(m - 2)*(f*Cos[a + b*x])^n*(g*Sin[c + d*x])^ p, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && NeQ[m + n + 2*p, 0] && IntegersQ[2*m, 2*n, 2*p]
+Int[(e_.*cos[a_. + b_.*x_])^m_*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -f*(e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n - 1)*(g*Sin[c + d*x])^p/(b*e*(m + n + 2*p)) + 2*f*g*(n + p - 1)/(e*(m + n + 2*p))* Int[(e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n - 1)*(g* Sin[c + d*x])^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n + 2*p, 0] && IntegersQ[2*m, 2*n, 2*p]
+Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := f*(e*Sin[a + b*x])^(m + 1)*(f*Cos[a + b*x])^(n - 1)*(g*Sin[c + d*x])^p/(b*e*(m + n + 2*p)) + 2*f*g*(n + p - 1)/(e*(m + n + 2*p))* Int[(e*Sin[a + b*x])^(m + 1)*(f*Cos[a + b*x])^(n - 1)*(g* Sin[c + d*x])^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n + 2*p, 0] && IntegersQ[2*m, 2*n, 2*p]
+Int[(e_.*cos[a_. + b_.*x_])^m_*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -(e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*e*f*(m + p + 1)) + f*(m + n + 2*p + 2)/(2*e*g*(m + p + 1))* Int[(e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n - 1)*(g* Sin[c + d*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && GtQ[n, 0] && LtQ[p, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
+Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := (e*Sin[a + b*x])^(m + 1)*(f*Cos[a + b*x])^(n + 1)*(g*Sin[c + d*x])^ p/(b*e*f*(m + p + 1)) + f*(m + n + 2*p + 2)/(2*e*g*(m + p + 1))* Int[(e*Sin[a + b*x])^(m + 1)*(f*Cos[a + b*x])^(n - 1)*(g* Sin[c + d*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && GtQ[n, 0] && LtQ[p, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
+Int[(e_.*cos[a_. + b_.*x_])^m_*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := -(e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*e*f*(m + p + 1)) + (m + n + 2*p + 2)/(e^2*(m + p + 1))* Int[(e*Cos[a + b*x])^(m + 2)*(f*Sin[a + b*x])^n*(g*Sin[c + d*x])^ p, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
+Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := (e*Sin[a + b*x])^(m + 1)*(f*Cos[a + b*x])^(n + 1)*(g*Sin[c + d*x])^ p/(b*e*f*(m + p + 1)) + (m + n + 2*p + 2)/(e^2*(m + p + 1))* Int[(e*Sin[a + b*x])^(m + 2)*(f*Cos[a + b*x])^n*(g*Sin[c + d*x])^ p, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
+(* Int[(e_.*cos[a_.+b_.*x_])^m_*(f_.*sin[a_.+b_.*x_])^n_.*(g_.*sin[c_. +d_.*x_])^p_,x_Symbol] := -(e*Cos[a+b*x])^(m+1)*(f*Sin[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*e*f*( m+p+1)*(Sin[a+b*x]^2)^((n+p+1)/2))* Hypergeometric2F1[-(n+p-1)/2,(m+p+1)/2,(m+p+3)/2,Cos[a+b*x]^2] /; FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && EqQ[b*c-a*d,0] && EqQ[d/b,2] &&  Not[IntegerQ[p]] && Not[IntegerQ[m+p]] && Not[IntegerQ[n+p]] *)
+Int[(e_.*cos[a_. + b_.*x_])^m_.*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol] := (g*Sin[c + d*x])^p/((e*Cos[a + b*x])^p*(f*Sin[a + b*x])^p)* Int[(e*Cos[a + b*x])^(m + p)*(f*Sin[a + b*x])^(n + p), x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]]
+Int[(e_.*cos[a_. + b_.*x_])^m_.*sin[c_. + d_.*x_], x_Symbol] := -(m + 2)*(e*Cos[a + b*x])^(m + 1)* Cos[(m + 1)*(a + b*x)]/(d*e*(m + 1)) /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, Abs[m + 2]]
diff --git a/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m
new file mode 100755
index 0000000..3a7a435
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m	
@@ -0,0 +1,76 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.7.5 Inert trig functions *)
+Int[(a_.*F_[c_. + d_.*x_]^p_)^n_, x_Symbol] := With[{v = ActivateTrig[F[c + d*x]]}, a^IntPart[n]*(v/NonfreeFactors[v, x])^(p*IntPart[n])*(a*v^p)^ FracPart[n]/NonfreeFactors[v, x]^(p*FracPart[n])* Int[NonfreeFactors[v, x]^(n*p), x]] /; FreeQ[{a, c, d, n, p}, x] && InertTrigQ[F] && Not[IntegerQ[n]] && IntegerQ[p]
+Int[(a_.*(b_.*F_[c_. + d_.*x_])^p_)^n_., x_Symbol] := With[{v = ActivateTrig[F[c + d*x]]}, a^IntPart[n]*(a*(b*v)^p)^FracPart[n]/(b*v)^(p*FracPart[n])* Int[(b*v)^(n*p), x]] /; FreeQ[{a, b, c, d, n, p}, x] && InertTrigQ[F] && Not[IntegerQ[n]] && Not[IntegerQ[p]]
+Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])
+Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -d/(b*c)* Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])
+Int[u_*Cosh[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x]
+Int[u_*Sinh[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x]
+Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, 1/(b*c)* Subst[Int[SubstFor[1/x, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cot] || EqQ[F, cot])
+Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -1/(b*c)* Subst[Int[SubstFor[1/x, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Tan] || EqQ[F, tan])
+Int[u_*Coth[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, 1/(b*c)* Subst[Int[SubstFor[1/x, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x]
+Int[u_*Tanh[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, 1/(b*c)* Subst[Int[SubstFor[1/x, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x]
+Int[u_*F_[c_.*(a_. + b_.*x_)]^2, x_Symbol] := With[{d = FreeFactors[Tan[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d] /; FunctionOfQ[Tan[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Sec] || EqQ[F, sec])
+Int[u_/cos[c_.*(a_. + b_.*x_)]^2, x_Symbol] := With[{d = FreeFactors[Tan[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d] /; FunctionOfQ[Tan[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u]
+Int[u_*F_[c_.*(a_. + b_.*x_)]^2, x_Symbol] := With[{d = FreeFactors[Cot[c*(a + b*x)], x]}, -d/(b*c)* Subst[Int[SubstFor[1, Cot[c*(a + b*x)]/d, u, x], x], x, Cot[c*(a + b*x)]/d] /; FunctionOfQ[Cot[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Csc] || EqQ[F, csc])
+Int[u_/sin[c_.*(a_. + b_.*x_)]^2, x_Symbol] := With[{d = FreeFactors[Cot[c*(a + b*x)], x]}, -d/(b*c)* Subst[Int[SubstFor[1, Cot[c*(a + b*x)]/d, u, x], x], x, Cot[c*(a + b*x)]/d] /; FunctionOfQ[Cot[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u]
+Int[u_*Sech[c_.*(a_. + b_.*x_)]^2, x_Symbol] := With[{d = FreeFactors[Tanh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Tanh[c*(a + b*x)]/d, u, x], x], x, Tanh[c*(a + b*x)]/d] /; FunctionOfQ[Tanh[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u]
+Int[u_*Csch[c_.*(a_. + b_.*x_)]^2, x_Symbol] := With[{d = FreeFactors[Coth[c*(a + b*x)], x]}, -d/(b*c)* Subst[Int[SubstFor[1, Coth[c*(a + b*x)]/d, u, x], x], x, Coth[c*(a + b*x)]/d] /; FunctionOfQ[Coth[c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u]
+Int[u_*F_[c_.*(a_. + b_.*x_)]^n_., x_Symbol] := With[{d = FreeFactors[Tan[c*(a + b*x)], x]}, 1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[1/(x^n*(1 + d^2*x^2)), Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d] /; FunctionOfQ[Tan[c*(a + b*x)]/d, u, x, True] && TryPureTanSubst[ActivateTrig[u]*Cot[c*(a + b*x)]^n, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && (EqQ[F, Cot] || EqQ[F, cot])
+Int[u_*F_[c_.*(a_. + b_.*x_)]^n_., x_Symbol] := With[{d = FreeFactors[Cot[c*(a + b*x)], x]}, -1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[1/(x^n*(1 + d^2*x^2)), Cot[c*(a + b*x)]/d, u, x], x], x, Cot[c*(a + b*x)]/d] /; FunctionOfQ[Cot[c*(a + b*x)]/d, u, x, True] && TryPureTanSubst[ActivateTrig[u]*Tan[c*(a + b*x)]^n, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && (EqQ[F, Tan] || EqQ[F, tan])
+Int[u_*Coth[c_.*(a_. + b_.*x_)]^n_., x_Symbol] := With[{d = FreeFactors[Tanh[c*(a + b*x)], x]}, 1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[1/(x^n*(1 - d^2*x^2)), Tanh[c*(a + b*x)]/d, u, x], x], x, Tanh[c*(a + b*x)]/d] /; FunctionOfQ[Tanh[c*(a + b*x)]/d, u, x, True] && TryPureTanSubst[ActivateTrig[u]*Coth[c*(a + b*x)]^n, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[n]
+Int[u_*Tanh[c_.*(a_. + b_.*x_)]^n_., x_Symbol] := With[{d = FreeFactors[Coth[c*(a + b*x)], x]}, 1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[1/(x^n*(1 - d^2*x^2)), Coth[c*(a + b*x)]/d, u, x], x], x, Coth[c*(a + b*x)]/d] /; FunctionOfQ[Coth[c*(a + b*x)]/d, u, x, True] && TryPureTanSubst[ActivateTrig[u]*Tanh[c*(a + b*x)]^n, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[n]
+If[TrueQ[$LoadShowSteps], Int[u_, x_Symbol] := With[{v = FunctionOfTrig[u, x]}, ShowStep["", "Int[F[Cot[a+b*x]],x]", "-1/b*Subst[Int[F[x]/(1+x^2),x],x,Cot[a+b*x]]", Hold[ With[{d = FreeFactors[Cot[v], x]}, Dist[-d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Cot[v]/d, u, x], x], x, Cot[v]/d], x]]]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Cot[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x]] /; SimplifyFlag, Int[u_, x_Symbol] := With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors[Cot[v], x]}, Dist[-d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Cot[v]/d, u, x], x], x, Cot[v]/d], x]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Cot[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x]]]
+If[TrueQ[$LoadShowSteps], Int[u_, x_Symbol] := With[{v = FunctionOfTrig[u, x]}, ShowStep["", "Int[F[Tan[a+b*x]],x]", "1/b*Subst[Int[F[x]/(1+x^2),x],x,Tan[a+b*x]]", Hold[ With[{d = FreeFactors[Tan[v], x]}, Dist[d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]]]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Tan[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x]] /; SimplifyFlag, Int[u_, x_Symbol] := With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors[Tan[v], x]}, Dist[d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Tan[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x]]]
+Int[F_[a_. + b_.*x_]^p_.*G_[c_. + d_.*x_]^q_., x_Symbol] := Int[ExpandTrigReduce[ActivateTrig[F[a + b*x]^p*G[c + d*x]^q], x], x] /; FreeQ[{a, b, c, d}, x] && (EqQ[F, sin] || EqQ[F, cos]) && (EqQ[G, sin] || EqQ[G, cos]) && IGtQ[p, 0] && IGtQ[q, 0]
+Int[F_[a_. + b_.*x_]^p_.*G_[c_. + d_.*x_]^q_.*H_[e_. + f_.*x_]^r_., x_Symbol] := Int[ExpandTrigReduce[ ActivateTrig[F[a + b*x]^p*G[c + d*x]^q*H[e + f*x]^r], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && (EqQ[F, sin] || EqQ[F, cos]) && (EqQ[G, sin] || EqQ[G, cos]) && (EqQ[H, sin] || EqQ[H, cos]) && IGtQ[p, 0] && IGtQ[q, 0] && IGtQ[r, 0]
+Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])
+Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -d/(b*c)* Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])
+Int[u_*Cosh[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x]
+Int[u_*Sinh[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x]
+Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, 1/(b*c)* Subst[Int[SubstFor[1/x, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cot] || EqQ[F, cot])
+Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -1/(b*c)* Subst[Int[SubstFor[1/x, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Tan] || EqQ[F, tan])
+Int[u_*Coth[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, 1/(b*c)* Subst[Int[SubstFor[1/x, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x]
+Int[u_*Tanh[c_.*(a_. + b_.*x_)], x_Symbol] := With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, 1/(b*c)* Subst[Int[SubstFor[1/x, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x]
+Int[u_*F_[c_.*(a_. + b_.*x_)]^n_, x_Symbol] := With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[ SubstFor[(1 - d^2*x^2)^((n - 1)/2), Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Cos] || EqQ[F, cos])
+Int[u_*F_[c_.*(a_. + b_.*x_)]^n_, x_Symbol] := With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[ SubstFor[(1 - d^2*x^2)^((-n - 1)/2), Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Sec] || EqQ[F, sec])
+Int[u_*F_[c_.*(a_. + b_.*x_)]^n_, x_Symbol] := With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -d/(b*c)* Subst[Int[ SubstFor[(1 - d^2*x^2)^((n - 1)/2), Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Sin] || EqQ[F, sin])
+Int[u_*F_[c_.*(a_. + b_.*x_)]^n_, x_Symbol] := With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -d/(b*c)* Subst[Int[ SubstFor[(1 - d^2*x^2)^((-n - 1)/2), Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Csc] || EqQ[F, csc])
+Int[u_*Cosh[c_.*(a_. + b_.*x_)]^n_, x_Symbol] := With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[ SubstFor[(1 + d^2*x^2)^((n - 1)/2), Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]
+Int[u_*Sech[c_.*(a_. + b_.*x_)]^n_, x_Symbol] := With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[ SubstFor[(1 + d^2*x^2)^((-n - 1)/2), Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]
+Int[u_*Sinh[c_.*(a_. + b_.*x_)]^n_, x_Symbol] := With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[ SubstFor[(-1 + d^2*x^2)^((n - 1)/2), Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]
+Int[u_*Csch[c_.*(a_. + b_.*x_)]^n_, x_Symbol] := With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[ SubstFor[(-1 + d^2*x^2)^((-n - 1)/2), Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]
+Int[u_*F_[c_.*(a_. + b_.*x_)]^n_, x_Symbol] := With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, 1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[(1 - d^2*x^2)^((n - 1)/2)/x^n, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Cot] || EqQ[F, cot])
+Int[u_*F_[c_.*(a_. + b_.*x_)]^n_, x_Symbol] := With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[(1 - d^2*x^2)^((n - 1)/2)/x^n, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Tan] || EqQ[F, tan])
+Int[u_*Coth[c_.*(a_. + b_.*x_)]^n_, x_Symbol] := With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, 1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[(1 + d^2*x^2)^((n - 1)/2)/x^n, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]
+Int[u_*Tanh[c_.*(a_. + b_.*x_)]^n_, x_Symbol] := With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, 1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[(-1 + d^2*x^2)^((n - 1)/2)/x^n, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]
+Int[u_*(v_ + d_.*F_[c_.*(a_. + b_.*x_)]^n_.), x_Symbol] := With[{e = FreeFactors[Sin[c*(a + b*x)], x]}, Int[ActivateTrig[u*v], x] + d*Int[ActivateTrig[u]*Cos[c*(a + b*x)]^n, x] /; FunctionOfQ[Sin[c*(a + b*x)]/e, u, x]] /; FreeQ[{a, b, c, d}, x] && Not[FreeQ[v, x]] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Cos] || EqQ[F, cos])
+Int[u_*(v_ + d_.*F_[c_.*(a_. + b_.*x_)]^n_.), x_Symbol] := With[{e = FreeFactors[Cos[c*(a + b*x)], x]}, Int[ActivateTrig[u*v], x] + d*Int[ActivateTrig[u]*Sin[c*(a + b*x)]^n, x] /; FunctionOfQ[Cos[c*(a + b*x)]/e, u, x]] /; FreeQ[{a, b, c, d}, x] && Not[FreeQ[v, x]] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Sin] || EqQ[F, sin])
+If[TrueQ[$LoadShowSteps], Int[u_, x_Symbol] := With[{v = FunctionOfTrig[u, x]}, ShowStep["", "Int[F[Sin[a+b*x]]*Cos[a+b*x],x]", "Subst[Int[F[x],x],x,Sin[a+b*x]]/b", Hold[ With[{d = FreeFactors[Sin[v], x]}, Dist[d/Coefficient[v, x, 1], Subst[Int[SubstFor[1, Sin[v]/d, u/Cos[v], x], x], x, Sin[v]/d], x]]]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Sin[v], x], u/Cos[v], x]] /; SimplifyFlag, Int[u_, x_Symbol] := With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors[Sin[v], x]}, Dist[d/Coefficient[v, x, 1], Subst[Int[SubstFor[1, Sin[v]/d, u/Cos[v], x], x], x, Sin[v]/d], x]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Sin[v], x], u/Cos[v], x]]]
+If[TrueQ[$LoadShowSteps], Int[u_, x_Symbol] := With[{v = FunctionOfTrig[u, x]}, ShowStep["", "Int[F[Cos[a+b*x]]*Sin[a+b*x],x]", "-Subst[Int[F[x],x],x,Cos[a+b*x]]/b", Hold[ With[{d = FreeFactors[Cos[v], x]}, Dist[-d/Coefficient[v, x, 1], Subst[Int[SubstFor[1, Cos[v]/d, u/Sin[v], x], x], x, Cos[v]/d], x]]]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Cos[v], x], u/Sin[v], x]] /; SimplifyFlag, Int[u_, x_Symbol] := With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors[Cos[v], x]}, Dist[-d/Coefficient[v, x, 1], Subst[Int[SubstFor[1, Cos[v]/d, u/Sin[v], x], x], x, Cos[v]/d], x]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Cos[v], x], u/Sin[v], x]]]
+Int[u_.*(a_. + b_.*cos[d_. + e_.*x_]^2 + c_.*sin[d_. + e_.*x_]^2)^p_., x_Symbol] := (a + c)^p*Int[ActivateTrig[u], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b - c, 0]
+Int[u_.*(a_. + b_.*tan[d_. + e_.*x_]^2 + c_.*sec[d_. + e_.*x_]^2)^p_., x_Symbol] := (a + c)^p*Int[ActivateTrig[u], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b + c, 0]
+Int[u_.*(a_. + b_.*cot[d_. + e_.*x_]^2 + c_.*csc[d_. + e_.*x_]^2)^p_., x_Symbol] := (a + c)^p*Int[ActivateTrig[u], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b + c, 0]
+Int[u_/y_, x_Symbol] := With[{q = DerivativeDivides[ActivateTrig[y], ActivateTrig[u], x]}, q*Log[RemoveContent[ActivateTrig[y], x]] /; Not[FalseQ[q]]] /; Not[InertTrigFreeQ[u]]
+Int[u_/(y_*w_), x_Symbol] := With[{q = DerivativeDivides[ActivateTrig[y*w], ActivateTrig[u], x]}, q*Log[RemoveContent[ActivateTrig[y*w], x]] /; Not[FalseQ[q]]] /; Not[InertTrigFreeQ[u]]
+Int[u_*y_^m_., x_Symbol] := With[{q = DerivativeDivides[ActivateTrig[y], ActivateTrig[u], x]}, q*ActivateTrig[y^(m + 1)]/(m + 1) /; Not[FalseQ[q]]] /; FreeQ[m, x] && NeQ[m, -1] && Not[InertTrigFreeQ[u]]
+Int[u_*y_^m_.*z_^n_., x_Symbol] := With[{q = DerivativeDivides[ActivateTrig[y*z], ActivateTrig[u*z^(n - m)], x]}, q*ActivateTrig[y^(m + 1)*z^(m + 1)]/(m + 1) /; Not[FalseQ[q]]] /; FreeQ[{m, n}, x] && NeQ[m, -1] && Not[InertTrigFreeQ[u]]
+Int[u_.*(a_.*F_[c_. + d_.*x_]^p_)^n_, x_Symbol] := With[{v = ActivateTrig[F[c + d*x]]}, a^IntPart[n]*(v/NonfreeFactors[v, x])^(p*IntPart[n])*(a*v^p)^ FracPart[n]/NonfreeFactors[v, x]^(p*FracPart[n])* Int[ActivateTrig[u]*NonfreeFactors[v, x]^(n*p), x]] /; FreeQ[{a, c, d, n, p}, x] && InertTrigQ[F] && Not[IntegerQ[n]] && IntegerQ[p]
+Int[u_.*(a_.*(b_.*F_[c_. + d_.*x_])^p_)^n_., x_Symbol] := With[{v = ActivateTrig[F[c + d*x]]}, a^IntPart[n]*(a*(b*v)^p)^FracPart[n]/(b*v)^(p*FracPart[n])* Int[ActivateTrig[u]*(b*v)^(n*p), x]] /; FreeQ[{a, b, c, d, n, p}, x] && InertTrigQ[F] && Not[IntegerQ[n]] && Not[IntegerQ[p]]
+If[TrueQ[$LoadShowSteps], Int[u_, x_Symbol] := With[{v = FunctionOfTrig[u, x]}, ShowStep["", "Int[F[Tan[a+b*x]],x]", "1/b*Subst[Int[F[x]/(1+x^2),x],x,Tan[a+b*x]]", Hold[ With[{d = FreeFactors[Tan[v], x]}, Dist[d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]]]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Tan[v], x], u, x]] /; SimplifyFlag && InverseFunctionFreeQ[u, x] && Not[ MatchQ[u, v_.*(c_.*tan[w_]^n_.*tan[z_]^n_.)^p_. /; FreeQ[{c, p}, x] && IntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]], Int[u_, x_Symbol] := With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors[Tan[v], x]}, Dist[d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] && Not[ MatchQ[u, v_.*(c_.*tan[w_]^n_.*tan[z_]^n_.)^p_. /; FreeQ[{c, p}, x] && IntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]]]
+Int[u_*(c_.*sin[v_])^m_, x_Symbol] := With[{w = FunctionOfTrig[u*Sin[v/2]^(2*m)/(c*Tan[v/2])^m, x]}, (c*Sin[v])^m*(c*Tan[v/2])^m/Sin[v/2]^(2*m)* Int[u*Sin[v/2]^(2*m)/(c*Tan[v/2])^m, x] /; Not[FalseQ[w]] && FunctionOfQ[NonfreeFactors[Tan[w], x], u*Sin[v/2]^(2*m)/(c*Tan[v/2])^m, x]] /; FreeQ[c, x] && LinearQ[v, x] && IntegerQ[m + 1/2] && Not[SumQ[u]] && InverseFunctionFreeQ[u, x]
+Int[u_.*(a_.*tan[c_. + d_.*x_]^n_. + b_.*sec[c_. + d_.*x_]^n_.)^p_, x_Symbol] := Int[ActivateTrig[u]*Sec[c + d*x]^(n*p)*(b + a*Sin[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]
+Int[u_.*(a_.*cot[c_. + d_.*x_]^n_. + b_.*csc[c_. + d_.*x_]^n_.)^p_, x_Symbol] := Int[ActivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]
+Int[u_*(a_*F_[c_. + d_.*x_]^p_. + b_.*F_[c_. + d_.*x_]^q_.)^n_., x_Symbol] := Int[ActivateTrig[u*F[c + d*x]^(n*p)*(a + b*F[c + d*x]^(q - p))^n], x] /; FreeQ[{a, b, c, d, p, q}, x] && InertTrigQ[F] && IntegerQ[n] && PosQ[q - p]
+Int[u_*(a_*F_[d_. + e_.*x_]^p_. + b_.*F_[d_. + e_.*x_]^q_. + c_.*F_[d_. + e_.*x_]^r_.)^n_., x_Symbol] := Int[ActivateTrig[ u*F[d + e*x]^(n*p)*(a + b*F[d + e*x]^(q - p) + c*F[d + e*x]^(r - p))^n], x] /; FreeQ[{a, b, c, d, e, p, q, r}, x] && InertTrigQ[F] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]
+Int[u_*(a_ + b_.*F_[d_. + e_.*x_]^p_. + c_.*F_[d_. + e_.*x_]^q_.)^n_., x_Symbol] := Int[ActivateTrig[ u*F[d + e*x]^(n*p)*(b + a*F[d + e*x]^(-p) + c*F[d + e*x]^(q - p))^ n], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && InertTrigQ[F] && IntegerQ[n] && NegQ[p]
+Int[u_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_., x_Symbol] := Int[ActivateTrig[u]*(a*E^(-a/b*(c + d*x)))^n, x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 + b^2, 0]
+Int[u_, x_Symbol] := Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]
+Int[u_.*(a_*v_)^p_, x_Symbol] := With[{uu = ActivateTrig[u], vv = ActivateTrig[v]}, a^IntPart[p]*(a*vv)^FracPart[p]/(vv^FracPart[p])* Int[uu*vv^p, x]] /; FreeQ[{a, p}, x] && Not[IntegerQ[p]] && Not[InertTrigFreeQ[v]]
+Int[u_.*(v_^m_)^p_, x_Symbol] := With[{uu = ActivateTrig[u], vv = ActivateTrig[v]}, (vv^m)^FracPart[p]/(vv^(m*FracPart[p]))*Int[uu*vv^(m*p), x]] /; FreeQ[{m, p}, x] && Not[IntegerQ[p]] && Not[InertTrigFreeQ[v]]
+Int[u_.*(v_^m_.*w_^n_.)^p_, x_Symbol] := With[{uu = ActivateTrig[u], vv = ActivateTrig[v], ww = ActivateTrig[w]}, (vv^m*ww^n)^FracPart[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p]))* Int[uu*vv^(m*p)*ww^(n*p), x]] /; FreeQ[{m, n, p}, x] && Not[IntegerQ[p]] && (Not[InertTrigFreeQ[v]] || Not[InertTrigFreeQ[w]])
+Int[u_, x_Symbol] := With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /; Not[InertTrigFreeQ[u]]
+If[TrueQ[$LoadShowSteps], Int[u_, x_Symbol] := With[{w = Block[{$ShowSteps = False, $StepCounter = Null}, Int[SubstFor[ 1/(1 + FreeFactors[Tan[FunctionOfTrig[u, x]/2], x]^2*x^2), Tan[FunctionOfTrig[u, x]/2]/ FreeFactors[Tan[FunctionOfTrig[u, x]/2], x], u, x], x]]}, ShowStep["", "Int[F[Sin[a+b*x],Cos[a+b*x]],x]", "2/b*Subst[Int[1/(1+x^2)*F[2*x/(1+x^2),(1-x^2)/(1+x^2)],x],x, Tan[(a+b*x)/2]]", Hold[ Module[{v = FunctionOfTrig[u, x], d}, d = FreeFactors[Tan[v/2], x]; Dist[2*d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v/2]/d, u, x], x], x, Tan[v/2]/d], x]]]] /; CalculusFreeQ[w, x]] /; SimplifyFlag && InverseFunctionFreeQ[u, x] && Not[FalseQ[FunctionOfTrig[u, x]]], Int[u_, x_Symbol] := With[{w = Block[{$ShowSteps = False, $StepCounter = Null}, Int[SubstFor[ 1/(1 + FreeFactors[Tan[FunctionOfTrig[u, x]/2], x]^2*x^2), Tan[FunctionOfTrig[u, x]/2]/ FreeFactors[Tan[FunctionOfTrig[u, x]/2], x], u, x], x]]}, Module[{v = FunctionOfTrig[u, x], d}, d = FreeFactors[Tan[v/2], x]; Dist[2*d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v/2]/d, u, x], x], x, Tan[v/2]/d], x]] /; CalculusFreeQ[w, x]] /; InverseFunctionFreeQ[u, x] && Not[FalseQ[FunctionOfTrig[u, x]]]]
+(* If[TrueQ[$LoadShowSteps], Int[u_,x_Symbol] := With[{v=FunctionOfTrig[u,x]}, ShowStep["","Int[F[Sin[a+b*x],Cos[a+b*x]],x]","2/b*Subst[Int[1/(1+x^ 2)*F[2*x/(1+x^2),(1-x^2)/(1+x^2)],x],x,Tan[(a+b*x)/2]]",Hold[ With[{d=FreeFactors[Tan[v/2],x]}, Dist[2*d/Coefficient[v,x,1],Subst[Int[SubstFor[1/(1+d^2*x^2),Tan[v/ 2]/d,u,x],x],x,Tan[v/2]/d],x]]]] /; Not[FalseQ[v]]] /; SimplifyFlag && InverseFunctionFreeQ[u,x], Int[u_,x_Symbol] := With[{v=FunctionOfTrig[u,x]}, With[{d=FreeFactors[Tan[v/2],x]}, Dist[2*d/Coefficient[v,x,1],Subst[Int[SubstFor[1/(1+d^2*x^2),Tan[v/ 2]/d,u,x],x],x,Tan[v/2]/d],x]] /; Not[FalseQ[v]]] /; InverseFunctionFreeQ[u,x]] *)
+Int[u_, x_Symbol] := With[{v = ActivateTrig[u]}, CannotIntegrate[v, x]] /; Not[InertTrigFreeQ[u]]
diff --git a/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m
new file mode 100755
index 0000000..bb1370d
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m	
@@ -0,0 +1,31 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p *)
+Int[(c_. + d_.*x_)^m_.*Sin[a_. + b_.*x_]^n_.*Cos[a_. + b_.*x_], x_Symbol] := (c + d*x)^m*Sin[a + b*x]^(n + 1)/(b*(n + 1)) - d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n + 1), x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(c_. + d_.*x_)^m_.*Sin[a_. + b_.*x_]*Cos[a_. + b_.*x_]^n_., x_Symbol] := -(c + d*x)^m*Cos[a + b*x]^(n + 1)/(b*(n + 1)) + d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Cos[a + b*x]^(n + 1), x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(c_. + d_.*x_)^m_.*Sin[a_. + b_.*x_]^n_.*Cos[a_. + b_.*x_]^p_., x_Symbol] := Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(c_. + d_.*x_)^m_.*Sin[a_. + b_.*x_]^n_.*Tan[a_. + b_.*x_]^p_., x_Symbol] := -Int[(c + d*x)^m*Sin[a + b*x]^n*Tan[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Sin[a + b*x]^(n - 2)*Tan[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(c_. + d_.*x_)^m_.*Cos[a_. + b_.*x_]^n_.*Cot[a_. + b_.*x_]^p_., x_Symbol] := -Int[(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(c_. + d_.*x_)^m_.*Sec[a_. + b_.*x_]^n_.*Tan[a_. + b_.*x_]^p_., x_Symbol] := (c + d*x)^m*Sec[a + b*x]^n/(b*n) - d*m/(b*n)*Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*Csc[a_. + b_.*x_]^n_.*Cot[a_. + b_.*x_]^p_., x_Symbol] := -(c + d*x)^m*Csc[a + b*x]^n/(b*n) + d*m/(b*n)*Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*Sec[a_. + b_.*x_]^2*Tan[a_. + b_.*x_]^n_., x_Symbol] := (c + d*x)^m*Tan[a + b*x]^(n + 1)/(b*(n + 1)) - d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Tan[a + b*x]^(n + 1), x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(c_. + d_.*x_)^m_.*Csc[a_. + b_.*x_]^2*Cot[a_. + b_.*x_]^n_., x_Symbol] := -(c + d*x)^m*Cot[a + b*x]^(n + 1)/(b*(n + 1)) + d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Cot[a + b*x]^(n + 1), x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(c_. + d_.*x_)^m_.*Sec[a_. + b_.*x_]*Tan[a_. + b_.*x_]^p_, x_Symbol] := -Int[(c + d*x)^m*Sec[a + b*x]*Tan[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Sec[a + b*x]^3*Tan[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 0]
+Int[(c_. + d_.*x_)^m_.*Sec[a_. + b_.*x_]^n_.*Tan[a_. + b_.*x_]^p_, x_Symbol] := -Int[(c + d*x)^m*Sec[a + b*x]^n*Tan[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Sec[a + b*x]^(n + 2)*Tan[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p/2, 0]
+Int[(c_. + d_.*x_)^m_.*Csc[a_. + b_.*x_]*Cot[a_. + b_.*x_]^p_, x_Symbol] := -Int[(c + d*x)^m*Csc[a + b*x]*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csc[a + b*x]^3*Cot[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 0]
+Int[(c_. + d_.*x_)^m_.*Csc[a_. + b_.*x_]^n_.*Cot[a_. + b_.*x_]^p_, x_Symbol] := -Int[(c + d*x)^m*Csc[a + b*x]^n*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csc[a + b*x]^(n + 2)*Cot[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p/2, 0]
+Int[(c_. + d_.*x_)^m_.*Sec[a_. + b_.*x_]^n_.*Tan[a_. + b_.*x_]^p_., x_Symbol] := Module[{u = IntHide[Sec[a + b*x]^n*Tan[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[(c + d*x)^(m - 1)*u, x]] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 0] && (IntegerQ[n/2] || IntegerQ[(p - 1)/2])
+Int[(c_. + d_.*x_)^m_.*Csc[a_. + b_.*x_]^n_.*Cot[a_. + b_.*x_]^p_., x_Symbol] := Module[{u = IntHide[Csc[a + b*x]^n*Cot[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[(c + d*x)^(m - 1)*u, x]] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 0] && (IntegerQ[n/2] || IntegerQ[(p - 1)/2])
+Int[(c_. + d_.*x_)^m_.*Csc[a_. + b_.*x_]^n_.*Sec[a_. + b_.*x_]^n_., x_Symbol] := 2^n*Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n, x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
+Int[(c_. + d_.*x_)^m_.*Csc[a_. + b_.*x_]^n_.*Sec[a_. + b_.*x_]^p_., x_Symbol] := Module[{u = IntHide[Csc[a + b*x]^n*Sec[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[(c + d*x)^(m - 1)*u, x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]
+Int[u_^m_.*F_[v_]^n_.*G_[w_]^p_., x_Symbol] := Int[ExpandToSum[u, x]^m*F[ExpandToSum[v, x]]^n* G[ExpandToSum[v, x]]^p, x] /; FreeQ[{m, n, p}, x] && TrigQ[F] && TrigQ[G] && EqQ[v, w] && LinearQ[{u, v, w}, x] && Not[LinearMatchQ[{u, v, w}, x]]
+Int[(e_. + f_.*x_)^m_.* Cos[c_. + d_.*x_]*(a_ + b_.*Sin[c_. + d_.*x_])^n_., x_Symbol] := (e + f*x)^m*(a + b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)) - f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Sin[c + d*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(e_. + f_.*x_)^m_.* Sin[c_. + d_.*x_]*(a_ + b_.*Cos[c_. + d_.*x_])^n_., x_Symbol] := -(e + f*x)^m*(a + b*Cos[c + d*x])^(n + 1)/(b*d*(n + 1)) + f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Cos[c + d*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(e_. + f_.*x_)^m_.* Sec[c_. + d_.*x_]^2*(a_ + b_.*Tan[c_. + d_.*x_])^n_., x_Symbol] := (e + f*x)^m*(a + b*Tan[c + d*x])^(n + 1)/(b*d*(n + 1)) - f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Tan[c + d*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(e_. + f_.*x_)^m_.* Csc[c_. + d_.*x_]^2*(a_ + b_.*Cot[c_. + d_.*x_])^n_., x_Symbol] := -(e + f*x)^m*(a + b*Cot[c + d*x])^(n + 1)/(b*d*(n + 1)) + f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Cot[c + d*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(e_. + f_.*x_)^m_.*Sec[c_. + d_.*x_]* Tan[c_. + d_.*x_]*(a_ + b_.*Sec[c_. + d_.*x_])^n_., x_Symbol] := (e + f*x)^m*(a + b*Sec[c + d*x])^(n + 1)/(b*d*(n + 1)) - f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Sec[c + d*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(e_. + f_.*x_)^m_.*Csc[c_. + d_.*x_]* Cot[c_. + d_.*x_]*(a_ + b_.*Csc[c_. + d_.*x_])^n_., x_Symbol] := -(e + f*x)^m*(a + b*Csc[c + d*x])^(n + 1)/(b*d*(n + 1)) + f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Csc[c + d*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(e_. + f_.*x_)^m_.*Sin[a_. + b_.*x_]^p_.*Sin[c_. + d_.*x_]^q_., x_Symbol] := Int[ExpandTrigReduce[(e + f*x)^m, Sin[a + b*x]^p*Sin[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IGtQ[q, 0] && IntegerQ[m]
+Int[(e_. + f_.*x_)^m_.*Cos[a_. + b_.*x_]^p_.*Cos[c_. + d_.*x_]^q_., x_Symbol] := Int[ExpandTrigReduce[(e + f*x)^m, Cos[a + b*x]^p*Cos[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IGtQ[q, 0] && IntegerQ[m]
+Int[(e_. + f_.*x_)^m_.*Sin[a_. + b_.*x_]^p_.*Cos[c_. + d_.*x_]^q_., x_Symbol] := Int[ExpandTrigReduce[(e + f*x)^m, Sin[a + b*x]^p*Cos[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IGtQ[q, 0]
+Int[(e_. + f_.*x_)^m_.*F_[a_. + b_.*x_]^p_.*G_[c_. + d_.*x_]^q_., x_Symbol] := Int[ExpandTrigExpand[(e + f*x)^m*G[c + d*x]^q, F, c + d*x, p, b/d, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && MemberQ[{Sin, Cos}, F] && MemberQ[{Sec, Csc}, G] && IGtQ[p, 0] && IGtQ[q, 0] && EqQ[b*c - a*d, 0] && IGtQ[b/d, 1]
diff --git a/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m
new file mode 100755
index 0000000..8faae2e
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m	
@@ -0,0 +1,53 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.7.7 F^(c (a+b x)) trig(d+e x)^n *)
+Int[F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_], x_Symbol] := b*c*Log[F]*F^(c*(a + b*x))* Sin[d + e*x]/(e^2 + b^2*c^2*Log[F]^2) - e*F^(c*(a + b*x))*Cos[d + e*x]/(e^2 + b^2*c^2*Log[F]^2) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*Cos[d_. + e_.*x_], x_Symbol] := b*c*Log[F]*F^(c*(a + b*x))* Cos[d + e*x]/(e^2 + b^2*c^2*Log[F]^2) + e*F^(c*(a + b*x))*Sin[d + e*x]/(e^2 + b^2*c^2*Log[F]^2) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_]^n_, x_Symbol] := b*c*Log[F]*F^(c*(a + b*x))* Sin[d + e*x]^n/(e^2*n^2 + b^2*c^2*Log[F]^2) - e*n*F^(c*(a + b*x))*Cos[d + e*x]* Sin[d + e*x]^(n - 1)/(e^2*n^2 + b^2*c^2*Log[F]^2) + (n*(n - 1)*e^2)/(e^2*n^2 + b^2*c^2*Log[F]^2)* Int[F^(c*(a + b*x))*Sin[d + e*x]^(n - 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && GtQ[n, 1]
+Int[F_^(c_.*(a_. + b_.*x_))*Cos[d_. + e_.*x_]^m_, x_Symbol] := b*c*Log[F]*F^(c*(a + b*x))* Cos[d + e*x]^m/(e^2*m^2 + b^2*c^2*Log[F]^2) + e*m*F^(c*(a + b*x))*Sin[d + e*x]* Cos[d + e*x]^(m - 1)/(e^2*m^2 + b^2*c^2*Log[F]^2) + (m*(m - 1)*e^2)/(e^2*m^2 + b^2*c^2*Log[F]^2)* Int[F^(c*(a + b*x))*Cos[d + e*x]^(m - 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*m^2 + b^2*c^2*Log[F]^2, 0] && GtQ[m, 1]
+Int[F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Sin[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) + F^(c*(a + b*x))*Cos[d + e*x]*Sin[d + e*x]^(n + 1)/(e*(n + 1)) /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n + 2)^2 + b^2*c^2*Log[F]^2, 0] && NeQ[n, -1] && NeQ[n, -2]
+Int[F_^(c_.*(a_. + b_.*x_))*Cos[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Cos[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) - F^(c*(a + b*x))*Sin[d + e*x]*Cos[d + e*x]^(n + 1)/(e*(n + 1)) /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n + 2)^2 + b^2*c^2*Log[F]^2, 0] && NeQ[n, -1] && NeQ[n, -2]
+Int[F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Sin[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) + F^(c*(a + b*x))*Cos[d + e*x]*Sin[d + e*x]^(n + 1)/(e*(n + 1)) + (e^2*(n + 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n + 1)*(n + 2))* Int[F^(c*(a + b*x))*Sin[d + e*x]^(n + 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n + 2)^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1] && NeQ[n, -2]
+Int[F_^(c_.*(a_. + b_.*x_))*Cos[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Cos[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) - F^(c*(a + b*x))*Sin[d + e*x]*Cos[d + e*x]^(n + 1)/(e*(n + 1)) + (e^2*(n + 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n + 1)*(n + 2))* Int[F^(c*(a + b*x))*Cos[d + e*x]^(n + 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n + 2)^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1] && NeQ[n, -2]
+Int[F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_]^n_, x_Symbol] := E^(I*n*(d + e*x))*Sin[d + e*x]^n/(-1 + E^(2*I*(d + e*x)))^n* Int[F^(c*(a + b*x))*(-1 + E^(2*I*(d + e*x)))^n/E^(I*n*(d + e*x)), x] /; FreeQ[{F, a, b, c, d, e, n}, x] && Not[IntegerQ[n]]
+Int[F_^(c_.*(a_. + b_.*x_))*Cos[d_. + e_.*x_]^n_, x_Symbol] := E^(I*n*(d + e*x))*Cos[d + e*x]^n/(1 + E^(2*I*(d + e*x)))^n* Int[F^(c*(a + b*x))*(1 + E^(2*I*(d + e*x)))^n/E^(I*n*(d + e*x)), x] /; FreeQ[{F, a, b, c, d, e, n}, x] && Not[IntegerQ[n]]
+Int[F_^(c_.*(a_. + b_.*x_))*Tan[d_. + e_.*x_]^n_., x_Symbol] := I^n*Int[ ExpandIntegrand[ F^(c*(a + b*x))*(1 - E^(2*I*(d + e*x)))^ n/(1 + E^(2*I*(d + e*x)))^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
+Int[F_^(c_.*(a_. + b_.*x_))*Cot[d_. + e_.*x_]^n_., x_Symbol] := (-I)^n* Int[ExpandIntegrand[ F^(c*(a + b*x))*(1 + E^(2*I*(d + e*x)))^ n/(1 - E^(2*I*(d + e*x)))^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
+Int[F_^(c_.*(a_. + b_.*x_))*Sec[d_. + e_.*x_]^n_, x_Symbol] := b*c*Log[F]* F^(c*(a + b*x))*(Sec[d + e x]^n/(e^2*n^2 + b^2*c^2*Log[F]^2)) - e*n*F^(c*(a + b*x))* Sec[d + e x]^(n + 1)*(Sin[d + e x]/(e^2*n^2 + b^2*c^2*Log[F]^2)) + e^2*n*((n + 1)/(e^2*n^2 + b^2*c^2*Log[F]^2))* Int[F^(c*(a + b*x))*Sec[d + e x]^(n + 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1]
+Int[F_^(c_.*(a_. + b_.*x_))*Csc[d_. + e_.*x_]^n_, x_Symbol] := b*c*Log[F]* F^(c*(a + b*x))*(Csc[d + e x]^n/(e^2*n^2 + b^2*c^2*Log[F]^2)) + e*n*F^(c*(a + b*x))* Csc[d + e x]^(n + 1)*(Cos[d + e x]/(e^2*n^2 + b^2*c^2*Log[F]^2)) + e^2*n*((n + 1)/(e^2*n^2 + b^2*c^2*Log[F]^2))* Int[F^(c*(a + b*x))*Csc[d + e x]^(n + 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1]
+Int[F_^(c_.*(a_. + b_.*x_))*Sec[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Sec[d + e x]^(n - 2)/(e^2*(n - 1)*(n - 2)) + F^(c*(a + b*x))*Sec[d + e x]^(n - 1)*Sin[d + e x]/(e*(n - 1)) /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && NeQ[n, 1] && NeQ[n, 2]
+Int[F_^(c_.*(a_. + b_.*x_))*Csc[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Csc[d + e x]^(n - 2)/(e^2*(n - 1)*(n - 2)) + F^(c*(a + b*x))*Csc[d + e x]^(n - 1)*Cos[d + e x]/(e*(n - 1)) /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && NeQ[n, 1] && NeQ[n, 2]
+Int[F_^(c_.*(a_. + b_.*x_))*Sec[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Sec[d + e x]^(n - 2)/(e^2*(n - 1)*(n - 2)) + F^(c*(a + b*x))*Sec[d + e x]^(n - 1)*Sin[d + e x]/(e*(n - 1)) + (e^2*(n - 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n - 1)*(n - 2))* Int[F^(c*(a + b*x))*Sec[d + e x]^(n - 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && GtQ[n, 1] && NeQ[n, 2]
+Int[F_^(c_.*(a_. + b_.*x_))*Csc[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Csc[d + e x]^(n - 2)/(e^2*(n - 1)*(n - 2)) - F^(c*(a + b*x))*Csc[d + e x]^(n - 1)*Cos[d + e x]/(e*(n - 1)) + (e^2*(n - 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n - 1)*(n - 2))* Int[F^(c*(a + b*x))*Csc[d + e x]^(n - 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && GtQ[n, 1] && NeQ[n, 2]
+(* Int[F_^(c_.*(a_.+b_.*x_))*Sec[d_.+e_.*x_]^n_.,x_Symbol] := 2^n*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(I*n*(d+e*x))/(1+E^(2*I*( d+e*x)))^n,x],x] /; FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] *)
+(* Int[F_^(c_.*(a_.+b_.*x_))*Csc[d_.+e_.*x_]^n_.,x_Symbol] := (2*I)^n*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(-I*n*(d+e*x))/(1-E^(- 2*I*(d+e*x)))^n,x],x] /; FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] *)
+Int[F_^(c_.*(a_. + b_.*x_))*Sec[d_. + k_.*Pi + e_.*x_]^n_., x_Symbol] := 2^n*E^(I*k*n*Pi)*E^(I*n*(d + e*x))* F^(c*(a + b*x))/(I*e*n + b*c*Log[F])* Hypergeometric2F1[n, n/2 - I*b*c*Log[F]/(2*e), 1 + n/2 - I*b*c*Log[F]/(2*e), -E^(2*I*k*Pi)*E^(2*I*(d + e*x))] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[4*k] && IntegerQ[n]
+Int[F_^(c_.*(a_. + b_.*x_))*Sec[d_. + e_.*x_]^n_., x_Symbol] := 2^n*E^(I*n*(d + e*x))*F^(c*(a + b*x))/(I*e*n + b*c*Log[F])* Hypergeometric2F1[n, n/2 - I*b*c*Log[F]/(2*e), 1 + n/2 - I*b*c*Log[F]/(2*e), -E^(2*I*(d + e*x))] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
+Int[F_^(c_.*(a_. + b_.*x_))*Csc[d_. + k_.*Pi + e_.*x_]^n_., x_Symbol] := (-2*I)^n*E^(I*k*n*Pi)* E^(I*n*(d + e*x))*(F^(c*(a + b*x))/(I*e*n + b*c*Log[F]))* Hypergeometric2F1[n, n/2 - I*b*c*Log[F]/(2*e), 1 + n/2 - I*b*c*Log[F]/(2*e), E^(2*I*k*Pi)*E^(2*I*(d + e*x))] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[4*k] && IntegerQ[n]
+Int[F_^(c_.*(a_. + b_.*x_))*Csc[d_. + e_.*x_]^n_., x_Symbol] := (-2*I)^n*E^(I*n*(d + e*x))*(F^(c*(a + b*x))/(I*e*n + b*c*Log[F]))* Hypergeometric2F1[n, n/2 - I*b*c*Log[F]/(2*e), 1 + n/2 - I*b*c*Log[F]/(2*e), E^(2*I*(d + e*x))] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
+Int[F_^(c_.*(a_. + b_.*x_))*Sec[d_. + e_.*x_]^n_., x_Symbol] := (1 + E^(2*I*(d + e*x)))^n*Sec[d + e*x]^n/E^(I*n*(d + e*x))* Int[SimplifyIntegrand[ F^(c*(a + b*x))*E^(I*n*(d + e*x))/(1 + E^(2*I*(d + e*x)))^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && Not[IntegerQ[n]]
+Int[F_^(c_.*(a_. + b_.*x_))*Csc[d_. + e_.*x_]^n_., x_Symbol] := (1 - E^(-2*I*(d + e*x)))^n*Csc[d + e*x]^n/E^(-I*n*(d + e*x))* Int[SimplifyIntegrand[ F^(c*(a + b*x))*E^(-I*n*(d + e*x))/(1 - E^(-2*I*(d + e*x)))^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && Not[IntegerQ[n]]
+Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Sin[d_. + e_.*x_])^n_., x_Symbol] := 2^n*f^n* Int[F^(c*(a + b*x))*Cos[d/2 - f*Pi/(4*g) + e*x/2]^(2*n), x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f^2 - g^2, 0] && ILtQ[n, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Cos[d_. + e_.*x_])^n_., x_Symbol] := 2^n*f^n*Int[F^(c*(a + b*x))*Cos[d/2 + e*x/2]^(2*n), x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f - g, 0] && ILtQ[n, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Cos[d_. + e_.*x_])^n_., x_Symbol] := 2^n*f^n*Int[F^(c*(a + b*x))*Sin[d/2 + e*x/2]^(2*n), x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f + g, 0] && ILtQ[n, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Sin[d_. + e_.*x_])^n_., x_Symbol] := (f + g*Sin[d + e*x])^n/Cos[d/2 - f*Pi/(4*g) + e*x/2]^(2*n)* Int[F^(c*(a + b*x))*Cos[d/2 - f*Pi/(4*g) + e*x/2]^(2*n), x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && EqQ[f^2 - g^2, 0] && Not[IntegerQ[n]]
+Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Cos[d_. + e_.*x_])^n_., x_Symbol] := (f + g*Cos[d + e*x])^n/Cos[d/2 + e*x/2]^(2*n)* Int[F^(c*(a + b*x))*Cos[d/2 + e*x/2]^(2*n), x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && EqQ[f - g, 0] && Not[IntegerQ[n]]
+Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Cos[d_. + e_.*x_])^n_., x_Symbol] := (f + g*Cos[d + e*x])^n/Sin[d/2 + e*x/2]^(2*n)* Int[F^(c*(a + b*x))*Sin[d/2 + e*x/2]^(2*n), x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && EqQ[f + g, 0] && Not[IntegerQ[n]]
+Int[F_^(c_.*(a_. + b_.*x_))* Cos[d_. + e_.*x_]^m_.*(f_ + g_.*Sin[d_. + e_.*x_])^n_., x_Symbol] := g^n*Int[F^(c*(a + b*x))*Tan[f*Pi/(4*g) - d/2 - e*x/2]^m, x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f^2 - g^2, 0] && IntegersQ[m, n] && EqQ[m + n, 0]
+Int[F_^(c_.*(a_. + b_.*x_))* Sin[d_. + e_.*x_]^m_.*(f_ + g_.*Cos[d_. + e_.*x_])^n_., x_Symbol] := f^n*Int[F^(c*(a + b*x))*Tan[d/2 + e*x/2]^m, x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f - g, 0] && IntegersQ[m, n] && EqQ[m + n, 0]
+Int[F_^(c_.*(a_. + b_.*x_))* Sin[d_. + e_.*x_]^m_.*(f_ + g_.*Cos[d_. + e_.*x_])^n_., x_Symbol] := f^n*Int[F^(c*(a + b*x))*Cot[d/2 + e*x/2]^m, x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f + g, 0] && IntegersQ[m, n] && EqQ[m + n, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*(h_ + i_.*Cos[d_. + e_.*x_])/(f_ + g_.*Sin[d_. + e_.*x_]), x_Symbol] := 2*i*Int[F^(c*(a + b*x))*(Cos[d + e*x]/(f + g*Sin[d + e*x])), x] + Int[F^(c*(a + b*x))*((h - i*Cos[d + e*x])/(f + g*Sin[d + e*x])), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, i}, x] && EqQ[f^2 - g^2, 0] && EqQ[h^2 - i^2, 0] && EqQ[g*h - f*i, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*(h_ + i_.*Sin[d_. + e_.*x_])/(f_ + g_.*Cos[d_. + e_.*x_]), x_Symbol] := 2*i*Int[F^(c*(a + b*x))*(Sin[d + e*x]/(f + g*Cos[d + e*x])), x] + Int[F^(c*(a + b*x))*((h - i*Sin[d + e*x])/(f + g*Cos[d + e*x])), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, i}, x] && EqQ[f^2 - g^2, 0] && EqQ[h^2 - i^2, 0] && EqQ[g*h + f*i, 0]
+Int[F_^(c_.*u_)*G_[v_]^n_., x_Symbol] := Int[F^(c*ExpandToSum[u, x])*G[ExpandToSum[v, x]]^n, x] /; FreeQ[{F, c, n}, x] && TrigQ[G] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
+Int[(f_.*x_)^m_.*F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_]^n_., x_Symbol] := Module[{u = IntHide[F^(c*(a + b*x))*Sin[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - f*m*Int[(f*x)^(m - 1)*u, x]] /; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]
+Int[(f_.*x_)^m_.*F_^(c_.*(a_. + b_.*x_))*Cos[d_. + e_.*x_]^n_., x_Symbol] := Module[{u = IntHide[F^(c*(a + b*x))*Cos[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - f*m*Int[(f*x)^(m - 1)*u, x]] /; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]
+Int[(f_.*x_)^m_*F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_], x_Symbol] := (f*x)^(m + 1)/(f*(m + 1))*F^(c*(a + b*x))*Sin[d + e*x] - e/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Cos[d + e*x], x] - b*c*Log[F]/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Sin[d + e*x], x] /; FreeQ[{F, a, b, c, d, e, f, m}, x] && (LtQ[m, -1] || SumSimplerQ[m, 1])
+Int[(f_.*x_)^m_*F_^(c_.*(a_. + b_.*x_))*Cos[d_. + e_.*x_], x_Symbol] := (f*x)^(m + 1)/(f*(m + 1))*F^(c*(a + b*x))*Cos[d + e*x] + e/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Sin[d + e*x], x] - b*c*Log[F]/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Cos[d + e*x], x] /; FreeQ[{F, a, b, c, d, e, f, m}, x] && (LtQ[m, -1] || SumSimplerQ[m, 1])
+(* Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Sin[d_.+e_.*x_]^n_.,x_ Symbol] := I^n/2^n*Int[ExpandIntegrand[(f*x)^m*F^(c*(a+b*x)),(E^(-I*(d+e*x))-E^ (I*(d+e*x)))^n,x],x] /; FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] *)
+(* Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Cos[d_.+e_.*x_]^n_.,x_ Symbol] := 1/2^n*Int[ExpandIntegrand[(f*x)^m*F^(c*(a+b*x)),(E^(-I*(d+e*x))+E^( I*(d+e*x)))^n,x],x] /; FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] *)
+Int[F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_]^m_.* Cos[f_. + g_.*x_]^n_., x_Symbol] := Int[ExpandTrigReduce[F^(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[x_^p_.*F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_]^m_.* Cos[f_. + g_.*x_]^n_., x_Symbol] := Int[ExpandTrigReduce[x^p*F^(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*G_[d_. + e_.*x_]^m_.*H_[d_. + e_.*x_]^n_., x_Symbol] := Int[ExpandTrigToExp[F^(c*(a + b*x)), G[d + e*x]^m*H[d + e*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IGtQ[m, 0] && IGtQ[n, 0] && TrigQ[G] && TrigQ[H]
+Int[F_^u_*Sin[v_]^n_., x_Symbol] := Int[ExpandTrigToExp[F^u, Sin[v]^n, x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]
+Int[F_^u_*Cos[v_]^n_., x_Symbol] := Int[ExpandTrigToExp[F^u, Cos[v]^n, x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]
+Int[F_^u_*Sin[v_]^m_.*Cos[v_]^n_., x_Symbol] := Int[ExpandTrigToExp[F^u, Sin[v]^m*Cos[v]^n, x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[m, 0] && IGtQ[n, 0]
diff --git a/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m
new file mode 100755
index 0000000..f7ff512
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m	
@@ -0,0 +1,55 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.7.8 u trig(a+b log(c x^n))^p *)
+Int[Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := x*Sin[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 + 1) - b*d*n*x*Cos[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 + 1) /; FreeQ[{a, b, c, d, n}, x] && NeQ[b^2*d^2*n^2 + 1, 0]
+Int[Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := x*Cos[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 + 1) + b*d*n*x*Sin[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 + 1) /; FreeQ[{a, b, c, d, n}, x] && NeQ[b^2*d^2*n^2 + 1, 0]
+Int[Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol] := x*Sin[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*n^2*p^2 + 1) - b*d*n*p*x*Cos[d*(a + b*Log[c*x^n])]* Sin[d*(a + b*Log[c*x^n])]^(p - 1)/(b^2*d^2*n^2*p^2 + 1) + b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 + 1)* Int[Sin[d*(a + b*Log[c*x^n])]^(p - 2), x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + 1, 0]
+Int[Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol] := x*Cos[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*n^2*p^2 + 1) + b*d*n*p*x*Cos[d*(a + b*Log[c*x^n])]^(p - 1)* Sin[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2*p^2 + 1) + b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 + 1)* Int[Cos[d*(a + b*Log[c*x^n])]^(p - 2), x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + 1, 0]
+Int[Sin[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := 1/(2^p*b^p*d^p*p^p)* Int[ExpandIntegrand[(E^(a*b*d^2*p)*x^(-1/p) - E^(-a*b*d^2*p)*x^(1/p))^p, x], x] /; FreeQ[{a, b, d}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + 1, 0]
+Int[Cos[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := 1/2^p* Int[ExpandIntegrand[(E^(a*b*d^2*p)*x^(-1/p) + E^(-a*b*d^2*p)*x^(1/p))^p, x], x] /; FreeQ[{a, b, d}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + 1, 0]
+(* Int[Sin[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol] := 1/((-2*I)^p*E^(I*a*d*p))*Int[(1-E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d* p),x] /; FreeQ[{a,b,d},x] && IntegerQ[p] *)
+(* Int[Cos[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol] := 1/(2^p*E^(I*a*d*p))*Int[(1+E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p),x]  /; FreeQ[{a,b,d},x] && IntegerQ[p] *)
+Int[Sin[d_.*(a_. + b_.*Log[x_])]^p_, x_Symbol] := Sin[d*(a + b*Log[x])]^p*x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p* Int[(1 - E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p), x] /; FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]
+Int[Cos[d_.*(a_. + b_.*Log[x_])]^p_, x_Symbol] := Cos[d*(a + b*Log[x])]^p*x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p* Int[(1 + E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p), x] /; FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]
+Int[Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Sin[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Cos[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[(e_.*x_)^m_.*Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := (m + 1)*(e*x)^(m + 1)* Sin[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 + e*(m + 1)^2) - b*d*n*(e*x)^(m + 1)* Cos[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 + e*(m + 1)^2) /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 + (m + 1)^2, 0]
+Int[(e_.*x_)^m_.*Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := (m + 1)*(e*x)^(m + 1)* Cos[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 + e*(m + 1)^2) + b*d*n*(e*x)^(m + 1)* Sin[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 + e*(m + 1)^2) /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 + (m + 1)^2, 0]
+Int[(e_.*x_)^m_.*Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol] := (m + 1)*(e*x)^(m + 1)* Sin[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2) - b*d*n*p*(e*x)^(m + 1)*Cos[d*(a + b*Log[c*x^n])]* Sin[d*(a + b*Log[c*x^n])]^(p - 1)/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2) + b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 + (m + 1)^2)* Int[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^(p - 2), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + (m + 1)^2, 0]
+Int[(e_.*x_)^m_.*Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol] := (m + 1)*(e*x)^(m + 1)* Cos[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2) + b*d*n*p*(e*x)^(m + 1)*Sin[d*(a + b*Log[c*x^n])]* Cos[d*(a + b*Log[c*x^n])]^(p - 1)/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2) + b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 + (m + 1)^2)* Int[(e*x)^m*Cos[d*(a + b*Log[c*x^n])]^(p - 2), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + (m + 1)^2, 0]
+Int[(e_.*x_)^m_.*Sin[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := (m + 1)^p/(2^p*b^p*d^p*p^p)* Int[ ExpandIntegrand[(e*x)^ m*(E^(a*b*d^2*p/(m + 1))*x^(-(m + 1)/p) - E^(-a*b*d^2*p/(m + 1))*x^((m + 1)/p))^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + (m + 1)^2, 0]
+Int[(e_.*x_)^m_.*Cos[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := 1/2^p* Int[ExpandIntegrand[(e*x)^ m*(E^(a*b*d^2*p/(m + 1))*x^(-(m + 1)/p) + E^(-a*b*d^2*p/(m + 1))*x^((m + 1)/p))^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + (m + 1)^2, 0]
+(* Int[(e_.*x_)^m_.*Sin[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol] := 1/((-2*I)^p*E^(I*a*d*p))*Int[(e*x)^m*(1-E^(2*I*a*d)*x^(2*I*b*d))^p/ x^(I*b*d*p),x] /; FreeQ[{a,b,d,e,m},x] && IntegerQ[p] *)
+(* Int[(e_.*x_)^m_.*Cos[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol] := 1/(2^p*E^(I*a*d*p))*Int[(e*x)^m*(1+E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I* b*d*p),x] /; FreeQ[{a,b,d,e,m},x] && IntegerQ[p] *)
+Int[(e_.*x_)^m_.*Sin[d_.*(a_. + b_.*Log[x_])]^p_, x_Symbol] := Sin[d*(a + b*Log[x])]^p*x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p* Int[(e*x)^m*(1 - E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p), x] /; FreeQ[{a, b, d, e, m, p}, x] && Not[IntegerQ[p]]
+Int[(e_.*x_)^m_.*Cos[d_.*(a_. + b_.*Log[x_])]^p_, x_Symbol] := Cos[d*(a + b*Log[x])]^p*x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p* Int[(e*x)^m*(1 + E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p), x] /; FreeQ[{a, b, d, e, m, p}, x] && Not[IntegerQ[p]]
+Int[(e_.*x_)^m_.*Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := (e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Sin[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[(e_.*x_)^m_.*Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := (e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Cos[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[(h_.*(e_. + f_.*Log[g_.*x_^m_.]))^q_.* Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := I*E^(-I*a*d)*(c*x^n)^(-I*b*d)/(2*x^(-I*b*d*n))* Int[x^(-I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] - I*E^(I*a*d)*(c*x^n)^(I*b*d)/(2*x^(I*b*d*n))* Int[x^(I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, q}, x]
+Int[(h_.*(e_. + f_.*Log[g_.*x_^m_.]))^q_.* Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := E^(-I*a*d)*(c*x^n)^(-I*b*d)/(2*x^(-I*b*d*n))* Int[x^(-I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] + E^(I*a*d)*(c*x^n)^(I*b*d)/(2*x^(I*b*d*n))* Int[x^(I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, q}, x]
+Int[(i_.*x_)^r_.*(h_.*(e_. + f_.*Log[g_.*x_^m_.]))^q_.* Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := I*E^(-I*a*d)*(i*x)^r*(c*x^n)^(-I*b*d)/(2*x^(r - I*b*d*n))* Int[x^(r - I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] - I*E^(I*a*d)*(i*x)^r*(c*x^n)^(I*b*d)/(2*x^(r + I*b*d*n))* Int[x^(r + I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, i, m, n, q, r}, x]
+Int[(i_.*x_)^r_.*(h_.*(e_. + f_.*Log[g_.*x_^m_.]))^q_.* Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := E^(-I*a*d)*(i*x)^r*(c*x^n)^(-I*b*d)/(2*x^(r - I*b*d*n))* Int[x^(r - I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] + E^(I*a*d)*(i*x)^r*(c*x^n)^(I*b*d)/(2*x^(r + I*b*d*n))* Int[x^(r + I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, i, m, n, q, r}, x]
+Int[Tan[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Int[((I - I*E^(2*I*a*d)*x^(2*I*b*d))/(1 + E^(2*I*a*d)*x^(2*I*b*d)))^p, x] /; FreeQ[{a, b, d, p}, x]
+Int[Cot[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Int[((-I - I*E^(2*I*a*d)*x^(2*I*b*d))/(1 - E^(2*I*a*d)*x^(2*I*b*d)))^p, x] /; FreeQ[{a, b, d, p}, x]
+Int[Tan[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Tan[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[Cot[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Cot[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[(e_.*x_)^m_.*Tan[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Int[(e*x)^ m*((I - I*E^(2*I*a*d)*x^(2*I*b*d))/(1 + E^(2*I*a*d)*x^(2*I*b*d)))^p, x] /; FreeQ[{a, b, d, e, m, p}, x]
+Int[(e_.*x_)^m_.*Cot[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Int[(e*x)^ m*((-I - I*E^(2*I*a*d)*x^(2*I*b*d))/(1 - E^(2*I*a*d)*x^(2*I*b*d)))^p, x] /; FreeQ[{a, b, d, e, m, p}, x]
+Int[(e_.*x_)^m_.*Tan[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := (e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Tan[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[(e_.*x_)^m_.*Cot[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := (e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Cot[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[Sec[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := 2^p*E^(I*a*d*p)* Int[x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p, x] /; FreeQ[{a, b, d}, x] && IntegerQ[p]
+Int[Csc[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := (-2*I)^p*E^(I*a*d*p)* Int[x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, x] /; FreeQ[{a, b, d}, x] && IntegerQ[p]
+Int[Sec[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Sec[d*(a + b*Log[x])]^p*(1 + E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p)* Int[x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p, x] /; FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]
+Int[Csc[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Csc[d*(a + b*Log[x])]^p*(1 - E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p)* Int[x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, x] /; FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]
+Int[Sec[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Sec[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[Csc[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Csc[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[(e_.*x_)^m_.*Sec[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := 2^p*E^(I*a*d*p)* Int[(e*x)^m*x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p, x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]
+Int[(e_.*x_)^m_.*Csc[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := (-2*I)^p*E^(I*a*d*p)* Int[(e*x)^m*x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]
+Int[(e_.*x_)^m_.*Sec[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Sec[d*(a + b*Log[x])]^p*(1 + E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p)* Int[(e*x)^m*x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p, x] /; FreeQ[{a, b, d, e, m, p}, x] && Not[IntegerQ[p]]
+Int[(e_.*x_)^m_.*Csc[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Csc[d*(a + b*Log[x])]^p*(1 - E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p)* Int[(e*x)^m*x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, x] /; FreeQ[{a, b, d, e, m, p}, x] && Not[IntegerQ[p]]
+Int[(e_.*x_)^m_.*Sec[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := (e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Sec[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[(e_.*x_)^m_.*Csc[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := (e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Csc[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[Sin[a_.*x_*Log[b_.*x_]]*Log[b_.*x_], x_Symbol] := -Cos[a*x*Log[b*x]]/a - Int[Sin[a*x*Log[b*x]], x] /; FreeQ[{a, b}, x]
+Int[Cos[a_.*x_*Log[b_.*x_]]*Log[b_.*x_], x_Symbol] := Sin[a*x*Log[b*x]]/a - Int[Cos[a*x*Log[b*x]], x] /; FreeQ[{a, b}, x]
+Int[x_^m_.*Sin[a_.*x_^n_.*Log[b_.*x_]]*Log[b_.*x_], x_Symbol] := -Cos[a*x^n*Log[b*x]]/(a*n) - 1/n*Int[x^m*Sin[a*x^n*Log[b*x]], x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
+Int[x_^m_.*Cos[a_.*x_^n_.*Log[b_.*x_]]*Log[b_.*x_], x_Symbol] := Sin[a*x^n*Log[b*x]]/(a*n) - 1/n*Int[x^m*Cos[a*x^n*Log[b*x]], x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
diff --git a/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m
new file mode 100755
index 0000000..17b3fe2
--- /dev/null
+++ b/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m	
@@ -0,0 +1,107 @@
+
+(* ::Subsection::Closed:: *)
+(* 4.7.9 Active trig functions *)
+Int[(e_. + f_.*x_)^m_.* Sin[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Sin[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Sin[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Cos[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Cos[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Cos[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]/(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := -I*(e + f*x)^(m + 1)/(b*f*(m + 1)) + 2*Int[(e + f*x)^m*E^(I*(c + d*x))/(a - I*b*E^(I*(c + d*x))), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]
+Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]/(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := I*(e + f*x)^(m + 1)/(b*f*(m + 1)) - 2*I*Int[(e + f*x)^m*E^(I*(c + d*x))/(a + b*E^(I*(c + d*x))), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]
+Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]/(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := -I*(e + f*x)^(m + 1)/(b*f*(m + 1)) + Int[(e + f*x)^m* E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x))), x] + Int[(e + f*x)^m* E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x))), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]
+Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]/(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := I*(e + f*x)^(m + 1)/(b*f*(m + 1)) - I*Int[(e + f*x)^m* E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] + b*E^(I*(c + d*x))), x] - I*Int[(e + f*x)^m* E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] + b*E^(I*(c + d*x))), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]
+Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]/(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := -I*(e + f*x)^(m + 1)/(b*f*(m + 1)) + I*Int[(e + f*x)^m* E^(I*(c + d*x))/(I*a - Rt[-a^2 + b^2, 2] + b*E^(I*(c + d*x))), x] + I*Int[(e + f*x)^m* E^(I*(c + d*x))/(I*a + Rt[-a^2 + b^2, 2] + b*E^(I*(c + d*x))), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]
+Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]/(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := I*(e + f*x)^(m + 1)/(b*f*(m + 1)) + Int[(e + f*x)^m* E^(I*(c + d*x))/(I*a - Rt[-a^2 + b^2, 2] + I*b*E^(I*(c + d*x))), x] + Int[(e + f*x)^m* E^(I*(c + d*x))/(I*a + Rt[-a^2 + b^2, 2] + I*b*E^(I*(c + d*x))), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]
+Int[(e_. + f_.*x_)^m_.* Cos[c_. + d_.*x_]^n_/(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x] - 1/b*Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)*Sin[c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 - b^2, 0]
+Int[(e_. + f_.*x_)^m_.* Sin[c_. + d_.*x_]^n_/(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Sin[c + d*x]^(n - 2), x] - 1/b*Int[(e + f*x)^m*Sin[c + d*x]^(n - 2)*Cos[c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 - b^2, 0]
+Int[(e_. + f_.*x_)^m_.* Cos[c_. + d_.*x_]^n_/(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := a/b^2*Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x] - 1/b*Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)*Sin[c + d*x], x] - (a^2 - b^2)/b^2* Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)/(a + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.* Sin[c_. + d_.*x_]^n_/(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := a/b^2*Int[(e + f*x)^m*Sin[c + d*x]^(n - 2), x] - 1/b*Int[(e + f*x)^m*Sin[c + d*x]^(n - 2)*Cos[c + d*x], x] - (a^2 - b^2)/b^2* Int[(e + f*x)^m*Sin[c + d*x]^(n - 2)/(a + b*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.* Tan[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Sec[c + d*x]*Tan[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Sec[c + d*x]* Tan[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Cot[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Csc[c + d*x]*Cot[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Csc[c + d*x]* Cot[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Cot[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Cot[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Cos[c + d*x]* Cot[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Tan[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Tan[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Sin[c + d*x]* Tan[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Sec[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Sec[c + d*x]^(n + 2), x] - 1/b*Int[(e + f*x)^m*Sec[c + d*x]^(n + 1)*Tan[c + d*x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]
+Int[(e_. + f_.*x_)^m_.* Csc[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Csc[c + d*x]^(n + 2), x] - 1/b*Int[(e + f*x)^m*Csc[c + d*x]^(n + 1)*Cot[c + d*x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]
+Int[(e_. + f_.*x_)^m_.* Sec[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := -b^2/(a^2 - b^2)* Int[(e + f*x)^m*Sec[c + d*x]^(n - 2)/(a + b*Sin[c + d*x]), x] + 1/(a^2 - b^2)* Int[(e + f*x)^m*Sec[c + d*x]^n*(a - b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Csc[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := -b^2/(a^2 - b^2)* Int[(e + f*x)^m*Csc[c + d*x]^(n - 2)/(a + b*Cos[c + d*x]), x] + 1/(a^2 - b^2)* Int[(e + f*x)^m*Csc[c + d*x]^n*(a - b*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Csc[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Csc[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Csc[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Sec[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Sec[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Sec[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := Unintegrable[(e + f*x)^m*F[c + d*x]^n/(a + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && TrigQ[F]
+Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := Unintegrable[(e + f*x)^m*F[c + d*x]^n/(a + b*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && TrigQ[F]
+Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]^p_.* Sin[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Cos[c + d*x]^p*Sin[c + d*x]^(n - 1), x] - a/b* Int[(e + f*x)^m*Cos[c + d*x]^p* Sin[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]^p_.* Cos[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Sin[c + d*x]^p*Cos[c + d*x]^(n - 1), x] - a/b* Int[(e + f*x)^m*Sin[c + d*x]^p* Cos[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]^p_.* Tan[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Cos[c + d*x]^(p - 1)*Tan[c + d*x]^(n - 1), x] - a/b* Int[(e + f*x)^m*Cos[c + d*x]^(p - 1)* Tan[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]^p_.* Cot[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Sin[c + d*x]^(p - 1)*Cot[c + d*x]^(n - 1), x] - a/b* Int[(e + f*x)^m*Sin[c + d*x]^(p - 1)* Cot[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]^p_.* Cot[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Cos[c + d*x]^p*Cot[c + d*x]^n, x] - b/a* Int[(e + f*x)^m*Cos[c + d*x]^(p + 1)* Cot[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]^p_.* Tan[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Sin[c + d*x]^p*Tan[c + d*x]^n, x] - b/a* Int[(e + f*x)^m*Sin[c + d*x]^(p + 1)* Tan[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]^p_.* Csc[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Cos[c + d*x]^p*Csc[c + d*x]^n, x] - b/a* Int[(e + f*x)^m*Cos[c + d*x]^p* Csc[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]^p_.* Sec[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Sin[c + d*x]^p*Sec[c + d*x]^n, x] - b/a* Int[(e + f*x)^m*Sin[c + d*x]^p* Sec[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]^p_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol] := Unintegrable[(e + f*x)^m*Cos[c + d*x]^p* F[c + d*x]^n/(a + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && TrigQ[F]
+Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]^p_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol] := Unintegrable[(e + f*x)^m*Sin[c + d*x]^p* F[c + d*x]^n/(a + b*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && TrigQ[F]
+Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Sec[c_. + d_.*x_]), x_Symbol] := Int[(e + f*x)^m*Cos[c + d*x]*F[c + d*x]^n/(b + a*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && TrigQ[F] && IntegersQ[m, n]
+Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Csc[c_. + d_.*x_]), x_Symbol] := Int[(e + f*x)^m*Sin[c + d*x]*F[c + d*x]^n/(b + a*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && TrigQ[F] && IntegersQ[m, n]
+Int[(e_. + f_.*x_)^m_.*F_[c_. + d_.*x_]^n_.* G_[c_. + d_.*x_]^p_./(a_ + b_.*Sec[c_. + d_.*x_]), x_Symbol] := Int[(e + f*x)^m*Cos[c + d*x]*F[c + d*x]^n* G[c + d*x]^p/(b + a*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && TrigQ[F] && TrigQ[G] && IntegersQ[m, n, p]
+Int[(e_. + f_.*x_)^m_.*F_[c_. + d_.*x_]^n_.* G_[c_. + d_.*x_]^p_./(a_ + b_.*Csc[c_. + d_.*x_]), x_Symbol] := Int[(e + f*x)^m*Sin[c + d*x]*F[c + d*x]^n* G[c + d*x]^p/(b + a*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && TrigQ[F] && TrigQ[G] && IntegersQ[m, n, p]
+Int[Sin[a_. + b_.*x_]^p_.*Sin[c_. + d_.*x_]^q_., x_Symbol] := 1/2^(p + q)* Int[ExpandIntegrand[(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^ q, (I/E^(I*(a + b*x)) - I*E^(I*(a + b*x)))^p, x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]
+Int[Cos[a_. + b_.*x_]^p_.*Cos[c_. + d_.*x_]^q_., x_Symbol] := 1/2^(p + q)* Int[ExpandIntegrand[(E^(-I*(c + d*x)) + E^(I*(c + d*x)))^ q, (E^(-I*(a + b*x)) + E^(I*(a + b*x)))^p, x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]
+Int[Sin[a_. + b_.*x_]^p_.*Cos[c_. + d_.*x_]^q_., x_Symbol] := 1/2^(p + q)* Int[ExpandIntegrand[(E^(-I*(c + d*x)) + E^(I*(c + d*x)))^ q, (I/E^(I*(a + b*x)) - I*E^(I*(a + b*x)))^p, x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]
+Int[Cos[a_. + b_.*x_]^p_.*Sin[c_. + d_.*x_]^q_., x_Symbol] := 1/2^(p + q)* Int[ExpandIntegrand[(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^ q, (E^(-I*(a + b*x)) + E^(I*(a + b*x)))^p, x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]
+Int[Sin[a_. + b_.*x_]*Tan[c_. + d_.*x_], x_Symbol] := Int[E^(-I*(a + b*x))/2 - E^(I*(a + b*x))/2 - E^(-I*(a + b*x))/(1 + E^(2*I*(c + d*x))) + E^(I*(a + b*x))/(1 + E^(2*I*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
+Int[Cos[a_. + b_.*x_]*Cot[c_. + d_.*x_], x_Symbol] := Int[I*E^(-I*(a + b*x))/2 + I*E^(I*(a + b*x))/2 - I*E^(-I*(a + b*x))/(1 - E^(2*I*(c + d*x))) - I*E^(I*(a + b*x))/(1 - E^(2*I*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
+Int[Sin[a_. + b_.*x_]*Cot[c_. + d_.*x_], x_Symbol] := Int[-E^(-I*(a + b*x))/2 + E^(I*(a + b*x))/2 + E^(-I*(a + b*x))/(1 - E^(2*I*(c + d*x))) - E^(I*(a + b*x))/(1 - E^(2*I*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
+Int[Cos[a_. + b_.*x_]*Tan[c_. + d_.*x_], x_Symbol] := Int[-I*E^(-I*(a + b*x))/2 - I*E^(I*(a + b*x))/2 + I*E^(-I*(a + b*x))/(1 + E^(2*I*(c + d*x))) + I*E^(I*(a + b*x))/(1 + E^(2*I*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
+Int[Sin[a_./(c_. + d_.*x_)]^n_., x_Symbol] := -1/d*Subst[Int[Sin[a*x]^n/x^2, x], x, 1/(c + d*x)] /; FreeQ[{a, c, d}, x] && IGtQ[n, 0]
+Int[Cos[a_./(c_. + d_.*x_)]^n_., x_Symbol] := -1/d*Subst[Int[Cos[a*x]^n/x^2, x], x, 1/(c + d*x)] /; FreeQ[{a, c, d}, x] && IGtQ[n, 0]
+Int[Sin[e_.*(a_. + b_.*x_)/(c_. + d_.*x_)]^n_., x_Symbol] := -1/d*Subst[Int[Sin[b*e/d - e*(b*c - a*d)*x/d]^n/x^2, x], x, 1/(c + d*x)] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - a*d, 0]
+Int[Cos[e_.*(a_. + b_.*x_)/(c_. + d_.*x_)]^n_., x_Symbol] := -1/d*Subst[Int[Cos[b*e/d - e*(b*c - a*d)*x/d]^n/x^2, x], x, 1/(c + d*x)] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - a*d, 0]
+Int[Sin[u_]^n_., x_Symbol] := Module[{lst = QuotientOfLinearsParts[u, x]}, Int[Sin[(lst[[1]] + lst[[2]]*x)/(lst[[3]] + lst[[4]]*x)]^n, x]] /; IGtQ[n, 0] && QuotientOfLinearsQ[u, x]
+Int[Cos[u_]^n_., x_Symbol] := Module[{lst = QuotientOfLinearsParts[u, x]}, Int[Cos[(lst[[1]] + lst[[2]]*x)/(lst[[3]] + lst[[4]]*x)]^n, x]] /; IGtQ[n, 0] && QuotientOfLinearsQ[u, x]
+Int[u_.*Sin[v_]^p_.*Sin[w_]^q_., x_Symbol] := Int[u*Sin[v]^(p + q), x] /; EqQ[w, v]
+Int[u_.*Cos[v_]^p_.*Cos[w_]^q_., x_Symbol] := Int[u*Cos[v]^(p + q), x] /; EqQ[w, v]
+Int[Sin[v_]^p_.*Sin[w_]^q_., x_Symbol] := Int[ExpandTrigReduce[Sin[v]^p*Sin[w]^q, x], x] /; (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x]) && IGtQ[p, 0] && IGtQ[q, 0]
+Int[Cos[v_]^p_.*Cos[w_]^q_., x_Symbol] := Int[ExpandTrigReduce[Cos[v]^p*Cos[w]^q, x], x] /; (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x]) && IGtQ[p, 0] && IGtQ[q, 0]
+Int[x_^m_.*Sin[v_]^p_.*Sin[w_]^q_., x_Symbol] := Int[ExpandTrigReduce[x^m, Sin[v]^p*Sin[w]^q, x], x] /; IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])
+Int[x_^m_.*Cos[v_]^p_.*Cos[w_]^q_., x_Symbol] := Int[ExpandTrigReduce[x^m, Cos[v]^p*Cos[w]^q, x], x] /; IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])
+Int[u_.*Sin[v_]^p_.*Cos[w_]^p_., x_Symbol] := 1/2^p*Int[u*Sin[2*v]^p, x] /; EqQ[w, v] && IntegerQ[p]
+Int[Sin[v_]^p_.*Cos[w_]^q_., x_Symbol] := Int[ExpandTrigReduce[Sin[v]^p*Cos[w]^q, x], x] /; IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])
+Int[x_^m_.*Sin[v_]^p_.*Cos[w_]^q_., x_Symbol] := Int[ExpandTrigReduce[x^m, Sin[v]^p*Cos[w]^q, x], x] /; IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])
+Int[Sin[v_]*Tan[w_]^n_., x_Symbol] := -Int[Cos[v]*Tan[w]^(n - 1), x] + Cos[v - w]*Int[Sec[w]*Tan[w]^(n - 1), x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
+Int[Cos[v_]*Cot[w_]^n_., x_Symbol] := -Int[Sin[v]*Cot[w]^(n - 1), x] + Cos[v - w]*Int[Csc[w]*Cot[w]^(n - 1), x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
+Int[Sin[v_]*Cot[w_]^n_., x_Symbol] := Int[Cos[v]*Cot[w]^(n - 1), x] + Sin[v - w]*Int[Csc[w]*Cot[w]^(n - 1), x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
+Int[Cos[v_]*Tan[w_]^n_., x_Symbol] := Int[Sin[v]*Tan[w]^(n - 1), x] - Sin[v - w]*Int[Sec[w]*Tan[w]^(n - 1), x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
+Int[Sin[v_]*Sec[w_]^n_., x_Symbol] := Cos[v - w]*Int[Tan[w]*Sec[w]^(n - 1), x] + Sin[v - w]*Int[Sec[w]^(n - 1), x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
+Int[Cos[v_]*Csc[w_]^n_., x_Symbol] := Cos[v - w]*Int[Cot[w]*Csc[w]^(n - 1), x] - Sin[v - w]*Int[Csc[w]^(n - 1), x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
+Int[Sin[v_]*Csc[w_]^n_., x_Symbol] := Sin[v - w]*Int[Cot[w]*Csc[w]^(n - 1), x] + Cos[v - w]*Int[Csc[w]^(n - 1), x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
+Int[Cos[v_]*Sec[w_]^n_., x_Symbol] := -Sin[v - w]*Int[Tan[w]*Sec[w]^(n - 1), x] + Cos[v - w]*Int[Sec[w]^(n - 1), x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
+Int[(e_. + f_.*x_)^m_.*(a_ + b_.*Sin[c_. + d_.*x_]*Cos[c_. + d_.*x_])^ n_., x_Symbol] := Int[(e + f*x)^m*(a + b*Sin[2*c + 2*d*x]/2)^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
+Int[x_^m_.*(a_ + b_.*Sin[c_. + d_.*x_]^2)^n_, x_Symbol] := 1/2^n*Int[x^m*(2*a + b - b*Cos[2*c + 2*d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a + b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1] || EqQ[m, 1] && EqQ[n, -2])
+Int[x_^m_.*(a_ + b_.*Cos[c_. + d_.*x_]^2)^n_, x_Symbol] := 1/2^n*Int[x^m*(2*a + b + b*Cos[2*c + 2*d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a + b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1] || EqQ[m, 1] && EqQ[n, -2])
+Int[(f_. + g_.*x_)^ m_./(a_. + b_.*Cos[d_. + e_.*x_]^2 + c_.*Sin[d_. + e_.*x_]^2), x_Symbol] := 2*Int[(f + g*x)^m/(2*a + b + c + (b - c)*Cos[2*d + 2*e*x]), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]
+Int[(f_. + g_.*x_)^m_.* Sec[d_. + e_.*x_]^2/(b_ + c_.*Tan[d_. + e_.*x_]^2), x_Symbol] := 2*Int[(f + g*x)^m/(b + c + (b - c)*Cos[2*d + 2*e*x]), x] /; FreeQ[{b, c, d, e, f, g}, x] && IGtQ[m, 0]
+Int[(f_. + g_.*x_)^m_.* Sec[d_. + e_.*x_]^2/(b_. + a_.*Sec[d_. + e_.*x_]^2 + c_.*Tan[d_. + e_.*x_]^2), x_Symbol] := 2*Int[(f + g*x)^m/(2*a + b + c + (b - c)*Cos[2*d + 2*e*x]), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]
+Int[(f_. + g_.*x_)^m_.* Csc[d_. + e_.*x_]^2/(c_ + b_.*Cot[d_. + e_.*x_]^2), x_Symbol] := 2*Int[(f + g*x)^m/(b + c + (b - c)*Cos[2*d + 2*e*x]), x] /; FreeQ[{b, c, d, e, f, g}, x] && IGtQ[m, 0]
+Int[(f_. + g_.*x_)^m_.* Csc[d_. + e_.*x_]^2/(c_. + b_.*Cot[d_. + e_.*x_]^2 + a_.*Csc[d_. + e_.*x_]^2), x_Symbol] := 2*Int[(f + g*x)^m/(2*a + b + c + (b - c)*Cos[2*d + 2*e*x]), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]
+Int[(e_. + f_.*x_)*(A_ + B_.*Sin[c_. + d_.*x_])/(a_ + b_.*Sin[c_. + d_.*x_])^2, x_Symbol] := -B*(e + f*x)*Cos[c + d*x]/(a*d*(a + b*Sin[c + d*x])) + B*f/(a*d)*Int[Cos[c + d*x]/(a + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[a*A - b*B, 0]
+Int[(e_. + f_.*x_)*(A_ + B_.*Cos[c_. + d_.*x_])/(a_ + b_.*Cos[c_. + d_.*x_])^2, x_Symbol] := B*(e + f*x)*Sin[c + d*x]/(a*d*(a + b*Cos[c + d*x])) - B*f/(a*d)*Int[Sin[c + d*x]/(a + b*Cos[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[a*A - b*B, 0]
+Int[x_^2/(c_.*Sin[a_.*x_] + d_.*x_*Cos[a_.*x_])^2, x_Symbol] := x/(a*d*Sin[a*x]*(c*Sin[a*x] + d*x*Cos[a*x])) + 1/d^2*Int[1/Sin[a*x]^2, x] /; FreeQ[{a, c, d}, x] && EqQ[a*c + d, 0]
+Int[x_^2/(c_.*Cos[a_.*x_] + d_.*x_*Sin[a_.*x_])^2, x_Symbol] := -x/(a*d*Cos[a*x]*(c*Cos[a*x] + d*x*Sin[a*x])) + 1/d^2*Int[1/Cos[a*x]^2, x] /; FreeQ[{a, c, d}, x] && EqQ[a*c - d, 0]
+Int[Sin[a_.*x_]^2/(c_.*Sin[a_.*x_] + d_.*x_*Cos[a_.*x_])^2, x_Symbol] := 1/(d^2*x) + Sin[a*x]/(a*d*x*(d*x*Cos[a*x] + c*Sin[a*x])) /; FreeQ[{a, c, d}, x] && EqQ[a*c + d, 0]
+Int[Cos[a_.*x_]^2/(c_.*Cos[a_.*x_] + d_.*x_*Sin[a_.*x_])^2, x_Symbol] := 1/(d^2*x) - Cos[a*x]/(a*d*x*(d*x*Sin[a*x] + c*Cos[a*x])) /; FreeQ[{a, c, d}, x] && EqQ[a*c - d, 0]
+Int[(b_.*x_)^m_* Sin[a_.*x_]^n_/(c_.*Sin[a_.*x_] + d_.*x_*Cos[a_.*x_])^2, x_Symbol] := b*(b*x)^(m - 1)* Sin[a*x]^(n - 1)/(a*d*(c*Sin[a*x] + d*x*Cos[a*x])) - b^2*(n - 1)/d^2*Int[(b*x)^(m - 2)*Sin[a*x]^(n - 2), x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c + d, 0] && EqQ[m, 2 - n]
+Int[(b_.*x_)^m_* Cos[a_.*x_]^n_/(c_.*Cos[a_.*x_] + d_.*x_*Sin[a_.*x_])^2, x_Symbol] := -b*(b*x)^(m - 1)* Cos[a*x]^(n - 1)/(a*d*(c*Cos[a*x] + d*x*Sin[a*x])) - b^2*(n - 1)/d^2*Int[(b*x)^(m - 2)*Cos[a*x]^(n - 2), x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c - d, 0] && EqQ[m, 2 - n]
+Int[(b_.*x_)^m_.* Csc[a_.*x_]^n_./(c_.*Sin[a_.*x_] + d_.*x_*Cos[a_.*x_])^2, x_Symbol] := b*(b*x)^(m - 1)* Csc[a*x]^(n + 1)/(a*d*(c*Sin[a*x] + d*x*Cos[a*x])) + b^2*(n + 1)/d^2*Int[(b*x)^(m - 2)*Csc[a*x]^(n + 2), x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c + d, 0] && EqQ[m, n + 2]
+Int[(b_.*x_)^m_.* Sec[a_.*x_]^n_./(c_.*Cos[a_.*x_] + d_.*x_*Sin[a_.*x_])^2, x_Symbol] := -b*(b*x)^(m - 1)* Sec[a*x]^(n + 1)/(a*d*(c*Cos[a*x] + d*x*Sin[a*x])) + b^2*(n + 1)/d^2*Int[(b*x)^(m - 2)*Sec[a*x]^(n + 2), x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c - d, 0] && EqQ[m, n + 2]
+Int[(g_. + h_.*x_)^p_.*(a_ + b_.*Sin[e_. + f_.*x_])^ m_.*(c_ + d_.*Sin[e_. + f_.*x_])^n_., x_Symbol] := a^m*c^m* Int[(g + h*x)^p*Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && IGtQ[n - m, 0]
+Int[(g_. + h_.*x_)^p_.*(a_ + b_.*Cos[e_. + f_.*x_])^ m_.*(c_ + d_.*Cos[e_. + f_.*x_])^n_., x_Symbol] := a^m*c^m* Int[(g + h*x)^p*Sin[e + f*x]^(2*m)*(c + d*Cos[e + f*x])^(n - m), x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && IGtQ[n - m, 0]
+Int[(g_. + h_.*x_)^p_.*(a_ + b_.*Sin[e_. + f_.*x_])^ m_*(c_ + d_.*Sin[e_. + f_.*x_])^n_, x_Symbol] := a^IntPart[m]* c^IntPart[m]*(a + b*Sin[e + f*x])^ FracPart[m]*(c + d*Sin[e + f*x])^FracPart[m]/ Cos[e + f*x]^(2*FracPart[m])* Int[(g + h*x)^p* Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p] && IntegerQ[2*m] && IGeQ[n - m, 0]
+Int[(g_. + h_.*x_)^p_.*(a_ + b_.*Cos[e_. + f_.*x_])^ m_*(c_ + d_.*Cos[e_. + f_.*x_])^n_, x_Symbol] := a^IntPart[m]* c^IntPart[m]*(a + b*Cos[e + f*x])^ FracPart[m]*(c + d*Cos[e + f*x])^FracPart[m]/ Sin[e + f*x]^(2*FracPart[m])* Int[(g + h*x)^p* Sin[e + f*x]^(2*m)*(c + d*Cos[e + f*x])^(n - m), x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p] && IntegerQ[2*m] && IGeQ[n - m, 0]
+Int[Sec[v_]^m_.*(a_ + b_.*Tan[v_])^n_., x_Symbol] := Int[(a*Cos[v] + b*Sin[v])^n, x] /; FreeQ[{a, b}, x] && IntegerQ[(m - 1)/2] && EqQ[m + n, 0]
+Int[Csc[v_]^m_.*(a_ + b_.*Cot[v_])^n_., x_Symbol] := Int[(b*Cos[v] + a*Sin[v])^n, x] /; FreeQ[{a, b}, x] && IntegerQ[(m - 1)/2] && EqQ[m + n, 0]
+Int[u_.*Sin[a_. + b_.*x_]^m_.*Sin[c_. + d_.*x_]^n_., x_Symbol] := Int[ExpandTrigReduce[u, Sin[a + b*x]^m*Sin[c + d*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[u_.*Cos[a_. + b_.*x_]^m_.*Cos[c_. + d_.*x_]^n_., x_Symbol] := Int[ExpandTrigReduce[u, Cos[a + b*x]^m*Cos[c + d*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[Sec[a_. + b_.*x_]*Sec[c_ + d_.*x_], x_Symbol] := -Csc[(b*c - a*d)/d]*Int[Tan[a + b*x], x] + Csc[(b*c - a*d)/b]*Int[Tan[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
+Int[Csc[a_. + b_.*x_]*Csc[c_ + d_.*x_], x_Symbol] := Csc[(b*c - a*d)/b]*Int[Cot[a + b*x], x] - Csc[(b*c - a*d)/d]*Int[Cot[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
+Int[Tan[a_. + b_.*x_]*Tan[c_ + d_.*x_], x_Symbol] := -b*x/d + b/d*Cos[(b*c - a*d)/d]*Int[Sec[a + b*x]*Sec[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
+Int[Cot[a_. + b_.*x_]*Cot[c_ + d_.*x_], x_Symbol] := -b*x/d + Cos[(b*c - a*d)/d]*Int[Csc[a + b*x]*Csc[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
+Int[u_.*(a_.*Cos[v_] + b_.*Sin[v_])^n_., x_Symbol] := Int[u*(a*E^(-a/b*v))^n, x] /; FreeQ[{a, b, n}, x] && EqQ[a^2 + b^2, 0]
+Int[Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol] := I/2*Int[E^(-I*d*(a + b*Log[c*x^n])^2), x] - I/2*Int[E^(I*d*(a + b*Log[c*x^n])^2), x] /; FreeQ[{a, b, c, d, n}, x]
+Int[Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol] := 1/2*Int[E^(-I*d*(a + b*Log[c*x^n])^2), x] + 1/2*Int[E^(I*d*(a + b*Log[c*x^n])^2), x] /; FreeQ[{a, b, c, d, n}, x]
+Int[(e_.*x_)^m_.*Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol] := I/2*Int[(e*x)^m*E^(-I*d*(a + b*Log[c*x^n])^2), x] - I/2*Int[(e*x)^m*E^(I*d*(a + b*Log[c*x^n])^2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[(e_.*x_)^m_.*Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol] := 1/2*Int[(e*x)^m*E^(-I*d*(a + b*Log[c*x^n])^2), x] + 1/2*Int[(e*x)^m*E^(I*d*(a + b*Log[c*x^n])^2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
diff --git a/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.1 (a+b arcsin(c x))^n.m b/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.1 (a+b arcsin(c x))^n.m
new file mode 100755
index 0000000..9602c39
--- /dev/null
+++ b/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.1 (a+b arcsin(c x))^n.m	
@@ -0,0 +1,9 @@
+
+(* ::Subsection::Closed:: *)
+(* 5.1.1 (a+b arcsin(c x))^n *)
+Int[(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := x*(a + b*ArcSin[c*x])^n - b*c*n*Int[x*(a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
+Int[(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := x*(a + b*ArcCos[c*x])^n + b*c*n*Int[x*(a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
+Int[(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol] := Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1)) + c/(b*(n + 1))* Int[x*(a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
+Int[(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol] := -Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1)) - c/(b*(n + 1))* Int[x*(a + b*ArcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
+Int[(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol] := 1/(b*c)*Subst[Int[x^n*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]] /; FreeQ[{a, b, c, n}, x]
+Int[(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol] := -1/(b*c)*Subst[Int[x^n*Sin[-a/b + x/b], x], x, a + b*ArcCos[c*x]] /; FreeQ[{a, b, c, n}, x]
diff --git a/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m b/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m
new file mode 100755
index 0000000..5db4a82
--- /dev/null
+++ b/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m	
@@ -0,0 +1,19 @@
+
+(* ::Subsection::Closed:: *)
+(* 5.1.2 (d x)^m (a+b arcsin(c x))^n *)
+(* Int[(a_.+b_.*ArcSin[c_.*x_])^n_./x_,x_Symbol] := 1/b*Subst[Int[x^n*Cot[-a/b+x/b],x],x,a+b*ArcSin[c*x]] /; FreeQ[{a,b,c},x] && IGtQ[n,0] *)
+(* Int[(a_.+b_.*ArcCos[c_.*x_])^n_./x_,x_Symbol] := -1/b*Subst[Int[x^n*Tan[-a/b+x/b],x],x,a+b*ArcCos[c*x]] /; FreeQ[{a,b,c},x] && IGtQ[n,0] *)
+Int[(a_. + b_.*ArcSin[c_.*x_])^n_./x_, x_Symbol] := Subst[Int[(a + b*x)^n*Cot[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
+Int[(a_. + b_.*ArcCos[c_.*x_])^n_./x_, x_Symbol] := -Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcCos[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := (d*x)^(m + 1)*(a + b*ArcSin[c*x])^n/(d*(m + 1)) - b*c*n/(d*(m + 1))* Int[(d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := (d*x)^(m + 1)*(a + b*ArcCos[c*x])^n/(d*(m + 1)) + b*c*n/(d*(m + 1))* Int[(d*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
+Int[x_^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol] := x^(m + 1)*(a + b*ArcSin[c*x])^n/(m + 1) - b*c*n/(m + 1)* Int[x^(m + 1)*(a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
+Int[x_^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol] := x^(m + 1)*(a + b*ArcCos[c*x])^n/(m + 1) + b*c*n/(m + 1)* Int[x^(m + 1)*(a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
+Int[x_^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol] := x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1)) - 1/(b^2*c^(m + 1)*(n + 1))* Subst[Int[ ExpandTrigReduce[x^(n + 1), Sin[-a/b + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
+Int[x_^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol] := -x^m*Sqrt[ 1 - c^2*x^2]*(a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1)) - 1/(b^2*c^(m + 1)*(n + 1))* Subst[Int[ ExpandTrigReduce[x^(n + 1), Cos[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cos[-a/b + x/b]^2), x], x], x, a + b*ArcCos[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
+Int[x_^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol] := x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1)) - m/(b*c*(n + 1))* Int[x^(m - 1)*(a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2], x] + c*(m + 1)/(b*(n + 1))* Int[x^(m + 1)*(a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
+Int[x_^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol] := -x^m*Sqrt[ 1 - c^2*x^2]*(a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1)) + m/(b*c*(n + 1))* Int[x^(m - 1)*(a + b*ArcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2], x] - c*(m + 1)/(b*(n + 1))* Int[x^(m + 1)*(a + b*ArcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
+Int[x_^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol] := 1/(b*c^(m + 1))* Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
+Int[x_^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol] := -1/(b*c^(m + 1))* Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b], x], x, a + b*ArcCos[c*x]] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := Unintegrable[(d*x)^m*(a + b*ArcSin[c*x])^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := Unintegrable[(d*x)^m*(a + b*ArcCos[c*x])^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
diff --git a/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m b/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m
new file mode 100755
index 0000000..fcbdfa0
--- /dev/null
+++ b/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m	
@@ -0,0 +1,37 @@
+
+(* ::Subsection::Closed:: *)
+(* 5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n *)
+(* Int[(a_.+b_.*ArcSin[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := 1/c*Simp[Sqrt[1-c^2*x^2]/Sqrt[d+e*x^2]]*Subst[Int[(a+b*x)^n,x],x, ArcSin[c*x]] /; FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e,0] *)
+(* Int[(a_.+b_.*ArcCos[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := -1/c*Simp[Sqrt[1-c^2*x^2]/Sqrt[d+e*x^2]]*Subst[Int[(a+b*x)^n,x],x, ArcCos[c*x]] /; FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e,0] *)
+Int[1/(Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSin[c_.*x_])), x_Symbol] := 1/(b*c)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]* Log[a + b*ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
+Int[1/(Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCos[c_.*x_])), x_Symbol] := -1/(b*c)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]* Log[a + b*ArcCos[c*x]]/(b*c*Sqrt[d]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
+Int[(a_. + b_.*ArcSin[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol] := 1/(b*c*(n + 1))* Simp[Sqrt[1 - c^2*x^2]/ Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1) /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
+Int[(a_. + b_.*ArcCos[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol] := -1/(b*c*(n + 1))* Simp[Sqrt[1 - c^2*x^2]/ Sqrt[d + e*x^2]]*(a + b*ArcCos[c*x])^(n + 1) /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
+Int[Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := x*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n/2 - b*c*n/2*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]* Int[x*(a + b*ArcSin[c*x])^(n - 1), x] + 1/2*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]* Int[(a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
+Int[Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := x*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n/2 + b*c*n/2*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]* Int[x*(a + b*ArcCos[c*x])^(n - 1), x] + 1/2*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]* Int[(a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := x*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n/(2*p + 1) + 2*d*p/(2*p + 1)* Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x] - b*c*n/(2*p + 1)*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := x*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n/(2*p + 1) + 2*d*p/(2*p + 1)* Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x] + b*c*n/(2*p + 1)*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
+Int[(a_. + b_.*ArcSin[c_.*x_])^n_./(d_ + e_.*x_^2)^(3/2), x_Symbol] := x*(a + b*ArcSin[c*x])^n/(d*Sqrt[d + e*x^2]) - b*c*n/d*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]* Int[x*(a + b*ArcSin[c*x])^(n - 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
+Int[(a_. + b_.*ArcCos[c_.*x_])^n_./(d_ + e_.*x_^2)^(3/2), x_Symbol] := x*(a + b*ArcCos[c*x])^n/(d*Sqrt[d + e*x^2]) + b*c*n/d*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]* Int[x*(a + b*ArcCos[c*x])^(n - 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
+Int[(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := -x*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n/(2*d*(p + 1)) + (2*p + 3)/(2*d*(p + 1))* Int[(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x] + b*c*n/(2*(p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
+Int[(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := -x*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n/(2*d*(p + 1)) + (2*p + 3)/(2*d*(p + 1))* Int[(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x] - b*c*n/(2*(p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
+Int[(a_. + b_.*ArcSin[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol] := 1/(c*d)*Subst[Int[(a + b*x)^n*Sec[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
+Int[(a_. + b_.*ArcCos[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol] := -1/(c*d)*Subst[Int[(a + b*x)^n*Csc[x], x], x, ArcCos[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
+(* Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol] := d^p*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n+1)/(b*c*(n+1)) + c*d^p*(2*p+1)/(b*(n+1))*Int[x*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^ (n+1),x] /; FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e,0] && LtQ[n,-1] && (IntegerQ[p]  || GtQ[d,0]) *)
+(* Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol] := -d^p*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n+1)/(b*c*(n+1)) - c*d^p*(2*p+1)/(b*(n+1))*Int[x*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^ (n+1),x] /; FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e,0] && LtQ[n,-1] && (IntegerQ[p]  || GtQ[d,0]) *)
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol] := Sqrt[1 - c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1)) + c*(2*p + 1)/(b*(n + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol] := -Sqrt[1 - c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1)) - c*(2*p + 1)/(b*(n + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := 1/(b*c)*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Subst[Int[x^n*Cos[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSin[c*x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := -1/(b*c)*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Subst[Int[x^n*Sin[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCos[c*x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || IGtQ[n, 0])
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || IGtQ[n, 0])
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := Unintegrable[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x] /; FreeQ[{a, b, c, d, e, n, p}, x]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := Unintegrable[(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x] /; FreeQ[{a, b, c, d, e, n, p}, x]
+Int[(d_ + e_.*x_)^p_*(f_ + g_.*x_)^q_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := (-d^2*g/e)^q* Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]
+Int[(d_ + e_.*x_)^p_*(f_ + g_.*x_)^q_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := (-d^2*g/e)^q* Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcCos[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]
+Int[(d_ + e_.*x_)^p_*(f_ + g_.*x_)^q_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := (d + e*x)^q*(f + g*x)^q/(1 - c^2*x^2)^q* Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
+Int[(d_ + e_.*x_)^p_*(f_ + g_.*x_)^q_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := (d + e*x)^q*(f + g*x)^q/(1 - c^2*x^2)^q* Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcCos[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
diff --git a/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m b/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m
new file mode 100755
index 0000000..831a2ba
--- /dev/null
+++ b/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m	
@@ -0,0 +1,65 @@
+
+(* ::Subsection::Closed:: *)
+(* 5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n *)
+Int[x_*(a_. + b_.*ArcSin[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol] := -1/e*Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
+Int[x_*(a_. + b_.*ArcCos[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol] := 1/e*Subst[Int[(a + b*x)^n*Cot[x], x], x, ArcCos[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
+Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := (d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n/(2*e*(p + 1)) + b*n/(2*c*(p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
+Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := (d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n/(2*e*(p + 1)) - b*n/(2*c*(p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
+Int[(a_. + b_.*ArcSin[c_.*x_])^n_./(x_*(d_ + e_.*x_^2)), x_Symbol] := 1/d*Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
+Int[(a_. + b_.*ArcCos[c_.*x_])^n_./(x_*(d_ + e_.*x_^2)), x_Symbol] := -1/d*Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, ArcCos[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^ n/(d*f*(m + 1)) - b*c*n/(f*(m + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^ n/(d*f*(m + 1)) + b*c*n/(f*(m + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])/x_, x_Symbol] := (d + e*x^2)^p*(a + b*ArcSin[c*x])/(2*p) - b*c*d^p/(2*p)*Int[(1 - c^2*x^2)^(p - 1/2), x] + d*Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])/x, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])/x_, x_Symbol] := (d + e*x^2)^p*(a + b*ArcCos[c*x])/(2*p) + b*c*d^p/(2*p)*Int[(1 - c^2*x^2)^(p - 1/2), x] + d*Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])/x, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSin[c*x])/(f*(m + 1)) - b*c*d^p/(f*(m + 1))* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2), x] - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcCos[c*x])/(f*(m + 1)) + b*c*d^p/(f*(m + 1))* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2), x] - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
+Int[x_^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]* Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -1/2] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
+Int[x_^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]* Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -1/2] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
+Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n/(f*(m + 1)) - b*c*n/(f*(m + 1))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]* Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x] + c^2/(f^2*(m + 1))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]* Int[(f*x)^(m + 2)*(a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n/(f*(m + 1)) + b*c*n/(f*(m + 1))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]* Int[(f*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1), x] + c^2/(f^2*(m + 1))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]* Int[(f*x)^(m + 2)*(a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n/(f*(m + 2)) - b*c*n/(f*(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]* Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x] + 1/(m + 2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]* Int[(f*x)^m*(a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
+Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n/(f*(m + 2)) + b*c*n/(f*(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]* Int[(f*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1), x] + 1/(m + 2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]* Int[(f*x)^m*(a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n/(f*(m + 1)) - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x] - b*c*n/(f*(m + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n/(f*(m + 1)) - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x] + b*c*n/(f*(m + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcSin[c*x])^n/(f*(m + 2*p + 1)) + 2*d*p/(m + 2*p + 1)* Int[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x] - b*c*n/(f*(m + 2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && Not[LtQ[m, -1]]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcCos[c*x])^n/(f*(m + 2*p + 1)) + 2*d*p/(m + 2*p + 1)* Int[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x] + b*c*n/(f*(m + 2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && Not[LtQ[m, -1]]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^ n/(d*f*(m + 1)) + c^2*(m + 2*p + 3)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x] - b*c*n/(f*(m + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^ n/(d*f*(m + 1)) + c^2*(m + 2*p + 3)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x] + b*c*n/(f*(m + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^ n/(2*e*(p + 1)) - f^2*(m - 1)/(2*e*(p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x] + b*f*n/(2*c*(p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^ n/(2*e*(p + 1)) - f^2*(m - 1)/(2*e*(p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x] - b*f*n/(2*c*(p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := -(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^ n/(2*d*f*(p + 1)) + (m + 2*p + 3)/(2*d*(p + 1))* Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x] + b*c*n/(2*f*(p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && Not[GtQ[m, 1]] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := -(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^ n/(2*d*f*(p + 1)) + (m + 2*p + 3)/(2*d*(p + 1))* Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x] - b*c*n/(2*f*(p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && Not[GtQ[m, 1]] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^ n/(e*(m + 2*p + 1)) + f^2*(m - 1)/(c^2*(m + 2*p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x] + b*f*n/(c*(m + 2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^ n/(e*(m + 2*p + 1)) + f^2*(m - 1)/(c^2*(m + 2*p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x] - b*f*n/(c*(m + 2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol] := (f*x)^m* Sqrt[1 - c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1)) - f*m/(b*c*(n + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol] := -(f*x)^m* Sqrt[1 - c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1)) + f*m/(b*c*(n + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol] := (f*x)^m* Sqrt[1 - c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1)) - f*m/(b*c*(n + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x] + c*(m + 2*p + 1)/(b*f*(n + 1))* Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol] := -(f*x)^m* Sqrt[1 - c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1)) + f*m/(b*c*(n + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n + 1), x] - c*(m + 2*p + 1)/(b*f*(n + 1))* Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]
+(* Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_ Symbol] := (f*x)^m*Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcSin[c*x])^(n+1)/(b*c*(n+ 1)) - f*m/(b*c*(n+1))*Simp[(d+e*x^2)^p/(1-c^2*x^2)^p]*Int[(f*x)^(m-1)*(1- c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n+1),x] + c*(2*p+1)/(b*f*(n+1))*Simp[(d+e*x^2)^p/(1-c^2*x^2)^p]*Int[(f*x)^(m+ 1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n+1),x] /; FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e,0] && LtQ[n,-1] &&  IntegerQ[2*p] && NeQ[p,-1/2] && IGtQ[m,-3] *)
+(* Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_ Symbol] := -(f*x)^m*Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcCos[c*x])^(n+1)/(b*c*( n+1)) + f*m/(b*c*(n+1))*Simp[(d+e*x^2)^p/(1-c^2*x^2)^p]*Int[(f*x)^(m-1)*(1- c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n+1),x] - c*(2*p+1)/(b*f*(n+1))*Simp[(d+e*x^2)^p/(1-c^2*x^2)^p]*Int[(f*x)^(m+ 1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n+1),x] /; FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e,0] && LtQ[n,-1] &&  IntegerQ[2*p] && NeQ[p,-1/2] && IGtQ[m,-3] *)
+Int[(f_.*x_)^m_*(a_. + b_.*ArcSin[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol] := f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n/(e*m) + b*f*n/(c*m)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]* Int[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x] + f^2*(m - 1)/(c^2*m)* Int[((f*x)^(m - 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCos[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol] := f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n/(e*m) - b*f*n/(c*m)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]* Int[(f*x)^(m - 1)*(a + b*ArcCos[c*x])^(n - 1), x] + f^2*(m - 1)/(c^2*m)* Int[((f*x)^(m - 2)*(a + b*ArcCos[c*x])^n)/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1]
+Int[x_^m_*(a_. + b_.*ArcSin[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol] := 1/c^(m + 1)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]* Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
+Int[x_^m_*(a_. + b_.*ArcCos[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol] := -1/c^(m + 1)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]* Subst[Int[(a + b*x)^n*Cos[x]^m, x], x, ArcCos[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcSin[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol] := (f*x)^(m + 1)/(f*(m + 1))* Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])* Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] - b*c*(f*x)^(m + 2)/(f^2*(m + 1)*(m + 2))* Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]* HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && Not[IntegerQ[m]]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCos[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol] := (f*x)^(m + 1)/(f*(m + 1))*(a + b*ArcCos[c*x])* Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]* Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] + b*c*(f*x)^(m + 2)/(f^2*(m + 1)*(m + 2))* Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]* HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && Not[IntegerQ[m]]
+Int[(f_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_/Sqrt[d_ + e_.*x_^2], x_Symbol] := (f*x)^m/(b*c*(n + 1))* Simp[Sqrt[1 - c^2*x^2]/ Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1) - f*m/(b*c*(n + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]* Int[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]
+Int[(f_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_/Sqrt[d_ + e_.*x_^2], x_Symbol] := -(f*x)^m/(b*c*(n + 1))* Simp[Sqrt[1 - c^2*x^2]/ Sqrt[d + e*x^2]]*(a + b*ArcCos[c*x])^(n + 1) + f*m/(b*c*(n + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]* Int[(f*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]
+Int[x_^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := 1/(b*c^(m + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSin[c*x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
+Int[x_^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := -1/(b*c^(m + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCos[c*x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n/ Sqrt[d + e*x^2], (f*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && Not[IGtQ[(m + 1)/2, 0]] && (EqQ[m, -1] || EqQ[m, -2])
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n/ Sqrt[d + e*x^2], (f*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && Not[IGtQ[(m + 1)/2, 0]] && (EqQ[m, -1] || EqQ[m, -2])
+Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol] := (d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])/(2*e*(p + 1)) - b*c/(2*e*(p + 1))*Int[(d + e*x^2)^(p + 1)/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]
+Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol] := (d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])/(2*e*(p + 1)) + b*c/(2*e*(p + 1))*Int[(d + e*x^2)^(p + 1)/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := Unintegrable[(f*x)^m*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := Unintegrable[(f*x)^m*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
+Int[(h_.*x_)^m_.*(d_ + e_.*x_)^p_*(f_ + g_.*x_)^ q_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := (-d^2*g/e)^q* Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^ n, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]
+Int[(h_.*x_)^m_.*(d_ + e_.*x_)^p_*(f_ + g_.*x_)^ q_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := (-d^2*g/e)^q* Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcCos[c*x])^ n, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]
+Int[(h_.*x_)^m_.*(d_ + e_.*x_)^p_*(f_ + g_.*x_)^ q_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := (-d^2*g/e)^IntPart[q]*(d + e*x)^ FracPart[q]*(f + g*x)^FracPart[q]/(1 - c^2*x^2)^FracPart[q]* Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^ q*(a + b*ArcSin[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
+Int[(h_.*x_)^m_.*(d_ + e_.*x_)^p_*(f_ + g_.*x_)^ q_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := (-d^2*g/e)^IntPart[q]*(d + e*x)^ FracPart[q]*(f + g*x)^FracPart[q]/(1 - c^2*x^2)^FracPart[q]* Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^ q*(a + b*ArcCos[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
diff --git a/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m b/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m
new file mode 100755
index 0000000..4adf45a
--- /dev/null
+++ b/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m	
@@ -0,0 +1,67 @@
+
+(* ::Subsection::Closed:: *)
+(* 5.1.5 u (a+b arcsin(c x))^n *)
+Int[(a_. + b_.*ArcSin[c_.*x_])^n_./(d_ + e_.*x_), x_Symbol] := Subst[Int[(a + b*x)^n*Cos[x]/(c*d + e*Sin[x]), x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
+Int[(a_. + b_.*ArcCos[c_.*x_])^n_./(d_ + e_.*x_), x_Symbol] := -Subst[Int[(a + b*x)^n*Sin[x]/(c*d + e*Cos[x]), x], x, ArcCos[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
+Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := (d + e*x)^(m + 1)*(a + b*ArcSin[c*x])^n/(e*(m + 1)) - b*c*n/(e*(m + 1))* Int[(d + e*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1)/ Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
+Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := (d + e*x)^(m + 1)*(a + b*ArcCos[c*x])^n/(e*(m + 1)) + b*c*n/(e*(m + 1))* Int[(d + e*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1)/ Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
+Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
+Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol] := 1/c^(m + 1)* Subst[Int[(a + b*x)^n*Cos[x]*(c*d + e*Sin[x])^m, x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol] := -1/c^(m + 1)* Subst[Int[(a + b*x)^n*Sin[x]*(c*d + e*Cos[x])^m, x], x, ArcCos[c*x]] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]
+Int[Px_*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol] := With[{u = IntHide[ExpandExpression[Px, x], x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c}, x] && PolynomialQ[Px, x]
+Int[Px_*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol] := With[{u = IntHide[ExpandExpression[Px, x], x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c}, x] && PolynomialQ[Px, x]
+(* Int[Px_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol] := With[{u=IntHide[Px,x]}, Dist[(a+b*ArcSin[c*x])^n,u,x] -  b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcSin[c*x])^(n-1)/Sqrt[1-c^2*x^2], x],x]] /; FreeQ[{a,b,c},x] && PolynomialQ[Px,x] && IGtQ[n,0] *)
+(* Int[Px_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol] := With[{u=IntHide[Px,x]}, Dist[(a+b*ArcCos[c*x])^n,u,x] +  b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcCos[c*x])^(n-1)/Sqrt[1-c^2*x^2], x],x]] /; FreeQ[{a,b,c},x] && PolynomialQ[Px,x] && IGtQ[n,0] *)
+Int[Px_*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[Px*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, n}, x] && PolynomialQ[Px, x]
+Int[Px_*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[Px*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, n}, x] && PolynomialQ[Px, x]
+Int[Px_*(d_. + e_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol] := With[{u = IntHide[Px*(d + e*x)^m, x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]
+Int[Px_*(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol] := With[{u = IntHide[Px*(d + e*x)^m, x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]
+Int[(f_. + g_.*x_)^p_.*(d_ + e_.*x_)^m_*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol] := With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcSin[c*x])^n, u, x] - b*c*n*Int[ SimplifyIntegrand[ u*(a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]
+Int[(f_. + g_.*x_)^p_.*(d_ + e_.*x_)^m_*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol] := With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcCos[c*x])^n, u, x] + b*c*n*Int[ SimplifyIntegrand[ u*(a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]
+Int[(f_. + g_.*x_ + h_.*x_^2)^ p_.*(a_. + b_.*ArcSin[c_.*x_])^n_/(d_ + e_.*x_)^2, x_Symbol] := With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcSin[c*x])^n, u, x] - b*c*n*Int[ SimplifyIntegrand[ u*(a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]
+Int[(f_. + g_.*x_ + h_.*x_^2)^ p_.*(a_. + b_.*ArcCos[c_.*x_])^n_/(d_ + e_.*x_)^2, x_Symbol] := With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcCos[c*x])^n, u, x] + b*c*n*Int[ SimplifyIntegrand[ u*(a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]
+Int[Px_*(d_ + e_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[Px*(d + e*x)^m*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && IGtQ[n, 0] && IntegerQ[m]
+Int[Px_*(d_ + e_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[Px*(d + e*x)^m*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && IGtQ[n, 0] && IntegerQ[m]
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol] := With[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[Dist[1/Sqrt[1 - c^2*x^2], u, x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol] := With[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[Dist[1/Sqrt[1 - c^2*x^2], u, x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^ n_., x_Symbol] := Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^ m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (m == 1 || p > 0 || n == 1 && p > -1 || m == 2 && p < -2)
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^ n_., x_Symbol] := Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, (f + g*x)^ m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (m == 1 || p > 0 || n == 1 && p > -1 || m == 2 && p < -2)
+Int[(f_ + g_.*x_)^m_* Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := (f + g*x)^ m*(d + e*x^2)*(a + b*ArcSin[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)) - 1/(b*c*Sqrt[d]*(n + 1))* Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^m_* Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := -(f + g*x)^m*(d + e*x^2)*(a + b*ArcCos[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)) + 1/(b*c*Sqrt[d]*(n + 1))* Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^ n_., x_Symbol] := Int[ExpandIntegrand[ Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n, (f + g*x)^ m*(d + e*x^2)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^ n_., x_Symbol] := Int[ExpandIntegrand[ Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n, (f + g*x)^ m*(d + e*x^2)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^ n_., x_Symbol] := (f + g*x)^ m*(d + e*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n + 1)/(b*c* Sqrt[d]*(n + 1)) - 1/(b*c*Sqrt[d]*(n + 1))* Int[ ExpandIntegrand[(f + g*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), (d*g*m + e*f*(2*p + 1)*x + e*g*(m + 2*p + 1)*x^2)*(d + e*x^2)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && IGtQ[p - 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^ n_., x_Symbol] := -(f + g*x)^ m*(d + e*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n + 1)/(b*c* Sqrt[d]*(n + 1)) + 1/(b*c*Sqrt[d]*(n + 1))* Int[ ExpandIntegrand[(f + g*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), (d*g*m + e*f*(2*p + 1)*x + e*g*(m + 2*p + 1)*x^2)*(d + e*x^2)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && IGtQ[p - 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_/ Sqrt[d_ + e_.*x_^2], x_Symbol] := (f + g*x)^m*(a + b*ArcSin[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)) - g*m/(b*c*Sqrt[d]*(n + 1))* Int[(f + g*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && GtQ[d, 0] && LtQ[n, -1]
+Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_/ Sqrt[d_ + e_.*x_^2], x_Symbol] := -(f + g*x)^m*(a + b*ArcCos[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)) + g*m/(b*c*Sqrt[d]*(n + 1))* Int[(f + g*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && GtQ[d, 0] && LtQ[n, -1]
+Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_./ Sqrt[d_ + e_.*x_^2], x_Symbol] := 1/(c^(m + 1)*Sqrt[d])* Subst[Int[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])
+Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_./ Sqrt[d_ + e_.*x_^2], x_Symbol] := -1/(c^(m + 1)*Sqrt[d])* Subst[Int[(a + b*x)^n*(c*f + g*Cos[x])^m, x], x, ArcCos[c*x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^ n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n/ Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^ n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n/ Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^ n_., x_Symbol] := Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p - 1/2] && Not[GtQ[d, 0]]
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^ n_., x_Symbol] := Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcCos[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p - 1/2] && Not[GtQ[d, 0]]
+Int[Log[h_.*(f_. + g_.*x_)^m_.]*(a_. + b_.*ArcSin[c_.*x_])^n_./ Sqrt[d_ + e_.*x_^2], x_Symbol] := Log[h*(f + g*x)^m]*(a + b*ArcSin[c*x])^(n + 1)/(b*c* Sqrt[d]*(n + 1)) - g*m/(b*c*Sqrt[d]*(n + 1))* Int[(a + b*ArcSin[c*x])^(n + 1)/(f + g*x), x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IGtQ[n, 0]
+Int[Log[h_.*(f_. + g_.*x_)^m_.]*(a_. + b_.*ArcCos[c_.*x_])^n_./ Sqrt[d_ + e_.*x_^2], x_Symbol] := -Log[h*(f + g*x)^m]*(a + b*ArcCos[c*x])^(n + 1)/(b*c* Sqrt[d]*(n + 1)) + g*m/(b*c*Sqrt[d]*(n + 1))* Int[(a + b*ArcCos[c*x])^(n + 1)/(f + g*x), x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IGtQ[n, 0]
+Int[Log[h_.*(f_. + g_.*x_)^m_.]*(d_ + e_.*x_^2)^ p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[Log[h*(f + g*x)^m]*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && Not[GtQ[d, 0]]
+Int[Log[h_.*(f_. + g_.*x_)^m_.]*(d_ + e_.*x_^2)^ p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]* Int[Log[h*(f + g*x)^m]*(1 - c^2*x^2)^p*(a + b*ArcCos[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && Not[GtQ[d, 0]]
+Int[(d_ + e_.*x_)^m_*(f_ + g_.*x_)^m_*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x)^m*(f + g*x)^m, x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[Dist[1/Sqrt[1 - c^2*x^2], u, x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m + 1/2, 0]
+Int[(d_ + e_.*x_)^m_*(f_ + g_.*x_)^m_*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x)^m*(f + g*x)^m, x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[Dist[1/Sqrt[1 - c^2*x^2], u, x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m + 1/2, 0]
+Int[(d_ + e_.*x_)^m_.*(f_ + g_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^ n_., x_Symbol] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^m*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && IntegerQ[m]
+Int[(d_ + e_.*x_)^m_.*(f_ + g_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^ n_., x_Symbol] := Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^m*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && IntegerQ[m]
+Int[u_*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol] := With[{v = IntHide[u, x]}, Dist[a + b*ArcSin[c*x], v, x] - b*c*Int[SimplifyIntegrand[v/Sqrt[1 - c^2*x^2], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
+Int[u_*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol] := With[{v = IntHide[u, x]}, Dist[a + b*ArcCos[c*x], v, x] + b*c*Int[SimplifyIntegrand[v/Sqrt[1 - c^2*x^2], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
+Int[Px_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := With[{u = ExpandIntegrand[Px*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && PolynomialQ[Px, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
+Int[Px_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := With[{u = ExpandIntegrand[Px*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && PolynomialQ[Px, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
+Int[Px_.*(f_ + g_.*(d_ + e_.*x_^2)^p_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^ n_., x_Symbol] := With[{u = ExpandIntegrand[Px*(f + g*(d + e*x^2)^p)^m*(a + b*ArcSin[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, g}, x] && PolynomialQ[Px, x] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && IntegersQ[m, n]
+Int[Px_.*(f_ + g_.*(d_ + e_.*x_^2)^p_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^ n_., x_Symbol] := With[{u = ExpandIntegrand[Px*(f + g*(d + e*x^2)^p)^m*(a + b*ArcCos[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, g}, x] && PolynomialQ[Px, x] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && IntegersQ[m, n]
+Int[RFx_*ArcSin[c_.*x_]^n_., x_Symbol] := With[{u = ExpandIntegrand[ArcSin[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[c, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
+Int[RFx_*ArcCos[c_.*x_]^n_., x_Symbol] := With[{u = ExpandIntegrand[ArcCos[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[c, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
+Int[RFx_*(a_ + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[RFx*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
+Int[RFx_*(a_ + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[RFx*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
+Int[RFx_*(d_ + e_.*x_^2)^p_*ArcSin[c_.*x_]^n_., x_Symbol] := With[{u = ExpandIntegrand[(d + e*x^2)^p*ArcSin[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
+Int[RFx_*(d_ + e_.*x_^2)^p_*ArcCos[c_.*x_]^n_., x_Symbol] := With[{u = ExpandIntegrand[(d + e*x^2)^p*ArcCos[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
+Int[RFx_*(d_ + e_.*x_^2)^p_*(a_ + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
+Int[RFx_*(d_ + e_.*x_^2)^p_*(a_ + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
+Int[u_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol] := Unintegrable[u*(a + b*ArcSin[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
+Int[u_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol] := Unintegrable[u*(a + b*ArcCos[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
diff --git a/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m b/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m
new file mode 100755
index 0000000..064ecd5
--- /dev/null
+++ b/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m	
@@ -0,0 +1,46 @@
+
+(* ::Subsection::Closed:: *)
+(* 5.1.6 Miscellaneous inverse sine *)
+Int[(a_. + b_.*ArcSin[c_ + d_.*x_])^n_., x_Symbol] := 1/d*Subst[Int[(a + b*ArcSin[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, n}, x]
+Int[(a_. + b_.*ArcCos[c_ + d_.*x_])^n_., x_Symbol] := 1/d*Subst[Int[(a + b*ArcCos[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, n}, x]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSin[c_ + d_.*x_])^n_., x_Symbol] := 1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcSin[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCos[c_ + d_.*x_])^n_., x_Symbol] := 1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcCos[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
+Int[(A_. + B_.*x_ + C_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_ + d_.*x_])^n_., x_Symbol] := 1/d*Subst[Int[(-C/d^2 + C/d^2*x^2)^p*(a + b*ArcSin[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, A, B, C, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[(A_. + B_.*x_ + C_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_ + d_.*x_])^n_., x_Symbol] := 1/d*Subst[Int[(-C/d^2 + C/d^2*x^2)^p*(a + b*ArcCos[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, A, B, C, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[(e_. + f_.*x_)^m_.*(A_. + B_.*x_ + C_.*x_^2)^ p_.*(a_. + b_.*ArcSin[c_ + d_.*x_])^n_., x_Symbol] := 1/d*Subst[ Int[((d*e - c*f)/d + f*x/d)^m*(-C/d^2 + C/d^2*x^2)^ p*(a + b*ArcSin[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[(e_. + f_.*x_)^m_.*(A_. + B_.*x_ + C_.*x_^2)^ p_.*(a_. + b_.*ArcCos[c_ + d_.*x_])^n_., x_Symbol] := 1/d*Subst[ Int[((d*e - c*f)/d + f*x/d)^m*(-C/d^2 + C/d^2*x^2)^ p*(a + b*ArcCos[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[Sqrt[a_. + b_.*ArcSin[c_ + d_.*x_^2]], x_Symbol] := x*Sqrt[a + b*ArcSin[c + d*x^2]] - Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])* FresnelC[Sqrt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]]/ (Sqrt[ c/b]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])) + Sqrt[Pi]*x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])* FresnelS[Sqrt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]]/ (Sqrt[ c/b]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]
+Int[Sqrt[a_. + b_.*ArcCos[1 + d_.*x_^2]], x_Symbol] := -2*Sqrt[a + b*ArcCos[1 + d*x^2]]*Sin[ArcCos[1 + d*x^2]/2]^2/(d*x) - 2*Sqrt[Pi]*Sin[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]* FresnelC[ Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]]/(Sqrt[1/b]*d* x) + 2*Sqrt[Pi]*Cos[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]* FresnelS[ Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]]/(Sqrt[1/b]*d*x) /; FreeQ[{a, b, d}, x]
+Int[Sqrt[a_. + b_.*ArcCos[-1 + d_.*x_^2]], x_Symbol] := 2*Sqrt[a + b*ArcCos[-1 + d*x^2]]* Cos[(1/2)*ArcCos[-1 + d*x^2]]^2/(d*x) - 2*Sqrt[Pi]*Cos[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]* FresnelC[ Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]]/(Sqrt[1/b]*d*x) - 2*Sqrt[Pi]*Sin[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]* FresnelS[ Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]]/(Sqrt[1/b]*d*x) /; FreeQ[{a, b, d}, x]
+Int[(a_. + b_.*ArcSin[c_ + d_.*x_^2])^n_, x_Symbol] := x*(a + b*ArcSin[c + d*x^2])^n + 2*b*n* Sqrt[-2*c*d*x^2 - d^2*x^4]*(a + b*ArcSin[c + d*x^2])^(n - 1)/(d*x) - 4*b^2*n*(n - 1)*Int[(a + b*ArcSin[c + d*x^2])^(n - 2), x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && GtQ[n, 1]
+Int[(a_. + b_.*ArcCos[c_ + d_.*x_^2])^n_, x_Symbol] := x*(a + b*ArcCos[c + d*x^2])^n - 2*b*n* Sqrt[-2*c*d*x^2 - d^2*x^4]*(a + b*ArcCos[c + d*x^2])^(n - 1)/(d*x) - 4*b^2*n*(n - 1)*Int[(a + b*ArcCos[c + d*x^2])^(n - 2), x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && GtQ[n, 1]
+Int[1/(a_. + b_.*ArcSin[c_ + d_.*x_^2]), x_Symbol] := -x*(c*Cos[a/(2*b)] - Sin[a/(2*b)])* CosIntegral[(c/(2*b))*(a + b*ArcSin[c + d*x^2])]/ (2* b*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])) - x*(c*Cos[a/(2*b)] + Sin[a/(2*b)])* SinIntegral[(c/(2*b))*(a + b*ArcSin[c + d*x^2])]/ (2* b*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]
+Int[1/(a_. + b_.*ArcCos[1 + d_.*x_^2]), x_Symbol] := x*Cos[a/(2*b)]* CosIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[-d*x^2]) + x*Sin[a/(2*b)]* SinIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[-d*x^2]) /; FreeQ[{a, b, d}, x]
+Int[1/(a_. + b_.*ArcCos[-1 + d_.*x_^2]), x_Symbol] := x*Sin[a/(2*b)]* CosIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[d*x^2]) - x*Cos[a/(2*b)]* SinIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[d*x^2]) /; FreeQ[{a, b, d}, x]
+Int[1/Sqrt[a_. + b_.*ArcSin[c_ + d_.*x_^2]], x_Symbol] := -Sqrt[Pi]*x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])* FresnelC[1/(Sqrt[b*c]*Sqrt[Pi])*Sqrt[a + b*ArcSin[c + d*x^2]]]/ (Sqrt[ b*c]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])) - Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])* FresnelS[(1/(Sqrt[b*c]*Sqrt[Pi]))*Sqrt[a + b*ArcSin[c + d*x^2]]]/ (Sqrt[ b*c]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]
+Int[1/Sqrt[a_. + b_.*ArcCos[1 + d_.*x_^2]], x_Symbol] := -2*Sqrt[Pi/b]*Cos[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]* FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]]/(d*x) - 2*Sqrt[Pi/b]*Sin[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]* FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]]/(d*x) /; FreeQ[{a, b, d}, x]
+Int[1/Sqrt[a_. + b_.*ArcCos[-1 + d_.*x_^2]], x_Symbol] := 2*Sqrt[Pi/b]*Sin[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]* FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]]/(d*x) - 2*Sqrt[Pi/b]*Cos[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]* FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]]/(d*x) /; FreeQ[{a, b, d}, x]
+Int[1/(a_. + b_.*ArcSin[c_ + d_.*x_^2])^(3/2), x_Symbol] := -Sqrt[-2*c*d*x^2 - d^2*x^4]/(b*d*x*Sqrt[a + b*ArcSin[c + d*x^2]]) - (c/b)^(3/2)*Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])* FresnelC[Sqrt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]]/ (Cos[(1/2)*ArcSin[c + d*x^2]] - c*Sin[ArcSin[c + d*x^2]/2]) + (c/b)^(3/2)*Sqrt[Pi]*x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])* FresnelS[Sqrt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]]/ (Cos[(1/2)*ArcSin[c + d*x^2]] - c*Sin[ArcSin[c + d*x^2]/2]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]
+Int[1/(a_. + b_.*ArcCos[1 + d_.*x_^2])^(3/2), x_Symbol] := Sqrt[-2*d*x^2 - d^2*x^4]/(b*d*x*Sqrt[a + b*ArcCos[1 + d*x^2]]) - 2*(1/b)^(3/2)*Sqrt[Pi]*Sin[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]* FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]]/(d*x) + 2*(1/b)^(3/2)*Sqrt[Pi]*Cos[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]* FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]]/(d*x) /; FreeQ[{a, b, d}, x]
+Int[1/(a_. + b_.*ArcCos[-1 + d_.*x_^2])^(3/2), x_Symbol] := Sqrt[2*d*x^2 - d^2*x^4]/(b*d*x*Sqrt[a + b*ArcCos[-1 + d*x^2]]) - 2*(1/b)^(3/2)*Sqrt[Pi]*Cos[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]* FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]]/(d*x) - 2*(1/b)^(3/2)*Sqrt[Pi]*Sin[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]* FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]]/(d*x) /; FreeQ[{a, b, d}, x]
+Int[1/(a_. + b_.*ArcSin[c_ + d_.*x_^2])^2, x_Symbol] := -Sqrt[-2*c*d*x^2 - d^2*x^4]/(2*b*d*x*(a + b*ArcSin[c + d*x^2])) - x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])* CosIntegral[(c/(2*b))*(a + b*ArcSin[c + d*x^2])]/ (4* b^2*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])) + x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])* SinIntegral[(c/(2*b))*(a + b*ArcSin[c + d*x^2])]/ (4* b^2*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]
+Int[1/(a_. + b_.*ArcCos[1 + d_.*x_^2])^2, x_Symbol] := Sqrt[-2*d*x^2 - d^2*x^4]/(2*b*d*x*(a + b*ArcCos[1 + d*x^2])) + x*Sin[a/(2*b)]* CosIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[(-d)*x^2]) - x*Cos[a/(2*b)]* SinIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[(-d)*x^2]) /; FreeQ[{a, b, d}, x]
+Int[1/(a_. + b_.*ArcCos[-1 + d_.*x_^2])^2, x_Symbol] := Sqrt[2*d*x^2 - d^2*x^4]/(2*b*d*x*(a + b*ArcCos[-1 + d*x^2])) - x*Cos[a/(2*b)]* CosIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[d*x^2]) - x*Sin[a/(2*b)]* SinIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[d*x^2]) /; FreeQ[{a, b, d}, x]
+Int[(a_. + b_.*ArcSin[c_ + d_.*x_^2])^n_, x_Symbol] := x*(a + b*ArcSin[c + d*x^2])^(n + 2)/(4*b^2*(n + 1)*(n + 2)) + Sqrt[-2*c*d*x^2 - d^2*x^4]*(a + b*ArcSin[c + d*x^2])^(n + 1)/(2*b*d*(n + 1)*x) - 1/(4*b^2*(n + 1)*(n + 2))* Int[(a + b*ArcSin[c + d*x^2])^(n + 2), x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && LtQ[n, -1] && NeQ[n, -2]
+Int[(a_. + b_.*ArcCos[c_ + d_.*x_^2])^n_, x_Symbol] := x*(a + b*ArcCos[c + d*x^2])^(n + 2)/(4*b^2*(n + 1)*(n + 2)) - Sqrt[-2*c*d*x^2 - d^2*x^4]*(a + b*ArcCos[c + d*x^2])^(n + 1)/(2*b*d*(n + 1)*x) - 1/(4*b^2*(n + 1)*(n + 2))* Int[(a + b*ArcCos[c + d*x^2])^(n + 2), x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && LtQ[n, -1] && NeQ[n, -2]
+Int[ArcSin[a_.*x_^p_]^n_./x_, x_Symbol] := 1/p*Subst[Int[x^n*Cot[x], x], x, ArcSin[a*x^p]] /; FreeQ[{a, p}, x] && IGtQ[n, 0]
+Int[ArcCos[a_.*x_^p_]^n_./x_, x_Symbol] := -1/p*Subst[Int[x^n*Tan[x], x], x, ArcCos[a*x^p]] /; FreeQ[{a, p}, x] && IGtQ[n, 0]
+Int[u_.*ArcSin[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol] := Int[u*ArcCsc[a/c + b*x^n/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
+Int[u_.*ArcCos[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol] := Int[u*ArcSec[a/c + b*x^n/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
+Int[ArcSin[Sqrt[1 + b_.*x_^2]]^n_./Sqrt[1 + b_.*x_^2], x_Symbol] := Sqrt[-b*x^2]/(b*x)* Subst[Int[ArcSin[x]^n/Sqrt[1 - x^2], x], x, Sqrt[1 + b*x^2]] /; FreeQ[{b, n}, x]
+Int[ArcCos[Sqrt[1 + b_.*x_^2]]^n_./Sqrt[1 + b_.*x_^2], x_Symbol] := Sqrt[-b*x^2]/(b*x)* Subst[Int[ArcCos[x]^n/Sqrt[1 - x^2], x], x, Sqrt[1 + b*x^2]] /; FreeQ[{b, n}, x]
+Int[u_.*f_^(c_.*ArcSin[a_. + b_.*x_]^n_.), x_Symbol] := 1/b*Subst[ Int[ReplaceAll[u, x -> -a/b + Sin[x]/b]*f^(c*x^n)*Cos[x], x], x, ArcSin[a + b*x]] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
+Int[u_.*f_^(c_.*ArcCos[a_. + b_.*x_]^n_.), x_Symbol] := -1/b*Subst[ Int[ReplaceAll[u, x -> -a/b + Cos[x]/b]*f^(c*x^n)*Sin[x], x], x, ArcCos[a + b*x]] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
+Int[ArcSin[a_.*x_^2 + b_.*Sqrt[c_ + d_.*x_^2]], x_Symbol] := x*ArcSin[a*x^2 + b*Sqrt[c + d*x^2]] - x*Sqrt[b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2]]/ Sqrt[(-x^2)*(b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2])]* Int[ x*(b*d + 2*a*Sqrt[c + d*x^2])/(Sqrt[c + d*x^2]* Sqrt[b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2*c, 1]
+Int[ArcCos[a_.*x_^2 + b_.*Sqrt[c_ + d_.*x_^2]], x_Symbol] := x*ArcCos[a*x^2 + b*Sqrt[c + d*x^2]] + x*Sqrt[b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2]]/ Sqrt[(-x^2)*(b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2])]* Int[ x*(b*d + 2*a*Sqrt[c + d*x^2])/(Sqrt[c + d*x^2]* Sqrt[b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2*c, 1]
+Int[ArcSin[u_], x_Symbol] := x*ArcSin[u] - Int[SimplifyIntegrand[x*D[u, x]/Sqrt[1 - u^2], x], x] /; InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]
+Int[ArcCos[u_], x_Symbol] := x*ArcCos[u] + Int[SimplifyIntegrand[x*D[u, x]/Sqrt[1 - u^2], x], x] /; InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcSin[u_]), x_Symbol] := (c + d*x)^(m + 1)*(a + b*ArcSin[u])/(d*(m + 1)) - b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/Sqrt[1 - u^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcCos[u_]), x_Symbol] := (c + d*x)^(m + 1)*(a + b*ArcCos[u])/(d*(m + 1)) + b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/Sqrt[1 - u^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]
+Int[v_*(a_. + b_.*ArcSin[u_]), x_Symbol] := With[{w = IntHide[v, x]}, Dist[(a + b*ArcSin[u]), w, x] - b*Int[SimplifyIntegrand[w*D[u, x]/Sqrt[1 - u^2], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_. /; FreeQ[{c, d, m}, x]]]
+Int[v_*(a_. + b_.*ArcCos[u_]), x_Symbol] := With[{w = IntHide[v, x]}, Dist[(a + b*ArcCos[u]), w, x] + b*Int[SimplifyIntegrand[w*D[u, x]/Sqrt[1 - u^2], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_. /; FreeQ[{c, d, m}, x]]]
diff --git a/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 (a+b arctan(c x^n))^p.m b/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 (a+b arctan(c x^n))^p.m
new file mode 100755
index 0000000..c0b54a3
--- /dev/null
+++ b/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 (a+b arctan(c x^n))^p.m	
@@ -0,0 +1,13 @@
+
+(* ::Subsection::Closed:: *)
+(* 5.3.1 (a+b arctan(c x^n))^p *)
+Int[(a_. + b_.*ArcTan[c_.*x_^n_.])^p_., x_Symbol] := x*(a + b*ArcTan[c*x^n])^p - b*c*n*p* Int[x^n*(a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n)), x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
+Int[(a_. + b_.*ArcCot[c_.*x_^n_.])^p_., x_Symbol] := x*(a + b*ArcCot[c*x^n])^p + b*c*n*p* Int[x^n*(a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n)), x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
+Int[(a_. + b_.*ArcTan[c_.*x_^n_])^p_, x_Symbol] := Int[ExpandIntegrand[(a + (I*b*Log[1 - I*c*x^n])/ 2 - (I*b*Log[1 + I*c*x^n])/2)^p, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_^n_])^p_, x_Symbol] := Int[ExpandIntegrand[(a + (I*b*Log[1 - I*x^(-n)/c])/ 2 - (I*b*Log[1 + I*x^(-n)/c])/2)^p, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_^n_])^p_, x_Symbol] := Int[(a + b*ArcCot[x^(-n)/c])^p, x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && ILtQ[n, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_^n_])^p_, x_Symbol] := Int[(a + b*ArcTan[x^(-n)/c])^p, x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && ILtQ[n, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_^n_])^p_, x_Symbol] := With[{k = Denominator[n]}, k*Subst[Int[x^(k - 1)*(a + b*ArcTan[c*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && FractionQ[n]
+Int[(a_. + b_.*ArcCot[c_.*x_^n_])^p_, x_Symbol] := With[{k = Denominator[n]}, k*Subst[Int[x^(k - 1)*(a + b*ArcCot[c*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && FractionQ[n]
+Int[(a_. + b_.*ArcTan[c_.*x_^n_.])^p_, x_Symbol] := Unintegrable[(a + b*ArcTan[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p}, x]
+Int[(a_. + b_.*ArcCot[c_.*x_^n_.])^p_, x_Symbol] := Unintegrable[(a + b*ArcCot[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p}, x]
diff --git a/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 (d x)^m (a+b arctan(c x^n))^p.m b/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 (d x)^m (a+b arctan(c x^n))^p.m
new file mode 100755
index 0000000..20f06b8
--- /dev/null
+++ b/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 (d x)^m (a+b arctan(c x^n))^p.m	
@@ -0,0 +1,27 @@
+
+(* ::Subsection::Closed:: *)
+(* 5.3.2 (d x)^m (a+b arctan(c x^n))^p *)
+Int[(a_. + b_.*ArcTan[c_.*x_])/x_, x_Symbol] := a*Log[x] + I*b/2*Int[Log[1 - I*c*x]/x, x] - I*b/2*Int[Log[1 + I*c*x]/x, x] /; FreeQ[{a, b, c}, x]
+Int[(a_. + b_.*ArcCot[c_.*x_])/x_, x_Symbol] := a*Log[x] + I*b/2*Int[Log[1 - I/(c*x)]/x, x] - I*b/2*Int[Log[1 + I/(c*x)]/x, x] /; FreeQ[{a, b, c}, x]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_/x_, x_Symbol] := 2*(a + b*ArcTan[c*x])^p*ArcTanh[1 - 2/(1 + I*c*x)] - 2*b*c*p* Int[(a + b*ArcTan[c*x])^(p - 1)* ArcTanh[1 - 2/(1 + I*c*x)]/(1 + c^2*x^2), x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_/x_, x_Symbol] := 2*(a + b*ArcCot[c*x])^p*ArcCoth[1 - 2/(1 + I*c*x)] + 2*b*c*p* Int[(a + b*ArcCot[c*x])^(p - 1)* ArcCoth[1 - 2/(1 + I*c*x)]/(1 + c^2*x^2), x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
+Int[(a_. + b_.*ArcTan[c_.*x_^n_])^p_./x_, x_Symbol] := 1/n*Subst[Int[(a + b*ArcTan[c*x])^p/x, x], x, x^n] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_^n_])^p_./x_, x_Symbol] := 1/n*Subst[Int[(a + b*ArcCot[c*x])^p/x, x], x, x^n] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]
+Int[x_^m_.*(a_. + b_.*ArcTan[c_.*x_^n_.])^p_., x_Symbol] := x^(m + 1)*(a + b*ArcTan[c*x^n])^p/(m + 1) - b*c*n*p/(m + 1)* Int[x^(m + n)*(a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n)), x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || EqQ[n, 1] && IntegerQ[m]) && NeQ[m, -1]
+Int[x_^m_.*(a_. + b_.*ArcCot[c_.*x_^n_.])^p_., x_Symbol] := x^(m + 1)*(a + b*ArcCot[c*x^n])^p/(m + 1) + b*c*n*p/(m + 1)* Int[x^(m + n)*(a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n)), x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || EqQ[n, 1] && IntegerQ[m]) && NeQ[m, -1]
+Int[x_^m_.*(a_. + b_.*ArcTan[c_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcTan[c*x])^p, x], x, x^n] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simplify[(m + 1)/n]]
+Int[x_^m_.*(a_. + b_.*ArcCot[c_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcCot[c*x])^p, x], x, x^n] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simplify[(m + 1)/n]]
+Int[x_^m_.*(a_. + b_.*ArcTan[c_.*x_^n_])^p_, x_Symbol] := Int[ExpandIntegrand[ x^m*(a + (I*b*Log[1 - I*c*x^n])/2 - (I*b*Log[1 + I*c*x^n])/2)^p, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0] && IntegerQ[m]
+Int[x_^m_.*(a_. + b_.*ArcCot[c_.*x_^n_])^p_, x_Symbol] := Int[ExpandIntegrand[ x^m*(a + (I*b*Log[1 - I*x^(-n)/c])/2 - (I*b*Log[1 + I*x^(-n)/c])/ 2)^p, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0] && IntegerQ[m]
+Int[x_^m_.*(a_. + b_.*ArcTan[c_.*x_^n_])^p_, x_Symbol] := With[{k = Denominator[m]}, k*Subst[Int[x^(k*(m + 1) - 1)*(a + b*ArcTan[c*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0] && FractionQ[m]
+Int[x_^m_.*(a_. + b_.*ArcCot[c_.*x_^n_])^p_, x_Symbol] := With[{k = Denominator[m]}, k*Subst[Int[x^(k*(m + 1) - 1)*(a + b*ArcCot[c*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0] && FractionQ[m]
+Int[x_^m_.*(a_. + b_.*ArcTan[c_.*x_^n_])^p_, x_Symbol] := Int[x^m*(a + b*ArcCot[x^(-n)/c])^p, x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && ILtQ[n, 0]
+Int[x_^m_.*(a_. + b_.*ArcCot[c_.*x_^n_])^p_, x_Symbol] := Int[x^m*(a + b*ArcTan[x^(-n)/c])^p, x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && ILtQ[n, 0]
+Int[x_^m_.*(a_. + b_.*ArcTan[c_.*x_^n_])^p_, x_Symbol] := With[{k = Denominator[n]}, k*Subst[Int[x^(k*(m + 1) - 1)*(a + b*ArcTan[c*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && FractionQ[n]
+Int[x_^m_.*(a_. + b_.*ArcCot[c_.*x_^n_])^p_, x_Symbol] := With[{k = Denominator[n]}, k*Subst[Int[x^(k*(m + 1) - 1)*(a + b*ArcCot[c*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && FractionQ[n]
+Int[(d_*x_)^m_*(a_. + b_.*ArcTan[c_.*x_^n_.]), x_Symbol] := (d*x)^(m + 1)*(a + b*ArcTan[c*x^n])/(d*(m + 1)) - b*c*n/(d^n*(m + 1))*Int[(d*x)^(m + n)/(1 + c^2*x^(2*n)), x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[n] && NeQ[m, -1]
+Int[(d_*x_)^m_*(a_. + b_.*ArcCot[c_.*x_^n_.]), x_Symbol] := (d*x)^(m + 1)*(a + b*ArcCot[c*x^n])/(d*(m + 1)) + b*c*n/(d^n*(m + 1))*Int[(d*x)^(m + n)/(1 + c^2*x^(2*n)), x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[n] && NeQ[m, -1]
+Int[(d_.*x_)^m_*(a_. + b_.*ArcTan[c_.*x_^n_])^p_., x_Symbol] := d^IntPart[m]*(d*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*ArcTan[c*x])^p, x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || RationalQ[m, n])
+Int[(d_.*x_)^m_*(a_. + b_.*ArcCot[c_.*x_^n_])^p_., x_Symbol] := d^IntPart[m]*(d*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*ArcCot[c*x])^p, x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || RationalQ[m, n])
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcTan[c_.*x_^n_.])^p_., x_Symbol] := Unintegrable[(d*x)^m*(a + b*ArcTan[c*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcCot[c_.*x_^n_.])^p_., x_Symbol] := Unintegrable[(d*x)^m*(a + b*ArcCot[c*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
diff --git a/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 (d+e x)^m (a+b arctan(c x^n))^p.m b/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 (d+e x)^m (a+b arctan(c x^n))^p.m
new file mode 100755
index 0000000..24f2a87
--- /dev/null
+++ b/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 (d+e x)^m (a+b arctan(c x^n))^p.m	
@@ -0,0 +1,25 @@
+
+(* ::Subsection::Closed:: *)
+(* 5.3.3 (d+e x)^m (a+b arctan(c x^n))^p *)
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := -(a + b*ArcTan[c*x])^p*Log[2/(1 + e*x/d)]/e + b*c*p/e* Int[(a + b*ArcTan[c*x])^(p - 1)*Log[2/(1 + e*x/d)]/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := -(a + b*ArcCot[c*x])^p*Log[2/(1 + e*x/d)]/e - b*c*p/e* Int[(a + b*ArcCot[c*x])^(p - 1)*Log[2/(1 + e*x/d)]/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_])/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)]/e + b*c/e*Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x] + (a + b*ArcTan[c*x])* Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e - b*c/e* Int[Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcCot[c*x])*Log[2/(1 - I*c*x)]/e - b*c/e*Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x] + (a + b*ArcCot[c*x])* Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e + b*c/e* Int[Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_])^2/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)]/e + I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)]/e - b^2*PolyLog[3, 1 - 2/(1 - I*c*x)]/(2*e) + (a + b*ArcTan[c*x])^2* Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e - I*b*(a + b*ArcTan[c*x])* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e + b^2* PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(2*e) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])^2/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcCot[c*x])^2*Log[2/(1 - I*c*x)]/e - I*b*(a + b*ArcCot[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)]/e - b^2*PolyLog[3, 1 - 2/(1 - I*c*x)]/(2*e) + (a + b*ArcCot[c*x])^2* Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e + I*b*(a + b*ArcCot[c*x])* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e + b^2* PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(2*e) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_])^3/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcTan[c*x])^3*Log[2/(1 - I*c*x)]/e + 3*I*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 - I*c*x)]/(2*e) - 3*b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - 2/(1 - I*c*x)]/(2*e) - 3*I*b^3*PolyLog[4, 1 - 2/(1 - I*c*x)]/(4*e) + (a + b*ArcTan[c*x])^3* Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e - 3*I*b*(a + b*ArcTan[c*x])^2* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(2*e) + 3*b^2*(a + b*ArcTan[c*x])* PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(2*e) + 3*I*b^3* PolyLog[4, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(4*e) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])^3/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcCot[c*x])^3*Log[2/(1 - I*c*x)]/e - 3*I*b*(a + b*ArcCot[c*x])^2*PolyLog[2, 1 - 2/(1 - I*c*x)]/(2*e) - 3*b^2*(a + b*ArcCot[c*x])*PolyLog[3, 1 - 2/(1 - I*c*x)]/(2*e) + 3*I*b^3*PolyLog[4, 1 - 2/(1 - I*c*x)]/(4*e) + (a + b*ArcCot[c*x])^3* Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e + 3*I*b*(a + b*ArcCot[c*x])^2* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(2*e) + 3*b^2*(a + b*ArcCot[c*x])* PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(2*e) - 3*I*b^3* PolyLog[4, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(4*e) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol] := (d + e*x)^(q + 1)*(a + b*ArcTan[c*x])/(e*(q + 1)) - b*c/(e*(q + 1))*Int[(d + e*x)^(q + 1)/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol] := (d + e*x)^(q + 1)*(a + b*ArcCot[c*x])/(e*(q + 1)) + b*c/(e*(q + 1))*Int[(d + e*x)^(q + 1)/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_, x_Symbol] := (d + e*x)^(q + 1)*(a + b*ArcTan[c*x])^p/(e*(q + 1)) - b*c*p/(e*(q + 1))* Int[ExpandIntegrand[(a + b*ArcTan[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_, x_Symbol] := (d + e*x)^(q + 1)*(a + b*ArcCot[c*x])^p/(e*(q + 1)) + b*c*p/(e*(q + 1))* Int[ExpandIntegrand[(a + b*ArcCot[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
+Int[(a_. + b_.*ArcTan[c_.*x_^n_])/(d_ + e_.*x_), x_Symbol] := Log[d + e*x]*(a + b*ArcTan[c*x^n])/e - b*c*n/e*Int[x^(n - 1)*Log[d + e*x]/(1 + c^2*x^(2*n)), x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[n]
+Int[(a_. + b_.*ArcCot[c_.*x_^n_])/(d_ + e_.*x_), x_Symbol] := Log[d + e*x]*(a + b*ArcCot[c*x^n])/e + b*c*n/e*Int[x^(n - 1)*Log[d + e*x]/(1 + c^2*x^(2*n)), x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[n]
+Int[(a_. + b_.*ArcTan[c_.*x_^n_])/(d_ + e_.*x_), x_Symbol] := With[{k = Denominator[n]}, k*Subst[Int[x^(k - 1)*(a + b*ArcTan[c*x^(k*n)])/(d + e*x^k), x], x, x^(1/k)]] /; FreeQ[{a, b, c, d, e}, x] && FractionQ[n]
+Int[(a_. + b_.*ArcCot[c_.*x_^n_])/(d_ + e_.*x_), x_Symbol] := With[{k = Denominator[n]}, k*Subst[Int[x^(k - 1)*(a + b*ArcCot[c*x^(k*n)])/(d + e*x^k), x], x, x^(1/k)]] /; FreeQ[{a, b, c, d, e}, x] && FractionQ[n]
+Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcTan[c_.*x_^n_]), x_Symbol] := (d + e*x)^(m + 1)*(a + b*ArcTan[c*x^n])/(e*(m + 1)) - b*c*n/(e*(m + 1))* Int[x^(n - 1)*(d + e*x)^(m + 1)/(1 + c^2*x^(2*n)), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
+Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcCot[c_.*x_^n_]), x_Symbol] := (d + e*x)^(m + 1)*(a + b*ArcCot[c*x^n])/(e*(m + 1)) + b*c*n/(e*(m + 1))* Int[x^(n - 1)*(d + e*x)^(m + 1)/(1 + c^2*x^(2*n)), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
+Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcTan[c_.*x_^n_])^p_, x_Symbol] := Int[ExpandIntegrand[(a + b*ArcTan[c*x^n])^p, (d + e*x)^m, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 1] && IGtQ[m, 0]
+Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcCot[c_.*x_^n_])^p_, x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCot[c*x^n])^p, (d + e*x)^m, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 1] && IGtQ[m, 0]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcTan[c_.*x_^n_])^p_., x_Symbol] := Unintegrable[(d + e*x)^m*(a + b*ArcTan[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCot[c_.*x_^n_])^p_., x_Symbol] := Unintegrable[(d + e*x)^m*(a + b*ArcCot[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
diff --git a/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 u (a+b arctan(c x))^p.m b/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 u (a+b arctan(c x))^p.m
new file mode 100755
index 0000000..177e1c7
--- /dev/null
+++ b/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 u (a+b arctan(c x))^p.m	
@@ -0,0 +1,166 @@
+
+(* ::Subsection::Closed:: *)
+(* 5.3.4 u (a+b arctan(c x))^p *)
+Int[(f_.*x_)^m_.*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := f/e*Int[(f*x)^(m - 1)*(a + b*ArcTan[c*x])^p, x] - d*f/e*Int[(f*x)^(m - 1)*(a + b*ArcTan[c*x])^p/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && GtQ[m, 0]
+Int[(f_.*x_)^m_.*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := f/e*Int[(f*x)^(m - 1)*(a + b*ArcCot[c*x])^p, x] - d*f/e*Int[(f*x)^(m - 1)*(a + b*ArcCot[c*x])^p/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && GtQ[m, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_./(x_*(d_ + e_.*x_)), x_Symbol] := (a + b*ArcTan[c*x])^p*Log[2 - 2/(1 + e*x/d)]/d - b*c*p/d* Int[(a + b*ArcTan[c*x])^(p - 1)* Log[2 - 2/(1 + e*x/d)]/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_./(x_*(d_ + e_.*x_)), x_Symbol] := (a + b*ArcCot[c*x])^p*Log[2 - 2/(1 + e*x/d)]/d + b*c*p/d* Int[(a + b*ArcCot[c*x])^(p - 1)* Log[2 - 2/(1 + e*x/d)]/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := 1/d*Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x] - e/(d*f)*Int[(f*x)^(m + 1)*(a + b*ArcTan[c*x])^p/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := 1/d*Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x] - e/(d*f)*Int[(f*x)^(m + 1)*(a + b*ArcCot[c*x])^p/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_.*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcTan[c*x]), u] - b*c*Int[SimplifyIntegrand[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[ 2*m] && (IGtQ[m, 0] && IGtQ[q, 0] || ILtQ[m + q + 1, 0] && LtQ[m*q, 0])
+Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_.*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcCot[c*x]), u] + b*c*Int[SimplifyIntegrand[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[ 2*m] && (IGtQ[m, 0] && IGtQ[q, 0] || ILtQ[m + q + 1, 0] && LtQ[m*q, 0])
+Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_, x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcTan[c*x])^p, u] - b*c*p*Int[ ExpandIntegrand[(a + b*ArcTan[c*x])^(p - 1), u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 + e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]
+Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_, x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcCot[c*x])^p, u] + b*c*p*Int[ ExpandIntegrand[(a + b*ArcCot[c*x])^(p - 1), u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 + e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCot[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol] := -b*(d + e*x^2)^q/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcTan[c*x])/(2*q + 1) + 2*d*q/(2*q + 1)*Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0]
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol] := b*(d + e*x^2)^q/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcCot[c*x])/(2*q + 1) + 2*d*q/(2*q + 1)*Int[(d + e*x^2)^(q - 1)*(a + b*ArcCot[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0]
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_, x_Symbol] := -b*p*(d + e*x^2)^ q*(a + b*ArcTan[c*x])^(p - 1)/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p/(2*q + 1) + 2*d*q/(2*q + 1)* Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x] + b^2*d*p*(p - 1)/(2*q*(2*q + 1))* Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^(p - 2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_, x_Symbol] := b*p*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p - 1)/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcCot[c*x])^p/(2*q + 1) + 2*d*q/(2*q + 1)* Int[(d + e*x^2)^(q - 1)*(a + b*ArcCot[c*x])^p, x] + b^2*d*p*(p - 1)/(2*q*(2*q + 1))* Int[(d + e*x^2)^(q - 1)*(a + b*ArcCot[c*x])^(p - 2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]
+(* Int[(a_.+b_.*ArcTan[c_.*x_])^p_./(d_+e_.*x_^2),x_Symbol] := 1/(c*d)*Subst[Int[(a+b*x)^p,x],x,ArcTan[c*x]] /; FreeQ[{a,b,c,d,e,p},x] && EqQ[e,c^2*d] *)
+(* Int[(a_.+b_.*ArcCot[c_.*x_])^p_./(d_+e_.*x_^2),x_Symbol] := -1/(c*d)*Subst[Int[(a+b*x)^p,x],x,ArcCot[c*x]] /; FreeQ[{a,b,c,d,e,p},x] && EqQ[e,c^2*d] *)
+Int[1/((d_ + e_.*x_^2)*(a_. + b_.*ArcTan[c_.*x_])), x_Symbol] := Log[RemoveContent[a + b*ArcTan[c*x], x]]/(b*c*d) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
+Int[1/((d_ + e_.*x_^2)*(a_. + b_.*ArcCot[c_.*x_])), x_Symbol] := -Log[RemoveContent[a + b*ArcCot[c*x], x]]/(b*c*d) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)) /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := -(a + b*ArcCot[c*x])^(p + 1)/(b*c*d*(p + 1)) /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
+Int[(a_. + b_.*ArcTan[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol] := -2*I*(a + b*ArcTan[c*x])* ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d]) + I*b* PolyLog[2, -I*Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d]) - I*b*PolyLog[2, I*Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol] := -2*I*(a + b*ArcCot[c*x])* ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d]) - I*b* PolyLog[2, -I*Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d]) + I*b*PolyLog[2, I*Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := 1/(c*Sqrt[d])*Subst[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := -x*Sqrt[1 + 1/(c^2*x^2)]/Sqrt[d + e*x^2]* Subst[Int[(a + b*x)^p*Csc[x], x], x, ArcCot[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcTan[c*x])^p/Sqrt[1 + c^2*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Not[GtQ[d, 0]]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcCot[c*x])^p/Sqrt[1 + c^2*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Not[GtQ[d, 0]]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := x*(a + b*ArcTan[c*x])^p/(2*d*(d + e*x^2)) + (a + b*ArcTan[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)) - b*c*p/2*Int[x*(a + b*ArcTan[c*x])^(p - 1)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := x*(a + b*ArcCot[c*x])^p/(2*d*(d + e*x^2)) - (a + b*ArcCot[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)) + b*c*p/2*Int[x*(a + b*ArcCot[c*x])^(p - 1)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_])/(d_ + e_.*x_^2)^(3/2), x_Symbol] := b/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcTan[c*x])/(d*Sqrt[d + e*x^2]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
+Int[(a_. + b_.*ArcCot[c_.*x_])/(d_ + e_.*x_^2)^(3/2), x_Symbol] := -b/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcCot[c*x])/(d*Sqrt[d + e*x^2]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol] := b*(d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -3/2]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol] := -b*(d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -3/2]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_/(d_ + e_.*x_^2)^(3/2), x_Symbol] := b*p*(a + b*ArcTan[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcTan[c*x])^p/(d*Sqrt[d + e*x^2]) - b^2*p*(p - 1)* Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(3/2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 1]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_/(d_ + e_.*x_^2)^(3/2), x_Symbol] := -b*p*(a + b*ArcCot[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcCot[c*x])^p/(d*Sqrt[d + e*x^2]) - b^2*p*(p - 1)* Int[(a + b*ArcCot[c*x])^(p - 2)/(d + e*x^2)^(3/2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 1]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_, x_Symbol] := b*p*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p - 1)/(4*c* d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x] - b^2*p*(p - 1)/(4*(q + 1)^2)* Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_, x_Symbol] := -b*p*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^(p - 1)/(4*c* d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p, x] - b^2*p*(p - 1)/(4*(q + 1)^2)* Int[(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p - 2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_, x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)) - 2*c*(q + 1)/(b*(p + 1))* Int[x*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && LtQ[p, -1]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_, x_Symbol] := -(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^(p + 1)/(b*c* d*(p + 1)) + 2*c*(q + 1)/(b*(p + 1))* Int[x*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && LtQ[p, -1]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol] := d^q/c* Subst[Int[(a + b*x)^p/Cos[x]^(2*(q + 1)), x], x, ArcTan[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol] := d^(q + 1/2)*Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]* Int[(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] && Not[IntegerQ[q] || GtQ[d, 0]]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol] := -d^q/c* Subst[Int[(a + b*x)^p/Sin[x]^(2*(q + 1)), x], x, ArcCot[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] && IntegerQ[q]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol] := -d^(q + 1/2)*x*Sqrt[(1 + c^2*x^2)/(c^2*x^2)]/Sqrt[d + e*x^2]* Subst[Int[(a + b*x)^p/Sin[x]^(2*(q + 1)), x], x, ArcCot[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] && Not[IntegerQ[q]]
+Int[ArcTan[c_.*x_]/(d_. + e_.*x_^2), x_Symbol] := I/2*Int[Log[1 - I*c*x]/(d + e*x^2), x] - I/2*Int[Log[1 + I*c*x]/(d + e*x^2), x] /; FreeQ[{c, d, e}, x]
+Int[ArcCot[c_.*x_]/(d_. + e_.*x_^2), x_Symbol] := I/2*Int[Log[1 - I/(c*x)]/(d + e*x^2), x] - I/2*Int[Log[1 + I/(c*x)]/(d + e*x^2), x] /; FreeQ[{c, d, e}, x]
+Int[(a_ + b_.*ArcTan[c_.*x_])/(d_. + e_.*x_^2), x_Symbol] := a*Int[1/(d + e*x^2), x] + b*Int[ArcTan[c*x]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x]
+Int[(a_ + b_.*ArcCot[c_.*x_])/(d_. + e_.*x_^2), x_Symbol] := a*Int[1/(d + e*x^2), x] + b*Int[ArcCot[c*x]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x]
+Int[(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - b*c*Int[u/(1 + c^2*x^2), x]] /; FreeQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
+Int[(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^q, x]}, Dist[a + b*ArcCot[c*x], u, x] + b*c*Int[u/(1 + c^2*x^2), x]] /; FreeQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (d + e*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCot[c*x])^p, (d + e*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := f^2/e*Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x] - d*f^2/e*Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := f^2/e*Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x])^p, x] - d*f^2/e*Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/d*Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x] - e/(d*f^2)* Int[(f*x)^(m + 2)*(a + b*ArcTan[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/d*Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x] - e/(d*f^2)* Int[(f*x)^(m + 2)*(a + b*ArcCot[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
+Int[x_*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := -I*(a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1)) - 1/(c*d)*Int[(a + b*ArcTan[c*x])^p/(I - c*x), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
+Int[x_*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := I*(a + b*ArcCot[c*x])^(p + 1)/(b*e*(p + 1)) - 1/(c*d)*Int[(a + b*ArcCot[c*x])^p/(I - c*x), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
+Int[x_*(a_. + b_.*ArcTan[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol] := x*(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)) - 1/(b*c*d*(p + 1))*Int[(a + b*ArcTan[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && Not[IGtQ[p, 0]] && NeQ[p, -1]
+Int[x_*(a_. + b_.*ArcCot[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol] := -x*(a + b*ArcCot[c*x])^(p + 1)/(b*c*d*(p + 1)) + 1/(b*c*d*(p + 1))*Int[(a + b*ArcCot[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && Not[IGtQ[p, 0]] && NeQ[p, -1]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_./(x_*(d_ + e_.*x_^2)), x_Symbol] := -I*(a + b*ArcTan[c*x])^(p + 1)/(b*d*(p + 1)) + I/d*Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_./(x_*(d_ + e_.*x_^2)), x_Symbol] := I*(a + b*ArcCot[c*x])^(p + 1)/(b*d*(p + 1)) + I/d*Int[(a + b*ArcCot[c*x])^p/(x*(I + c*x)), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTan[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol] := (f*x)^m*(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)) - f*m/(b*c*d*(p + 1))* Int[(f*x)^(m - 1)*(a + b*ArcTan[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[p, -1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCot[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol] := -(f*x)^m*(a + b*ArcCot[c*x])^(p + 1)/(b*c*d*(p + 1)) + f*m/(b*c*d*(p + 1))* Int[(f*x)^(m - 1)*(a + b*ArcCot[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[p, -1]
+Int[x_^m_.*(a_. + b_.*ArcTan[c_.*x_])/(d_ + e_.*x_^2), x_Symbol] := Int[ExpandIntegrand[(a + b*ArcTan[c*x]), x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && Not[EqQ[m, 1] && NeQ[a, 0]]
+Int[x_^m_.*(a_. + b_.*ArcCot[c_.*x_])/(d_ + e_.*x_^2), x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCot[c*x]), x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && Not[EqQ[m, 1] && NeQ[a, 0]]
+Int[x_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p/(2*e*(q + 1)) - b*p/(2*c*(q + 1))* Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
+Int[x_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p/(2*e*(q + 1)) + b*p/(2*c*(q + 1))* Int[(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p - 1), x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
+Int[x_*(a_. + b_.*ArcTan[c_.*x_])^p_/(d_ + e_.*x_^2)^2, x_Symbol] := x*(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)*(d + e*x^2)) - (1 - c^2*x^2)*(a + b*ArcTan[c*x])^(p + 2)/(b^2* e*(p + 1)*(p + 2)*(d + e*x^2)) - 4/(b^2*(p + 1)*(p + 2))* Int[x*(a + b*ArcTan[c*x])^(p + 2)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[p, -1] && NeQ[p, -2]
+Int[x_*(a_. + b_.*ArcCot[c_.*x_])^p_/(d_ + e_.*x_^2)^2, x_Symbol] := -x*(a + b*ArcCot[c*x])^(p + 1)/(b*c*d*(p + 1)*(d + e*x^2)) - (1 - c^2*x^2)*(a + b*ArcCot[c*x])^(p + 2)/(b^2* e*(p + 1)*(p + 2)*(d + e*x^2)) - 4/(b^2*(p + 1)*(p + 2))* Int[x*(a + b*ArcCot[c*x])^(p + 2)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[p, -1] && NeQ[p, -2]
+Int[x_^2*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol] := -b*(d + e*x^2)^(q + 1)/(4*c^3*d*(q + 1)^2) + x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])/(2*c^2*d*(q + 1)) - 1/(2*c^2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -5/2]
+Int[x_^2*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol] := b*(d + e*x^2)^(q + 1)/(4*c^3*d*(q + 1)^2) + x*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])/(2*c^2*d*(q + 1)) - 1/(2*c^2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -5/2]
+Int[x_^2*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := (a + b*ArcTan[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)) - x*(a + b*ArcTan[c*x])^p/(2*c^2*d*(d + e*x^2)) + b*p/(2*c)*Int[x*(a + b*ArcTan[c*x])^(p - 1)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
+Int[x_^2*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := -(a + b*ArcCot[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)) - x*(a + b*ArcCot[c*x])^p/(2*c^2*d*(d + e*x^2)) - b*p/(2*c)*Int[x*(a + b*ArcCot[c*x])^(p - 1)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol] := b*(f*x)^m*(d + e*x^2)^(q + 1)/(c*d*m^2) - f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])/(c^2*d* m) + f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol] := -b*(f*x)^m*(d + e*x^2)^(q + 1)/(c*d*m^2) - f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])/(c^2*d* m) + f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_, x_Symbol] := b*p*(f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p - 1)/(c*d*m^2) - f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^ p/(c^2*d*m) - b^2*p*(p - 1)/m^2* Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 2), x] + f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_, x_Symbol] := -b*p*(f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^(p - 1)/(c*d*m^2) - f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^ p/(c^2*d*m) - b^2*p*(p - 1)/m^2* Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p - 2), x] + f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_, x_Symbol] := (f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p + 1)/(b*c* d*(p + 1)) - f*m/(b*c*(p + 1))* Int[(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_, x_Symbol] := -(f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^(p + 1)/(b*c* d*(p + 1)) + f*m/(b*c*(p + 1))* Int[(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^ p/(d*f*(m + 1)) - b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^ p/(d*f*(m + 1)) + b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
+Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol] := (f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x])/(f*(m + 2)) - b*c*d/(f*(m + 2))*Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x] + d/(m + 2)*Int[(f*x)^m*(a + b*ArcTan[c*x])/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && NeQ[m, -2]
+Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol] := (f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcCot[c*x])/(f*(m + 2)) + b*c*d/(f*(m + 2))*Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x] + d/(m + 2)*Int[(f*x)^m*(a + b*ArcCot[c*x])/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && NeQ[m, -2]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcCot[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol] := d*Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x] + c^2*d/f^2* Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || EqQ[p, 1] && IntegerQ[q])
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol] := d*Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcCot[c*x])^p, x] + c^2*d/f^2* Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(a + b*ArcCot[c*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || EqQ[p, 1] && IntegerQ[q])
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTan[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x])^p/(c^2*d*m) - b*f*p/(c*m)* Int[(f*x)^(m - 1)*(a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2], x] - f^2*(m - 1)/(c^2*m)* Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && GtQ[m, 1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCot[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCot[c*x])^p/(c^2*d*m) + b*f*p/(c*m)* Int[(f*x)^(m - 1)*(a + b*ArcCot[c*x])^(p - 1)/Sqrt[d + e*x^2], x] - f^2*(m - 1)/(c^2*m)* Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x])^p/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && GtQ[m, 1]
+Int[(a_. + b_.*ArcTan[c_.*x_])/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := -2/Sqrt[d]*(a + b*ArcTan[c*x])* ArcTanh[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]] + I*b/Sqrt[d]*PolyLog[2, -Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]] - I*b/Sqrt[d]*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := -2/Sqrt[d]*(a + b*ArcCot[c*x])* ArcTanh[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]] - I*b/Sqrt[d]*PolyLog[2, -Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]] + I*b/Sqrt[d]*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := 1/Sqrt[d]*Subst[Int[(a + b*x)^p*Csc[x], x], x, ArcTan[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := -c*x*Sqrt[1 + 1/(c^2*x^2)]/Sqrt[d + e*x^2]* Subst[Int[(a + b*x)^p*Sec[x], x], x, ArcCot[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_./(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcTan[c*x])^p/(x*Sqrt[1 + c^2*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Not[GtQ[d, 0]]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_./(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcCot[c*x])^p/(x*Sqrt[1 + c^2*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Not[GtQ[d, 0]]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_./(x_^2*Sqrt[d_ + e_.*x_^2]), x_Symbol] := -Sqrt[d + e*x^2]*(a + b*ArcTan[c*x])^p/(d*x) + b*c*p*Int[(a + b*ArcTan[c*x])^(p - 1)/(x*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_./(x_^2*Sqrt[d_ + e_.*x_^2]), x_Symbol] := -Sqrt[d + e*x^2]*(a + b*ArcCot[c*x])^p/(d*x) - b*c*p*Int[(a + b*ArcCot[c*x])^(p - 1)/(x*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTan[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := (f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x])^p/(d*f*(m + 1)) - b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2], x] - c^2*(m + 2)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(a + b*ArcTan[c*x])^p/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCot[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := (f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcCot[c*x])^p/(d*f*(m + 1)) + b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(a + b*ArcCot[c*x])^(p - 1)/Sqrt[d + e*x^2], x] - c^2*(m + 2)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(a + b*ArcCot[c*x])^p/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
+Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol] := 1/e*Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x] - d/e*Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
+Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol] := 1/e*Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p, x] - d/e*Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^p, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
+Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol] := 1/d*Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x] - e/d*Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
+Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol] := 1/d*Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p, x] - e/d*Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^p, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol] := x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p + 1)/(b*c* d*(p + 1)) - m/(b*c*(p + 1))* Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x] - c*(m + 2*q + 2)/(b*(p + 1))* Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[e, c^2*d] && IntegerQ[m] && LtQ[q, -1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol] := -x^m*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^(p + 1)/(b*c* d*(p + 1)) + m/(b*c*(p + 1))* Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p + 1), x] + c*(m + 2*q + 2)/(b*(p + 1))* Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[e, c^2*d] && IntegerQ[m] && LtQ[q, -1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol] := d^q/c^(m + 1)* Subst[Int[(a + b*x)^p*Sin[x]^m/Cos[x]^(m + 2*(q + 1)), x], x, ArcTan[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q] || GtQ[d, 0])
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol] := d^(q + 1/2)*Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]* Int[x^m*(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && Not[IntegerQ[q] || GtQ[d, 0]]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol] := -d^q/c^(m + 1)* Subst[Int[(a + b*x)^p*Cos[x]^m/Sin[x]^(m + 2*(q + 1)), x], x, ArcCot[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && IntegerQ[q]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol] := -d^(q + 1/2)*x*Sqrt[(1 + c^2*x^2)/(c^2*x^2)]/(c^m*Sqrt[d + e*x^2])* Subst[Int[(a + b*x)^p*Cos[x]^m/Sin[x]^(m + 2*(q + 1)), x], x, ArcCot[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && Not[IntegerQ[q]]
+Int[x_*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])/(2*e*(q + 1)) - b*c/(2*e*(q + 1))*Int[(d + e*x^2)^(q + 1)/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
+Int[x_*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])/(2*e*(q + 1)) + b*c/(2*e*(q + 1))*Int[(d + e*x^2)^(q + 1)/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
+Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ( IGtQ[q, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[q, 0] && GtQ[m + 2*q + 3, 0]] || ILtQ[(m + 2*q + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]] )
+Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcCot[c*x], u, x] + b*c*Int[SimplifyIntegrand[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ( IGtQ[q, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[q, 0] && GtQ[m + 2*q + 3, 0]] || ILtQ[(m + 2*q + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]] )
+Int[x_*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcTan[c*x])^p/(1 - Rt[-e/d, 2]*x)^2, x] - 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcTan[c*x])^p/(1 + Rt[-e/d, 2]*x)^2, x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0]
+Int[x_*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcCot[c*x])^p/(1 - Rt[-e/d, 2]*x)^2, x] - 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcCot[c*x])^p/(1 + Rt[-e/d, 2]*x)^2, x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol] := With[{u = ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (EqQ[p, 1] && GtQ[q, 0] || IntegerQ[m])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol] := With[{u = ExpandIntegrand[(a + b*ArcCot[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (EqQ[p, 1] && GtQ[q, 0] || IntegerQ[m])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*ArcTan[c_.*x_]), x_Symbol] := a*Int[(f*x)^m*(d + e*x^2)^q, x] + b*Int[(f*x)^m*(d + e*x^2)^q*ArcTan[c*x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*ArcCot[c_.*x_]), x_Symbol] := a*Int[(f*x)^m*(d + e*x^2)^q, x] + b*Int[(f*x)^m*(d + e*x^2)^q*ArcCot[c*x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x]
+Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]
+Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCot[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]
+Int[ArcTanh[u_]*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/2*Int[Log[1 + u]*(a + b*ArcTan[c*x])^p/(d + e*x^2), x] - 1/2*Int[Log[1 - u]*(a + b*ArcTan[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I + c*x))^2, 0]
+Int[ArcCoth[u_]*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/2*Int[ Log[SimplifyIntegrand[1 + 1/u, x]]*(a + b*ArcCot[c*x])^ p/(d + e*x^2), x] - 1/2* Int[Log[SimplifyIntegrand[1 - 1/u, x]]*(a + b*ArcCot[c*x])^ p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I + c*x))^2, 0]
+Int[ArcTanh[u_]*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/2*Int[Log[1 + u]*(a + b*ArcTan[c*x])^p/(d + e*x^2), x] - 1/2*Int[Log[1 - u]*(a + b*ArcTan[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I - c*x))^2, 0]
+Int[ArcCoth[u_]*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/2*Int[ Log[SimplifyIntegrand[1 + 1/u, x]]*(a + b*ArcCot[c*x])^ p/(d + e*x^2), x] - 1/2* Int[Log[SimplifyIntegrand[1 - 1/u, x]]*(a + b*ArcCot[c*x])^ p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I - c*x))^2, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_.*Log[f_ + g_.*x_]/(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcTan[c*x])^(p + 1)*Log[f + g*x]/(b*c*d*(p + 1)) - g/(b*c*d*(p + 1))*Int[(a + b*ArcTan[c*x])^(p + 1)/(f + g*x), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[c^2*f^2 + g^2, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_.*Log[f_ + g_.*x_]/(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcCot[c*x])^(p + 1)*Log[f + g*x]/(b*c*d*(p + 1)) - g/(b*c*d*(p + 1))*Int[(a + b*ArcCot[c*x])^(p + 1)/(f + g*x), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[c^2*f^2 + g^2, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol] := I*(a + b*ArcTan[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) - b*p*I/2* Int[(a + b*ArcTan[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - 2*I/(I + c*x))^2, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol] := I*(a + b*ArcCot[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) + b*p*I/2* Int[(a + b*ArcCot[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - 2*I/(I + c*x))^2, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol] := -I*(a + b*ArcTan[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) + b*p*I/2* Int[(a + b*ArcTan[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - 2*I/(I - c*x))^2, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol] := -I*(a + b*ArcCot[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) - b*p*I/2* Int[(a + b*ArcCot[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - 2*I/(I - c*x))^2, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol] := -I*(a + b*ArcTan[c*x])^p*PolyLog[k + 1, u]/(2*c*d) + b*p*I/2* Int[(a + b*ArcTan[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I + c*x))^2, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol] := -I*(a + b*ArcCot[c*x])^p*PolyLog[k + 1, u]/(2*c*d) - b*p*I/2* Int[(a + b*ArcCot[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I + c*x))^2, 0]
+Int[(a_. + b_.*ArcTan[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol] := I*(a + b*ArcTan[c*x])^p*PolyLog[k + 1, u]/(2*c*d) - b*p*I/2* Int[(a + b*ArcTan[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I - c*x))^2, 0]
+Int[(a_. + b_.*ArcCot[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol] := I*(a + b*ArcCot[c*x])^p*PolyLog[k + 1, u]/(2*c*d) + b*p*I/2* Int[(a + b*ArcCot[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I - c*x))^2, 0]
+Int[1/((d_ + e_.*x_^2)*(a_. + b_.*ArcCot[c_.*x_])*(a_. + b_.*ArcTan[c_.*x_])), x_Symbol] := (-Log[a + b*ArcCot[c*x]] + Log[a + b*ArcTan[c*x]])/(b*c* d*(2*a + b*ArcCot[c*x] + b*ArcTan[c*x])) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
+Int[(a_. + b_.*ArcCot[c_.*x_])^ q_.*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := -(a + b*ArcCot[c*x])^(q + 1)*(a + b*ArcTan[c*x])^p/(b*c*d*(q + 1)) + p/(q + 1)* Int[(a + b*ArcCot[c*x])^(q + 1)*(a + b*ArcTan[c*x])^(p - 1)/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && IGeQ[q, p]
+Int[(a_. + b_.*ArcTan[c_.*x_])^ q_.*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcTan[c*x])^(q + 1)*(a + b*ArcCot[c*x])^ p/(b*c*d*(q + 1)) + p/(q + 1)* Int[(a + b*ArcTan[c*x])^(q + 1)*(a + b*ArcCot[c*x])^(p - 1)/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && IGeQ[q, p]
+Int[ArcTan[a_.*x_]/(c_ + d_.*x_^n_.), x_Symbol] := I/2*Int[Log[1 - I*a*x]/(c + d*x^n), x] - I/2*Int[Log[1 + I*a*x]/(c + d*x^n), x] /; FreeQ[{a, c, d}, x] && IntegerQ[n] && Not[EqQ[n, 2] && EqQ[d, a^2*c]]
+Int[ArcCot[a_.*x_]/(c_ + d_.*x_^n_.), x_Symbol] := I/2*Int[Log[1 - I/(a*x)]/(c + d*x^n), x] - I/2*Int[Log[1 + I/(a*x)]/(c + d*x^n), x] /; FreeQ[{a, c, d}, x] && IntegerQ[n] && Not[EqQ[n, 2] && EqQ[d, a^2*c]]
+Int[Log[d_.*x_^m_.]*ArcTan[c_.*x_^n_.]/x_, x_Symbol] := I/2*Int[Log[d*x^m]*Log[1 - I*c*x^n]/x, x] - I/2*Int[Log[d*x^m]*Log[1 + I*c*x^n]/x, x] /; FreeQ[{c, d, m, n}, x]
+Int[Log[d_.*x_^m_.]*ArcCot[c_.*x_^n_.]/x_, x_Symbol] := I/2*Int[Log[d*x^m]*Log[1 - I/(c*x^n)]/x, x] - I/2*Int[Log[d*x^m]*Log[1 + I/(c*x^n)]/x, x] /; FreeQ[{c, d, m, n}, x]
+Int[Log[d_.*x_^m_.]*(a_ + b_.*ArcTan[c_.*x_^n_.])/x_, x_Symbol] := a*Int[Log[d*x^m]/x, x] + b*Int[(Log[d*x^m]*ArcTan[c*x^n])/x, x] /; FreeQ[{a, b, c, d, m, n}, x]
+Int[Log[d_.*x_^m_.]*(a_ + b_.*ArcCot[c_.*x_^n_.])/x_, x_Symbol] := a*Int[Log[d*x^m]/x, x] + b*Int[(Log[d*x^m]*ArcCot[c*x^n])/x, x] /; FreeQ[{a, b, c, d, m, n}, x]
+Int[(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol] := x*(d + e*Log[f + g*x^2])*(a + b*ArcTan[c*x]) - 2*e*g*Int[x^2*(a + b*ArcTan[c*x])/(f + g*x^2), x] - b*c*Int[x*(d + e*Log[f + g*x^2])/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x]
+Int[(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol] := x*(d + e*Log[f + g*x^2])*(a + b*ArcCot[c*x]) - 2*e*g*Int[x^2*(a + b*ArcCot[c*x])/(f + g*x^2), x] + b*c*Int[x*(d + e*Log[f + g*x^2])/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x]
+Int[Log[f_. + g_.*x_^2]*ArcTan[c_.*x_]/x_, x_Symbol] := (Log[f + g*x^2] - Log[1 - I*c*x] - Log[1 + I*c*x])* Int[ArcTan[c*x]/x, x] + I/2*Int[Log[1 - I*c*x]^2/x, x] - I/2*Int[Log[1 + I*c*x]^2/x, x] /; FreeQ[{c, f, g}, x] && EqQ[g, c^2*f]
+Int[Log[f_. + g_.*x_^2]*ArcCot[c_.*x_]/x_, x_Symbol] := (Log[f + g*x^2] - Log[c^2*x^2] - Log[1 - I/(c*x)] - Log[1 + I/(c*x)])*Int[ArcCot[c*x]/x, x] + Int[Log[c^2*x^2]*ArcCot[c*x]/x, x] + I/2*Int[Log[1 - I/(c*x)]^2/x, x] - I/2*Int[Log[1 + I/(c*x)]^2/x, x] /; FreeQ[{c, f, g}, x] && EqQ[g, c^2*f]
+Int[Log[f_. + g_.*x_^2]*(a_ + b_.*ArcTan[c_.*x_])/x_, x_Symbol] := a*Int[Log[f + g*x^2]/x, x] + b*Int[Log[f + g*x^2]*ArcTan[c*x]/x, x] /; FreeQ[{a, b, c, f, g}, x]
+Int[Log[f_. + g_.*x_^2]*(a_ + b_.*ArcCot[c_.*x_])/x_, x_Symbol] := a*Int[Log[f + g*x^2]/x, x] + b*Int[Log[f + g*x^2]*ArcCot[c*x]/x, x] /; FreeQ[{a, b, c, f, g}, x]
+Int[(d_ + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTan[c_.*x_])/x_, x_Symbol] := d*Int[(a + b*ArcTan[c*x])/x, x] + e*Int[Log[f + g*x^2]*(a + b*ArcTan[c*x])/x, x] /; FreeQ[{a, b, c, d, e, f, g}, x]
+Int[(d_ + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCot[c_.*x_])/x_, x_Symbol] := d*Int[(a + b*ArcCot[c*x])/x, x] + e*Int[Log[f + g*x^2]*(a + b*ArcCot[c*x])/x, x] /; FreeQ[{a, b, c, d, e, f, g}, x]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol] := x^(m + 1)*(d + e*Log[f + g*x^2])*(a + b*ArcTan[c*x])/(m + 1) - 2*e*g/(m + 1)*Int[x^(m + 2)*(a + b*ArcTan[c*x])/(f + g*x^2), x] - b*c/(m + 1)* Int[x^(m + 1)*(d + e*Log[f + g*x^2])/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m/2, 0]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol] := x^(m + 1)*(d + e*Log[f + g*x^2])*(a + b*ArcCot[c*x])/(m + 1) - 2*e*g/(m + 1)*Int[x^(m + 2)*(a + b*ArcCot[c*x])/(f + g*x^2), x] + b*c/(m + 1)* Int[x^(m + 1)*(d + e*Log[f + g*x^2])/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m/2, 0]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcTan[c*x], u, x] - b*c*Int[ExpandIntegrand[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcCot[c*x], u, x] + b*c*Int[ExpandIntegrand[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(a + b*ArcTan[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - 2*e*g*Int[ExpandIntegrand[x*u/(f + g*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(a + b*ArcCot[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - 2*e*g*Int[ExpandIntegrand[x*u/(f + g*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]
+Int[x_*(d_. + e_.*Log[f_ + g_.*x_^2])*(a_. + b_.*ArcTan[c_.*x_])^2, x_Symbol] := (f + g*x^2)*(d + e*Log[f + g*x^2])*(a + b*ArcTan[c*x])^2/(2*g) - e*x^2*(a + b*ArcTan[c*x])^2/2 - b/c*Int[(d + e*Log[f + g*x^2])*(a + b*ArcTan[c*x]), x] + b*c*e*Int[x^2*(a + b*ArcTan[c*x])/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[g, c^2*f]
+Int[x_*(d_. + e_.*Log[f_ + g_.*x_^2])*(a_. + b_.*ArcCot[c_.*x_])^2, x_Symbol] := (f + g*x^2)*(d + e*Log[f + g*x^2])*(a + b*ArcCot[c*x])^2/(2*g) - e*x^2*(a + b*ArcCot[c*x])^2/2 + b/c*Int[(d + e*Log[f + g*x^2])*(a + b*ArcCot[c*x]), x] - b*c*e*Int[x^2*(a + b*ArcCot[c*x])/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[g, c^2*f]
+Int[u_.*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol] := Unintegrable[u*(a + b*ArcTan[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || MatchQ[u, (d_. + e_.*x)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x)^q_. /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[u, (d_. + e_.*x^2)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x^2)^q_. /; FreeQ[{d, e, f, m, q}, x]])
+Int[u_.*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol] := Unintegrable[u*(a + b*ArcCot[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || MatchQ[u, (d_. + e_.*x)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x)^q_. /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[u, (d_. + e_.*x^2)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x^2)^q_. /; FreeQ[{d, e, f, m, q}, x]])
diff --git a/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.5 u (a+b arctan(c+d x))^p.m b/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.5 u (a+b arctan(c+d x))^p.m
new file mode 100755
index 0000000..ec06db2
--- /dev/null
+++ b/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.5 u (a+b arctan(c+d x))^p.m	
@@ -0,0 +1,23 @@
+
+(* ::Subsection::Closed:: *)
+(* 5.3.5 u (a+b arctan(c+d x))^p *)
+Int[(a_. + b_.*ArcTan[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(a + b*ArcTan[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*ArcCot[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(a + b*ArcCot[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*ArcTan[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(a + b*ArcTan[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]
+Int[(a_. + b_.*ArcCot[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(a + b*ArcCot[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcTan[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(f*x/d)^m*(a + b*ArcTan[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCot[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(f*x/d)^m*(a + b*ArcCot[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_*(a_. + b_.*ArcTan[c_ + d_.*x_])^p_., x_Symbol] := (e + f*x)^(m + 1)*(a + b*ArcTan[c + d*x])^p/(f*(m + 1)) - b*d*p/(f*(m + 1))* Int[(e + f*x)^(m + 1)*(a + b*ArcTan[c + d*x])^(p - 1)/(1 + (c + d*x)^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[m, -1]
+Int[(e_. + f_.*x_)^m_*(a_. + b_.*ArcCot[c_ + d_.*x_])^p_., x_Symbol] := (e + f*x)^(m + 1)*(a + b*ArcCot[c + d*x])^p/(f*(m + 1)) + b*d*p/(f*(m + 1))* Int[(e + f*x)^(m + 1)*(a + b*ArcCot[c + d*x])^(p - 1)/(1 + (c + d*x)^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[m, -1]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcTan[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcTan[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCot[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcCot[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcTan[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(e + f*x)^m*(a + b*ArcTan[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && Not[IGtQ[p, 0]]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCot[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(e + f*x)^m*(a + b*ArcCot[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && Not[IGtQ[p, 0]]
+Int[ArcTan[a_ + b_.*x_]/(c_ + d_.*x_^n_.), x_Symbol] := I/2*Int[Log[1 - I*a - I*b*x]/(c + d*x^n), x] - I/2*Int[Log[1 + I*a + I*b*x]/(c + d*x^n), x] /; FreeQ[{a, b, c, d}, x] && RationalQ[n]
+Int[ArcCot[a_ + b_.*x_]/(c_ + d_.*x_^n_.), x_Symbol] := I/2*Int[Log[(-I + a + b*x)/(a + b*x)]/(c + d*x^n), x] - I/2*Int[Log[(I + a + b*x)/(a + b*x)]/(c + d*x^n), x] /; FreeQ[{a, b, c, d}, x] && RationalQ[n]
+Int[ArcTan[a_ + b_.*x_]/(c_ + d_.*x_^n_), x_Symbol] := Unintegrable[ArcTan[a + b*x]/(c + d*x^n), x] /; FreeQ[{a, b, c, d, n}, x] && Not[RationalQ[n]]
+Int[ArcCot[a_ + b_.*x_]/(c_ + d_.*x_^n_), x_Symbol] := Unintegrable[ArcCot[a + b*x]/(c + d*x^n), x] /; FreeQ[{a, b, c, d, n}, x] && Not[RationalQ[n]]
+Int[(A_. + B_.*x_ + C_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(C/d^2 + C/d^2*x^2)^q*(a + b*ArcTan[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, A, B, C, p, q}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[(A_. + B_.*x_ + C_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(C/d^2 + C/d^2*x^2)^q*(a + b*ArcCot[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, A, B, C, p, q}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[(e_. + f_.*x_)^m_.*(A_. + B_.*x_ + C_.*x_^2)^ q_.*(a_. + b_.*ArcTan[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[ Int[((d*e - c*f)/d + f*x/d)^m*(C/d^2 + C/d^2*x^2)^ q*(a + b*ArcTan[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[(e_. + f_.*x_)^m_.*(A_. + B_.*x_ + C_.*x_^2)^ q_.*(a_. + b_.*ArcCot[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[ Int[((d*e - c*f)/d + f*x/d)^m*(C/d^2 + C/d^2*x^2)^ q*(a + b*ArcCot[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
diff --git a/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.6 Exponentials of inverse tangent.m b/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.6 Exponentials of inverse tangent.m
new file mode 100755
index 0000000..23165c0
--- /dev/null
+++ b/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.6 Exponentials of inverse tangent.m	
@@ -0,0 +1,80 @@
+
+(* ::Subsection::Closed:: *)
+(* 5.3.6 Exponentials of inverse tangent *)
+Int[E^(n_*ArcTan[a_.*x_]), x_Symbol] := Int[((1 - I*a*x)^((I*n + 1)/2)/((1 + I*a*x)^((I*n - 1)/2)* Sqrt[1 + a^2*x^2])), x] /; FreeQ[a, x] && IntegerQ[(I*n - 1)/2]
+Int[x_^m_.*E^(n_*ArcTan[a_.*x_]), x_Symbol] := Int[x^ m*((1 - I*a*x)^((I*n + 1)/2)/((1 + I*a*x)^((I*n - 1)/2)* Sqrt[1 + a^2*x^2])), x] /; FreeQ[{a, m}, x] && IntegerQ[(I*n - 1)/2]
+Int[E^(n_.*ArcTan[a_.*x_]), x_Symbol] := Int[(1 - I*a*x)^(I*n/2)/(1 + I*a*x)^(I*n/2), x] /; FreeQ[{a, n}, x] && Not[IntegerQ[(I*n - 1)/2]]
+Int[x_^m_.*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := Int[x^m*(1 - I*a*x)^(I*n/2)/(1 + I*a*x)^(I*n/2), x] /; FreeQ[{a, m, n}, x] && Not[IntegerQ[(I*n - 1)/2]]
+Int[u_.*(c_ + d_.*x_)^p_.*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := c^p*Int[u*(1 + d*x/c)^p*(1 - I*a*x)^(I*n/2)/(1 + I*a*x)^(I*n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 + d^2, 0] && (IntegerQ[p] || GtQ[c, 0])
+Int[u_.*(c_ + d_.*x_)^p_.*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := Int[u*(c + d*x)^p*(1 - I*a*x)^(I*n/2)/(1 + I*a*x)^(I*n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 + d^2, 0] && Not[IntegerQ[p] || GtQ[c, 0]]
+Int[u_.*(c_ + d_./x_)^p_.*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := d^p*Int[u/x^p*(1 + c*x/d)^p*E^(n*ArcTan[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[c^2 + a^2*d^2, 0] && IntegerQ[p]
+Int[u_.*(c_ + d_./x_)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := (-1)^(n/2)*c^p* Int[u*(1 + d/(c*x))^ p*(1 - 1/(I*a*x))^(I*n/2)/(1 + 1/(I*a*x))^(I*n/2), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 + a^2*d^2, 0] && Not[IntegerQ[p]] && IntegerQ[I*n/2] && GtQ[c, 0]
+Int[u_.*(c_ + d_./x_)^p_*E^(n_*ArcTan[a_.*x_]), x_Symbol] := Int[u*(c + d/x)^p*(1 - I*a*x)^(I*n/2)/(1 + I*a*x)^(I*n/2), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 + a^2*d^2, 0] && Not[IntegerQ[p]] && IntegerQ[I*n/2] && Not[GtQ[c, 0]]
+Int[u_.*(c_ + d_./x_)^p_*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := x^p*(c + d/x)^p/(1 + c*x/d)^p* Int[u/x^p*(1 + c*x/d)^p*E^(n*ArcTan[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && Not[IntegerQ[p]]
+Int[E^(n_.*ArcTan[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol] := (n + a*x)*E^(n*ArcTan[a*x])/(a*c*(n^2 + 1)*Sqrt[c + d*x^2]) /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && Not[IntegerQ[I*n]]
+Int[(c_ + d_.*x_^2)^p_*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := (n - 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)* E^(n*ArcTan[a*x])/(a*c*(n^2 + 4*(p + 1)^2)) + 2*(p + 1)*(2*p + 3)/(c*(n^2 + 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1] && Not[IntegerQ[I*n]] && NeQ[n^2 + 4*(p + 1)^2, 0] && IntegerQ[2*p]
+Int[E^(n_.*ArcTan[a_.*x_])/(c_ + d_.*x_^2), x_Symbol] := E^(n*ArcTan[a*x])/(a*c*n) /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c]
+Int[(c_ + d_.*x_^2)^p_.*E^(n_*ArcTan[a_.*x_]), x_Symbol] := c^p*Int[(1 + a^2*x^2)^(p - I*n/2)*(1 - I*a*x)^(I*n), x] /; FreeQ[{a, c, d, p}, x] && EqQ[d, a^2*c] && IntegerQ[p] && IntegerQ[(I*n + 1)/2] && Not[IntegerQ[p - I*n/2]]
+Int[(c_ + d_.*x_^2)^p_.*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := c^p*Int[(1 - I*a*x)^(p + I*n/2)*(1 + I*a*x)^(p - I*n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0])
+Int[(c_ + d_.*x_^2)^p_*E^(n_*ArcTan[a_.*x_]), x_Symbol] := c^(I*n/2)*Int[(c + d*x^2)^(p - I*n/2)*(1 - I*a*x)^(I*n), x] /; FreeQ[{a, c, d, p}, x] && EqQ[d, a^2*c] && Not[IntegerQ[p] || GtQ[c, 0]] && IGtQ[I*n/2, 0]
+Int[(c_ + d_.*x_^2)^p_*E^(n_*ArcTan[a_.*x_]), x_Symbol] := 1/c^(I*n/2)*Int[(c + d*x^2)^(p + I*n/2)/(1 + I*a*x)^(I*n), x] /; FreeQ[{a, c, d, p}, x] && EqQ[d, a^2*c] && Not[IntegerQ[p] || GtQ[c, 0]] && ILtQ[I*n/2, 0]
+Int[(c_ + d_.*x_^2)^p_*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := c^IntPart[p]*(c + d*x^2)^FracPart[p]/(1 + a^2*x^2)^FracPart[p]* Int[(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && Not[IntegerQ[p] || GtQ[c, 0]]
+Int[x_*E^(n_.*ArcTan[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol] := -(1 - a*n*x)*E^(n*ArcTan[a*x])/(d*(n^2 + 1)*Sqrt[c + d*x^2]) /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && Not[IntegerQ[I*n]]
+Int[x_*(c_ + d_.*x_^2)^p_*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := (c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x])/(2*d*(p + 1)) - a*c*n/(2*d*(p + 1))*Int[(c + d*x^2)^p*E^(n*ArcTan[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1] && Not[IntegerQ[I*n]] && IntegerQ[2*p]
+(* Int[x_*(c_+d_.*x_^2)^p_*E^(n_.*ArcTan[a_.*x_]),x_Symbol] := (2*(p+1)+a*n*x)*(c+d*x^2)^(p+1)*E^(n*ArcTan[a*x])/(a^2*c*(n^2+4*(p+ 1)^2)) - n*(2*p+3)/(a*c*(n^2+4*(p+1)^2))*Int[(c+d*x^2)^(p+1)*E^(n*ArcTan[a*x] ),x] /; FreeQ[{a,c,d,n},x] && EqQ[d,a^2*c] && LtQ[p,-1] &&  NeQ[n^2+4*(p+1)^2,0] && Not[IntegerQ[I*n]] *)
+Int[x_^2*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := -(1 - a*n*x)*(c + d*x^2)^(p + 1)* E^(n*ArcTan[a*x])/(a*d*n*(n^2 + 1)) /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && EqQ[n^2 - 2*(p + 1), 0] && Not[IntegerQ[I*n]]
+Int[x_^2*(c_ + d_.*x_^2)^p_*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := -(n - 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)* E^(n*ArcTan[a*x])/(a*d*(n^2 + 4*(p + 1)^2)) + (n^2 - 2*(p + 1))/(d*(n^2 + 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1] && Not[IntegerQ[I*n]] && NeQ[n^2 + 4*(p + 1)^2, 0] && IntegerQ[2*p]
+Int[x_^m_.*(c_ + d_.*x_^2)^p_.*E^(n_*ArcTan[a_.*x_]), x_Symbol] := c^p*Int[x^m*(1 + a^2*x^2)^(p - I*n/2)*(1 - I*a*x)^(I*n), x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0]) && IntegerQ[(I*n + 1)/2] && Not[IntegerQ[p - I*n/2]]
+Int[x_^m_.*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := c^p*Int[x^m*(1 - I*a*x)^(p + I*n/2)*(1 + I*a*x)^(p - I*n/2), x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0])
+Int[x_^m_.*(c_ + d_.*x_^2)^p_*E^(n_*ArcTan[a_.*x_]), x_Symbol] := c^(I*n/2)*Int[x^m*(c + d*x^2)^(p - I*n/2)*(1 - I*a*x)^(I*n), x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[d, a^2*c] && Not[IntegerQ[p] || GtQ[c, 0]] && IGtQ[I*n/2, 0]
+Int[x_^m_.*(c_ + d_.*x_^2)^p_*E^(n_*ArcTan[a_.*x_]), x_Symbol] := 1/c^(I*n/2)*Int[x^m*(c + d*x^2)^(p + I*n/2)/(1 + I*a*x)^(I*n), x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[d, a^2*c] && Not[IntegerQ[p] || GtQ[c, 0]] && ILtQ[I*n/2, 0]
+Int[x_^m_.*(c_ + d_.*x_^2)^p_*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := c^IntPart[p]*(c + d*x^2)^FracPart[p]/(1 + a^2*x^2)^FracPart[p]* Int[x^m*(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && Not[IntegerQ[p] || GtQ[c, 0]]
+Int[u_*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := c^p*Int[u*(1 - I*a*x)^(p + I*n/2)*(1 + I*a*x)^(p - I*n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0])
+Int[u_*(c_ + d_.*x_^2)^p_.*E^(n_*ArcTan[a_.*x_]), x_Symbol] := c^IntPart[p]*(c + d*x^2)^ FracPart[p]/((1 - I*a*x)^FracPart[p]*(1 + I*a*x)^FracPart[p])* Int[u*(1 - I*a*x)^(p + I*n/2)*(1 + I*a*x)^(p - I*n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0]) && IntegerQ[I*n/2]
+Int[u_*(c_ + d_.*x_^2)^p_*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := c^IntPart[p]*(c + d*x^2)^FracPart[p]/(1 + a^2*x^2)^FracPart[p]* Int[u*(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && Not[IntegerQ[p] || GtQ[c, 0]] && Not[IntegerQ[I*n/2]]
+Int[u_.*(c_ + d_./x_^2)^p_.*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := d^p*Int[u/x^(2*p)*(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[c - a^2*d, 0] && IntegerQ[p]
+Int[u_.*(c_ + d_./x_^2)^p_*E^(n_*ArcTan[a_.*x_]), x_Symbol] := c^p*Int[u*(1 - I/(a*x))^p*(1 + I/(a*x))^p*E^(n*ArcTan[a*x]), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c - a^2*d, 0] && Not[IntegerQ[p]] && IntegerQ[I*n/2] && GtQ[c, 0]
+Int[u_.*(c_ + d_./x_^2)^p_*E^(n_*ArcTan[a_.*x_]), x_Symbol] := x^(2*p)*(c + d/x^2)^p/((1 - I*a*x)^p*(1 + I*a*x)^p)* Int[u/x^(2*p)*(1 - I*a*x)^p*(1 + I*a*x)^p*E^(n*ArcTan[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c - a^2*d, 0] && Not[IntegerQ[p]] && IntegerQ[I*n/2] && Not[GtQ[c, 0]]
+Int[u_.*(c_ + d_./x_^2)^p_*E^(n_.*ArcTan[a_.*x_]), x_Symbol] := x^(2*p)*(c + d/x^2)^p/(1 + a^2*x^2)^p* Int[u/x^(2*p)*(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c - a^2*d, 0] && Not[IntegerQ[p]] && Not[IntegerQ[I*n/2]]
+Int[E^(n_.*ArcTan[c_.*(a_ + b_.*x_)]), x_Symbol] := Int[(1 - I*a*c - I*b*c*x)^(I*n/2)/(1 + I*a*c + I*b*c*x)^(I*n/2), x] /; FreeQ[{a, b, c, n}, x]
+Int[x_^m_*E^(n_*ArcTan[c_.*(a_ + b_.*x_)]), x_Symbol] := 4/(I^m*n*b^(m + 1)*c^(m + 1))* Subst[ Int[x^(2/(I*n))*(1 - I*a*c - (1 + I*a*c)*x^(2/(I*n)))^ m/(1 + x^(2/(I*n)))^(m + 2), x], x, (1 - I*c*(a + b*x))^(I*n/2)/(1 + I*c*(a + b*x))^(I*n/2)] /; FreeQ[{a, b, c}, x] && ILtQ[m, 0] && LtQ[-1, I*n, 1]
+Int[(d_. + e_.*x_)^m_.*E^(n_.*ArcTan[c_.*(a_ + b_.*x_)]), x_Symbol] := Int[(d + e*x)^ m*(1 - I*a*c - I*b*c*x)^(I*n/2)/(1 + I*a*c + I*b*c*x)^(I*n/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcTan[a_ + b_.*x_]), x_Symbol] := (c/(1 + a^2))^p* Int[u*(1 - I*a - I*b*x)^(p + I*n/2)*(1 + I*a + I*b*x)^(p - I*n/2), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, 2*a*e] && EqQ[b^2*c - e (1 + a^2), 0] && (IntegerQ[p] || GtQ[c/(1 + a^2), 0])
+Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcTan[a_ + b_.*x_]), x_Symbol] := (c + d*x + e*x^2)^p/(1 + a^2 + 2*a*b*x + b^2*x^2)^p* Int[u*(1 + a^2 + 2*a*b*x + b^2*x^2)^p*E^(n*ArcTan[a*x]), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, 2*a*e] && EqQ[b^2*c - e (1 + a^2), 0] && Not[IntegerQ[p] || GtQ[c/(1 + a^2), 0]]
+Int[u_.*E^(n_.*ArcTan[c_./(a_. + b_.*x_)]), x_Symbol] := Int[u*E^(n*ArcCot[a/c + b*x/c]), x] /; FreeQ[{a, b, c, n}, x]
+Int[u_.*E^(n_*ArcCot[a_.*x_]), x_Symbol] := (-1)^(I*n/2)*Int[u*E^(-n*ArcTan[a*x]), x] /; FreeQ[a, x] && IntegerQ[I*n/2]
+Int[E^(n_*ArcCot[a_.*x_]), x_Symbol] := -Subst[ Int[(1 - I*x/a)^((I*n + 1)/2)/(x^2*(1 + I*x/a)^((I*n - 1)/2)* Sqrt[1 + x^2/a^2]), x], x, 1/x] /; FreeQ[a, x] && IntegerQ[(I*n - 1)/2]
+Int[x_^m_.*E^(n_*ArcCot[a_.*x_]), x_Symbol] := -Subst[ Int[(1 - I*x/a)^((I*n + 1)/ 2)/(x^(m + 2)*(1 + I*x/a)^((I*n - 1)/2)*Sqrt[1 + x^2/a^2]), x], x, 1/x] /; FreeQ[a, x] && IntegerQ[(I*n - 1)/2] && IntegerQ[m]
+Int[E^(n_.*ArcCot[a_.*x_]), x_Symbol] := -Subst[Int[(1 - I*x/a)^(I*n/2)/(x^2*(1 + I*x/a)^(I*n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && Not[IntegerQ[I*n]]
+Int[x_^m_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := -Subst[Int[(1 - I*x/a)^(n/2)/(x^(m + 2)*(1 + I*x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && Not[IntegerQ[I*n]] && IntegerQ[m]
+Int[x_^m_*E^(n_*ArcCot[a_.*x_]), x_Symbol] := -x^m*(1/x)^m* Subst[Int[(1 - I*x/a)^((I*n + 1)/ 2)/(x^(m + 2)*(1 + I*x/a)^((I*n - 1)/2)*Sqrt[1 + x^2/a^2]), x], x, 1/x] /; FreeQ[{a, m}, x] && IntegerQ[(I*n - 1)/2] && Not[IntegerQ[m]]
+Int[x_^m_*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := -Subst[Int[(1 - I*x/a)^(n/2)/(x^(m + 2)*(1 + I*x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, m, n}, x] && Not[IntegerQ[I*n/2]] && Not[IntegerQ[m]]
+Int[u_.*(c_ + d_.*x_)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := d^p*Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCot[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c^2 + d^2, 0] && Not[IntegerQ[I*n/2]] && IntegerQ[p]
+Int[u_.*(c_ + d_.*x_)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := (c + d*x)^p/(x^p*(1 + c/(d*x))^p)* Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCot[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 + d^2, 0] && Not[IntegerQ[I*n/2]] && Not[IntegerQ[p]]
+Int[(c_ + d_./x_)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := -c^p*Subst[ Int[(1 + d*x/c)^p*(1 - I*x/a)^(I*n/2)/(x^2*(1 + I*x/a)^(I*n/2)), x], x, 1/x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && Not[IntegerQ[I*n/2]] && (IntegerQ[p] || GtQ[c, 0])
+Int[x_^m_.*(c_ + d_./x_)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := -c^p*Subst[ Int[(1 + d*x/c)^ p*(1 - I*x/a)^(I*n/2)/(x^(m + 2)*(1 + I*x/a)^(I*n/2)), x], x, 1/x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && Not[IntegerQ[I*n/2]] && (IntegerQ[p] || GtQ[c, 0]) && IntegerQ[m]
+Int[(c_ + d_./x_)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := (c + d/x)^p/(1 + d/(c*x))^p* Int[(1 + d/(c*x))^p*E^(n*ArcCot[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && Not[IntegerQ[I*n/2]] && Not[IntegerQ[p] || GtQ[c, 0]]
+Int[x_^m_*(c_ + d_./x_)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := -c^p*x^m*(1/x)^m* Subst[Int[(1 + d*x/c)^ p*(1 - I*x/a)^(I*n/2)/(x^(m + 2)*(1 + I*x/a)^(I*n/2)), x], x, 1/x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && Not[IntegerQ[I*n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegerQ[m]]
+Int[u_.*(c_ + d_./x_)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := (c + d/x)^p/(1 + d/(c*x))^p* Int[u*(1 + d/(c*x))^p*E^(n*ArcCot[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && Not[IntegerQ[I*n/2]] && Not[IntegerQ[p] || GtQ[c, 0]]
+Int[E^(n_.*ArcCot[a_.*x_])/(c_ + d_.*x_^2), x_Symbol] := -E^(n*ArcCot[a*x])/(a*c*n) /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c]
+Int[E^(n_.*ArcCot[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol] := -(n - a*x)*E^(n*ArcCot[a*x])/(a*c*(n^2 + 1)*Sqrt[c + d*x^2]) /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && Not[IntegerQ[(I*n - 1)/2]]
+Int[(c_ + d_.*x_^2)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := -(n + 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCot[a*x])/(a*c*(n^2 + 4*(p + 1)^2)) + 2*(p + 1)*(2*p + 3)/(c*(n^2 + 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 + 4*(p + 1)^2, 0] && Not[IntegerQ[p] && IntegerQ[I*n/2]] && Not[Not[IntegerQ[p]] && IntegerQ[(I*n - 1)/2]]
+Int[x_*E^(n_.*ArcCot[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol] := -(1 + a*n*x)*E^(n*ArcCot[a*x])/(a^2*c*(n^2 + 1)*Sqrt[c + d*x^2]) /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && Not[IntegerQ[(I*n - 1)/2]]
+Int[x_*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := (2*(p + 1) - a*n*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCot[a*x])/(a^2*c*(n^2 + 4*(p + 1)^2)) + n*(2*p + 3)/(a*c*(n^2 + 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LeQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 + 4*(p + 1)^2, 0] && Not[IntegerQ[p] && IntegerQ[I*n/2]] && Not[Not[IntegerQ[p]] && IntegerQ[(I*n - 1)/2]]
+Int[x_^2*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := (n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCot[a*x])/(a^3*c*n^2*(n^2 + 1)) /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && EqQ[n^2 - 2*(p + 1), 0] && NeQ[n^2 + 1, 0]
+Int[x_^2*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := (n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCot[a*x])/(a^3*c*(n^2 + 4*(p + 1)^2)) + (n^2 - 2*(p + 1))/(a^2*c*(n^2 + 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LeQ[p, -1] && NeQ[n^2 - 2*(p + 1), 0] && NeQ[n^2 + 4*(p + 1)^2, 0] && Not[IntegerQ[p] && IntegerQ[I*n/2]] && Not[Not[IntegerQ[p]] && IntegerQ[(I*n - 1)/2]]
+Int[x_^m_.*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := -c^p/a^(m + 1)* Subst[Int[E^(n*x)*Cot[x]^(m + 2*(p + 1))/Cos[x]^(2*(p + 1)), x], x, ArcCot[a*x]] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && IntegerQ[m] && LeQ[3, m, -2 (p + 1)] && IntegerQ[p]
+Int[u_.*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := d^p*Int[u*x^(2*p)*(1 + 1/(a^2*x^2))^p*E^(n*ArcCot[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && Not[IntegerQ[I*n/2]] && IntegerQ[p]
+Int[u_.*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := (c + d*x^2)^p/(x^(2*p)*(1 + 1/(a^2*x^2))^p)* Int[u*x^(2*p)*(1 + 1/(a^2*x^2))^p*E^(n*ArcCot[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && Not[IntegerQ[I*n/2]] && Not[IntegerQ[p]]
+Int[u_.*(c_ + d_./x_^2)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := c^p/(I*a)^(2*p)* Int[u/x^(2*p)*(-1 + I*a*x)^(p - I*n/2)*(1 + I*a*x)^(p + I*n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c, a^2*d] && Not[IntegerQ[I*n/2]] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + I*n/2]
+Int[(c_ + d_./x_^2)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := -c^p*Subst[ Int[(1 - I*x/a)^(p + I*n/2)*(1 + I*x/a)^(p - I*n/2)/x^2, x], x, 1/x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c, a^2*d] && Not[IntegerQ[I*n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegerQ[2*p] && IntegerQ[p + I*n/2]]
+Int[x_^m_.*(c_ + d_./x_^2)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := -c^p*Subst[ Int[(1 - I*x/a)^(p + I*n/2)*(1 + I*x/a)^(p - I*n/2)/x^(m + 2), x], x, 1/x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c, a^2*d] && Not[IntegerQ[I*n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegerQ[2*p] && IntegerQ[p + I*n/2]] && IntegerQ[m]
+Int[x_^m_*(c_ + d_./x_^2)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := -c^p*x^m*(1/x)^m* Subst[Int[(1 - I*x/a)^(p + I*n/2)*(1 + I*x/a)^(p - I*n/2)/ x^(m + 2), x], x, 1/x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[c, a^2*d] && Not[IntegerQ[I*n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegerQ[2*p] && IntegerQ[p + I*n/2]] && Not[IntegerQ[m]]
+Int[u_.*(c_ + d_./x_^2)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol] := (c + d/x^2)^p/(1 + 1/(a^2*x^2))^p* Int[u*(1 + 1/(a^2*x^2))^p*E^(n*ArcCot[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c, a^2*d] && Not[IntegerQ[I*n/2]] && Not[IntegerQ[p] || GtQ[c, 0]]
+Int[u_.*E^(n_*ArcCot[c_.*(a_ + b_.*x_)]), x_Symbol] := (-1)^(I*n/2)*Int[u*E^(-n*ArcTan[c*(a + b*x)]), x] /; FreeQ[{a, b, c}, x] && IntegerQ[I*n/2]
+Int[E^(n_.*ArcCot[c_.*(a_ + b_.*x_)]), x_Symbol] := (I*c*(a + b*x))^(I* n/2)*(1 + 1/(I*c*(a + b*x)))^(I*n/2)/(1 + I*a*c + I*b*c*x)^(I* n/2)* Int[(1 + I*a*c + I*b*c*x)^(I*n/2)/(-1 + I*a*c + I*b*c*x)^(I* n/2), x] /; FreeQ[{a, b, c, n}, x] && Not[IntegerQ[I*n/2]]
+Int[x_^m_*E^(n_*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol] := 4/(I^m*n*b^(m + 1)*c^(m + 1))* Subst[ Int[x^(2/(I*n))*(1 + I*a*c + (1 - I*a*c)*x^(2/(I*n)))^ m/(-1 + x^(2/(I*n)))^(m + 2), x], x, (1 + 1/(I*c*(a + b*x)))^(I*n/2)/(1 - 1/(I*c*(a + b*x)))^(I* n/2)] /; FreeQ[{a, b, c}, x] && ILtQ[m, 0] && LtQ[-1, I*n, 1]
+Int[(d_. + e_.*x_)^m_.*E^(n_.*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol] := (I*c*(a + b*x))^(I* n/2)*(1 + 1/(I*c*(a + b*x)))^(I*n/2)/(1 + I*a*c + I*b*c*x)^(I* n/2)* Int[(d + e*x)^ m*(1 + I*a*c + I*b*c*x)^(I*n/2)/(-1 + I*a*c + I*b*c*x)^(I*n/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && Not[IntegerQ[I*n/2]]
+Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcCot[a_ + b_.*x_]), x_Symbol] := (c/(1 + a^2))^ p*((I*a + I*b*x)/(1 + I*a + I*b*x))^(I* n/2)*((1 + I*a + I*b*x)/(I*a + I*b*x))^(I*n/2)* ((1 - I*a - I*b*x)^(I*n/2)/(-1 + I*a + I*b*x)^(I*n/2))* Int[ u*(1 - I*a - I*b*x)^(p - I*n/2)*(1 + I*a + I*b*x)^(p + I*n/2), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && Not[IntegerQ[I*n/2]] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c - e (1 + a^2), 0] && (IntegerQ[p] || GtQ[c/(1 + a^2), 0])
+Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcCot[a_ + b_.*x_]), x_Symbol] := (c + d*x + e*x^2)^p/(1 + a^2 + 2*a*b*x + b^2*x^2)^p* Int[u*(1 + a^2 + 2*a*b*x + b^2*x^2)^p*E^(n*ArcCot[a*x]), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && Not[IntegerQ[I*n/2]] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c - e (1 + a^2), 0] && Not[IntegerQ[p] || GtQ[c/(1 + a^2), 0]]
+Int[u_.*E^(n_.*ArcCot[c_./(a_. + b_.*x_)]), x_Symbol] := Int[u*E^(n*ArcTan[a/c + b*x/c]), x] /; FreeQ[{a, b, c, n}, x]
diff --git a/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.7 Miscellaneous inverse tangent.m b/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.7 Miscellaneous inverse tangent.m
new file mode 100755
index 0000000..c608bfd
--- /dev/null
+++ b/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.7 Miscellaneous inverse tangent.m	
@@ -0,0 +1,82 @@
+
+(* ::Subsection::Closed:: *)
+(* 5.3.7 Miscellaneous inverse tangent *)
+Int[ArcTan[a_ + b_.*x_^n_], x_Symbol] := x*ArcTan[a + b*x^n] - b*n*Int[x^n/(1 + a^2 + 2*a*b*x^n + b^2*x^(2*n)), x] /; FreeQ[{a, b, n}, x]
+Int[ArcCot[a_ + b_.*x_^n_], x_Symbol] := x*ArcCot[a + b*x^n] + b*n*Int[x^n/(1 + a^2 + 2*a*b*x^n + b^2*x^(2*n)), x] /; FreeQ[{a, b, n}, x]
+Int[ArcTan[a_. + b_.*x_^n_]/x_, x_Symbol] := I/2*Int[Log[1 - I*a - I*b*x^n]/x, x] - I/2*Int[Log[1 + I*a + I*b*x^n]/x, x] /; FreeQ[{a, b, n}, x]
+Int[ArcCot[a_. + b_.*x_^n_]/x_, x_Symbol] := I/2*Int[Log[1 - I/(a + b*x^n)]/x, x] - I/2*Int[Log[1 + I/(a + b*x^n)]/x, x] /; FreeQ[{a, b, n}, x]
+Int[x_^m_.*ArcTan[a_ + b_.*x_^n_], x_Symbol] := x^(m + 1)*ArcTan[a + b*x^n]/(m + 1) - b*n/(m + 1)* Int[x^(m + n)/(1 + a^2 + 2*a*b*x^n + b^2*x^(2*n)), x] /; FreeQ[{a, b}, x] && RationalQ[m, n] && m + 1 != 0 && m + 1 != n
+Int[x_^m_.*ArcCot[a_ + b_.*x_^n_], x_Symbol] := x^(m + 1)*ArcCot[a + b*x^n]/(m + 1) + b*n/(m + 1)* Int[x^(m + n)/(1 + a^2 + 2*a*b*x^n + b^2*x^(2*n)), x] /; FreeQ[{a, b}, x] && RationalQ[m, n] && m + 1 != 0 && m + 1 != n
+Int[ArcTan[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol] := I/2*Int[Log[1 - I*a - I*b*f^(c + d*x)], x] - I/2*Int[Log[1 + I*a + I*b*f^(c + d*x)], x] /; FreeQ[{a, b, c, d, f}, x]
+Int[ArcCot[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol] := I/2*Int[Log[1 - I/(a + b*f^(c + d*x))], x] - I/2*Int[Log[1 + I/(a + b*f^(c + d*x))], x] /; FreeQ[{a, b, c, d, f}, x] 
+Int[x_^m_.*ArcTan[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol] := I/2*Int[x^m*Log[1 - I*a - I*b*f^(c + d*x)], x] - I/2*Int[x^m*Log[1 + I*a + I*b*f^(c + d*x)], x] /; FreeQ[{a, b, c, d, f}, x] && IntegerQ[m] && m > 0
+Int[x_^m_.*ArcCot[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol] := I/2*Int[x^m*Log[1 - I/(a + b*f^(c + d*x))], x] - I/2*Int[x^m*Log[1 + I/(a + b*f^(c + d*x))], x] /; FreeQ[{a, b, c, d, f}, x] && IntegerQ[m] && m > 0
+Int[u_.*ArcTan[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol] := Int[u*ArcCot[a/c + b*x^n/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
+Int[u_.*ArcCot[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol] := Int[u*ArcTan[a/c + b*x^n/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
+Int[ArcTan[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol] := x*ArcTan[(c*x)/Sqrt[a + b*x^2]] - c*Int[x/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]
+Int[ArcCot[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol] := x*ArcCot[(c*x)/Sqrt[a + b*x^2]] + c*Int[x/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]
+Int[ArcTan[c_.*x_/Sqrt[a_. + b_.*x_^2]]/x_, x_Symbol] := ArcTan[c*x/Sqrt[a + b*x^2]]*Log[x] - c*Int[Log[x]/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]
+Int[ArcCot[c_.*x_/Sqrt[a_. + b_.*x_^2]]/x_, x_Symbol] := ArcCot[c*x/Sqrt[a + b*x^2]]*Log[x] + c*Int[Log[x]/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]
+Int[(d_.*x_)^m_.*ArcTan[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol] := (d*x)^(m + 1)*ArcTan[(c*x)/Sqrt[a + b*x^2]]/(d*(m + 1)) - c/(d*(m + 1))*Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]
+Int[(d_.*x_)^m_.*ArcCot[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol] := (d*x)^(m + 1)*ArcCot[(c*x)/Sqrt[a + b*x^2]]/(d*(m + 1)) + c/(d*(m + 1))*Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]
+Int[1/(Sqrt[a_. + b_.*x_^2]*ArcTan[c_.*x_/Sqrt[a_. + b_.*x_^2]]), x_Symbol] := 1/c*Log[ArcTan[c*x/Sqrt[a + b*x^2]]] /; FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]
+Int[1/(Sqrt[a_. + b_.*x_^2]*ArcCot[c_.*x_/Sqrt[a_. + b_.*x_^2]]), x_Symbol] := -1/c*Log[ArcCot[c*x/Sqrt[a + b*x^2]]] /; FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]
+Int[ArcTan[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[a_. + b_.*x_^2], x_Symbol] := ArcTan[c*x/Sqrt[a + b*x^2]]^(m + 1)/(c*(m + 1)) /; FreeQ[{a, b, c, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]
+Int[ArcCot[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[a_. + b_.*x_^2], x_Symbol] := -ArcCot[c*x/Sqrt[a + b*x^2]]^(m + 1)/(c*(m + 1)) /; FreeQ[{a, b, c, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]
+Int[ArcTan[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[d_. + e_.*x_^2], x_Symbol] := Sqrt[a + b*x^2]/Sqrt[d + e*x^2]* Int[ArcTan[c*x/Sqrt[a + b*x^2]]^m/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b + c^2, 0] && EqQ[b*d - a*e, 0]
+Int[ArcCot[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[d_. + e_.*x_^2], x_Symbol] := Sqrt[a + b*x^2]/Sqrt[d + e*x^2]* Int[ArcCot[c*x/Sqrt[a + b*x^2]]^m/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b + c^2, 0] && EqQ[b*d - a*e, 0]
+Int[u_.*ArcTan[v_ + s_.*Sqrt[w_]], x_Symbol] := Pi*s/4*Int[u, x] + 1/2*Int[u*ArcTan[v], x] /; EqQ[s^2, 1] && EqQ[w, v^2 + 1]
+Int[u_.*ArcCot[v_ + s_.*Sqrt[w_]], x_Symbol] := Pi*s/4*Int[u, x] - 1/2*Int[u*ArcTan[v], x] /; EqQ[s^2, 1] && EqQ[w, v^2 + 1]
+If[TrueQ[$LoadShowSteps], Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, ShowStep["", "Int[f[x,ArcTan[a+b*x]]/(1+(a+b*x)^2),x]", "Subst[Int[f[-a/b+Tan[x]/b,x],x],x,ArcTan[a+b*x]]/b", Hold[ (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[Int[ SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*Sec[x]^(2*(n + 1)), x], x], x, tmp]]] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcTan] && EqQ[Discriminant[v, x]*tmp[[1]]^2 + D[v, x]^2, 0]] /; SimplifyFlag && QuadraticQ[v, x] && ILtQ[n, 0] && NegQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]], Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[ Int[SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*Sec[x]^(2*(n + 1)), x], x], x, tmp] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcTan] && EqQ[Discriminant[v, x]*tmp[[1]]^2 + D[v, x]^2, 0]] /; QuadraticQ[v, x] && ILtQ[n, 0] && NegQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]]]
+If[TrueQ[$LoadShowSteps], Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, ShowStep["", "Int[f[x,ArcCot[a+b*x]]/(1+(a+b*x)^2),x]", "-Subst[Int[f[-a/b+Cot[x]/b,x],x],x,ArcCot[a+b*x]]/b", Hold[ -(-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[Int[ SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*Csc[x]^(2*(n + 1)), x], x], x, tmp]]] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcCot] && EqQ[Discriminant[v, x]*tmp[[1]]^2 + D[v, x]^2, 0]] /; SimplifyFlag && QuadraticQ[v, x] && ILtQ[n, 0] && NegQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]], Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, -(-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[ Int[SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*Csc[x]^(2*(n + 1)), x], x], x, tmp] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcCot] && EqQ[Discriminant[v, x]*tmp[[1]]^2 + D[v, x]^2, 0]] /; QuadraticQ[v, x] && ILtQ[n, 0] && NegQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]]]
+Int[ArcTan[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := x*ArcTan[c + d*Tan[a + b*x]] - I*b*Int[x/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c + I*d)^2, -1]
+Int[ArcCot[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := x*ArcCot[c + d*Tan[a + b*x]] + I*b*Int[x/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c + I*d)^2, -1]
+Int[ArcTan[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := x*ArcTan[c + d*Cot[a + b*x]] - I*b*Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, -1]
+Int[ArcCot[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := x*ArcCot[c + d*Cot[a + b*x]] + I*b*Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, -1]
+Int[ArcTan[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := x*ArcTan[c + d*Tan[a + b*x]] - b*(1 + I*c + d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + I*c - d + (1 + I*c + d)*E^(2*I*a + 2*I*b*x)), x] + b*(1 - I*c - d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - I*c + d + (1 - I*c - d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, -1]
+Int[ArcCot[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := x*ArcCot[c + d*Tan[a + b*x]] + b*(1 + I*c + d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + I*c - d + (1 + I*c + d)*E^(2*I*a + 2*I*b*x)), x] - b*(1 - I*c - d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - I*c + d + (1 - I*c - d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, -1]
+Int[ArcTan[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := x*ArcTan[c + d*Cot[a + b*x]] + b*(1 + I*c - d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + I*c + d - (1 + I*c - d)*E^(2*I*a + 2*I*b*x)), x] - b*(1 - I*c + d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - I*c - d - (1 - I*c + d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, -1]
+Int[ArcCot[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := x*ArcCot[c + d*Cot[a + b*x]] - b*(1 + I*c - d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + I*c + d - (1 + I*c - d)*E^(2*I*a + 2*I*b*x)), x] + b*(1 - I*c + d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - I*c - d - (1 - I*c + d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - I*d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTan[c + d*Tan[a + b*x]]/(f*(m + 1)) - I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c + I*d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCot[c + d*Tan[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c + I*d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTan[c + d*Cot[a + b*x]]/(f*(m + 1)) - I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCot[c + d*Cot[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTan[c + d*Tan[a + b*x]]/(f*(m + 1)) - b*(1 + I*c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + I*c - d + (1 + I*c + d)*E^(2*I*a + 2*I*b*x)), x] + b*(1 - I*c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - I*c + d + (1 - I*c - d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c + I*d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCot[c + d*Tan[a + b*x]]/(f*(m + 1)) + b*(1 + I*c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + I*c - d + (1 + I*c + d)*E^(2*I*a + 2*I*b*x)), x] - b*(1 - I*c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - I*c + d + (1 - I*c - d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c + I*d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTan[c + d*Cot[a + b*x]]/(f*(m + 1)) + b*(1 + I*c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + I*c + d - (1 + I*c - d)*E^(2*I*a + 2*I*b*x)), x] - b*(1 - I*c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - I*c - d - (1 - I*c + d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - I*d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCot[c + d*Cot[a + b*x]]/(f*(m + 1)) - b*(1 + I*c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + I*c + d - (1 + I*c - d)*E^(2*I*a + 2*I*b*x)), x] + b*(1 - I*c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - I*c - d - (1 - I*c + d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - I*d)^2, -1]
+Int[ArcTan[Tanh[a_. + b_.*x_]], x_Symbol] := x*ArcTan[Tanh[a + b*x]] - b*Int[x*Sech[2*a + 2*b*x], x] /; FreeQ[{a, b}, x]
+Int[ArcCot[Tanh[a_. + b_.*x_]], x_Symbol] := x*ArcCot[Tanh[a + b*x]] + b*Int[x*Sech[2*a + 2*b*x], x] /; FreeQ[{a, b}, x]
+Int[ArcTan[Coth[a_. + b_.*x_]], x_Symbol] := x*ArcTan[Coth[a + b*x]] + b*Int[x*Sech[2*a + 2*b*x], x] /; FreeQ[{a, b}, x]
+Int[ArcCot[Coth[a_. + b_.*x_]], x_Symbol] := x*ArcCot[Coth[a + b*x]] - b*Int[x*Sech[2*a + 2*b*x], x] /; FreeQ[{a, b}, x]
+Int[(e_. + f_.*x_)^m_.*ArcTan[Tanh[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTan[Tanh[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sech[2*a + 2*b*x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.*ArcCot[Tanh[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCot[Tanh[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sech[2*a + 2*b*x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.*ArcTan[Coth[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTan[Coth[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sech[2*a + 2*b*x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.*ArcCot[Coth[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCot[Coth[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sech[2*a + 2*b*x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
+Int[ArcTan[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := x*ArcTan[c + d*Tanh[a + b*x]] - b*Int[x/(c - d + c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, -1]
+Int[ArcCot[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := x*ArcCot[c + d*Tanh[a + b*x]] + b*Int[x/(c - d + c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, -1]
+Int[ArcTan[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := x*ArcTan[c + d*Coth[a + b*x]] - b*Int[x/(c - d - c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, -1]
+Int[ArcCot[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := x*ArcCot[c + d*Coth[a + b*x]] + b*Int[x/(c - d - c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, -1]
+Int[ArcTan[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := x*ArcTan[c + d*Tanh[a + b*x]] + I*b*(I - c - d)* Int[x*E^(2*a + 2*b*x)/(I - c + d + (I - c - d)*E^(2*a + 2*b*x)), x] - I*b*(I + c + d)* Int[x*E^(2*a + 2*b*x)/(I + c - d + (I + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, -1]
+Int[ArcCot[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := x*ArcCot[c + d*Tanh[a + b*x]] - I*b*(I - c - d)* Int[x*E^(2*a + 2*b*x)/(I - c + d + (I - c - d)*E^(2*a + 2*b*x)), x] + I*b*(I + c + d)* Int[x*E^(2*a + 2*b*x)/(I + c - d + (I + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, -1]
+Int[ArcTan[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := x*ArcTan[c + d*Coth[a + b*x]] - I*b*(I - c - d)* Int[x*E^(2*a + 2*b*x)/(I - c + d - (I - c - d)*E^(2*a + 2*b*x)), x] + I*b*(I + c + d)* Int[x*E^(2*a + 2*b*x)/(I + c - d - (I + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, -1]
+Int[ArcCot[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := x*ArcCot[c + d*Coth[a + b*x]] + I*b*(I - c - d)* Int[x*E^(2*a + 2*b*x)/(I - c + d - (I - c - d)*E^(2*a + 2*b*x)), x] - I*b*(I + c + d)* Int[x*E^(2*a + 2*b*x)/(I + c - d - (I + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTan[c + d*Tanh[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d + c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCot[c + d*Tanh[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d + c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTan[c + d*Coth[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCot[c + d*Coth[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTan[c + d*Tanh[a + b*x]]/(f*(m + 1)) + I*b*(I - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I - c + d + (I - c - d)*E^(2*a + 2*b*x)), x] - I*b*(I + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I + c - d + (I + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCot[c + d*Tanh[a + b*x]]/(f*(m + 1)) - I*b*(I - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I - c + d + (I - c - d)*E^(2*a + 2*b*x)), x] + I*b*(I + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I + c - d + (I + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTan[c + d*Coth[a + b*x]]/(f*(m + 1)) - I*b*(I - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I - c + d - (I - c - d)*E^(2*a + 2*b*x)), x] + I*b*(I + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I + c - d - (I + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, -1]
+Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCot[c + d*Coth[a + b*x]]/(f*(m + 1)) + I*b*(I - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I - c + d - (I - c - d)*E^(2*a + 2*b*x)), x] - I*b*(I + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I + c - d - (I + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, -1]
+Int[ArcTan[u_], x_Symbol] := x*ArcTan[u] - Int[SimplifyIntegrand[x*D[u, x]/(1 + u^2), x], x] /; InverseFunctionFreeQ[u, x]
+Int[ArcCot[u_], x_Symbol] := x*ArcCot[u] + Int[SimplifyIntegrand[x*D[u, x]/(1 + u^2), x], x] /; InverseFunctionFreeQ[u, x]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcTan[u_]), x_Symbol] := (c + d*x)^(m + 1)*(a + b*ArcTan[u])/(d*(m + 1)) - b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(1 + u^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && FalseQ[PowerVariableExpn[u, m + 1, x]]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcCot[u_]), x_Symbol] := (c + d*x)^(m + 1)*(a + b*ArcCot[u])/(d*(m + 1)) + b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(1 + u^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && FalseQ[PowerVariableExpn[u, m + 1, x]]
+Int[v_*(a_. + b_.*ArcTan[u_]), x_Symbol] := With[{w = IntHide[v, x]}, Dist[(a + b*ArcTan[u]), w, x] - b*Int[SimplifyIntegrand[w*D[u, x]/(1 + u^2), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_. /; FreeQ[{c, d, m}, x]]] && FalseQ[FunctionOfLinear[v*(a + b*ArcTan[u]), x]]
+Int[v_*(a_. + b_.*ArcCot[u_]), x_Symbol] := With[{w = IntHide[v, x]}, Dist[(a + b*ArcCot[u]), w, x] + b*Int[SimplifyIntegrand[w*D[u, x]/(1 + u^2), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_. /; FreeQ[{c, d, m}, x]]] && FalseQ[FunctionOfLinear[v*(a + b*ArcCot[u]), x]]
+Int[ArcTan[v_]*Log[w_]/(a_. + b_.*x_), x_Symbol] := I/2*Int[Log[1 - I*v]*Log[w]/(a + b*x), x] - I/2*Int[Log[1 + I*v]*Log[w]/(a + b*x), x] /; FreeQ[{a, b}, x] && LinearQ[v, x] && LinearQ[w, x] && EqQ[Simplify[D[v/(a + b*x), x]], 0] && EqQ[Simplify[D[w/(a + b*x), x]], 0]
+Int[ArcTan[v_]*Log[w_], x_Symbol] := x*ArcTan[v]*Log[w] - Int[SimplifyIntegrand[x*Log[w]*D[v, x]/(1 + v^2), x], x] - Int[SimplifyIntegrand[x*ArcTan[v]*D[w, x]/w, x], x] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]
+Int[ArcCot[v_]*Log[w_], x_Symbol] := x*ArcCot[v]*Log[w] + Int[SimplifyIntegrand[x*Log[w]*D[v, x]/(1 + v^2), x], x] - Int[SimplifyIntegrand[x*ArcCot[v]*D[w, x]/w, x], x] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]
+Int[u_*ArcTan[v_]*Log[w_], x_Symbol] := With[{z = IntHide[u, x]}, Dist[ArcTan[v]*Log[w], z, x] - Int[SimplifyIntegrand[z*Log[w]*D[v, x]/(1 + v^2), x], x] - Int[SimplifyIntegrand[z*ArcTan[v]*D[w, x]/w, x], x] /; InverseFunctionFreeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]
+Int[u_*ArcCot[v_]*Log[w_], x_Symbol] := With[{z = IntHide[u, x]}, Dist[ArcCot[v]*Log[w], z, x] + Int[SimplifyIntegrand[z*Log[w]*D[v, x]/(1 + v^2), x], x] - Int[SimplifyIntegrand[z*ArcCot[v]*D[w, x]/w, x], x] /; InverseFunctionFreeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]
diff --git a/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m b/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m
new file mode 100755
index 0000000..dafd2d6
--- /dev/null
+++ b/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m	
@@ -0,0 +1,39 @@
+
+(* ::Subsection::Closed:: *)
+(* 5.5.1 u (a+b arcsec(c x))^n *)
+Int[ArcSec[c_.*x_], x_Symbol] := x*ArcSec[c*x] - 1/c*Int[1/(x*Sqrt[1 - 1/(c^2*x^2)]), x] /; FreeQ[c, x]
+Int[ArcCsc[c_.*x_], x_Symbol] := x*ArcCsc[c*x] + 1/c*Int[1/(x*Sqrt[1 - 1/(c^2*x^2)]), x] /; FreeQ[c, x]
+Int[(a_. + b_.*ArcSec[c_.*x_])^n_, x_Symbol] := 1/c*Subst[Int[(a + b*x)^n*Sec[x]*Tan[x], x], x, ArcSec[c*x]] /; FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]
+Int[(a_. + b_.*ArcCsc[c_.*x_])^n_, x_Symbol] := -1/c*Subst[Int[(a + b*x)^n*Csc[x]*Cot[x], x], x, ArcCsc[c*x]] /; FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]
+Int[(a_. + b_.*ArcSec[c_.*x_])/x_, x_Symbol] := -Subst[Int[(a + b*ArcCos[x/c])/x, x], x, 1/x] /; FreeQ[{a, b, c}, x]
+Int[(a_. + b_.*ArcCsc[c_.*x_])/x_, x_Symbol] := -Subst[Int[(a + b*ArcSin[x/c])/x, x], x, 1/x] /; FreeQ[{a, b, c}, x]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcSec[c_.*x_]), x_Symbol] := (d*x)^(m + 1)*(a + b*ArcSec[c*x])/(d*(m + 1)) - b*d/(c*(m + 1))*Int[(d*x)^(m - 1)/Sqrt[1 - 1/(c^2*x^2)], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcCsc[c_.*x_]), x_Symbol] := (d*x)^(m + 1)*(a + b*ArcCsc[c*x])/(d*(m + 1)) + b*d/(c*(m + 1))*Int[(d*x)^(m - 1)/Sqrt[1 - 1/(c^2*x^2)], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
+Int[x_^m_.*(a_. + b_.*ArcSec[c_.*x_])^n_, x_Symbol] := 1/c^(m + 1)* Subst[Int[(a + b*x)^n*Sec[x]^(m + 1)*Tan[x], x], x, ArcSec[c*x]] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n, 0] || LtQ[m, -1])
+Int[x_^m_.*(a_. + b_.*ArcCsc[c_.*x_])^n_, x_Symbol] := -1/c^(m + 1)* Subst[Int[(a + b*x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcCsc[c*x]] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n, 0] || LtQ[m, -1])
+Int[(a_. + b_.*ArcSec[c_.*x_])/(d_. + e_.*x_), x_Symbol] := (a + b*ArcSec[c*x])* Log[1 + (e - Sqrt[-c^2*d^2 + e^2])*E^(I*ArcSec[c*x])/(c*d)]/e + (a + b*ArcSec[c*x])* Log[1 + (e + Sqrt[-c^2*d^2 + e^2])*E^(I*ArcSec[c*x])/(c*d)]/e - (a + b*ArcSec[c*x])*Log[1 + E^(2*I*ArcSec[c*x])]/e - b/(c*e)* Int[Log[1 + (e - Sqrt[-c^2*d^2 + e^2])* E^(I*ArcSec[c*x])/(c*d)]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x] - b/(c*e)* Int[Log[1 + (e + Sqrt[-c^2*d^2 + e^2])* E^(I*ArcSec[c*x])/(c*d)]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x] + b/(c*e)* Int[Log[1 + E^(2*I*ArcSec[c*x])]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x] /; FreeQ[{a, b, c, d, e}, x]
+Int[(a_. + b_.*ArcCsc[c_.*x_])/(d_. + e_.*x_), x_Symbol] := (a + b*ArcCsc[c*x])* Log[1 - I*(e - Sqrt[-c^2*d^2 + e^2])*E^(I*ArcCsc[c*x])/(c*d)]/ e + (a + b*ArcCsc[c*x])* Log[1 - I*(e + Sqrt[-c^2*d^2 + e^2])*E^(I*ArcCsc[c*x])/(c*d)]/ e - (a + b*ArcCsc[c*x])*Log[1 - E^(2*I*ArcCsc[c*x])]/e + b/(c*e)* Int[Log[1 - I*(e - Sqrt[-c^2*d^2 + e^2])*E^(I*ArcCsc[c*x])/(c*d)]/(x^2* Sqrt[1 - 1/(c^2*x^2)]), x] + b/(c*e)* Int[Log[1 - I*(e + Sqrt[-c^2*d^2 + e^2])*E^(I*ArcCsc[c*x])/(c*d)]/(x^2* Sqrt[1 - 1/(c^2*x^2)]), x] - b/(c*e)* Int[Log[1 - E^(2*I*ArcCsc[c*x])]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x] /; FreeQ[{a, b, c, d, e}, x]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcSec[c_.*x_]), x_Symbol] := (d + e*x)^(m + 1)*(a + b*ArcSec[c*x])/(e*(m + 1)) - b/(c*e*(m + 1))* Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCsc[c_.*x_]), x_Symbol] := (d + e*x)^(m + 1)*(a + b*ArcCsc[c*x])/(e*(m + 1)) + b/(c*e*(m + 1))* Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
+Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSec[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[(a + b*ArcSec[c*x]), u, x] - b*c*x/Sqrt[c^2*x^2]* Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
+Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsc[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[(a + b*ArcCsc[c*x]), u, x] + b*c*x/Sqrt[c^2*x^2]* Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
+Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSec[c_.*x_])^n_., x_Symbol] := -Subst[Int[(e + d*x^2)^p*(a + b*ArcCos[x/c])^n/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[p]
+Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsc[c_.*x_])^n_., x_Symbol] := -Subst[Int[(e + d*x^2)^p*(a + b*ArcSin[x/c])^n/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[p]
+Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSec[c_.*x_])^n_., x_Symbol] := -Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcCos[x/c])^n/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
+Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsc[c_.*x_])^n_., x_Symbol] := -Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcSin[x/c])^n/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
+Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSec[c_.*x_])^n_., x_Symbol] := -Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcCos[x/c])^n/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]
+Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsc[c_.*x_])^n_., x_Symbol] := -Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcSin[x/c])^n/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]
+Int[x_*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSec[c_.*x_]), x_Symbol] := (d + e*x^2)^(p + 1)*(a + b*ArcSec[c*x])/(2*e*(p + 1)) - b*c*x/(2*e*(p + 1)*Sqrt[c^2*x^2])* Int[(d + e*x^2)^(p + 1)/(x*Sqrt[c^2*x^2 - 1]), x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
+Int[x_*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsc[c_.*x_]), x_Symbol] := (d + e*x^2)^(p + 1)*(a + b*ArcCsc[c*x])/(2*e*(p + 1)) + b*c*x/(2*e*(p + 1)*Sqrt[c^2*x^2])* Int[(d + e*x^2)^(p + 1)/(x*Sqrt[c^2*x^2 - 1]), x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
+Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSec[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[(a + b*ArcSec[c*x]), u, x] - b*c*x/Sqrt[c^2*x^2]* Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ( IGtQ[p, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]] || ILtQ[(m + 2*p + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]])
+Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsc[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[(a + b*ArcCsc[c*x]), u, x] + b*c*x/Sqrt[c^2*x^2]* Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ( IGtQ[p, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]] || ILtQ[(m + 2*p + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]])
+Int[x_^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSec[c_.*x_])^n_., x_Symbol] := -Subst[ Int[(e + d*x^2)^p*(a + b*ArcCos[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[m] && IntegerQ[p]
+Int[x_^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsc[c_.*x_])^n_., x_Symbol] := -Subst[ Int[(e + d*x^2)^p*(a + b*ArcSin[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[m] && IntegerQ[p]
+Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSec[c_.*x_])^n_., x_Symbol] := -Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcCos[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
+Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsc[c_.*x_])^n_., x_Symbol] := -Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcSin[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
+Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSec[c_.*x_])^n_., x_Symbol] := -Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcCos[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]
+Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsc[c_.*x_])^n_., x_Symbol] := -Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcSin[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]
+Int[u_*(a_. + b_.*ArcSec[c_.*x_]), x_Symbol] := With[{v = IntHide[u, x]}, Dist[(a + b*ArcSec[c*x]), v, x] - b/c* Int[SimplifyIntegrand[v/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
+Int[u_*(a_. + b_.*ArcCsc[c_.*x_]), x_Symbol] := With[{v = IntHide[u, x]}, Dist[(a + b*ArcCsc[c*x]), v, x] + b/c* Int[SimplifyIntegrand[v/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
+Int[u_.*(a_. + b_.*ArcSec[c_.*x_])^n_., x_Symbol] := Unintegrable[u*(a + b*ArcSec[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
+Int[u_.*(a_. + b_.*ArcCsc[c_.*x_])^n_., x_Symbol] := Unintegrable[u*(a + b*ArcCsc[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
diff --git a/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m b/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m
new file mode 100755
index 0000000..eb57b97
--- /dev/null
+++ b/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m	
@@ -0,0 +1,27 @@
+
+(* ::Subsection::Closed:: *)
+(* 5.5.2 Miscellaneous inverse secant *)
+Int[ArcSec[c_ + d_.*x_], x_Symbol] := (c + d*x)*ArcSec[c + d*x]/d - Int[1/((c + d*x)*Sqrt[1 - 1/(c + d*x)^2]), x] /; FreeQ[{c, d}, x]
+Int[ArcCsc[c_ + d_.*x_], x_Symbol] := (c + d*x)*ArcCsc[c + d*x]/d + Int[1/((c + d*x)*Sqrt[1 - 1/(c + d*x)^2]), x] /; FreeQ[{c, d}, x]
+Int[(a_. + b_.*ArcSec[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(a + b*ArcSec[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*ArcCsc[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(a + b*ArcCsc[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*ArcSec[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(a + b*ArcSec[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]
+Int[(a_. + b_.*ArcCsc[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(a + b*ArcCsc[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSec[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(f*x/d)^m*(a + b*ArcSec[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsc[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(f*x/d)^m*(a + b*ArcCsc[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSec[c_ + d_.*x_])^p_., x_Symbol] := 1/d^(m + 1)* Subst[Int[(a + b*x)^p*Sec[x]*Tan[x]*(d*e - c*f + f*Sec[x])^m, x], x, ArcSec[c + d*x]] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsc[c_ + d_.*x_])^p_., x_Symbol] := -1/d^(m + 1)* Subst[Int[(a + b*x)^p*Csc[x]*Cot[x]*(d*e - c*f + f*Csc[x])^m, x], x, ArcCsc[c + d*x]] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSec[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcSec[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsc[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcCsc[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSec[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(e + f*x)^m*(a + b*ArcSec[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && Not[IGtQ[p, 0]]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsc[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(e + f*x)^m*(a + b*ArcCsc[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && Not[IGtQ[p, 0]]
+Int[u_.*ArcSec[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol] := Int[u*ArcCos[a/c + b*x^n/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
+Int[u_.*ArcCsc[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol] := Int[u*ArcSin[a/c + b*x^n/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
+Int[u_.*f_^(c_.*ArcSec[a_. + b_.*x_]^n_.), x_Symbol] := 1/b*Subst[ Int[ReplaceAll[u, x -> -a/b + Sec[x]/b]*f^(c*x^n)*Sec[x]*Tan[x], x], x, ArcSec[a + b*x]] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
+Int[u_.*f_^(c_.*ArcCsc[a_. + b_.*x_]^n_.), x_Symbol] := -1/b*Subst[ Int[ReplaceAll[u, x -> -a/b + Csc[x]/b]*f^(c*x^n)*Csc[x]*Cot[x], x], x, ArcCsc[a + b*x]] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
+Int[ArcSec[u_], x_Symbol] := x*ArcSec[u] - u/Sqrt[u^2]* Int[SimplifyIntegrand[x*D[u, x]/(u*Sqrt[u^2 - 1]), x], x] /; InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]
+Int[ArcCsc[u_], x_Symbol] := x*ArcCsc[u] + u/Sqrt[u^2]* Int[SimplifyIntegrand[x*D[u, x]/(u*Sqrt[u^2 - 1]), x], x] /; InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcSec[u_]), x_Symbol] := (c + d*x)^(m + 1)*(a + b*ArcSec[u])/(d*(m + 1)) - b*u/(d*(m + 1)*Sqrt[u^2])* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(u*Sqrt[u^2 - 1]), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcCsc[u_]), x_Symbol] := (c + d*x)^(m + 1)*(a + b*ArcCsc[u])/(d*(m + 1)) + b*u/(d*(m + 1)*Sqrt[u^2])* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(u*Sqrt[u^2 - 1]), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]
+Int[v_*(a_. + b_.*ArcSec[u_]), x_Symbol] := With[{w = IntHide[v, x]}, Dist[(a + b*ArcSec[u]), w, x] - b*u/Sqrt[u^2]* Int[SimplifyIntegrand[w*D[u, x]/(u*Sqrt[u^2 - 1]), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_. /; FreeQ[{c, d, m}, x]]]
+Int[v_*(a_. + b_.*ArcCsc[u_]), x_Symbol] := With[{w = IntHide[v, x]}, Dist[(a + b*ArcCsc[u]), w, x] + b*u/Sqrt[u^2]* Int[SimplifyIntegrand[w*D[u, x]/(u*Sqrt[u^2 - 1]), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_. /; FreeQ[{c, d, m}, x]]]
diff --git a/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.10 (c+d x)^m (a+b sinh)^n.m b/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.10 (c+d x)^m (a+b sinh)^n.m
new file mode 100755
index 0000000..3d369c6
--- /dev/null
+++ b/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.10 (c+d x)^m (a+b sinh)^n.m	
@@ -0,0 +1,5 @@
+
+(* ::Subsection::Closed:: *)
+(* 6.1.10 (c+d x)^m (a+b sinh)^n *)
+Int[u_^m_.*(a_. + b_.*Sinh[v_])^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*(a + b*Sinh[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
+Int[u_^m_.*(a_. + b_.*Cosh[v_])^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*(a + b*Cosh[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
diff --git a/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m b/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m
new file mode 100755
index 0000000..3ac41bb
--- /dev/null
+++ b/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m	
@@ -0,0 +1,25 @@
+
+(* ::Subsection::Closed:: *)
+(* 6.1.11 (e x)^m (a+b x^n)^p sinh *)
+Int[(a_ + b_.*x_^n_)^p_.*Sinh[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Sinh[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
+Int[(a_ + b_.*x_^n_)^p_.*Cosh[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Cosh[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
+Int[(a_ + b_.*x_^n_)^p_*Sinh[c_. + d_.*x_], x_Symbol] := x^(-n + 1)*(a + b*x^n)^(p + 1)*Sinh[c + d*x]/(b*n*(p + 1)) - (-n + 1)/(b*n*(p + 1))* Int[x^(-n)*(a + b*x^n)^(p + 1)*Sinh[c + d*x], x] - d/(b*n*(p + 1))* Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Cosh[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[p] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[n, 2]
+Int[(a_ + b_.*x_^n_)^p_*Cosh[c_. + d_.*x_], x_Symbol] := x^(-n + 1)*(a + b*x^n)^(p + 1)*Cosh[c + d*x]/(b*n*(p + 1)) - (-n + 1)/(b*n*(p + 1))* Int[x^(-n)*(a + b*x^n)^(p + 1)*Cosh[c + d*x], x] - d/(b*n*(p + 1))* Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Sinh[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[p] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[n, 2]
+Int[(a_ + b_.*x_^n_)^p_*Sinh[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Sinh[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
+Int[(a_ + b_.*x_^n_)^p_*Cosh[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Cosh[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
+Int[(a_ + b_.*x_^n_)^p_*Sinh[c_. + d_.*x_], x_Symbol] := Int[x^(n*p)*(b + a*x^(-n))^p*Sinh[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && ILtQ[n, 0]
+Int[(a_ + b_.*x_^n_)^p_*Cosh[c_. + d_.*x_], x_Symbol] := Int[x^(n*p)*(b + a*x^(-n))^p*Cosh[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && ILtQ[n, 0]
+Int[(a_ + b_.*x_^n_)^p_*Sinh[c_. + d_.*x_], x_Symbol] := Unintegrable[(a + b*x^n)^p*Sinh[c + d*x], x] /; FreeQ[{a, b, c, d, n, p}, x]
+Int[(a_ + b_.*x_^n_)^p_*Cosh[c_. + d_.*x_], x_Symbol] := Unintegrable[(a + b*x^n)^p*Cosh[c + d*x], x] /; FreeQ[{a, b, c, d, n, p}, x]
+Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*Sinh[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Sinh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
+Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*Cosh[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Cosh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
+Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*Sinh[c_. + d_.*x_], x_Symbol] := e^m*(a + b*x^n)^(p + 1)*Sinh[c + d*x]/(b*n*(p + 1)) - d*e^m/(b*n*(p + 1))*Int[(a + b*x^n)^(p + 1)*Cosh[c + d*x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && EqQ[m - n + 1, 0] && LtQ[p, -1] && (IntegerQ[n] || GtQ[e, 0])
+Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*Cosh[c_. + d_.*x_], x_Symbol] := e^m*(a + b*x^n)^(p + 1)*Cosh[c + d*x]/(b*n*(p + 1)) - d*e^m/(b*n*(p + 1))*Int[(a + b*x^n)^(p + 1)*Sinh[c + d*x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && EqQ[m - n + 1, 0] && LtQ[p, -1] && (IntegerQ[n] || GtQ[e, 0])
+Int[x_^m_.*(a_ + b_.*x_^n_)^p_*Sinh[c_. + d_.*x_], x_Symbol] := x^(m - n + 1)*(a + b*x^n)^(p + 1)*Sinh[c + d*x]/(b*n*(p + 1)) - (m - n + 1)/(b*n*(p + 1))* Int[x^(m - n)*(a + b*x^n)^(p + 1)*Sinh[c + d*x], x] - d/(b*n*(p + 1))* Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Cosh[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1] && IGtQ[n, 0] && RationalQ[m] && (GtQ[m - n + 1, 0] || GtQ[n, 2])
+Int[x_^m_.*(a_ + b_.*x_^n_)^p_*Cosh[c_. + d_.*x_], x_Symbol] := x^(m - n + 1)*(a + b*x^n)^(p + 1)*Cosh[c + d*x]/(b*n*(p + 1)) - (m - n + 1)/(b*n*(p + 1))* Int[x^(m - n)*(a + b*x^n)^(p + 1)*Cosh[c + d*x], x] - d/(b*n*(p + 1))* Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Sinh[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1] && IGtQ[n, 0] && RationalQ[m] && (GtQ[m - n + 1, 0] || GtQ[n, 2])
+Int[x_^m_.*(a_ + b_.*x_^n_)^p_*Sinh[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Sinh[c + d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
+Int[x_^m_.*(a_ + b_.*x_^n_)^p_*Cosh[c_. + d_.*x_], x_Symbol] := Int[ExpandIntegrand[Cosh[c + d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
+Int[x_^m_.*(a_ + b_.*x_^n_)^p_*Sinh[c_. + d_.*x_], x_Symbol] := Int[x^(m + n*p)*(b + a*x^(-n))^p*Sinh[c + d*x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && ILtQ[n, 0]
+Int[x_^m_.*(a_ + b_.*x_^n_)^p_*Cosh[c_. + d_.*x_], x_Symbol] := Int[x^(m + n*p)*(b + a*x^(-n))^p*Cosh[c + d*x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && ILtQ[n, 0]
+Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*Sinh[c_. + d_.*x_], x_Symbol] := Unintegrable[(e*x)^m*(a + b*x^n)^p*Sinh[c + d*x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
+Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*Cosh[c_. + d_.*x_], x_Symbol] := Unintegrable[(e*x)^m*(a + b*x^n)^p*Cosh[c + d*x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
diff --git a/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m b/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m
new file mode 100755
index 0000000..6dd3104
--- /dev/null
+++ b/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m	
@@ -0,0 +1,79 @@
+
+(* ::Subsection::Closed:: *)
+(* 6.1.12 (e x)^m (a+b sinh(c+d x^n))^p *)
+Int[Sinh[c_. + d_.*x_^n_], x_Symbol] := 1/2*Int[E^(c + d*x^n), x] - 1/2*Int[E^(-c - d*x^n), x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]
+Int[Cosh[c_. + d_.*x_^n_], x_Symbol] := 1/2*Int[E^(c + d*x^n), x] + 1/2*Int[E^(-c - d*x^n), x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]
+Int[(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(a + b*Sinh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 1] && IGtQ[p, 1]
+Int[(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 1] && IGtQ[p, 1]
+Int[(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_., x_Symbol] := -Subst[Int[(a + b*Sinh[c + d*x^(-n)])^p/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n, 0] && IntegerQ[p]
+Int[(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_., x_Symbol] := -Subst[Int[(a + b*Cosh[c + d*x^(-n)])^p/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n, 0] && IntegerQ[p]
+Int[(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_., x_Symbol] := Module[{k = Denominator[n]}, k*Subst[Int[x^(k - 1)*(a + b*Sinh[c + d*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c, d}, x] && FractionQ[n] && IntegerQ[p]
+Int[(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_., x_Symbol] := Module[{k = Denominator[n]}, k*Subst[Int[x^(k - 1)*(a + b*Cosh[c + d*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c, d}, x] && FractionQ[n] && IntegerQ[p]
+Int[Sinh[c_. + d_.*x_^n_], x_Symbol] := 1/2*Int[E^(c + d*x^n), x] - 1/2*Int[E^(-c - d*x^n), x] /; FreeQ[{c, d, n}, x]
+Int[Cosh[c_. + d_.*x_^n_], x_Symbol] := 1/2*Int[E^(c + d*x^n), x] + 1/2*Int[E^(-c - d*x^n), x] /; FreeQ[{c, d, n}, x]
+Int[(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(a + b*Sinh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*Sinh[c_. + d_.*u_^n_])^p_., x_Symbol] := 1/Coefficient[u, x, 1]* Subst[Int[(a + b*Sinh[c + d*x^n])^p, x], x, u] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[p] && LinearQ[u, x] && NeQ[u, x]
+Int[(a_. + b_.*Cosh[c_. + d_.*u_^n_])^p_., x_Symbol] := 1/Coefficient[u, x, 1]* Subst[Int[(a + b*Cosh[c + d*x^n])^p, x], x, u] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[p] && LinearQ[u, x] && NeQ[u, x]
+Int[(a_. + b_.*Sinh[c_. + d_.*u_^n_])^p_, x_Symbol] := Unintegrable[(a + b*Sinh[c + d*u^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x]
+Int[(a_. + b_.*Cosh[c_. + d_.*u_^n_])^p_, x_Symbol] := Unintegrable[(a + b*Cosh[c + d*u^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x]
+Int[(a_. + b_.*Sinh[u_])^p_., x_Symbol] := Int[(a + b*Sinh[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[(a_. + b_.*Cosh[u_])^p_., x_Symbol] := Int[(a + b*Cosh[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[Sinh[d_.*x_^n_]/x_, x_Symbol] := SinhIntegral[d*x^n]/n /; FreeQ[{d, n}, x]
+Int[Cosh[d_.*x_^n_]/x_, x_Symbol] := CoshIntegral[d*x^n]/n /; FreeQ[{d, n}, x]
+Int[Sinh[c_ + d_.*x_^n_]/x_, x_Symbol] := Sinh[c]*Int[Cosh[d*x^n]/x, x] + Cosh[c]*Int[Sinh[d*x^n]/x, x] /; FreeQ[{c, d, n}, x]
+Int[Cosh[c_ + d_.*x_^n_]/x_, x_Symbol] := Cosh[c]*Int[Cosh[d*x^n]/x, x] + Sinh[c]*Int[Sinh[d*x^n]/x, x] /; FreeQ[{c, d, n}, x]
+Int[x_^m_.*(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sinh[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[ Simplify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0])
+Int[x_^m_.*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cosh[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[ Simplify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0])
+Int[(e_*x_)^m_*(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Sinh[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
+Int[(e_*x_)^m_*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Cosh[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
+Int[(e_.*x_)^m_.*Sinh[c_. + d_.*x_^n_], x_Symbol] := e^(n - 1)*(e*x)^(m - n + 1)*Cosh[c + d*x^n]/(d*n) - e^n*(m - n + 1)/(d*n)*Int[(e*x)^(m - n)*Cosh[c + d*x^n], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[0, n, m + 1]
+Int[(e_.*x_)^m_.*Cosh[c_. + d_.*x_^n_], x_Symbol] := e^(n - 1)*(e*x)^(m - n + 1)*Sinh[c + d*x^n]/(d*n) - e^n*(m - n + 1)/(d*n)*Int[(e*x)^(m - n)*Sinh[c + d*x^n], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[0, n, m + 1]
+Int[(e_.*x_)^m_*Sinh[c_. + d_.*x_^n_], x_Symbol] := (e*x)^(m + 1)*Sinh[c + d*x^n]/(e*(m + 1)) - d*n/(e^n*(m + 1))*Int[(e*x)^(m + n)*Cosh[c + d*x^n], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[m, -1]
+Int[(e_.*x_)^m_*Cosh[c_. + d_.*x_^n_], x_Symbol] := (e*x)^(m + 1)*Cosh[c + d*x^n]/(e*(m + 1)) - d*n/(e^n*(m + 1))*Int[(e*x)^(m + n)*Sinh[c + d*x^n], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[m, -1]
+Int[(e_.*x_)^m_.*Sinh[c_. + d_.*x_^n_], x_Symbol] := 1/2*Int[(e*x)^m*E^(c + d*x^n), x] - 1/2*Int[(e*x)^m*E^(-c - d*x^n), x] /; FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]
+Int[(e_.*x_)^m_.*Cosh[c_. + d_.*x_^n_], x_Symbol] := 1/2*Int[(e*x)^m*E^(c + d*x^n), x] + 1/2*Int[(e*x)^m*E^(-c - d*x^n), x] /; FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]
+Int[x_^m_.*Sinh[a_. + b_.*x_^n_]^p_, x_Symbol] := -Sinh[a + b*x^n]^p/((n - 1)*x^(n - 1)) + b*n*p/(n - 1)*Int[Sinh[a + b*x^n]^(p - 1)*Cosh[a + b*x^n], x] /; FreeQ[{a, b}, x] && IntegersQ[n, p] && EqQ[m + n, 0] && GtQ[p, 1] && NeQ[n, 1]
+Int[x_^m_.*Cosh[a_. + b_.*x_^n_]^p_, x_Symbol] := -Cosh[a + b*x^n]^p/((n - 1)*x^(n - 1)) + b*n*p/(n - 1)*Int[Cosh[a + b*x^n]^(p - 1)*Sinh[a + b*x^n], x] /; FreeQ[{a, b}, x] && IntegersQ[n, p] && EqQ[m + n, 0] && GtQ[p, 1] && NeQ[n, 1]
+Int[x_^m_.*Sinh[a_. + b_.*x_^n_]^p_, x_Symbol] := -n*Sinh[a + b*x^n]^p/(b^2*n^2*p^2) + x^n*Cosh[a + b*x^n]*Sinh[a + b*x^n]^(p - 1)/(b*n*p) - (p - 1)/p*Int[x^m*Sinh[a + b*x^n]^(p - 2), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1] && GtQ[p, 1]
+Int[x_^m_.*Cosh[a_. + b_.*x_^n_]^p_, x_Symbol] := -n*Cosh[a + b*x^n]^p/(b^2*n^2*p^2) + x^n*Sinh[a + b*x^n]*Cosh[a + b*x^n]^(p - 1)/(b*n*p) + (p - 1)/p*Int[x^m*Cosh[a + b*x^n]^(p - 2), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1] && GtQ[p, 1]
+Int[x_^m_.*Sinh[a_. + b_.*x_^n_]^p_, x_Symbol] := -(m - n + 1)*x^(m - 2*n + 1)*Sinh[a + b*x^n]^p/(b^2*n^2*p^2) + x^(m - n + 1)*Cosh[a + b*x^n]*Sinh[a + b*x^n]^(p - 1)/(b*n*p) - (p - 1)/p*Int[x^m*Sinh[a + b*x^n]^(p - 2), x] + (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*p^2)* Int[x^(m - 2*n)*Sinh[a + b*x^n]^p, x] /; FreeQ[{a, b}, x] && IntegersQ[m, n] && GtQ[p, 1] && LtQ[0, 2*n, m + 1]
+Int[x_^m_.*Cosh[a_. + b_.*x_^n_]^p_, x_Symbol] := -(m - n + 1)*x^(m - 2*n + 1)*Cosh[a + b*x^n]^p/(b^2*n^2*p^2) + x^(m - n + 1)*Sinh[a + b*x^n]*Cosh[a + b*x^n]^(p - 1)/(b*n*p) + (p - 1)/p*Int[x^m*Cosh[a + b*x^n]^(p - 2), x] + (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*p^2)* Int[x^(m - 2*n)*Cosh[a + b*x^n]^p, x] /; FreeQ[{a, b}, x] && IntegersQ[m, n] && GtQ[p, 1] && LtQ[0, 2*n, m + 1]
+Int[x_^m_.*Sinh[a_. + b_.*x_^n_]^p_, x_Symbol] := x^(m + 1)*Sinh[a + b*x^n]^p/(m + 1) - b*n*p*x^(m + n + 1)*Cosh[a + b*x^n]* Sinh[a + b*x^n]^(p - 1)/((m + 1)*(m + n + 1)) + b^2*n^2*p^2/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Sinh[a + b*x^n]^p, x] + b^2*n^2*p*(p - 1)/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Sinh[a + b*x^n]^(p - 2), x] /; FreeQ[{a, b}, x] && IntegersQ[m, n] && GtQ[p, 1] && LtQ[0, 2*n, 1 - m] && NeQ[m + n + 1, 0]
+Int[x_^m_.*Cosh[a_. + b_.*x_^n_]^p_, x_Symbol] := x^(m + 1)*Cosh[a + b*x^n]^p/(m + 1) - b*n*p*x^(m + n + 1)*Sinh[a + b*x^n]* Cosh[a + b*x^n]^(p - 1)/((m + 1)*(m + n + 1)) + b^2*n^2*p^2/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Cosh[a + b*x^n]^p, x] - b^2*n^2*p*(p - 1)/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Cosh[a + b*x^n]^(p - 2), x] /; FreeQ[{a, b}, x] && IntegersQ[m, n] && GtQ[p, 1] && LtQ[0, 2*n, 1 - m] && NeQ[m + n + 1, 0]
+Int[(e_.*x_)^m_*(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_., x_Symbol] := With[{k = Denominator[m]}, k/e* Subst[Int[x^(k*(m + 1) - 1)*(a + b*Sinh[c + d*x^(k*n)/e^n])^p, x], x, (e*x)^(1/k)]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && IGtQ[n, 0] && FractionQ[m]
+Int[(e_.*x_)^m_*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_., x_Symbol] := With[{k = Denominator[m]}, k/e* Subst[Int[x^(k*(m + 1) - 1)*(a + b*Cosh[c + d*x^(k*n)/e^n])^p, x], x, (e*x)^(1/k)]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && IGtQ[n, 0] && FractionQ[m]
+Int[(e_.*x_)^m_.*(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Sinh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]
+Int[(e_.*x_)^m_.*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]
+Int[x_^m_.*Sinh[a_. + b_.*x_^n_]^p_, x_Symbol] := x^n*Cosh[a + b*x^n]*Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1)) - n*Sinh[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) - (p + 2)/(p + 1)*Int[x^m*Sinh[a + b*x^n]^(p + 2), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && LtQ[p, -1] && NeQ[p, -2]
+Int[x_^m_.*Cosh[a_. + b_.*x_^n_]^p_, x_Symbol] := -x^n*Sinh[a + b*x^n]*Cosh[a + b*x^n]^(p + 1)/(b*n*(p + 1)) + n*Cosh[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) + (p + 2)/(p + 1)*Int[x^m*Cosh[a + b*x^n]^(p + 2), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && LtQ[p, -1] && NeQ[p, -2]
+Int[x_^m_.*Sinh[a_. + b_.*x_^n_]^p_, x_Symbol] := x^(m - n + 1)*Cosh[a + b*x^n]* Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1)) - (m - n + 1)*x^(m - 2*n + 1)* Sinh[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) - (p + 2)/(p + 1)*Int[x^m*Sinh[a + b*x^n]^(p + 2), x] + (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*(p + 1)*(p + 2))* Int[x^(m - 2*n)*Sinh[a + b*x^n]^(p + 2), x] /; FreeQ[{a, b}, x] && IntegersQ[m, n] && LtQ[p, -1] && NeQ[p, -2] && LtQ[0, 2*n, m + 1]
+Int[x_^m_.*Cosh[a_. + b_.*x_^n_]^p_, x_Symbol] := -x^(m - n + 1)*Sinh[a + b*x^n]* Cosh[a + b*x^n]^(p + 1)/(b*n*(p + 1)) + (m - n + 1)*x^(m - 2*n + 1)* Cosh[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) + (p + 2)/(p + 1)*Int[x^m*Cosh[a + b*x^n]^(p + 2), x] - (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*(p + 1)*(p + 2))* Int[x^(m - 2*n)*Cosh[a + b*x^n]^(p + 2), x] /; FreeQ[{a, b}, x] && IntegersQ[m, n] && LtQ[p, -1] && NeQ[p, -2] && LtQ[0, 2*n, m + 1]
+Int[x_^m_.*(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_., x_Symbol] := -Subst[Int[(a + b*Sinh[c + d*x^(-n)])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[p] && ILtQ[n, 0] && IntegerQ[m]
+Int[x_^m_.*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_., x_Symbol] := -Subst[Int[(a + b*Cosh[c + d*x^(-n)])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[p] && ILtQ[n, 0] && IntegerQ[m]
+Int[(e_.*x_)^m_*(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_., x_Symbol] := With[{k = Denominator[m]}, -k/e* Subst[Int[(a + b*Sinh[c + d/(e^n*x^(k*n))])^p/x^(k*(m + 1) + 1), x], x, 1/(e*x)^(1/k)]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && ILtQ[n, 0] && FractionQ[m]
+Int[(e_.*x_)^m_*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_., x_Symbol] := With[{k = Denominator[m]}, -k/e* Subst[Int[(a + b*Cosh[c + d/(e^n*x^(k*n))])^p/x^(k*(m + 1) + 1), x], x, 1/(e*x)^(1/k)]] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && ILtQ[n, 0] && FractionQ[m]
+Int[(e_.*x_)^m_*(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_., x_Symbol] := -(e*x)^m*(x^(-1))^m* Subst[Int[(a + b*Sinh[c + d*x^(-n)])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[p] && ILtQ[n, 0] && Not[RationalQ[m]]
+Int[(e_.*x_)^m_*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_., x_Symbol] := -(e*x)^m*(x^(-1))^m* Subst[Int[(a + b*Cosh[c + d*x^(-n)])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[p] && ILtQ[n, 0] && Not[RationalQ[m]]
+Int[x_^m_.*(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_., x_Symbol] := Module[{k = Denominator[n]}, k*Subst[Int[x^(k*(m + 1) - 1)*(a + b*Sinh[c + d*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p] && FractionQ[n]
+Int[x_^m_.*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_., x_Symbol] := Module[{k = Denominator[n]}, k*Subst[Int[x^(k*(m + 1) - 1)*(a + b*Cosh[c + d*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p] && FractionQ[n]
+Int[(e_*x_)^m_*(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Sinh[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[p] && FractionQ[n]
+Int[(e_*x_)^m_*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Cosh[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[p] && FractionQ[n]
+Int[x_^m_.*(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/(m + 1)* Subst[Int[(a + b*Sinh[c + d*x^Simplify[n/(m + 1)]])^p, x], x, x^(m + 1)] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && Not[IntegerQ[n]]
+Int[x_^m_.*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/(m + 1)* Subst[Int[(a + b*Cosh[c + d*x^Simplify[n/(m + 1)]])^p, x], x, x^(m + 1)] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && Not[IntegerQ[n]]
+Int[(e_*x_)^m_*(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Sinh[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && Not[IntegerQ[n]]
+Int[(e_*x_)^m_*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Cosh[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && Not[IntegerQ[n]]
+Int[(e_.*x_)^m_.*Sinh[c_. + d_.*x_^n_], x_Symbol] := 1/2*Int[(e*x)^m*E^(c + d*x^n), x] - 1/2*Int[(e*x)^m*E^(-c - d*x^n), x] /; FreeQ[{c, d, e, m, n}, x]
+Int[(e_.*x_)^m_.*Cosh[c_. + d_.*x_^n_], x_Symbol] := 1/2*Int[(e*x)^m*E^(c + d*x^n), x] + 1/2*Int[(e*x)^m*E^(-c - d*x^n), x] /; FreeQ[{c, d, e, m, n}, x]
+Int[(e_.*x_)^m_.*(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Sinh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
+Int[(e_.*x_)^m_.*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_, x_Symbol] := Int[ExpandTrigReduce[(e*x)^m, (a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
+Int[x_^m_.*(a_. + b_.*Sinh[c_. + d_.*u_^n_])^p_., x_Symbol] := 1/Coefficient[u, x, 1]^(m + 1)* Subst[Int[(x - Coefficient[u, x, 0])^m*(a + b*Sinh[c + d*x^n])^p, x], x, u] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x] && IntegerQ[m]
+Int[x_^m_.*(a_. + b_.*Cosh[c_. + d_.*u_^n_])^p_., x_Symbol] := 1/Coefficient[u, x, 1]^(m + 1)* Subst[Int[(x - Coefficient[u, x, 0])^m*(a + b*Cosh[c + d*x^n])^p, x], x, u] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x] && IntegerQ[m]
+Int[(e_.*x_)^m_.*(a_. + b_.*Sinh[c_. + d_.*u_^n_])^p_., x_Symbol] := Unintegrable[(e*x)^m*(a + b*Sinh[c + d*u^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && LinearQ[u, x]
+Int[(e_.*x_)^m_.*(a_. + b_.*Cosh[c_. + d_.*u_^n_])^p_., x_Symbol] := Unintegrable[(e*x)^m*(a + b*Cosh[c + d*u^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && LinearQ[u, x]
+Int[(e_*x_)^m_.*(a_. + b_.*Sinh[u_])^p_., x_Symbol] := Int[(e*x)^m*(a + b*Sinh[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[(e_*x_)^m_.*(a_. + b_.*Cosh[u_])^p_., x_Symbol] := Int[(e*x)^m*(a + b*Cosh[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[x_^m_.*Sinh[a_. + b_.*x_^n_]^p_.*Cosh[a_. + b_.*x_^n_.], x_Symbol] := Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1)) /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
+Int[x_^m_.*Cosh[a_. + b_.*x_^n_]^p_.*Sinh[a_. + b_.*x_^n_.], x_Symbol] := Cosh[a + b*x^n]^(p + 1)/(b*n*(p + 1)) /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
+Int[x_^m_.*Sinh[a_. + b_.*x_^n_.]^p_.*Cosh[a_. + b_.*x_^n_.], x_Symbol] := x^(m - n + 1)*Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1)) - (m - n + 1)/(b*n*(p + 1))* Int[x^(m - n)*Sinh[a + b*x^n]^(p + 1), x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
+Int[x_^m_.*Cosh[a_. + b_.*x_^n_.]^p_.*Sinh[a_. + b_.*x_^n_.], x_Symbol] := x^(m - n + 1)*Cosh[a + b*x^n]^(p + 1)/(b*n*(p + 1)) - (m - n + 1)/(b*n*(p + 1))* Int[x^(m - n)*Cosh[a + b*x^n]^(p + 1), x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
diff --git a/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.13 (d+e x)^m sinh(a+b x+c x^2)^n.m b/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.13 (d+e x)^m sinh(a+b x+c x^2)^n.m
new file mode 100755
index 0000000..22663b7
--- /dev/null
+++ b/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.13 (d+e x)^m sinh(a+b x+c x^2)^n.m	
@@ -0,0 +1,27 @@
+
+(* ::Subsection::Closed:: *)
+(* 6.1.13 (d+e x)^m sinh(a+b x+c x^2)^n *)
+Int[Sinh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := 1/2*Int[E^(a + b*x + c*x^2), x] - 1/2*Int[E^(-a - b*x - c*x^2), x] /; FreeQ[{a, b, c}, x]
+Int[Cosh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := 1/2*Int[E^(a + b*x + c*x^2), x] + 1/2*Int[E^(-a - b*x - c*x^2), x] /; FreeQ[{a, b, c}, x]
+Int[Sinh[a_. + b_.*x_ + c_.*x_^2]^n_, x_Symbol] := Int[ExpandTrigReduce[Sinh[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 1]
+Int[Cosh[a_. + b_.*x_ + c_.*x_^2]^n_, x_Symbol] := Int[ExpandTrigReduce[Cosh[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 1]
+Int[Sinh[v_]^n_., x_Symbol] := Int[Sinh[ExpandToSum[v, x]]^n, x] /; IGtQ[n, 0] && QuadraticQ[v, x] && Not[QuadraticMatchQ[v, x]]
+Int[Cosh[v_]^n_., x_Symbol] := Int[Cosh[ExpandToSum[v, x]]^n, x] /; IGtQ[n, 0] && QuadraticQ[v, x] && Not[QuadraticMatchQ[v, x]]
+Int[(d_. + e_.*x_)*Sinh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*Cosh[a + b*x + c*x^2]/(2*c) /; FreeQ[{a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]
+Int[(d_. + e_.*x_)*Cosh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*Sinh[a + b*x + c*x^2]/(2*c) /; FreeQ[{a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]
+Int[(d_. + e_.*x_)*Sinh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*Cosh[a + b*x + c*x^2]/(2*c) - (b*e - 2*c*d)/(2*c)*Int[Sinh[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0]
+Int[(d_. + e_.*x_)*Cosh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*Sinh[a + b*x + c*x^2]/(2*c) - (b*e - 2*c*d)/(2*c)*Int[Cosh[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0]
+Int[(d_. + e_.*x_)^m_*Sinh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*(d + e*x)^(m - 1)*Cosh[a + b*x + c*x^2]/(2*c) - e^2*(m - 1)/(2*c)* Int[(d + e*x)^(m - 2)*Cosh[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1] && EqQ[b*e - 2*c*d, 0]
+Int[(d_. + e_.*x_)^m_*Cosh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*(d + e*x)^(m - 1)*Sinh[a + b*x + c*x^2]/(2*c) - e^2*(m - 1)/(2*c)* Int[(d + e*x)^(m - 2)*Sinh[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1] && EqQ[b*e - 2*c*d, 0]
+Int[(d_. + e_.*x_)^m_*Sinh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*(d + e*x)^(m - 1)*Cosh[a + b*x + c*x^2]/(2*c) - (b*e - 2*c*d)/(2*c)* Int[(d + e*x)^(m - 1)*Sinh[a + b*x + c*x^2], x] - e^2*(m - 1)/(2*c)* Int[(d + e*x)^(m - 2)*Cosh[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1] && NeQ[b*e - 2*c*d, 0]
+Int[(d_. + e_.*x_)^m_*Cosh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*(d + e*x)^(m - 1)*Sinh[a + b*x + c*x^2]/(2*c) - (b*e - 2*c*d)/(2*c)* Int[(d + e*x)^(m - 1)*Cosh[a + b*x + c*x^2], x] - e^2*(m - 1)/(2*c)* Int[(d + e*x)^(m - 2)*Sinh[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1] && NeQ[b*e - 2*c*d, 0]
+Int[(d_. + e_.*x_)^m_*Sinh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := (d + e*x)^(m + 1)*Sinh[a + b*x + c*x^2]/(e*(m + 1)) - 2*c/(e^2*(m + 1))* Int[(d + e*x)^(m + 2)*Cosh[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1] && EqQ[b*e - 2*c*d, 0]
+Int[(d_. + e_.*x_)^m_*Cosh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := (d + e*x)^(m + 1)*Cosh[a + b*x + c*x^2]/(e*(m + 1)) - 2*c/(e^2*(m + 1))* Int[(d + e*x)^(m + 2)*Sinh[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1] && EqQ[b*e - 2*c*d, 0]
+Int[(d_. + e_.*x_)^m_*Sinh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := (d + e*x)^(m + 1)*Sinh[a + b*x + c*x^2]/(e*(m + 1)) - (b*e - 2*c*d)/(e^2*(m + 1))* Int[(d + e*x)^(m + 1)*Cosh[a + b*x + c*x^2], x] - 2*c/(e^2*(m + 1))* Int[(d + e*x)^(m + 2)*Cosh[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1] && NeQ[b*e - 2*c*d, 0]
+Int[(d_. + e_.*x_)^m_*Cosh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := (d + e*x)^(m + 1)*Cosh[a + b*x + c*x^2]/(e*(m + 1)) - (b*e - 2*c*d)/(e^2*(m + 1))* Int[(d + e*x)^(m + 1)*Sinh[a + b*x + c*x^2], x] - 2*c/(e^2*(m + 1))* Int[(d + e*x)^(m + 2)*Sinh[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1] && NeQ[b*e - 2*c*d, 0]
+Int[(d_. + e_.*x_)^m_.*Sinh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := Unintegrable[(d + e*x)^m*Sinh[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e, m}, x]
+Int[(d_. + e_.*x_)^m_.*Cosh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := Unintegrable[(d + e*x)^m*Cosh[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e, m}, x]
+Int[(d_. + e_.*x_)^m_.*Sinh[a_. + b_.*x_ + c_.*x_^2]^n_, x_Symbol] := Int[ExpandTrigReduce[(d + e*x)^m, Sinh[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]
+Int[(d_. + e_.*x_)^m_.*Cosh[a_. + b_.*x_ + c_.*x_^2]^n_, x_Symbol] := Int[ExpandTrigReduce[(d + e*x)^m, Cosh[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]
+Int[u_^m_.*Sinh[v_]^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*Sinh[ExpandToSum[v, x]]^n, x] /; FreeQ[m, x] && IGtQ[n, 0] && LinearQ[u, x] && QuadraticQ[v, x] && Not[LinearMatchQ[u, x] && QuadraticMatchQ[v, x]]
+Int[u_^m_.*Cosh[v_]^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*Cosh[ExpandToSum[v, x]]^n, x] /; FreeQ[m, x] && IGtQ[n, 0] && LinearQ[u, x] && QuadraticQ[v, x] && Not[LinearMatchQ[u, x] && QuadraticMatchQ[v, x]]
diff --git a/IntegrationRules/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.10 (c+d x)^m (a+b tanh)^n.m b/IntegrationRules/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.10 (c+d x)^m (a+b tanh)^n.m
new file mode 100755
index 0000000..739e215
--- /dev/null
+++ b/IntegrationRules/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.10 (c+d x)^m (a+b tanh)^n.m	
@@ -0,0 +1,5 @@
+
+(* ::Subsection::Closed:: *)
+(* 6.3.10 (c+d x)^m (a+b tanh)^n *)
+Int[u_^m_.*(a_. + b_.*Tanh[v_])^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*(a + b*Tanh[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
+Int[u_^m_.*(a_. + b_.*Coth[v_])^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*(a + b*Coth[ExpandToSum[v, x]])^n, x] /; FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
diff --git a/IntegrationRules/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.11 (e x)^m (a+b tanh(c+d x^n))^p.m b/IntegrationRules/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.11 (e x)^m (a+b tanh(c+d x^n))^p.m
new file mode 100755
index 0000000..388e7db
--- /dev/null
+++ b/IntegrationRules/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.11 (e x)^m (a+b tanh(c+d x^n))^p.m	
@@ -0,0 +1,23 @@
+
+(* ::Subsection::Closed:: *)
+(* 6.3.11 (e x)^m (a+b tanh(c+d x^n))^p *)
+Int[(a_. + b_.*Tanh[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[Int[x^(1/n - 1)*(a + b*Tanh[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
+Int[(a_. + b_.*Coth[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[Int[x^(1/n - 1)*(a + b*Coth[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
+Int[(a_. + b_.*Tanh[c_. + d_.*x_^n_])^p_., x_Symbol] := Integral[(a + b*Tanh[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
+Int[(a_. + b_.*Coth[c_. + d_.*x_^n_])^p_., x_Symbol] := Integral[(a + b*Coth[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
+Int[(a_. + b_.*Tanh[c_. + d_.*u_^n_])^p_., x_Symbol] := 1/Coefficient[u, x, 1]* Subst[Int[(a + b*Tanh[c + d*x^n])^p, x], x, u] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
+Int[(a_. + b_.*Coth[c_. + d_.*u_^n_])^p_., x_Symbol] := 1/Coefficient[u, x, 1]* Subst[Int[(a + b*Coth[c + d*x^n])^p, x], x, u] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
+Int[(a_. + b_.*Tanh[u_])^p_., x_Symbol] := Int[(a + b*Tanh[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[(a_. + b_.*Coth[u_])^p_., x_Symbol] := Int[(a + b*Coth[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[x_^m_.*(a_. + b_.*Tanh[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Tanh[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
+Int[x_^m_.*(a_. + b_.*Coth[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Coth[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
+Int[x_^m_.*Tanh[c_. + d_.*x_^n_]^2, x_Symbol] := -x^(m - n + 1)*Tanh[c + d*x^n]/(d*n) + Int[x^m, x] + (m - n + 1)/(d*n)*Int[x^(m - n)*Tanh[c + d*x^n], x] /; FreeQ[{c, d, m, n}, x]
+Int[x_^m_.*Coth[c_. + d_.*x_^n_]^2, x_Symbol] := -x^(m - n + 1)*Coth[c + d*x^n]/(d*n) + Int[x^m, x] + (m - n + 1)/(d*n)*Int[x^(m - n)*Coth[c + d*x^n], x] /; FreeQ[{c, d, m, n}, x]
+Int[x_^m_.*(a_. + b_.*Tanh[c_. + d_.*x_^n_])^p_., x_Symbol] := Integral[x^m*(a + b*Tanh[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
+Int[x_^m_.*(a_. + b_.*Coth[c_. + d_.*x_^n_])^p_., x_Symbol] := Integral[x^m*(a + b*Coth[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
+Int[(e_*x_)^m_.*(a_. + b_.*Tanh[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Tanh[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
+Int[(e_*x_)^m_.*(a_. + b_.*Coth[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Coth[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
+Int[(e_*x_)^m_.*(a_. + b_.*Tanh[u_])^p_., x_Symbol] := Int[(e*x)^m*(a + b*Tanh[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[(e_*x_)^m_.*(a_. + b_.*Coth[u_])^p_., x_Symbol] := Int[(e*x)^m*(a + b*Coth[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[x_^m_.*Sech[a_. + b_.*x_^n_.]^p_.*Tanh[a_. + b_.*x_^n_.]^q_., x_Symbol] := -x^(m - n + 1)*Sech[a + b*x^n]^p/(b*n*p) + (m - n + 1)/(b*n*p)*Int[x^(m - n)*Sech[a + b*x^n]^p, x] /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && EqQ[q, 1]
+Int[x_^m_.*Csch[a_. + b_.*x_^n_.]^p_.*Coth[a_. + b_.*x_^n_.]^q_., x_Symbol] := -x^(m - n + 1)*Csch[a + b*x^n]^p/(b*n*p) + (m - n + 1)/(b*n*p)*Int[x^(m - n)*Csch[a + b*x^n]^p, x] /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && EqQ[q, 1]
diff --git a/IntegrationRules/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.12 (d+e x)^m tanh(a+b x+c x^2)^n.m b/IntegrationRules/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.12 (d+e x)^m tanh(a+b x+c x^2)^n.m
new file mode 100755
index 0000000..d3f708f
--- /dev/null
+++ b/IntegrationRules/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.12 (d+e x)^m tanh(a+b x+c x^2)^n.m	
@@ -0,0 +1,11 @@
+
+(* ::Subsection::Closed:: *)
+(* 6.3.12 (d+e x)^m tanh(a+b x+c x^2)^n *)
+Int[Tanh[a_. + b_.*x_ + c_.*x_^2]^n_., x_Symbol] := Integral[Tanh[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, n}, x]
+Int[Coth[a_. + b_.*x_ + c_.*x_^2]^n_., x_Symbol] := Integral[Coth[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, n}, x]
+Int[(d_. + e_.*x_)*Tanh[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*Log[Cosh[a + b*x + c*x^2]]/(2*c) + (2*c*d - b*e)/(2*c)*Int[Tanh[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x]
+Int[(d_. + e_.*x_)*Coth[a_. + b_.*x_ + c_.*x_^2], x_Symbol] := e*Log[Sinh[a + b*x + c*x^2]]/(2*c) + (2*c*d - b*e)/(2*c)*Int[Coth[a + b*x + c*x^2], x] /; FreeQ[{a, b, c, d, e}, x]
+(* Int[x_^m_*Tanh[a_.+b_.*x_+c_.*x_^2],x_Symbol] := x^(m-1)*Log[Cosh[a+b*x+c*x^2]]/(2*c) - b/(2*c)*Int[x^(m-1)*Tanh[a+b*x+c*x^2],x] - (m-1)/(2*c)*Int[x^(m-2)*Log[Cosh[a+b*x+c*x^2]],x] /; FreeQ[{a,b,c},x] && GtQ[m,1] *)
+(* Int[x_^m_*Coth[a_.+b_.*x_+c_.*x_^2],x_Symbol] := x^(m-1)*Log[Sinh[a+b*x+c*x^2]]/(2*c) - b/(2*c)*Int[x^(m-1)*Coth[a+b*x+c*x^2],x] - (m-1)/(2*c)*Int[x^(m-2)*Log[Sinh[a+b*x+c*x^2]],x] /; FreeQ[{a,b,c},x] && GtQ[m,1] *)
+Int[(d_. + e_.*x_)^m_.*Tanh[a_. + b_.*x_ + c_.*x_^2]^n_., x_Symbol] := Integral[(d + e*x)^m*Tanh[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[(d_. + e_.*x_)^m_.*Coth[a_. + b_.*x_ + c_.*x_^2]^n_., x_Symbol] := Integral[(d + e*x)^m*Coth[a + b*x + c*x^2]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
diff --git a/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.10 (c+d x)^m (a+b sech)^n.m b/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.10 (c+d x)^m (a+b sech)^n.m
new file mode 100755
index 0000000..943cae1
--- /dev/null
+++ b/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.10 (c+d x)^m (a+b sech)^n.m	
@@ -0,0 +1,5 @@
+
+(* ::Subsection::Closed:: *)
+(* 6.5.10 (c+d x)^m (a+b sech)^n *)
+Int[u_^m_.*Sech[v_]^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*Sech[ExpandToSum[v, x]]^n, x] /; FreeQ[{m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
+Int[u_^m_.*Csch[v_]^n_., x_Symbol] := Int[ExpandToSum[u, x]^m*Csch[ExpandToSum[v, x]]^n, x] /; FreeQ[{m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
diff --git a/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m b/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m
new file mode 100755
index 0000000..92181cc
--- /dev/null
+++ b/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m	
@@ -0,0 +1,21 @@
+
+(* ::Subsection::Closed:: *)
+(* 6.5.11 (e x)^m (a+b sech(c+d x^n))^p *)
+Int[(a_. + b_.*Sech[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[Int[x^(1/n - 1)*(a + b*Sech[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
+Int[(a_. + b_.*Csch[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[Int[x^(1/n - 1)*(a + b*Csch[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]
+Int[(a_. + b_.*Sech[c_. + d_.*x_^n_])^p_., x_Symbol] := Unintegrable[(a + b*Sech[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
+Int[(a_. + b_.*Csch[c_. + d_.*x_^n_])^p_., x_Symbol] := Unintegrable[(a + b*Csch[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
+Int[(a_. + b_.*Sech[c_. + d_.*u_^n_])^p_., x_Symbol] := 1/Coefficient[u, x, 1]* Subst[Int[(a + b*Sech[c + d*x^n])^p, x], x, u] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
+Int[(a_. + b_.*Csch[c_. + d_.*u_^n_])^p_., x_Symbol] := 1/Coefficient[u, x, 1]* Subst[Int[(a + b*Csch[c + d*x^n])^p, x], x, u] /; FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]
+Int[(a_. + b_.*Sech[u_])^p_., x_Symbol] := Int[(a + b*Sech[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[(a_. + b_.*Csch[u_])^p_., x_Symbol] := Int[(a + b*Csch[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[x_^m_.*(a_. + b_.*Sech[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sech[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
+Int[x_^m_.*(a_. + b_.*Csch[c_. + d_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csch[c + d*x])^p, x], x, x^n] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
+Int[x_^m_.*(a_. + b_.*Sech[c_. + d_.*x_^n_])^p_., x_Symbol] := Unintegrable[x^m*(a + b*Sech[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
+Int[x_^m_.*(a_. + b_.*Csch[c_. + d_.*x_^n_])^p_., x_Symbol] := Unintegrable[x^m*(a + b*Csch[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
+Int[(e_*x_)^m_.*(a_. + b_.*Sech[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Sech[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
+Int[(e_*x_)^m_.*(a_. + b_.*Csch[c_. + d_.*x_^n_])^p_., x_Symbol] := e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Csch[c + d*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
+Int[(e_*x_)^m_.*(a_. + b_.*Sech[u_])^p_., x_Symbol] := Int[(e*x)^m*(a + b*Sech[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[(e_*x_)^m_.*(a_. + b_.*Csch[u_])^p_., x_Symbol] := Int[(e*x)^m*(a + b*Csch[ExpandToSum[u, x]])^p, x] /; FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]
+Int[x_^m_.*Sech[a_. + b_.*x_^n_.]^p_*Sinh[a_. + b_.*x_^n_.], x_Symbol] := -x^(m - n + 1)*Sech[a + b*x^n]^(p - 1)/(b*n*(p - 1)) + (m - n + 1)/(b*n*(p - 1))* Int[x^(m - n)*Sech[a + b*x^n]^(p - 1), x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]
+Int[x_^m_.*Csch[a_. + b_.*x_^n_.]^p_*Cosh[a_. + b_.*x_^n_.], x_Symbol] := -x^(m - n + 1)*Csch[a + b*x^n]^(p - 1)/(b*n*(p - 1)) + (m - n + 1)/(b*n*(p - 1))* Int[x^(m - n)*Csch[a + b*x^n]^(p - 1), x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]
diff --git a/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m b/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m
new file mode 100755
index 0000000..2397d56
--- /dev/null
+++ b/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m	
@@ -0,0 +1,31 @@
+
+(* ::Subsection::Closed:: *)
+(* 6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p *)
+Int[(c_. + d_.*x_)^m_.*Sinh[a_. + b_.*x_]^n_.*Cosh[a_. + b_.*x_], x_Symbol] := (c + d*x)^m*Sinh[a + b*x]^(n + 1)/(b*(n + 1)) - d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^(n + 1), x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(c_. + d_.*x_)^m_.*Sinh[a_. + b_.*x_]*Cosh[a_. + b_.*x_]^n_., x_Symbol] := (c + d*x)^m*Cosh[a + b*x]^(n + 1)/(b*(n + 1)) - d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Cosh[a + b*x]^(n + 1), x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(c_. + d_.*x_)^m_.*Sinh[a_. + b_.*x_]^n_.*Cosh[a_. + b_.*x_]^p_., x_Symbol] := Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(c_. + d_.*x_)^m_.*Sinh[a_. + b_.*x_]^n_.*Tanh[a_. + b_.*x_]^p_., x_Symbol] := Int[(c + d*x)^m*Sinh[a + b*x]^n*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sinh[a + b*x]^(n - 2)*Tanh[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(c_. + d_.*x_)^m_.*Cosh[a_. + b_.*x_]^n_.*Coth[a_. + b_.*x_]^p_., x_Symbol] := Int[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(c_. + d_.*x_)^m_.*Sech[a_. + b_.*x_]^n_.*Tanh[a_. + b_.*x_]^p_., x_Symbol] := -(c + d*x)^m*Sech[a + b*x]^n/(b*n) + d*m/(b*n)*Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*Csch[a_. + b_.*x_]^n_.*Coth[a_. + b_.*x_]^p_., x_Symbol] := -(c + d*x)^m*Csch[a + b*x]^n/(b*n) + d*m/(b*n)*Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*Sech[a_. + b_.*x_]^2*Tanh[a_. + b_.*x_]^n_., x_Symbol] := (c + d*x)^m*Tanh[a + b*x]^(n + 1)/(b*(n + 1)) - d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Tanh[a + b*x]^(n + 1), x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(c_. + d_.*x_)^m_.*Csch[a_. + b_.*x_]^2*Coth[a_. + b_.*x_]^n_., x_Symbol] := -(c + d*x)^m*Coth[a + b*x]^(n + 1)/(b*(n + 1)) + d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Coth[a + b*x]^(n + 1), x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(c_. + d_.*x_)^m_.*Sech[a_. + b_.*x_]*Tanh[a_. + b_.*x_]^p_, x_Symbol] := Int[(c + d*x)^m*Sech[a + b*x]*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sech[a + b*x]^3*Tanh[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 0]
+Int[(c_. + d_.*x_)^m_.*Sech[a_. + b_.*x_]^n_.*Tanh[a_. + b_.*x_]^p_, x_Symbol] := Int[(c + d*x)^m*Sech[a + b*x]^n*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sech[a + b*x]^(n + 2)*Tanh[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p/2, 0]
+Int[(c_. + d_.*x_)^m_.*Csch[a_. + b_.*x_]*Coth[a_. + b_.*x_]^p_, x_Symbol] := Int[(c + d*x)^m*Csch[a + b*x]*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csch[a + b*x]^3*Coth[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 0]
+Int[(c_. + d_.*x_)^m_.*Csch[a_. + b_.*x_]^n_.*Coth[a_. + b_.*x_]^p_, x_Symbol] := Int[(c + d*x)^m*Csch[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csch[a + b*x]^(n + 2)*Coth[a + b*x]^(p - 2), x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p/2, 0]
+Int[(c_. + d_.*x_)^m_.*Sech[a_. + b_.*x_]^n_.*Tanh[a_. + b_.*x_]^p_., x_Symbol] := With[{u = IntHide[Sech[a + b*x]^n*Tanh[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[(c + d*x)^(m - 1)*u, x]] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 0] && (IntegerQ[n/2] || IntegerQ[(p - 1)/2])
+Int[(c_. + d_.*x_)^m_.*Csch[a_. + b_.*x_]^n_.*Coth[a_. + b_.*x_]^p_., x_Symbol] := With[{u = IntHide[Csch[a + b*x]^n*Coth[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[(c + d*x)^(m - 1)*u, x]] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 0] && (IntegerQ[n/2] || IntegerQ[(p - 1)/2])
+Int[(c_. + d_.*x_)^m_.*Csch[a_. + b_.*x_]^n_.*Sech[a_. + b_.*x_]^n_., x_Symbol] := 2^n*Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
+Int[(c_. + d_.*x_)^m_.*Csch[a_. + b_.*x_]^n_.*Sech[a_. + b_.*x_]^p_., x_Symbol] := With[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[(c + d*x)^(m - 1)*u, x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]
+Int[u_^m_.*F_[v_]^n_.*G_[w_]^p_., x_Symbol] := Int[ExpandToSum[u, x]^m*F[ExpandToSum[v, x]]^n* G[ExpandToSum[v, x]]^p, x] /; FreeQ[{m, n, p}, x] && HyperbolicQ[F] && HyperbolicQ[G] && EqQ[v, w] && LinearQ[{u, v, w}, x] && Not[LinearMatchQ[{u, v, w}, x]]
+Int[(e_. + f_.*x_)^m_.* Cosh[c_. + d_.*x_]*(a_ + b_.*Sinh[c_. + d_.*x_])^n_., x_Symbol] := (e + f*x)^m*(a + b*Sinh[c + d*x])^(n + 1)/(b*d*(n + 1)) - f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Sinh[c + d*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(e_. + f_.*x_)^m_.* Sinh[c_. + d_.*x_]*(a_ + b_.*Cosh[c_. + d_.*x_])^n_., x_Symbol] := (e + f*x)^m*(a + b*Cosh[c + d*x])^(n + 1)/(b*d*(n + 1)) - f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Cosh[c + d*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(e_. + f_.*x_)^m_.* Sech[c_. + d_.*x_]^2*(a_ + b_.*Tanh[c_. + d_.*x_])^n_., x_Symbol] := (e + f*x)^m*(a + b*Tanh[c + d*x])^(n + 1)/(b*d*(n + 1)) - f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Tanh[c + d*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(e_. + f_.*x_)^m_.* Csch[c_. + d_.*x_]^2*(a_ + b_.*Coth[c_. + d_.*x_])^n_., x_Symbol] := -(e + f*x)^m*(a + b*Coth[c + d*x])^(n + 1)/(b*d*(n + 1)) + f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Coth[c + d*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(e_. + f_.*x_)^m_.*Sech[c_. + d_.*x_]* Tanh[c_. + d_.*x_]*(a_ + b_.*Sech[c_. + d_.*x_])^n_., x_Symbol] := -(e + f*x)^m*(a + b*Sech[c + d*x])^(n + 1)/(b*d*(n + 1)) + f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Sech[c + d*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(e_. + f_.*x_)^m_.*Csch[c_. + d_.*x_]* Coth[c_. + d_.*x_]*(a_ + b_.*Csch[c_. + d_.*x_])^n_., x_Symbol] := -(e + f*x)^m*(a + b*Csch[c + d*x])^(n + 1)/(b*d*(n + 1)) + f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Csch[c + d*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
+Int[(e_. + f_.*x_)^m_.*Sinh[a_. + b_.*x_]^p_.*Sinh[c_. + d_.*x_]^q_., x_Symbol] := Int[ExpandTrigReduce[(e + f*x)^m, Sinh[a + b*x]^p*Sinh[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IGtQ[q, 0] && IntegerQ[m]
+Int[(e_. + f_.*x_)^m_.*Cosh[a_. + b_.*x_]^p_.*Cosh[c_. + d_.*x_]^q_., x_Symbol] := Int[ExpandTrigReduce[(e + f*x)^m, Cosh[a + b*x]^p*Cosh[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IGtQ[q, 0] && IntegerQ[m]
+Int[(e_. + f_.*x_)^m_.*Sinh[a_. + b_.*x_]^p_.*Cosh[c_. + d_.*x_]^q_., x_Symbol] := Int[ExpandTrigReduce[(e + f*x)^m, Sinh[a + b*x]^p*Cosh[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IGtQ[q, 0]
+Int[(e_. + f_.*x_)^m_.*F_[a_. + b_.*x_]^p_.*G_[c_. + d_.*x_]^q_., x_Symbol] := Int[ExpandTrigExpand[(e + f*x)^m*G[c + d*x]^q, F, c + d*x, p, b/d, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && MemberQ[{Sinh, Cosh}, F] && MemberQ[{Sech, Csch}, G] && IGtQ[p, 0] && IGtQ[q, 0] && EqQ[b*c - a*d, 0] && IGtQ[b/d, 1]
diff --git a/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m b/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m
new file mode 100755
index 0000000..0ffe7ec
--- /dev/null
+++ b/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m	
@@ -0,0 +1,51 @@
+
+(* ::Subsection::Closed:: *)
+(* 6.7.7 F^(c (a+b x)) hyper(d+e x)^n *)
+Int[F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_], x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Sinh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2) + e*F^(c*(a + b*x))*Cosh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 - b^2*c^2*Log[F]^2, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*Cosh[d_. + e_.*x_], x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Cosh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2) + e*F^(c*(a + b*x))*Sinh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 - b^2*c^2*Log[F]^2, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Sinh[d + e*x]^n/(e^2*n^2 - b^2*c^2*Log[F]^2) + e*n*F^(c*(a + b*x))*Cosh[d + e*x]* Sinh[d + e*x]^(n - 1)/(e^2*n^2 - b^2*c^2*Log[F]^2) - n*(n - 1)*e^2/(e^2*n^2 - b^2*c^2*Log[F]^2)* Int[F^(c*(a + b*x))*Sinh[d + e*x]^(n - 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 - b^2*c^2*Log[F]^2, 0] && GtQ[n, 1]
+Int[F_^(c_.*(a_. + b_.*x_))*Cosh[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Cosh[d + e*x]^n/(e^2*n^2 - b^2*c^2*Log[F]^2) + e*n*F^(c*(a + b*x))*Sinh[d + e*x]* Cosh[d + e*x]^(n - 1)/(e^2*n^2 - b^2*c^2*Log[F]^2) + n*(n - 1)*e^2/(e^2*n^2 - b^2*c^2*Log[F]^2)* Int[F^(c*(a + b*x))*Cosh[d + e*x]^(n - 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 - b^2*c^2*Log[F]^2, 0] && GtQ[n, 1]
+Int[F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Sinh[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) + F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x]^(n + 1)/(e*(n + 1)) /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n + 2)^2 - b^2*c^2*Log[F]^2, 0] && NeQ[n, -1] && NeQ[n, -2]
+Int[F_^(c_.*(a_. + b_.*x_))*Cosh[d_. + e_.*x_]^n_, x_Symbol] := b*c*Log[F]*F^(c*(a + b*x))* Cosh[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) - F^(c*(a + b*x))*Sinh[d + e*x]*Cosh[d + e*x]^(n + 1)/(e*(n + 1)) /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n + 2)^2 - b^2*c^2*Log[F]^2, 0] && NeQ[n, -1] && NeQ[n, -2]
+Int[F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Sinh[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) + F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x]^(n + 1)/(e*(n + 1)) - (e^2*(n + 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n + 1)*(n + 2))* Int[F^(c*(a + b*x))*Sinh[d + e*x]^(n + 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n + 2)^2 - b^2*c^2*Log[F]^2, 0] && LtQ[n, -1] && NeQ[n, -2]
+Int[F_^(c_.*(a_. + b_.*x_))*Cosh[d_. + e_.*x_]^n_, x_Symbol] := b*c*Log[F]*F^(c*(a + b*x))* Cosh[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) - F^(c*(a + b*x))*Sinh[d + e*x]*Cosh[d + e*x]^(n + 1)/(e*(n + 1)) + (e^2*(n + 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n + 1)*(n + 2))* Int[F^(c*(a + b*x))*Cosh[d + e*x]^(n + 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n + 2)^2 - b^2*c^2*Log[F]^2, 0] && LtQ[n, -1] && NeQ[n, -2]
+Int[F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_]^n_, x_Symbol] := E^(n*(d + e*x))*Sinh[d + e*x]^n/(-1 + E^(2*(d + e*x)))^n* Int[F^(c*(a + b*x))*(-1 + E^(2*(d + e*x)))^n/E^(n*(d + e*x)), x] /; FreeQ[{F, a, b, c, d, e, n}, x] && Not[IntegerQ[n]]
+Int[F_^(c_.*(a_. + b_.*x_))*Cosh[d_. + e_.*x_]^n_, x_Symbol] := E^(n*(d + e*x))*Cosh[d + e*x]^n/(1 + E^(2*(d + e*x)))^n* Int[F^(c*(a + b*x))*(1 + E^(2*(d + e*x)))^n/E^(n*(d + e*x)), x] /; FreeQ[{F, a, b, c, d, e, n}, x] && Not[IntegerQ[n]]
+Int[F_^(c_.*(a_. + b_.*x_))*Tanh[d_. + e_.*x_]^n_., x_Symbol] := Int[ExpandIntegrand[ F^(c*(a + b*x))*(-1 + E^(2*(d + e*x)))^n/(1 + E^(2*(d + e*x)))^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
+Int[F_^(c_.*(a_. + b_.*x_))*Coth[d_. + e_.*x_]^n_., x_Symbol] := Int[ExpandIntegrand[ F^(c*(a + b*x))*(1 + E^(2*(d + e*x)))^n/(-1 + E^(2*(d + e*x)))^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
+Int[F_^(c_.*(a_. + b_.*x_))*Sech[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]* F^(c*(a + b*x))*(Sech[d + e*x]^n/(e^2*n^2 - b^2*c^2*Log[F]^2)) - e*n*F^(c*(a + b*x))* Sech[d + e*x]^(n + 1)*(Sinh[ d + e*x]/(e^2*n^2 - b^2*c^2*Log[F]^2)) + e^2*n*((n + 1)/(e^2*n^2 - b^2*c^2*Log[F]^2))* Int[F^(c*(a + b*x))*Sech[d + e*x]^(n + 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1]
+Int[F_^(c_.*(a_. + b_.*x_))*Csch[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]* F^(c*(a + b*x))*(Csch[d + e*x]^n/(e^2*n^2 - b^2*c^2*Log[F]^2)) - e*n*F^(c*(a + b*x))* Csch[d + e*x]^(n + 1)*(Cosh[ d + e*x]/(e^2*n^2 - b^2*c^2*Log[F]^2)) - e^2*n*((n + 1)/(e^2*n^2 - b^2*c^2*Log[F]^2))* Int[F^(c*(a + b*x))*Csch[d + e*x]^(n + 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1]
+Int[F_^(c_.*(a_. + b_.*x_))*Sech[d_. + e_.*x_]^n_, x_Symbol] := b*c*Log[F]*F^(c*(a + b*x))* Sech[d + e*x]^(n - 2)/(e^2*(n - 1)*(n - 2)) + F^(c*(a + b*x))*Sech[d + e*x]^(n - 1)*Sinh[d + e*x]/(e*(n - 1)) /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] && NeQ[n, 1] && NeQ[n, 2]
+Int[F_^(c_.*(a_. + b_.*x_))*Csch[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Csch[d + e*x]^(n - 2)/(e^2*(n - 1)*(n - 2)) - F^(c*(a + b*x))*Csch[d + e*x]^(n - 1)*Cosh[d + e*x]/(e*(n - 1)) /; FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] && NeQ[n, 1] && NeQ[n, 2]
+Int[F_^(c_.*(a_. + b_.*x_))*Sech[d_. + e_.*x_]^n_, x_Symbol] := b*c*Log[F]*F^(c*(a + b*x))* Sech[d + e*x]^(n - 2)/(e^2*(n - 1)*(n - 2)) + F^(c*(a + b*x))*Sech[d + e*x]^(n - 1)* Sinh[d + e*x]/(e*(n - 1)) + (e^2*(n - 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n - 1)*(n - 2))* Int[F^(c*(a + b*x))*Sech[d + e*x]^(n - 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] && GtQ[n, 1] && NeQ[n, 2]
+Int[F_^(c_.*(a_. + b_.*x_))*Csch[d_. + e_.*x_]^n_, x_Symbol] := -b*c*Log[F]*F^(c*(a + b*x))* Csch[d + e*x]^(n - 2)/(e^2*(n - 1)*(n - 2)) - F^(c*(a + b*x))*Csch[d + e*x]^(n - 1)* Cosh[d + e*x]/(e*(n - 1)) - (e^2*(n - 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n - 1)*(n - 2))* Int[F^(c*(a + b*x))*Csch[d + e*x]^(n - 2), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] && GtQ[n, 1] && NeQ[n, 2]
+(* Int[F_^(c_.*(a_.+b_.*x_))*Sech[d_.+e_.*x_]^n_.,x_Symbol] := 2^n*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(n*(d+e*x))/(1+E^(2*(d+e* x)))^n,x],x] /; FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] *)
+(* Int[F_^(c_.*(a_.+b_.*x_))*Csch[d_.+e_.*x_]^n_.,x_Symbol] := 2^n*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(-n*(d+e*x))/(1-E^(-2*(d+ e*x)))^n,x],x] /; FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] *)
+Int[F_^(c_.*(a_. + b_.*x_))*Sech[d_. + e_.*x_]^n_., x_Symbol] := 2^n*E^(n*(d + e*x))*F^(c*(a + b*x))/(e*n + b*c*Log[F])* Hypergeometric2F1[n, n/2 + b*c*Log[F]/(2*e), 1 + n/2 + b*c*Log[F]/(2*e), -E^(2*(d + e*x))] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
+Int[F_^(c_.*(a_. + b_.*x_))*Csch[d_. + e_.*x_]^n_., x_Symbol] := (-2)^n*E^(n*(d + e*x))*F^(c*(a + b*x))/(e*n + b*c*Log[F])* Hypergeometric2F1[n, n/2 + b*c*Log[F]/(2*e), 1 + n/2 + b*c*Log[F]/(2*e), E^(2*(d + e*x))] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
+Int[F_^(c_.*(a_. + b_.*x_))*Sech[d_. + e_.*x_]^n_., x_Symbol] := (1 + E^(2*(d + e*x)))^n*Sech[d + e*x]^n/E^(n*(d + e*x))* Int[SimplifyIntegrand[ F^(c*(a + b*x))*E^(n*(d + e*x))/(1 + E^(2*(d + e*x)))^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && Not[IntegerQ[n]]
+Int[F_^(c_.*(a_. + b_.*x_))*Csch[d_. + e_.*x_]^n_., x_Symbol] := (1 - E^(-2*(d + e*x)))^n*Csch[d + e*x]^n/E^(-n*(d + e*x))* Int[SimplifyIntegrand[ F^(c*(a + b*x))*E^(-n*(d + e*x))/(1 - E^(-2*(d + e*x)))^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && Not[IntegerQ[n]]
+Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Sinh[d_. + e_.*x_])^n_., x_Symbol] := 2^n*f^n* Int[F^(c*(a + b*x))*Cosh[d/2 - f*Pi/(4*g) + e*x/2]^(2*n), x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f^2 + g^2, 0] && ILtQ[n, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Cosh[d_. + e_.*x_])^n_., x_Symbol] := 2^n*g^n*Int[F^(c*(a + b*x))*Cosh[d/2 + e*x/2]^(2*n), x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f - g, 0] && ILtQ[n, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Cosh[d_. + e_.*x_])^n_., x_Symbol] := 2^n*g^n*Int[F^(c*(a + b*x))*Sinh[d/2 + e*x/2]^(2*n), x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f + g, 0] && ILtQ[n, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Sinh[d_. + e_.*x_])^n_., x_Symbol] := (f + g*Sinh[d + e*x])^n/Cosh[d/2 - f*Pi/(4*g) + e*x/2]^(2*n)* Int[F^(c*(a + b*x))*Cosh[d/2 - f*Pi/(4*g) + e*x/2]^(2*n), x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && EqQ[f^2 + g^2, 0] && Not[IntegerQ[n]]
+Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Cosh[d_. + e_.*x_])^n_., x_Symbol] := (f + g*Cosh[d + e*x])^n/Cosh[d/2 + e*x/2]^(2*n)* Int[F^(c*(a + b*x))*Cosh[d/2 + e*x/2]^(2*n), x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && EqQ[f - g, 0] && Not[IntegerQ[n]]
+Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Cosh[d_. + e_.*x_])^n_., x_Symbol] := (f + g*Cosh[d + e*x])^n/Sinh[d/2 + e*x/2]^(2*n)* Int[F^(c*(a + b*x))*Sinh[d/2 + e*x/2]^(2*n), x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && EqQ[f + g, 0] && Not[IntegerQ[n]]
+Int[F_^(c_.*(a_. + b_.*x_))* Cosh[d_. + e_.*x_]^m_.*(f_ + g_.*Sinh[d_. + e_.*x_])^n_., x_Symbol] := g^n*Int[F^(c*(a + b*x))*Tanh[d/2 + e*x/2 - f*Pi/(4*g)]^m, x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f^2 + g^2, 0] && IntegersQ[m, n] && EqQ[m + n, 0]
+Int[F_^(c_.*(a_. + b_.*x_))* Sinh[d_. + e_.*x_]^m_.*(f_ + g_.*Cosh[d_. + e_.*x_])^n_., x_Symbol] := g^n*Int[F^(c*(a + b*x))*Tanh[d/2 + e*x/2]^m, x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f - g, 0] && IntegersQ[m, n] && EqQ[m + n, 0]
+Int[F_^(c_.*(a_. + b_.*x_))* Sinh[d_. + e_.*x_]^m_.*(f_ + g_.*Cosh[d_. + e_.*x_])^n_., x_Symbol] := g^n*Int[F^(c*(a + b*x))*Coth[d/2 + e*x/2]^m, x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f + g, 0] && IntegersQ[m, n] && EqQ[m + n, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*(h_ + i_.*Cosh[d_. + e_.*x_])/(f_ + g_.*Sinh[d_. + e_.*x_]), x_Symbol] := 2*i*Int[F^(c*(a + b*x))*(Cosh[d + e*x]/(f + g*Sinh[d + e*x])), x] + Int[ F^(c*(a + b*x))*((h - i*Cosh[d + e*x])/(f + g*Sinh[d + e*x])), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, i}, x] && EqQ[f^2 + g^2, 0] && EqQ[h^2 - i^2, 0] && EqQ[g*h - f*i, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*(h_ + i_.*Sinh[d_. + e_.*x_])/(f_ + g_.*Cosh[d_. + e_.*x_]), x_Symbol] := 2*i*Int[F^(c*(a + b*x))*(Sinh[d + e*x]/(f + g*Cosh[d + e*x])), x] + Int[ F^(c*(a + b*x))*((h - i*Sinh[d + e*x])/(f + g*Cosh[d + e*x])), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, i}, x] && EqQ[f^2 - g^2, 0] && EqQ[h^2 + i^2, 0] && EqQ[g*h + f*i, 0]
+Int[F_^(c_.*u_)*G_[v_]^n_., x_Symbol] := Int[F^(c*ExpandToSum[u, x])*G[ExpandToSum[v, x]]^n, x] /; FreeQ[{F, c, n}, x] && HyperbolicQ[G] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]
+Int[(f_.*x_)^m_.*F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_]^n_., x_Symbol] := Module[{u = IntHide[F^(c*(a + b*x))*Sinh[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - f*m*Int[(f*x)^(m - 1)*u, x]] /; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]
+Int[(f_.*x_)^m_.*F_^(c_.*(a_. + b_.*x_))*Cosh[d_. + e_.*x_]^n_., x_Symbol] := Module[{u = IntHide[F^(c*(a + b*x))*Cosh[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - f*m*Int[(f*x)^(m - 1)*u, x]] /; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]
+Int[(f_.*x_)^m_*F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_], x_Symbol] := (f*x)^(m + 1)/(f*(m + 1))*F^(c*(a + b*x))*Sinh[d + e*x] - e/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Cosh[d + e*x], x] - b*c*Log[F]/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Sinh[d + e*x], x] /; FreeQ[{F, a, b, c, d, e, f, m}, x] && (LtQ[m, -1] || SumSimplerQ[m, 1])
+Int[(f_.*x_)^m_*F_^(c_.*(a_. + b_.*x_))*Cosh[d_. + e_.*x_], x_Symbol] := (f*x)^(m + 1)/(f*(m + 1))*F^(c*(a + b*x))*Cosh[d + e*x] - e/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Sinh[d + e*x], x] - b*c*Log[F]/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Cosh[d + e*x], x] /; FreeQ[{F, a, b, c, d, e, f, m}, x] && (LtQ[m, -1] || SumSimplerQ[m, 1])
+(* Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Sinh[d_.+e_.*x_]^n_.,x_ Symbol] := (-1)^n/2^n*Int[ExpandIntegrand[(f*x)^m*F^(c*(a+b*x)),(E^(-(d+e*x))- E^(d+e*x))^n,x],x] /; FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] *)
+(* Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Cosh[d_.+e_.*x_]^n_.,x_ Symbol] := 1/2^n*Int[ExpandIntegrand[(f*x)^m*F^(c*(a+b*x)),(E^(-(d+e*x))+E^(d+ e*x))^n,x],x] /; FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] *)
+Int[F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_]^m_.* Cosh[f_. + g_.*x_]^n_., x_Symbol] := Int[ExpandTrigReduce[F^(c*(a + b*x)), Sinh[d + e*x]^m*Cosh[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[x_^p_.*F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_]^m_.* Cosh[f_. + g_.*x_]^n_., x_Symbol] := Int[ExpandTrigReduce[x^p*F^(c*(a + b*x)), Sinh[d + e*x]^m*Cosh[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[F_^(c_.*(a_. + b_.*x_))*G_[d_. + e_.*x_]^m_.*H_[d_. + e_.*x_]^n_., x_Symbol] := Int[ExpandTrigToExp[F^(c*(a + b*x)), G[d + e*x]^m*H[d + e*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IGtQ[m, 0] && IGtQ[n, 0] && HyperbolicQ[G] && HyperbolicQ[H]
+Int[F_^u_*Sinh[v_]^n_., x_Symbol] := Int[ExpandTrigToExp[F^u, Sinh[v]^n, x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]
+Int[F_^u_*Cosh[v_]^n_., x_Symbol] := Int[ExpandTrigToExp[F^u, Cosh[v]^n, x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]
+Int[F_^u_*Sinh[v_]^m_.*Cosh[v_]^n_., x_Symbol] := Int[ExpandTrigToExp[F^u, Sinh[v]^m*Cosh[v]^n, x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[m, 0] && IGtQ[n, 0]
diff --git a/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m b/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m
new file mode 100755
index 0000000..388604d
--- /dev/null
+++ b/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m	
@@ -0,0 +1,57 @@
+
+(* ::Subsection::Closed:: *)
+(* 6.7.8 u hyper(a+b log(c x^n))^p *)
+Int[Sinh[b_.*Log[c_.*x_^n_.]]^p_., x_Symbol] := Int[((c*x^n)^b/2 - 1/(2*(c*x^n)^b))^p, x] /; FreeQ[c, x] && RationalQ[b, n, p]
+Int[Cosh[b_.*Log[c_.*x_^n_.]]^p_., x_Symbol] := Int[((c*x^n)^b/2 + 1/(2*(c*x^n)^b))^p, x] /; FreeQ[c, x] && RationalQ[b, n, p]
+Int[Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := -x*Sinh[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 - 1) + b*d*n*x*Cosh[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 - 1) /; FreeQ[{a, b, c, d, n}, x] && NeQ[b^2*d^2*n^2 - 1, 0]
+Int[Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := -x*Cosh[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 - 1) + b*d*n*x*Sinh[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 - 1) /; FreeQ[{a, b, c, d, n}, x] && NeQ[b^2*d^2*n^2 - 1, 0]
+Int[Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol] := -x*Sinh[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*n^2*p^2 - 1) + b*d*n*p*x*Cosh[d*(a + b*Log[c*x^n])]* Sinh[d*(a + b*Log[c*x^n])]^(p - 1)/(b^2*d^2*n^2*p^2 - 1) - b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 - 1)* Int[Sinh[d*(a + b*Log[c*x^n])]^(p - 2), x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - 1, 0]
+Int[Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol] := -x*Cosh[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*n^2*p^2 - 1) + b*d*n*p*x*Cosh[d*(a + b*Log[c*x^n])]^(p - 1)* Sinh[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2*p^2 - 1) + b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 - 1)* Int[Cosh[d*(a + b*Log[c*x^n])]^(p - 2), x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - 1, 0]
+Int[Sinh[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := 1/(2^p*b^p*d^p*p^p)* Int[ExpandIntegrand[(-E^(-a*b*d^2*p)*x^(-1/p) + E^(a*b*d^2*p)*x^(1/p))^p, x], x] /; FreeQ[{a, b, d}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 - 1, 0]
+Int[Cosh[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := 1/2^p* Int[ExpandIntegrand[(E^(-a*b*d^2*p)*x^(-1/p) + E^(a*b*d^2*p)*x^(1/p))^p, x], x] /; FreeQ[{a, b, d}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 - 1, 0]
+(* Int[Sinh[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol] := E^(a*d*p)/2^p*Int[x^(b*d*p)*(1-1/(E^(2*a*d)*x^(2*b*d)))^p,x] /; FreeQ[{a,b,d},x] && IntegerQ[p] *)
+(* Int[Cosh[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol] := E^(a*d*p)/2^p*Int[x^(b*d*p)*(1+1/(E^(2*a*d)*x^(2*b*d)))^p,x] /; FreeQ[{a,b,d},x] && IntegerQ[p] *)
+Int[Sinh[d_.*(a_. + b_.*Log[x_])]^p_, x_Symbol] := Sinh[d*(a + b*Log[x])]^ p/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p)* Int[x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p, x] /; FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]
+Int[Cosh[d_.*(a_. + b_.*Log[x_])]^p_, x_Symbol] := Cosh[d*(a + b*Log[x])]^ p/(x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p)* Int[x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p, x] /; FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]
+Int[Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Sinh[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Cosh[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[(e_.*x_)^m_.*Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := -(m + 1)*(e*x)^(m + 1)* Sinh[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 - e*(m + 1)^2) + b*d*n*(e*x)^(m + 1)* Cosh[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 - e*(m + 1)^2) /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 - (m + 1)^2, 0]
+Int[(e_.*x_)^m_.*Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := -(m + 1)*(e*x)^(m + 1)* Cosh[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 - e*(m + 1)^2) + b*d*n*(e*x)^(m + 1)* Sinh[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 - e*(m + 1)^2) /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 - (m + 1)^2, 0]
+Int[(e_.*x_)^m_.*Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol] := -(m + 1)*(e*x)^(m + 1)* Sinh[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*e*n^2*p^2 - e*(m + 1)^2) + b*d*n*p*(e*x)^(m + 1)*Cosh[d*(a + b*Log[c*x^n])]* Sinh[d*(a + b*Log[c*x^n])]^(p - 1)/(b^2*d^2*e*n^2*p^2 - e*(m + 1)^2) - b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 - (m + 1)^2)* Int[(e*x)^m*Sinh[d*(a + b*Log[c*x^n])]^(p - 2), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - (m + 1)^2, 0]
+Int[(e_.*x_)^m_.*Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol] := -(m + 1)*(e*x)^(m + 1)* Cosh[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*e*n^2*p^2 - e*(m + 1)^2) + b*d*n*p*(e*x)^(m + 1)*Sinh[d*(a + b*Log[c*x^n])]* Cosh[d*(a + b*Log[c*x^n])]^(p - 1)/(b^2*d^2*e*n^2*p^2 - e*(m + 1)^2) + b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 - (m + 1)^2)* Int[(e*x)^m*Cosh[d*(a + b*Log[c*x^n])]^(p - 2), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - (m + 1)^2, 0]
+Int[(e_.*x_)^m_.*Sinh[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := (m + 1)^p/(2^p*b^p*d^p*p^p)* Int[ ExpandIntegrand[(e*x)^ m*(-E^(-a*b*d^2*p/(m + 1))*x^(-(m + 1)/p) + E^(a*b*d^2*p/(m + 1))*x^((m + 1)/p))^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 - (m + 1)^2, 0]
+Int[(e_.*x_)^m_.*Cosh[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := 1/2^p* Int[ExpandIntegrand[(e*x)^ m*(E^(-a*b*d^2*p/(m + 1))*x^(-(m + 1)/p) + E^(a*b*d^2*p/(m + 1))*x^((m + 1)/p))^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 - (m + 1)^2, 0]
+(* Int[(e_.*x_)^m_.*Sinh[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol] := E^(a*d*p)/2^p*Int[(e*x)^m*x^(b*d*p)*(1-1/(E^(2*a*d)*x^(2*b*d)))^p,x]  /; FreeQ[{a,b,d,e,m},x] && IntegerQ[p] *)
+(* Int[(e_.*x_)^m_.*Cosh[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol] := E^(a*d*p)/2^p*Int[(e*x)^m*x^(b*d*p)*(1+1/(E^(2*a*d)*x^(2*b*d)))^p,x]  /; FreeQ[{a,b,d,e,m},x] && IntegerQ[p] *)
+Int[(e_.*x_)^m_.*Sinh[d_.*(a_. + b_.*Log[x_])]^p_, x_Symbol] := Sinh[d*(a + b*Log[x])]^ p/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p)* Int[(e*x)^m*x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p, x] /; FreeQ[{a, b, d, e, m, p}, x] && Not[IntegerQ[p]]
+Int[(e_.*x_)^m_.*Cosh[d_.*(a_. + b_.*Log[x_])]^p_, x_Symbol] := Cosh[d*(a + b*Log[x])]^ p/(x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p)* Int[(e*x)^m*x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p, x] /; FreeQ[{a, b, d, e, m, p}, x] && Not[IntegerQ[p]]
+Int[(e_.*x_)^m_.*Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := (e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Sinh[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[(e_.*x_)^m_.*Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := (e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Cosh[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[(h_.*(e_. + f_.*Log[g_.*x_^m_.]))^q_.* Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := -E^(-a*d)*(c*x^n)^(-b*d)/(2*x^(-b*d*n))* Int[x^(-b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] + E^(a*d)*(c*x^n)^(b*d)/(2*x^(b*d*n))* Int[x^(b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, q}, x]
+Int[(h_.*(e_. + f_.*Log[g_.*x_^m_.]))^q_.* Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := E^(-a*d)*(c*x^n)^(-b*d)/(2*x^(-b*d*n))* Int[x^(-b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] + E^(a*d)*(c*x^n)^(b*d)/(2*x^(b*d*n))* Int[x^(b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, q}, x]
+Int[(i_.*x_)^r_.*(h_.*(e_. + f_.*Log[g_.*x_^m_.]))^q_.* Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := -E^(-a*d)*(i*x)^r*(c*x^n)^(-b*d)/(2*x^(r - b*d*n))* Int[x^(r - b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] + E^(a*d)*(i*x)^r*(c*x^n)^(b*d)/(2*x^(r + b*d*n))* Int[x^(r + b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, i, m, n, q, r}, x]
+Int[(i_.*x_)^r_.*(h_.*(e_. + f_.*Log[g_.*x_^m_.]))^q_.* Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := E^(-a*d)*(i*x)^r*(c*x^n)^(-b*d)/(2*x^(r - b*d*n))* Int[x^(r - b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] + E^(a*d)*(i*x)^r*(c*x^n)^(b*d)/(2*x^(r + b*d*n))* Int[x^(r + b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, i, m, n, q, r}, x]
+Int[Tanh[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Int[(-1 + E^(2*a*d)*x^(2*b*d))^p/(1 + E^(2*a*d)*x^(2*b*d))^p, x] /; FreeQ[{a, b, d, p}, x]
+Int[Coth[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Int[(-1 - E^(2*a*d)*x^(2*b*d))^p/(1 - E^(2*a*d)*x^(2*b*d))^p, x] /; FreeQ[{a, b, d, p}, x]
+Int[Tanh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Tanh[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[Coth[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Coth[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[(e_.*x_)^m_.*Tanh[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Int[(e*x)^ m*(-1 + E^(2*a*d)*x^(2*b*d))^p/(1 + E^(2*a*d)*x^(2*b*d))^p, x] /; FreeQ[{a, b, d, e, m, p}, x]
+Int[(e_.*x_)^m_.*Coth[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Int[(e*x)^ m*(-1 - E^(2*a*d)*x^(2*b*d))^p/(1 - E^(2*a*d)*x^(2*b*d))^p, x] /; FreeQ[{a, b, d, e, m, p}, x]
+Int[(e_.*x_)^m_.*Tanh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := (e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Tanh[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[(e_.*x_)^m_.*Coth[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := (e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Coth[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[Sech[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := 2^p*E^(-a*d*p)*Int[x^(-b*d*p)/(1 + E^(-2*a*d)*x^(-2*b*d))^p, x] /; FreeQ[{a, b, d}, x] && IntegerQ[p]
+Int[Csch[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := 2^p*E^(-a*d*p)*Int[x^(-b*d*p)/(1 - E^(-2*a*d)*x^(-2*b*d))^p, x] /; FreeQ[{a, b, d}, x] && IntegerQ[p]
+Int[Sech[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Sech[d*(a + b*Log[x])]^p*(1 + E^(-2*a*d)*x^(-2*b*d))^p/x^(-b*d*p)* Int[x^(-b*d*p)/(1 + E^(-2*a*d)*x^(-2*b*d))^p, x] /; FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]
+Int[Csch[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Csch[d*(a + b*Log[x])]^p*(1 - E^(-2*a*d)*x^(-2*b*d))^p/x^(-b*d*p)* Int[x^(-b*d*p)/(1 - E^(-2*a*d)*x^(-2*b*d))^p, x] /; FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]
+Int[Sech[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Sech[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[Csch[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Csch[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[(e_.*x_)^m_.*Sech[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := 2^p*E^(-a*d*p)* Int[(e*x)^m*x^(-b*d*p)/(1 + E^(-2*a*d)*x^(-2*b*d))^p, x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]
+Int[(e_.*x_)^m_.*Csch[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := 2^p*E^(-a*d*p)* Int[(e*x)^m*x^(-b*d*p)/(1 - E^(-2*a*d)*x^(-2*b*d))^p, x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]
+Int[(e_.*x_)^m_.*Sech[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Sech[d*(a + b*Log[x])]^p*(1 + E^(-2*a*d)*x^(-2*b*d))^p/x^(-b*d*p)* Int[(e*x)^m*x^(-b*d*p)/(1 + E^(-2*a*d)*x^(-2*b*d))^p, x] /; FreeQ[{a, b, d, e, m, p}, x] && Not[IntegerQ[p]]
+Int[(e_.*x_)^m_.*Csch[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol] := Csch[d*(a + b*Log[x])]^p*(1 - E^(-2*a*d)*x^(-2*b*d))^p/x^(-b*d*p)* Int[(e*x)^m*x^(-b*d*p)/(1 - E^(-2*a*d)*x^(-2*b*d))^p, x] /; FreeQ[{a, b, d, e, m, p}, x] && Not[IntegerQ[p]]
+Int[(e_.*x_)^m_.*Sech[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := (e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Sech[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[(e_.*x_)^m_.*Csch[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol] := (e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Csch[d*(a + b*Log[x])]^p, x], x, c*x^n] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
+Int[Sinh[a_.*x_*Log[b_.*x_]]*Log[b_.*x_], x_Symbol] := Cosh[a*x*Log[b*x]]/a - Int[Sinh[a*x*Log[b*x]], x] /; FreeQ[{a, b}, x]
+Int[Cosh[a_.*x_*Log[b_.*x_]]*Log[b_.*x_], x_Symbol] := Sinh[a*x*Log[b*x]]/a - Int[Cosh[a*x*Log[b*x]], x] /; FreeQ[{a, b}, x]
+Int[x_^m_.*Sinh[a_.*x_^n_.*Log[b_.*x_]]*Log[b_.*x_], x_Symbol] := Cosh[a*x^n*Log[b*x]]/(a*n) - 1/n*Int[x^m*Sinh[a*x^n*Log[b*x]], x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
+Int[x_^m_.*Cosh[a_.*x_^n_.*Log[b_.*x_]]*Log[b_.*x_], x_Symbol] := Sinh[a*x^n*Log[b*x]]/(a*n) - 1/n*Int[x^m*Cosh[a*x^n*Log[b*x]], x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
diff --git a/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m b/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m
new file mode 100755
index 0000000..bf86294
--- /dev/null
+++ b/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m	
@@ -0,0 +1,99 @@
+
+(* ::Subsection::Closed:: *)
+(* 6.7.9 Active hyperbolic functions *)
+Int[(e_. + f_.*x_)^m_.* Sinh[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Sinh[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Sinh[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Cosh[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Cosh[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Cosh[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Cosh[c_. + d_.*x_]/(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := -(e + f*x)^(m + 1)/(b*f*(m + 1)) + 2*Int[(e + f*x)^m*E^(c + d*x)/(a + b*E^(c + d*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[a^2 + b^2, 0]
+Int[(e_. + f_.*x_)^m_.* Sinh[c_. + d_.*x_]/(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := -(e + f*x)^(m + 1)/(b*f*(m + 1)) + 2*Int[(e + f*x)^m*E^(c + d*x)/(a + b*E^(c + d*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]
+Int[(e_. + f_.*x_)^m_.* Cosh[c_. + d_.*x_]/(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := -(e + f*x)^(m + 1)/(b*f*(m + 1)) + Int[(e + f*x)^m* E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x] + Int[(e + f*x)^m* E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
+Int[(e_. + f_.*x_)^m_.* Sinh[c_. + d_.*x_]/(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := -(e + f*x)^(m + 1)/(b*f*(m + 1)) + Int[(e + f*x)^m* E^(c + d*x)/(a - Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x] + Int[(e + f*x)^m* E^(c + d*x)/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]
+Int[(e_. + f_.*x_)^m_.* Cosh[c_. + d_.*x_]^n_/(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x] + 1/b*Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2)*Sinh[c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 + b^2, 0]
+Int[(e_. + f_.*x_)^m_.* Sinh[c_. + d_.*x_]^n_/(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := -1/a*Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2), x] + 1/b*Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2)*Cosh[c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 - b^2, 0]
+Int[(e_. + f_.*x_)^m_.* Cosh[c_. + d_.*x_]^n_/(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := -a/b^2*Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x] + 1/b*Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2)*Sinh[c + d*x], x] + (a^2 + b^2)/b^2* Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.* Sinh[c_. + d_.*x_]^n_/(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := -a/b^2*Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2), x] + 1/b*Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2)*Cosh[c + d*x], x] + (a^2 - b^2)/b^2* Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2)/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.* Tanh[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Sech[c + d*x]*Tanh[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Sech[c + d*x]* Tanh[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Coth[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Csch[c + d*x]*Coth[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Csch[c + d*x]* Coth[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Coth[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Coth[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Cosh[c + d*x]* Coth[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Tanh[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Tanh[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Sinh[c + d*x]* Tanh[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Sech[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Sech[c + d*x]^(n + 2), x] + 1/b*Int[(e + f*x)^m*Sech[c + d*x]^(n + 1)*Tanh[c + d*x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 + b^2, 0]
+Int[(e_. + f_.*x_)^m_.* Csch[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := -1/a*Int[(e + f*x)^m*Csch[c + d*x]^(n + 2), x] + 1/b*Int[(e + f*x)^m*Csch[c + d*x]^(n + 1)*Coth[c + d*x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]
+Int[(e_. + f_.*x_)^m_.* Sech[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := b^2/(a^2 + b^2)* Int[(e + f*x)^m*Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x]), x] + 1/(a^2 + b^2)* Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Csch[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := b^2/(a^2 - b^2)* Int[(e + f*x)^m*Csch[c + d*x]^(n - 2)/(a + b*Cosh[c + d*x]), x] + 1/(a^2 - b^2)* Int[(e + f*x)^m*Csch[c + d*x]^n*(a - b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Csch[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Csch[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Csch[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* Sech[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Sech[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Sech[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := Unintegrable[(e + f*x)^m*F[c + d*x]^n/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && HyperbolicQ[F]
+Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := Unintegrable[(e + f*x)^m*F[c + d*x]^n/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && HyperbolicQ[F]
+Int[(e_. + f_.*x_)^m_.*Cosh[c_. + d_.*x_]^p_.* Sinh[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x] - a/b* Int[(e + f*x)^m*Cosh[c + d*x]^p* Sinh[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Sinh[c_. + d_.*x_]^p_.* Cosh[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Sinh[c + d*x]^p*Cosh[c + d*x]^(n - 1), x] - a/b* Int[(e + f*x)^m*Sinh[c + d*x]^p* Cosh[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Sinh[c_. + d_.*x_]^p_.* Tanh[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n, x] - a/b* Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)* Tanh[c + d*x]^n/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Cosh[c_. + d_.*x_]^p_.* Coth[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Cosh[c + d*x]^(p - 1)*Coth[c + d*x]^n, x] - a/b* Int[(e + f*x)^m*Cosh[c + d*x]^(p - 1)* Coth[c + d*x]^n/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Sech[c_. + d_.*x_]^p_.* Tanh[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*Tanh[c + d*x]^(n - 1), x] - a/b* Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)* Tanh[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Csch[c_. + d_.*x_]^p_.* Coth[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := 1/b*Int[(e + f*x)^m*Csch[c + d*x]^(p + 1)*Coth[c + d*x]^(n - 1), x] - a/b* Int[(e + f*x)^m*Csch[c + d*x]^(p + 1)* Coth[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Cosh[c_. + d_.*x_]^p_.* Coth[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Cosh[c + d*x]^p*Coth[c + d*x]^n, x] - b/a* Int[(e + f*x)^m*Cosh[c + d*x]^(p + 1)* Coth[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Sinh[c_. + d_.*x_]^p_.* Tanh[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Sinh[c + d*x]^p*Tanh[c + d*x]^n, x] - b/a* Int[(e + f*x)^m*Sinh[c + d*x]^(p + 1)* Tanh[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Csch[c_. + d_.*x_]^p_.* Coth[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Csch[c + d*x]^p*Coth[c + d*x]^n, x] - b/a* Int[(e + f*x)^m*Csch[c + d*x]^(p - 1)* Coth[c + d*x]^n/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Sech[c_. + d_.*x_]^p_.* Tanh[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Sech[c + d*x]^p*Tanh[c + d*x]^n, x] - b/a* Int[(e + f*x)^m*Sech[c + d*x]^(p - 1)* Tanh[c + d*x]^n/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Sech[c_. + d_.*x_]^p_.* Csch[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x] - b/a* Int[(e + f*x)^m*Sech[c + d*x]^p* Csch[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*Csch[c_. + d_.*x_]^p_.* Sech[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := 1/a*Int[(e + f*x)^m*Csch[c + d*x]^p*Sech[c + d*x]^n, x] - b/a* Int[(e + f*x)^m*Csch[c + d*x]^p* Sech[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*F_[c_. + d_.*x_]^n_.* G_[c_. + d_.*x_]^p_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol] := Unintegrable[(e + f*x)^m*F[c + d*x]^n* G[c + d*x]^p/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && HyperbolicQ[F] && HyperbolicQ[G]
+Int[(e_. + f_.*x_)^m_.*F_[c_. + d_.*x_]^n_.* G_[c_. + d_.*x_]^p_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol] := Unintegrable[(e + f*x)^m*F[c + d*x]^n* G[c + d*x]^p/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && HyperbolicQ[F] && HyperbolicQ[G]
+Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Sech[c_. + d_.*x_]), x_Symbol] := Int[(e + f*x)^m*Cosh[c + d*x]*F[c + d*x]^n/(b + a*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && IntegersQ[m, n]
+Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Csch[c_. + d_.*x_]), x_Symbol] := Int[(e + f*x)^m*Sinh[c + d*x]*F[c + d*x]^n/(b + a*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && IntegersQ[m, n]
+Int[(e_. + f_.*x_)^m_.*F_[c_. + d_.*x_]^n_.* G_[c_. + d_.*x_]^p_./(a_ + b_.*Sech[c_. + d_.*x_]), x_Symbol] := Int[(e + f*x)^m*Cosh[c + d*x]*F[c + d*x]^n* G[c + d*x]^p/(b + a*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && HyperbolicQ[G] && IntegersQ[m, n, p]
+Int[(e_. + f_.*x_)^m_.*F_[c_. + d_.*x_]^n_.* G_[c_. + d_.*x_]^p_./(a_ + b_.*Csch[c_. + d_.*x_]), x_Symbol] := Int[(e + f*x)^m*Sinh[c + d*x]*F[c + d*x]^n* G[c + d*x]^p/(b + a*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && HyperbolicQ[G] && IntegersQ[m, n, p]
+Int[Sinh[a_. + b_.*x_]^p_.*Sinh[c_. + d_.*x_]^q_., x_Symbol] := 1/2^(p + q)* Int[ExpandIntegrand[(-E^(-c - d*x) + E^(c + d*x))^ q, (-E^(-a - b*x) + E^(a + b*x))^p, x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]
+Int[Cosh[a_. + b_.*x_]^p_.*Cosh[c_. + d_.*x_]^q_., x_Symbol] := 1/2^(p + q)* Int[ExpandIntegrand[(E^(-c - d*x) + E^(c + d*x))^ q, (E^(-a - b*x) + E^(a + b*x))^p, x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]
+Int[Sinh[a_. + b_.*x_]^p_.*Cosh[c_. + d_.*x_]^q_., x_Symbol] := 1/2^(p + q)* Int[ExpandIntegrand[(E^(-c - d*x) + E^(c + d*x))^ q, (-E^(-a - b*x) + E^(a + b*x))^p, x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]
+Int[Cosh[a_. + b_.*x_]^p_.*Sinh[c_. + d_.*x_]^q_., x_Symbol] := 1/2^(p + q)* Int[ExpandIntegrand[(-E^(-c - d*x) + E^(c + d*x))^ q, (E^(-a - b*x) + E^(a + b*x))^p, x], x] /; FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]
+Int[Sinh[a_. + b_.*x_]*Tanh[c_. + d_.*x_], x_Symbol] := Int[-E^(-(a + b*x))/2 + E^(a + b*x)/2 + E^(-(a + b*x))/(1 + E^(2*(c + d*x))) - E^(a + b*x)/(1 + E^(2*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
+Int[Cosh[a_. + b_.*x_]*Coth[c_. + d_.*x_], x_Symbol] := Int[E^(-(a + b*x))/2 + E^(a + b*x)/2 - E^(-(a + b*x))/(1 - E^(2*(c + d*x))) - E^(a + b*x)/(1 - E^(2*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
+Int[Sinh[a_. + b_.*x_]*Coth[c_. + d_.*x_], x_Symbol] := Int[-E^(-(a + b*x))/2 + E^(a + b*x)/2 + E^(-(a + b*x))/(1 - E^(2*(c + d*x))) - E^(a + b*x)/(1 - E^(2*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
+Int[Cosh[a_. + b_.*x_]*Tanh[c_. + d_.*x_], x_Symbol] := Int[E^(-(a + b*x))/2 + E^(a + b*x)/2 - E^(-(a + b*x))/(1 + E^(2*(c + d*x))) - E^(a + b*x)/(1 + E^(2*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]
+Int[Sinh[a_./(c_. + d_.*x_)]^n_., x_Symbol] := -1/d*Subst[Int[Sinh[a*x]^n/x^2, x], x, 1/(c + d*x)] /; FreeQ[{a, c, d}, x] && IGtQ[n, 0]
+Int[Cosh[a_./(c_. + d_.*x_)]^n_., x_Symbol] := -1/d*Subst[Int[Cosh[a*x]^n/x^2, x], x, 1/(c + d*x)] /; FreeQ[{a, c, d}, x] && IGtQ[n, 0]
+Int[Sinh[e_.*(a_. + b_.*x_)/(c_. + d_.*x_)]^n_., x_Symbol] := -1/d*Subst[Int[Sinh[b*e/d - e*(b*c - a*d)*x/d]^n/x^2, x], x, 1/(c + d*x)] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - a*d, 0]
+Int[Cosh[e_.*(a_. + b_.*x_)/(c_. + d_.*x_)]^n_., x_Symbol] := -1/d*Subst[Int[Cosh[b*e/d - e*(b*c - a*d)*x/d]^n/x^2, x], x, 1/(c + d*x)] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - a*d, 0]
+Int[Sinh[u_]^n_., x_Symbol] := With[{lst = QuotientOfLinearsParts[u, x]}, Int[Sinh[(lst[[1]] + lst[[2]]*x)/(lst[[3]] + lst[[4]]*x)]^n, x]] /; IGtQ[n, 0] && QuotientOfLinearsQ[u, x]
+Int[Cosh[u_]^n_., x_Symbol] := With[{lst = QuotientOfLinearsParts[u, x]}, Int[Cosh[(lst[[1]] + lst[[2]]*x)/(lst[[3]] + lst[[4]]*x)]^n, x]] /; IGtQ[n, 0] && QuotientOfLinearsQ[u, x]
+Int[u_.*Sinh[v_]^p_.*Sinh[w_]^q_., x_Symbol] := Int[u*Sinh[v]^(p + q), x] /; EqQ[w, v]
+Int[u_.*Cosh[v_]^p_.*Cosh[w_]^q_., x_Symbol] := Int[u*Cosh[v]^(p + q), x] /; EqQ[w, v]
+Int[Sinh[v_]^p_.*Sinh[w_]^q_., x_Symbol] := Int[ExpandTrigReduce[Sinh[v]^p*Sinh[w]^q, x], x] /; IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])
+Int[Cosh[v_]^p_.*Cosh[w_]^q_., x_Symbol] := Int[ExpandTrigReduce[Cosh[v]^p*Cosh[w]^q, x], x] /; IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])
+Int[x_^m_.*Sinh[v_]^p_.*Sinh[w_]^q_., x_Symbol] := Int[ExpandTrigReduce[x^m, Sinh[v]^p*Sinh[w]^q, x], x] /; IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])
+Int[x_^m_.*Cosh[v_]^p_.*Cosh[w_]^q_., x_Symbol] := Int[ExpandTrigReduce[x^m, Cosh[v]^p*Cosh[w]^q, x], x] /; IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])
+Int[u_.*Sinh[v_]^p_.*Cosh[w_]^p_., x_Symbol] := 1/2^p*Int[u*Sinh[2*v]^p, x] /; EqQ[w, v] && IntegerQ[p]
+Int[Sinh[v_]^p_.*Cosh[w_]^q_., x_Symbol] := Int[ExpandTrigReduce[Sinh[v]^p*Cosh[w]^q, x], x] /; IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])
+Int[x_^m_.*Sinh[v_]^p_.*Cosh[w_]^q_., x_Symbol] := Int[ExpandTrigReduce[x^m, Sinh[v]^p*Cosh[w]^q, x], x] /; IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])
+Int[Sinh[v_]*Tanh[w_]^n_., x_Symbol] := Int[Cosh[v]*Tanh[w]^(n - 1), x] - Cosh[v - w]*Int[Sech[w]*Tanh[w]^(n - 1), x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
+Int[Cosh[v_]*Coth[w_]^n_., x_Symbol] := Int[Sinh[v]*Coth[w]^(n - 1), x] + Cosh[v - w]*Int[Csch[w]*Coth[w]^(n - 1), x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
+Int[Sinh[v_]*Coth[w_]^n_., x_Symbol] := Int[Cosh[v]*Coth[w]^(n - 1), x] + Sinh[v - w]*Int[Csch[w]*Coth[w]^(n - 1), x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
+Int[Cosh[v_]*Tanh[w_]^n_., x_Symbol] := Int[Sinh[v]*Tanh[w]^(n - 1), x] - Sinh[v - w]*Int[Sech[w]*Tanh[w]^(n - 1), x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
+Int[Sinh[v_]*Sech[w_]^n_., x_Symbol] := Cosh[v - w]*Int[Tanh[w]*Sech[w]^(n - 1), x] + Sinh[v - w]*Int[Sech[w]^(n - 1), x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
+Int[Cosh[v_]*Csch[w_]^n_., x_Symbol] := Cosh[v - w]*Int[Coth[w]*Csch[w]^(n - 1), x] + Sinh[v - w]*Int[Csch[w]^(n - 1), x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
+Int[Sinh[v_]*Csch[w_]^n_., x_Symbol] := Sinh[v - w]*Int[Coth[w]*Csch[w]^(n - 1), x] + Cosh[v - w]*Int[Csch[w]^(n - 1), x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
+Int[Cosh[v_]*Sech[w_]^n_., x_Symbol] := Sinh[v - w]*Int[Tanh[w]*Sech[w]^(n - 1), x] + Cosh[v - w]*Int[Sech[w]^(n - 1), x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
+Int[(e_. + f_.*x_)^ m_.*(a_ + b_.*Sinh[c_. + d_.*x_]*Cosh[c_. + d_.*x_])^n_., x_Symbol] := Int[(e + f*x)^m*(a + b*Sinh[2*c + 2*d*x]/2)^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
+Int[x_^m_.*(a_ + b_.*Sinh[c_. + d_.*x_]^2)^n_, x_Symbol] := 1/2^n*Int[x^m*(2*a - b + b*Cosh[2*c + 2*d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a - b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1] || EqQ[m, 1] && EqQ[n, -2])
+Int[x_^m_.*(a_ + b_.*Cosh[c_. + d_.*x_]^2)^n_, x_Symbol] := 1/2^n*Int[x^m*(2*a + b + b*Cosh[2*c + 2*d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a - b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1] || EqQ[m, 1] && EqQ[n, -2])
+Int[(f_. + g_.*x_)^ m_./(a_. + b_.*Cosh[d_. + e_.*x_]^2 + c_.*Sinh[d_. + e_.*x_]^2), x_Symbol] := 2*Int[(f + g*x)^m/(2*a + b - c + (b + c)*Cosh[2*d + 2*e*x]), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]
+Int[(f_. + g_.*x_)^m_.* Sech[d_. + e_.*x_]^2/(b_ + c_.*Tanh[d_. + e_.*x_]^2), x_Symbol] := 2*Int[(f + g*x)^m/(b - c + (b + c)*Cosh[2*d + 2*e*x]), x] /; FreeQ[{b, c, d, e, f, g}, x] && IGtQ[m, 0]
+Int[(f_. + g_.*x_)^m_.* Sech[d_. + e_.*x_]^2/(b_. + a_.*Sech[d_. + e_.*x_]^2 + c_.*Tanh[d_. + e_.*x_]^2), x_Symbol] := 2*Int[(f + g*x)^m/(2*a + b - c + (b + c)*Cosh[2*d + 2*e*x]), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]
+Int[(f_. + g_.*x_)^m_.* Csch[d_. + e_.*x_]^2/(c_ + b_.*Coth[d_. + e_.*x_]^2), x_Symbol] := 2*Int[(f + g*x)^m/(b - c + (b + c)*Cosh[2*d + 2*e*x]), x] /; FreeQ[{b, c, d, e, f, g}, x] && IGtQ[m, 0]
+Int[(f_. + g_.*x_)^m_.* Csch[d_. + e_.*x_]^2/(c_. + b_.*Coth[d_. + e_.*x_]^2 + a_.*Csch[d_. + e_.*x_]^2), x_Symbol] := 2*Int[(f + g*x)^m/(2*a + b - c + (b + c)*Cosh[2*d + 2*e*x]), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]
+Int[(e_. + f_.*x_)*(A_ + B_.*Sinh[c_. + d_.*x_])/(a_ + b_.*Sinh[c_. + d_.*x_])^2, x_Symbol] := B*(e + f*x)*Cosh[c + d*x]/(a*d*(a + b*Sinh[c + d*x])) - B*f/(a*d)*Int[Cosh[c + d*x]/(a + b*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[a*A + b*B, 0]
+Int[(e_. + f_.*x_)*(A_ + B_.*Cosh[c_. + d_.*x_])/(a_ + b_.*Cosh[c_. + d_.*x_])^2, x_Symbol] := B*(e + f*x)*Sinh[c + d*x]/(a*d*(a + b*Cosh[c + d*x])) - B*f/(a*d)*Int[Sinh[c + d*x]/(a + b*Cosh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[a*A - b*B, 0]
+Int[(e_. + f_.*x_)^m_.*Sinh[a_. + b_.*(c_ + d_.*x_)^n_]^p_., x_Symbol] := 1/d^(m + 1)* Subst[Int[(d*e - c*f + f*x)^m*Sinh[a + b*x^n]^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && RationalQ[p]
+Int[(e_. + f_.*x_)^m_.*Cosh[a_. + b_.*(c_ + d_.*x_)^n_]^p_., x_Symbol] := 1/d^(m + 1)* Subst[Int[(d*e - c*f + f*x)^m*Cosh[a + b*x^n]^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && RationalQ[p]
+Int[Sech[v_]^m_.*(a_ + b_.*Tanh[v_])^n_., x_Symbol] := Int[(a*Cosh[v] + b*Sinh[v])^n, x] /; FreeQ[{a, b}, x] && IntegerQ[(m - 1)/2] && EqQ[m + n, 0]
+Int[Csch[v_]^m_.*(a_ + b_.*Coth[v_])^n_., x_Symbol] := Int[(b*Cosh[v] + a*Sinh[v])^n, x] /; FreeQ[{a, b}, x] && IntegerQ[(m - 1)/2] && EqQ[m + n, 0]
+Int[u_.*Sinh[a_. + b_.*x_]^m_.*Sinh[c_. + d_.*x_]^n_., x_Symbol] := Int[ExpandTrigReduce[u, Sinh[a + b*x]^m*Sinh[c + d*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[u_.*Cosh[a_. + b_.*x_]^m_.*Cosh[c_. + d_.*x_]^n_., x_Symbol] := Int[ExpandTrigReduce[u, Cosh[a + b*x]^m*Cosh[c + d*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[Sech[a_. + b_.*x_]*Sech[c_ + d_.*x_], x_Symbol] := -Csch[(b*c - a*d)/d]*Int[Tanh[a + b*x], x] + Csch[(b*c - a*d)/b]*Int[Tanh[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
+Int[Csch[a_. + b_.*x_]*Csch[c_ + d_.*x_], x_Symbol] := Csch[(b*c - a*d)/b]*Int[Coth[a + b*x], x] - Csch[(b*c - a*d)/d]*Int[Coth[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
+Int[Tanh[a_. + b_.*x_]*Tanh[c_ + d_.*x_], x_Symbol] := b*x/d - b/d*Cosh[(b*c - a*d)/d]*Int[Sech[a + b*x]*Sech[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
+Int[Coth[a_. + b_.*x_]*Coth[c_ + d_.*x_], x_Symbol] := b*x/d + Cosh[(b*c - a*d)/d]*Int[Csch[a + b*x]*Csch[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
+Int[u_.*(a_.*Cosh[v_] + b_.*Sinh[v_])^n_., x_Symbol] := Int[u*(a*E^(a/b*v))^n, x] /; FreeQ[{a, b, n}, x] && EqQ[a^2 - b^2, 0]
+Int[Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol] := -1/2*Int[E^(-d*(a + b*Log[c*x^n])^2), x] + 1/2*Int[E^(d*(a + b*Log[c*x^n])^2), x] /; FreeQ[{a, b, c, d, n}, x]
+Int[Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol] := 1/2*Int[E^(-d*(a + b*Log[c*x^n])^2), x] + 1/2*Int[E^(d*(a + b*Log[c*x^n])^2), x] /; FreeQ[{a, b, c, d, n}, x]
+Int[(e_.*x_)^m_.*Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol] := -1/2*Int[(e*x)^m*E^(-d*(a + b*Log[c*x^n])^2), x] + 1/2*Int[(e*x)^m*E^(d*(a + b*Log[c*x^n])^2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[(e_.*x_)^m_.*Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol] := 1/2*Int[(e*x)^m*E^(-d*(a + b*Log[c*x^n])^2), x] + 1/2*Int[(e*x)^m*E^(d*(a + b*Log[c*x^n])^2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.1 (a+b arcsinh(c x))^n.m b/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.1 (a+b arcsinh(c x))^n.m
new file mode 100755
index 0000000..c56d5b5
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.1 (a+b arcsinh(c x))^n.m	
@@ -0,0 +1,6 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.1.1 (a+b arcsinh(c x))^n *)
+Int[(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := x*(a + b*ArcSinh[c*x])^n - b*c*n*Int[x*(a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
+Int[(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol] := Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1)) - c/(b*(n + 1))* Int[x*(a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
+Int[(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol] := 1/(b*c)* Subst[Int[x^n*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]] /; FreeQ[{a, b, c, n}, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m b/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m
new file mode 100755
index 0000000..f399210
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m	
@@ -0,0 +1,10 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.1.2 (d x)^m (a+b arcsinh(c x))^n *)
+Int[(a_. + b_.*ArcSinh[c_.*x_])^n_./x_, x_Symbol] := 1/b*Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := (d*x)^(m + 1)*(a + b*ArcSinh[c*x])^n/(d*(m + 1)) - b*c*n/(d*(m + 1))* Int[(d*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
+Int[x_^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol] := x^(m + 1)*(a + b*ArcSinh[c*x])^n/(m + 1) - b*c*n/(m + 1)* Int[x^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
+Int[x_^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol] := x^m*Sqrt[ 1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1)) - 1/(b^2*c^(m + 1)*(n + 1))* Subst[ Int[ExpandTrigReduce[x^(n + 1), Sinh[-a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSinh[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
+Int[x_^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol] := x^m*Sqrt[ 1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1)) - m/(b*c*(n + 1))* Int[x^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2], x] - c*(m + 1)/(b*(n + 1))* Int[x^(m + 1)*(a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
+Int[x_^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol] := 1/(b*c^(m + 1))* Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := Unintegrable[(d*x)^m*(a + b*ArcSinh[c*x])^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m b/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m
new file mode 100755
index 0000000..3172975
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m	
@@ -0,0 +1,20 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n *)
+(* Int[(a_.+b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := 1/c*Simp[Sqrt[1+c^2*x^2]/Sqrt[d+e*x^2]]*Subst[Int[(a+b*x)^n,x],x, ArcSinh[c*x]] /; FreeQ[{a,b,c,d,e,n},x] && EqQ[e,c^2*d] *)
+Int[1/(Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSinh[c_.*x_])), x_Symbol] := 1/(b*c)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]* Log[a + b*ArcSinh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
+Int[(a_. + b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol] := 1/(b*c*(n + 1))* Simp[Sqrt[1 + c^2*x^2]/ Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1) /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && NeQ[n, -1]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
+Int[Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := x*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n/2 - b*c*n/2*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]* Int[x*(a + b*ArcSinh[c*x])^(n - 1), x] + 1/2*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]* Int[(a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := x*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n/(2*p + 1) + 2*d*p/(2*p + 1)* Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x] - b*c*n/(2*p + 1)*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]
+Int[(a_. + b_.*ArcSinh[c_.*x_])^n_./(d_ + e_.*x_^2)^(3/2), x_Symbol] := x*(a + b*ArcSinh[c*x])^n/(d*Sqrt[d + e*x^2]) - b*c*n/d*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]* Int[x*(a + b*ArcSinh[c*x])^(n - 1)/(1 + c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
+Int[(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := -x*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n/(2*d*(p + 1)) + (2*p + 3)/(2*d*(p + 1))* Int[(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x] + b*c*n/(2*(p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[x*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
+Int[(a_. + b_.*ArcSinh[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol] := 1/(c*d)*Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
+(* Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol] := d^p*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n+1)/(b*c*(n+1)) - c*d^p*(2*p+1)/(b*(n+1))*Int[x*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x]) ^(n+1),x] /; FreeQ[{a,b,c,d,e,p},x] && EqQ[e,c^2*d] && LtQ[n,-1] && (IntegerQ[p] ||  GtQ[d,0]) *)
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol] := Simp[Sqrt[1 + c^2*x^2]*(d + e*x^2)^ p]*(a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1)) - c*(2*p + 1)/(b*(n + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && LtQ[n, -1]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := 1/(b*c)*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Subst[Int[x^n*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p, 0]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[e, c^2*d] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := Unintegrable[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x] /; FreeQ[{a, b, c, d, e, n, p}, x]
+Int[(d_ + e_.*x_)^p_*(f_ + g_.*x_)^q_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := (-d^2*g/e)^q* Int[(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]
+Int[(d_ + e_.*x_)^p_*(f_ + g_.*x_)^q_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := (d + e*x)^q*(f + g*x)^q/(1 + c^2*x^2)^q* Int[(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m b/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m
new file mode 100755
index 0000000..abeaa76
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m	
@@ -0,0 +1,34 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n *)
+Int[x_*(a_. + b_.*ArcSinh[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol] := 1/e*Subst[Int[(a + b*x)^n*Tanh[x], x], x, ArcSinh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
+Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := (d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n/(2*e*(p + 1)) - b*n/(2*c*(p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
+Int[(a_. + b_.*ArcSinh[c_.*x_])^n_./(x_*(d_ + e_.*x_^2)), x_Symbol] := 1/d*Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, ArcSinh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^ n/(d*f*(m + 1)) - b*c*n/(f*(m + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])/x_, x_Symbol] := (d + e*x^2)^p*(a + b*ArcSinh[c*x])/(2*p) - b*c*d^p/(2*p)*Int[(1 + c^2*x^2)^(p - 1/2), x] + d*Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])/x, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])/(f*(m + 1)) - b*c*d^p/(f*(m + 1))* Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2), x] - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
+Int[x_^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u] - b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]* Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -1/2] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
+Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n/(f*(m + 1)) - b*c*n/(f*(m + 1))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]* Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x] - c^2/(f^2*(m + 1))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]* Int[(f*x)^(m + 2)*(a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n/(f*(m + 2)) - b*c*n/(f*(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]* Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x] + 1/(m + 2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]* Int[(f*x)^m*(a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n/(f*(m + 1)) - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x] - b*c*n/(f*(m + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcSinh[c*x])^n/(f*(m + 2*p + 1)) + 2*d*p/(m + 2*p + 1)* Int[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x] - b*c*n/(f*(m + 2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && Not[LtQ[m, -1]]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^ n/(d*f*(m + 1)) - c^2*(m + 2*p + 3)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x] - b*c*n/(f*(m + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^ n/(2*e*(p + 1)) - f^2*(m - 1)/(2*e*(p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x] - b*f*n/(2*c*(p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := -(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^ n/(2*d*f*(p + 1)) + (m + 2*p + 3)/(2*d*(p + 1))* Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x] + b*c*n/(2*f*(p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && Not[GtQ[m, 1]] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^ n/(e*(m + 2*p + 1)) - f^2*(m - 1)/(c^2*(m + 2*p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x] - b*f*n/(c*(m + 2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol] := (f*x)^m* Sqrt[1 + c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1)) - f*m/(b*c*(n + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol] := (f*x)^m* Sqrt[1 + c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1)) - f*m/(b*c*(n + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1), x] - c*(m + 2*p + 1)/(b*f*(n + 1))* Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]
+(* Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_ Symbol] := (f*x)^m*Simp[Sqrt[1+c^2*x^2]*(d+e*x^2)^p]*(a+b*ArcSinh[c*x])^(n+1)/( b*c*(n+1)) - f*m/(b*c*(n+1))*Simp[(d+e*x^2)^p/(1+c^2*x^2)^p]*Int[(f*x)^(m-1)*(1+ c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n+1),x] - c*(2*p+1)/(b*f*(n+1))*Simp[(d+e*x^2)^p/(1+c^2*x^2)^p]*Int[(f*x)^(m+ 1)*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n+1),x] /; FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[e,c^2*d] && LtQ[n,-1] &&  IntegerQ[2*p] && NeQ[p,-1/2] && IGtQ[m,-3] *)
+Int[(f_.*x_)^m_*(a_. + b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol] := f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n/(e*m) - b*f*n/(c*m)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]* Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n - 1), x] - f^2*(m - 1)/(c^2*m)* Int[((f*x)^(m - 2)*(a + b*ArcSinh[c*x])^n)/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[m, 1]
+Int[x_^m_*(a_. + b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol] := 1/c^(m + 1)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]* Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcSinh[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol] := (f*x)^(m + 1)/(f*(m + 1))* Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])* Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -c^2*x^2] - b*c*(f*x)^(m + 2)/(f^2*(m + 1)*(m + 2))* Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]* HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -c^2*x^2] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && Not[IntegerQ[m]]
+Int[(f_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_/Sqrt[d_ + e_.*x_^2], x_Symbol] := (f*x)^m/(b*c*(n + 1))* Simp[Sqrt[1 + c^2*x^2]/ Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1) - f*m/(b*c*(n + 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]* Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[n, -1]
+Int[x_^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := 1/(b*c^(m + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Subst[ Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n/ Sqrt[d + e*x^2], (f*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[e, c^2*d] && IGtQ[p + 1/2, 0] && Not[IGtQ[(m + 1)/2, 0]] && (EqQ[m, -1] || EqQ[m, -2])
+Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol] := (d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])/(2*e*(p + 1)) - b*c/(2*e*(p + 1))*Int[(d + e*x^2)^(p + 1)/Sqrt[1 + c^2*x^2], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[e, c^2*d] && NeQ[p, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (GtQ[p, 0] || IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := Unintegrable[(f*x)^m*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
+Int[(h_.*x_)^m_.*(d_ + e_.*x_)^p_*(f_ + g_.*x_)^ q_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := (-d^2*g/e)^q* Int[(h*x)^m*(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^ n, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]
+Int[(h_.*x_)^m_.*(d_ + e_.*x_)^p_*(f_ + g_.*x_)^ q_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := (-d^2*g/e)^IntPart[q]*(d + e*x)^ FracPart[q]*(f + g*x)^FracPart[q]/(1 + c^2*x^2)^FracPart[q]* Int[(h*x)^m*(d + e*x)^(p - q)*(1 + c^2*x^2)^ q*(a + b*ArcSinh[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m b/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m
new file mode 100755
index 0000000..3724035
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m	
@@ -0,0 +1,35 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.1.5 u (a+b arcsinh(c x))^n *)
+Int[(a_. + b_.*ArcSinh[c_.*x_])^n_./(d_. + e_.*x_), x_Symbol] := Subst[Int[(a + b*x)^n*Cosh[x]/(c*d + e*Sinh[x]), x], x, ArcSinh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := (d + e*x)^(m + 1)*(a + b*ArcSinh[c*x])^n/(e*(m + 1)) - b*c*n/(e*(m + 1))* Int[(d + e*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1)/ Sqrt[1 + c^2*x^2], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
+Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := 1/c^(m + 1)* Subst[Int[(a + b*x)^n*Cosh[x]*(c*d + e*Sinh[x])^m, x], x, ArcSinh[c*x]] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]
+Int[Px_*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol] := With[{u = IntHide[ExpandExpression[Px, x], x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x]] /; FreeQ[{a, b, c}, x] && PolynomialQ[Px, x]
+(* Int[Px_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol] := With[{u=IntHide[Px,x]}, Dist[(a+b*ArcSinh[c*x])^n,u,x] -  b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcSinh[c*x])^(n-1)/Sqrt[1+c^2*x^2] ,x],x]] /; FreeQ[{a,b,c},x] && PolynomialQ[Px,x] && IGtQ[n,0] *)
+Int[Px_*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[Px*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, n}, x] && PolynomialQ[Px, x]
+Int[Px_*(d_. + e_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol] := With[{u = IntHide[Px*(d + e*x)^m, x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]
+Int[(f_. + g_.*x_)^p_.*(d_ + e_.*x_)^m_*(a_. + b_.*ArcSinh[c_.*x_])^ n_, x_Symbol] := With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcSinh[c*x])^n, u, x] - b*c*n*Int[ SimplifyIntegrand[ u*(a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]
+Int[(f_. + g_.*x_ + h_.*x_^2)^ p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_/(d_ + e_.*x_)^2, x_Symbol] := With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcSinh[c*x])^n, u, x] - b*c*n*Int[ SimplifyIntegrand[ u*(a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]
+Int[Px_*(d_ + e_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[Px*(d + e*x)^m*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && IGtQ[n, 0] && IntegerQ[m]
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol] := With[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[Dist[1/Sqrt[1 + c^2*x^2], u, x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^ n_., x_Symbol] := Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^ n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (EqQ[n, 1] && GtQ[p, -1] || GtQ[p, 0] || EqQ[m, 1] || EqQ[m, 2] && LtQ[p, -2])
+Int[(f_. + g_.*x_)^m_* Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := (f + g*x)^ m*(d + e*x^2)*(a + b*ArcSinh[c*x])^(n + 1)/(b*c* Sqrt[d]*(n + 1)) - 1/(b*c*Sqrt[d]*(n + 1))* Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^ n_., x_Symbol] := Int[ExpandIntegrand[ Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n, (f + g*x)^ m*(d + e*x^2)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^ n_., x_Symbol] := (f + g*x)^ m*(d + e*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n + 1)/(b*c* Sqrt[d]*(n + 1)) - 1/(b*c*Sqrt[d]*(n + 1))* Int[ ExpandIntegrand[(f + g*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), (d*g*m + e*f*(2*p + 1)*x + e*g*(m + 2*p + 1)*x^2)*(d + e*x^2)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && ILtQ[m, 0] && IGtQ[p - 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_/ Sqrt[d_ + e_.*x_^2], x_Symbol] := (f + g*x)^m*(a + b*ArcSinh[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)) - g*m/(b*c*Sqrt[d]*(n + 1))* Int[(f + g*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && GtQ[d, 0] && LtQ[n, -1]
+Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_./ Sqrt[d_ + e_.*x_^2], x_Symbol] := 1/(c^(m + 1)*Sqrt[d])* Subst[Int[(a + b*x)^n*(c*f + g*Sinh[x])^m, x], x, ArcSinh[c*x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^ n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n/ Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^ n_., x_Symbol] := Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[(f + g*x)^m*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IntegerQ[p - 1/2] && Not[GtQ[d, 0]]
+Int[Log[h_.*(f_. + g_.*x_)^m_.]*(a_. + b_.*ArcSinh[c_.*x_])^n_./ Sqrt[d_ + e_.*x_^2], x_Symbol] := Log[h*(f + g*x)^m]*(a + b*ArcSinh[c*x])^(n + 1)/(b*c* Sqrt[d]*(n + 1)) - g*m/(b*c*Sqrt[d]*(n + 1))* Int[(a + b*ArcSinh[c*x])^(n + 1)/(f + g*x), x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && IGtQ[n, 0]
+Int[Log[h_.*(f_. + g_.*x_)^m_.]*(d_ + e_.*x_^2)^ p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]* Int[Log[h*(f + g*x)^m]*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && Not[GtQ[d, 0]]
+Int[(d_ + e_.*x_)^m_*(f_ + g_.*x_)^m_*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x)^m*(f + g*x)^m, x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[Dist[1/Sqrt[1 + c^2*x^2], u, x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m + 1/2, 0]
+Int[(d_ + e_.*x_)^m_.*(f_ + g_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^ n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^ n, (d + e*x)^m*(f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && IntegerQ[m]
+Int[u_*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol] := With[{v = IntHide[u, x]}, Dist[a + b*ArcSinh[c*x], v, x] - b*c*Int[SimplifyIntegrand[v/Sqrt[1 + c^2*x^2], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
+Int[Px_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol] := With[{u = ExpandIntegrand[Px*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && PolynomialQ[Px, x] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]
+Int[Px_.*(f_ + g_.*(d_ + e_.*x_^2)^p_)^ m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := With[{u = ExpandIntegrand[ Px*(f + g*(d + e*x^2)^p)^m*(a + b*ArcSinh[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, g}, x] && PolynomialQ[Px, x] && EqQ[e, c^2*d] && IGtQ[p + 1/2, 0] && IntegersQ[m, n]
+Int[RFx_*ArcSinh[c_.*x_]^n_., x_Symbol] := With[{u = ExpandIntegrand[ArcSinh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[c, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
+Int[RFx_*(a_ + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[RFx*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
+Int[RFx_*(d_ + e_.*x_^2)^p_*ArcSinh[c_.*x_]^n_., x_Symbol] := With[{u = ExpandIntegrand[(d + e*x^2)^p*ArcSinh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]
+Int[RFx_*(d_ + e_.*x_^2)^p_*(a_ + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]
+Int[u_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol] := Unintegrable[u*(a + b*ArcSinh[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m b/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m
new file mode 100755
index 0000000..f03f7a4
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m	
@@ -0,0 +1,24 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.1.6 Miscellaneous inverse hyperbolic sine *)
+Int[(a_. + b_.*ArcSinh[c_ + d_.*x_])^n_., x_Symbol] := 1/d*Subst[Int[(a + b*ArcSinh[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, n}, x]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSinh[c_ + d_.*x_])^n_., x_Symbol] := 1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcSinh[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
+Int[(A_. + B_.*x_ + C_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_ + d_.*x_])^ n_., x_Symbol] := 1/d*Subst[Int[(C/d^2 + C/d^2*x^2)^p*(a + b*ArcSinh[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, A, B, C, n, p}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[(e_. + f_.*x_)^m_.*(A_. + B_.*x_ + C_.*x_^2)^ p_.*(a_. + b_.*ArcSinh[c_ + d_.*x_])^n_., x_Symbol] := 1/d*Subst[ Int[((d*e - c*f)/d + f*x/d)^m*(C/d^2 + C/d^2*x^2)^ p*(a + b*ArcSinh[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[Sqrt[a_. + b_.*ArcSinh[c_ + d_.*x_^2]], x_Symbol] := x*Sqrt[a + b*ArcSinh[c + d*x^2]] - Sqrt[Pi]*x*(Cosh[a/(2*b)] - c*Sinh[a/(2*b)])* FresnelC[Sqrt[-c/(Pi*b)]*Sqrt[a + b*ArcSinh[c + d*x^2]]]/ (Sqrt[-(c/b)]*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])) + Sqrt[Pi]*x*(Cosh[a/(2*b)] + c*Sinh[a/(2*b)])* FresnelS[Sqrt[-c/(Pi*b)]*Sqrt[a + b*ArcSinh[c + d*x^2]]]/ (Sqrt[-(c/b)]*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1]
+Int[(a_. + b_.*ArcSinh[c_ + d_.*x_^2])^n_, x_Symbol] := x*(a + b*ArcSinh[c + d*x^2])^n - 2*b*n* Sqrt[2*c*d*x^2 + d^2*x^4]*(a + b*ArcSinh[c + d*x^2])^(n - 1)/(d*x) + 4*b^2*n*(n - 1)*Int[(a + b*ArcSinh[c + d*x^2])^(n - 2), x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1] && GtQ[n, 1]
+Int[1/(a_. + b_.*ArcSinh[c_ + d_.*x_^2]), x_Symbol] := x*(c*Cosh[a/(2*b)] - Sinh[a/(2*b)])* CoshIntegral[(a + b*ArcSinh[c + d*x^2])/(2*b)]/ (2* b*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[(1/2)*ArcSinh[c + d*x^2]])) + x*(Cosh[a/(2*b)] - c*Sinh[a/(2*b)])* SinhIntegral[(a + b*ArcSinh[c + d*x^2])/(2*b)]/ (2* b*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[(1/2)*ArcSinh[c + d*x^2]])) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1]
+Int[1/Sqrt[a_. + b_.*ArcSinh[c_ + d_.*x_^2]], x_Symbol] := (c + 1)*Sqrt[Pi/2]*x*(Cosh[a/(2*b)] - Sinh[a/(2*b)])* Erfi[Sqrt[a + b*ArcSinh[c + d*x^2]]/Sqrt[2*b]]/ (2* Sqrt[b]*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])) + (c - 1)*Sqrt[Pi/2]*x*(Cosh[a/(2*b)] + Sinh[a/(2*b)])* Erf[Sqrt[a + b*ArcSinh[c + d*x^2]]/Sqrt[2*b]]/ (2* Sqrt[b]*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1]
+Int[1/(a_. + b_.*ArcSinh[c_ + d_.*x_^2])^(3/2), x_Symbol] := -Sqrt[2*c*d*x^2 + d^2*x^4]/(b*d*x*Sqrt[a + b*ArcSinh[c + d*x^2]]) - (-c/b)^(3/2)*Sqrt[Pi]*x*(Cosh[a/(2*b)] - c*Sinh[a/(2*b)])* FresnelC[Sqrt[-c/(Pi*b)]*Sqrt[a + b*ArcSinh[c + d*x^2]]]/ (Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2]) + (-c/b)^(3/2)*Sqrt[Pi]*x*(Cosh[a/(2*b)] + c*Sinh[a/(2*b)])* FresnelS[Sqrt[-c/(Pi*b)]*Sqrt[a + b*ArcSinh[c + d*x^2]]]/ (Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1]
+Int[1/(a_. + b_.*ArcSinh[c_ + d_.*x_^2])^2, x_Symbol] := -Sqrt[2*c*d*x^2 + d^2*x^4]/(2*b*d*x*(a + b*ArcSinh[c + d*x^2])) + x*(Cosh[a/(2*b)] - c*Sinh[a/(2*b)])* CoshIntegral[(a + b*ArcSinh[c + d*x^2])/(2*b)]/ (4* b^2*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])) + x*(c*Cosh[a/(2*b)] - Sinh[a/(2*b)])* SinhIntegral[(a + b*ArcSinh[c + d*x^2])/(2*b)]/ (4* b^2*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1]
+Int[(a_. + b_.*ArcSinh[c_ + d_.*x_^2])^n_, x_Symbol] := -x*(a + b*ArcSinh[c + d*x^2])^(n + 2)/(4*b^2*(n + 1)*(n + 2)) + Sqrt[ 2*c*d*x^2 + d^2*x^4]*(a + b*ArcSinh[c + d*x^2])^(n + 1)/(2*b*d*(n + 1)* x) + 1/(4*b^2*(n + 1)*(n + 2))* Int[(a + b*ArcSinh[c + d*x^2])^(n + 2), x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1] && LtQ[n, -1] && NeQ[n, -2]
+Int[ArcSinh[a_.*x_^p_]^n_./x_, x_Symbol] := 1/p*Subst[Int[x^n*Coth[x], x], x, ArcSinh[a*x^p]] /; FreeQ[{a, p}, x] && IGtQ[n, 0]
+Int[u_.*ArcSinh[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol] := Int[u*ArcCsch[a/c + b*x^n/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
+Int[ArcSinh[Sqrt[-1 + b_.*x_^2]]^n_./Sqrt[-1 + b_.*x_^2], x_Symbol] := Sqrt[b*x^2]/(b*x)* Subst[Int[ArcSinh[x]^n/Sqrt[1 + x^2], x], x, Sqrt[-1 + b*x^2]] /; FreeQ[{b, n}, x]
+Int[f_^(c_.*ArcSinh[a_. + b_.*x_]^n_.), x_Symbol] := 1/b*Subst[Int[f^(c*x^n)*Cosh[x], x], x, ArcSinh[a + b*x]] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
+Int[x_^m_.*f_^(c_.*ArcSinh[a_. + b_.*x_]^n_.), x_Symbol] := 1/b*Subst[Int[(-a/b + Sinh[x]/b)^m*f^(c*x^n)*Cosh[x], x], x, ArcSinh[a + b*x]] /; FreeQ[{a, b, c, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[ArcSinh[u_], x_Symbol] := x*ArcSinh[u] - Int[SimplifyIntegrand[x*D[u, x]/Sqrt[1 + u^2], x], x] /; InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcSinh[u_]), x_Symbol] := (c + d*x)^(m + 1)*(a + b*ArcSinh[u])/(d*(m + 1)) - b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/Sqrt[1 + u^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]
+Int[v_*(a_. + b_.*ArcSinh[u_]), x_Symbol] := With[{w = IntHide[v, x]}, Dist[(a + b*ArcSinh[u]), w, x] - b*Int[SimplifyIntegrand[w*D[u, x]/Sqrt[1 + u^2], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_. /; FreeQ[{c, d, m}, x]]]
+Int[E^(n_.*ArcSinh[u_]), x_Symbol] := Int[(u + Sqrt[1 + u^2])^n, x] /; IntegerQ[n] && PolyQ[u, x]
+Int[x_^m_.*E^(n_.*ArcSinh[u_]), x_Symbol] := Int[x^m*(u + Sqrt[1 + u^2])^n, x] /; RationalQ[m] && IntegerQ[n] && PolyQ[u, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.1 (a+b arccosh(c x))^n.m b/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.1 (a+b arccosh(c x))^n.m
new file mode 100755
index 0000000..cb56bfc
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.1 (a+b arccosh(c x))^n.m	
@@ -0,0 +1,6 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.2.1 (a+b arccosh(c x))^n *)
+Int[(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := x*(a + b*ArcCosh[c*x])^n - b*c*n* Int[x*(a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
+Int[(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol] := Sqrt[1 + c*x]* Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) - c/(b*(n + 1))* Int[x*(a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
+Int[(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol] := 1/(b*c)* Subst[Int[x^n*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]] /; FreeQ[{a, b, c, n}, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.2 (d x)^m (a+b arccosh(c x))^n.m b/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.2 (d x)^m (a+b arccosh(c x))^n.m
new file mode 100755
index 0000000..567483a
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.2 (d x)^m (a+b arccosh(c x))^n.m	
@@ -0,0 +1,10 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.2.2 (d x)^m (a+b arccosh(c x))^n *)
+Int[(a_. + b_.*ArcCosh[c_.*x_])^n_./x_, x_Symbol] := 1/b*Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (d*x)^(m + 1)*(a + b*ArcCosh[c*x])^n/(d*(m + 1)) - b*c*n/(d*(m + 1))* Int[(d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]* Sqrt[-1 + c*x]), x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
+Int[x_^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol] := x^(m + 1)*(a + b*ArcCosh[c*x])^n/(m + 1) - b*c*n/(m + 1)* Int[x^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]* Sqrt[-1 + c*x]), x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
+Int[x_^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol] := x^m*Sqrt[1 + c*x]* Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) + 1/(b^2*c^(m + 1)*(n + 1))* Subst[ Int[ExpandTrigReduce[x^(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
+Int[x_^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol] := x^m*Sqrt[1 + c*x]* Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) + m/(b*c*(n + 1))* Int[x^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]* Sqrt[-1 + c*x]), x] - c*(m + 1)/(b*(n + 1))* Int[x^(m + 1)*(a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]* Sqrt[-1 + c*x]), x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
+Int[x_^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol] := 1/(b*c^(m + 1))* Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Unintegrable[(d*x)^m*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.3 (d+e x^2)^p (a+b arccosh(c x))^n.m b/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.3 (d+e x^2)^p (a+b arccosh(c x))^n.m
new file mode 100755
index 0000000..fb56a6b
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.3 (d+e x^2)^p (a+b arccosh(c x))^n.m	
@@ -0,0 +1,28 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.2.3 (d+e x^2)^p (a+b arccosh(c x))^n *)
+Int[(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Int[(d1*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
+(* Int[(a_.+b_.*ArcCosh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol] := 1/c*Simp[Sqrt[1+c*x]*Sqrt[-1+c*x]/Sqrt[d+e*x^2]]*Subst[Int[(a+b*x)^ n,x],x,ArcCosh[c*x]] /; FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e,0] *)
+Int[1/(Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCosh[c_.*x_])), x_Symbol] := 1/(b*c)*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]/Sqrt[d + e*x^2]]* Log[a + b*ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
+Int[1/(Sqrt[d1_ + e1_.*x_]* Sqrt[d2_ + e2_.*x_]*(a_. + b_.*ArcCosh[c_.*x_])), x_Symbol] := 1/(b*c)*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]* Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*Log[a + b*ArcCosh[c*x]] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2]
+Int[(a_. + b_.*ArcCosh[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol] := 1/(b*c*(n + 1))* Simp[Sqrt[1 + c*x]* Sqrt[-1 + c*x]/Sqrt[d + e*x^2]]*(a + b*ArcCosh[c*x])^(n + 1) /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
+Int[(a_. + b_.*ArcCosh[c_.*x_])^ n_./(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol] := 1/(b*c*(n + 1))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]* Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[c*x])^(n + 1) /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && NeQ[n, -1]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
+Int[Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := x*Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])^n/2 - b*c*n/2*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]* Int[x*(a + b*ArcCosh[c*x])^(n - 1), x] - 1/2*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]* Int[(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
+Int[Sqrt[d1_ + e1_.*x_]* Sqrt[d2_ + e2_.*x_]*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := x*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n/2 - b*c*n/2*Simp[Sqrt[d1 + e1*x]/Sqrt[1 + c*x]]* Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]]* Int[x*(a + b*ArcCosh[c*x])^(n - 1), x] - 1/2*Simp[Sqrt[d1 + e1*x]/Sqrt[1 + c*x]]* Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]]* Int[(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := x*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n/(2*p + 1) + 2*d*p/(2*p + 1)* Int[(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x] - b*c*n/(2*p + 1)*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Int[ x*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
+Int[(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := x*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n/(2*p + 1) + 2*d1*d2*p/(2*p + 1)* Int[(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^ n, x] - b*c*n/(2*p + 1)*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Int[ x*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && GtQ[p, 0]
+Int[(a_. + b_.*ArcCosh[c_.*x_])^n_./(d_ + e_.*x_^2)^(3/2), x_Symbol] := x*(a + b*ArcCosh[c*x])^n/(d*Sqrt[d + e*x^2]) + b*c*n/d*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]/Sqrt[d + e*x^2]]* Int[x*(a + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
+Int[(a_. + b_.*ArcCosh[c_.*x_])^ n_./((d1_ + e1_.*x_)^(3/2)*(d2_ + e2_.*x_)^(3/2)), x_Symbol] := x*(a + b*ArcCosh[c*x])^ n/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]) + b*c*n/(d1*d2)*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]* Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]* Int[x*(a + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0]
+Int[(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := -x*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n/(2*d*(p + 1)) + (2*p + 3)/(2*d*(p + 1))* Int[(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x] - b*c*n/(2*(p + 1))* Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Int[ x*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
+Int[(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^p_*(a_. + b_.*ArcCosh[c_.*x_])^ n_., x_Symbol] := -x*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^ n/(2*d1*d2*(p + 1)) + (2*p + 3)/(2*d1*d2*(p + 1))* Int[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^ n, x] - b*c*n/(2*(p + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Int[ x*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
+Int[(a_. + b_.*ArcCosh[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol] := -1/(c*d)*Subst[Int[(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol] := Simp[Sqrt[1 + c*x]* Sqrt[-1 + c*x]*(d + e*x^2)^p]*(a + b*ArcCosh[c*x])^(n + 1)/(b* c*(n + 1)) - c*(2*p + 1)/(b*(n + 1))* Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Int[ x*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IntegerQ[2*p]
+Int[(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol] := Sqrt[1 + c*x]* Sqrt[-1 + c*x]*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) - c*(2*p + 1)/(b*(n + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Int[x*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && LtQ[n, -1] && IntegerQ[p + 1/2]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := 1/(b*c)*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Subst[Int[x^n*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0]
+Int[(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := 1/(b*c)*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Subst[Int[x^n*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && IGtQ[2*p, 0]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
+(* Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol] := 1/(b*c^(2*p+1))*Subst[Int[x^n*(c^2*d+e*Cosh[-a/b+x/b]^2)^p*Sinh[-a/ b+x/b],x],x,a+b*ArcCosh[c*x]] /; FreeQ[{a,b,c,d,e,n},x] && IGtQ[p,0] *)
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Unintegrable[(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d, e, n, p}, x]
+Int[(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Unintegrable[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n, p}, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.4 (f x)^m (d+e x^2)^p (a+b arccosh(c x))^n.m b/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.4 (f x)^m (d+e x^2)^p (a+b arccosh(c x))^n.m
new file mode 100755
index 0000000..06803d3
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.4 (f x)^m (d+e x^2)^p (a+b arccosh(c x))^n.m	
@@ -0,0 +1,58 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.2.4 (f x)^m (d+e x^2)^p (a+b arccosh(c x))^n *)
+Int[(f_.*x_)^m_.*(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Int[(f*x)^m*(d1*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
+Int[x_*(a_. + b_.*ArcCosh[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol] := 1/e*Subst[Int[(a + b*x)^n*Coth[x], x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
+Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n/(2*e*(p + 1)) - b*n/(2*c*(p + 1))* Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
+Int[x_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^ n/(2*e1*e2*(p + 1)) - b*n/(2*c*(p + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && NeQ[p, -1]
+Int[(a_. + b_.*ArcCosh[c_.*x_])^n_./(x_*(d_ + e_.*x_^2)), x_Symbol] := -1/d*Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^ n/(d*f*(m + 1)) + b*c*n/(f*(m + 1))* Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
+Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(d1*d2*f*(m + 1)) + b*c*n/(f*(m + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]
+Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])/x_, x_Symbol] := (d + e*x^2)^p*(a + b*ArcCosh[c*x])/(2*p) - b*c*(-d)^p/(2*p)* Int[(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2), x] + d*Int[(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])/x, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])/(f*(m + 1)) - b*c*(-d)^p/(f*(m + 1))* Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2), x] - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
+Int[x_^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u] - b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]* Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -1/2] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
+Int[x_^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(d1 + e1*x)^p*(d2 + e2*x)^p, x]}, Dist[a + b*ArcCosh[c*x], u] - b*c* Simp[Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]* Int[SimplifyIntegrand[u/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x], x]] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && IntegerQ[p - 1/2] && NeQ[p, -1/2] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
+Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])^n/(f*(m + 1)) - b*c*n/(f*(m + 1))* Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]* Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x] - c^2/(f^2*(m + 1))* Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]* Int[(f*x)^(m + 2)*(a + b*ArcCosh[c*x])^ n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_*Sqrt[d1_ + e1_.*x_]* Sqrt[d2_ + e2_.*x_]*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n/(f*(m + 1)) - b*c*n/(f*(m + 1))*Simp[Sqrt[d1 + e1*x]/Sqrt[1 + c*x]]* Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]]* Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x] - c^2/(f^2*(m + 1))*Simp[Sqrt[d1 + e1*x]/Sqrt[1 + c*x]]* Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]]* Int[((f*x)^(m + 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[1 + c*x]* Sqrt[-1 + c*x]), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])^n/(f*(m + 2)) - b*c*n/(f*(m + 2))* Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]* Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x] - 1/(m + 2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]* Int[(f*x)^ m*(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
+Int[(f_.*x_)^m_*Sqrt[d1_ + e1_.*x_]* Sqrt[d2_ + e2_.*x_]*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n/(f*(m + 2)) - b*c*n/(f*(m + 2))*Simp[Sqrt[d1 + e1*x]/Sqrt[1 + c*x]]* Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]]* Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x] - 1/(m + 2)*Simp[Sqrt[d1 + e1*x]/Sqrt[1 + c*x]]* Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]]* Int[(f*x)^ m*(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n/(f*(m + 1)) - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x] - b*c*n/(f*(m + 1))* Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^n/(f*(m + 1)) - 2*e1*e2*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^n, x] - b*c*n/(f*(m + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
+(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_ Symbol] := f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcCosh[c*x])^n/(e*(m+2*p+1)) + f^2*(m-1)/(c^2*(m+2*p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^p*(a+b*ArcCosh[ c*x])^n,x] - b*f*n/(c*(m+2*p+1))*Simp[(d+e*x^2)^p/((1+c*x)^p*(-1+c*x)^p)]* Int[(f*x)^(m-1)*(-1+c^2*x^2)^(p+1/2)*(a+b*ArcCosh[c*x])^(n-1),x] /; FreeQ[{a,b,c,d,e,f,p},x] && EqQ[c^2*d+e,0] && GtQ[n,0] && EqQ[n,1] &&  IGtQ[p+1/2,0] && IGtQ[(m-1)/2,0] *)
+(* Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_ Symbol] := f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*e*(p+1)) - f^2*(m-1)/(2*e*(p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcCosh[ c*x])^n,x] - b*f*n/(2*c*(p+1))*Simp[(d+e*x^2)^p/((1+c*x)^p*(-1+c*x)^p)]* Int[(f*x)^(m-1)*(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x] )^(n-1),x] /; FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e,0] && GtQ[n,0] && EqQ[n,1] &&  ILtQ[p-1/2,0] && IGtQ[(m-1)/2,0] *)
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcCosh[c*x])^n/(f*(m + 2*p + 1)) + 2*d*p/(m + 2*p + 1)* Int[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x] - b*c*n/(f*(m + 2*p + 1))* Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && Not[LtQ[m, -1]]
+Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^n/(f*(m + 2*p + 1)) + 2*d1*d2*p/(m + 2*p + 1)* Int[(f*x)^ m*(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^ n, x] - b*c*n/(f*(m + 2*p + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && GtQ[p, 0] && Not[LtQ[m, -1]]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^ n/(d*f*(m + 1)) + c^2*(m + 2*p + 3)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x] + b*c*n/(f*(m + 1))* Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
+Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(d1*d2*f*(m + 1)) + c^2*(m + 2*p + 3)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^n, x] + b*c*n/(f*(m + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && ILtQ[m, -1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^ n/(2*e*(p + 1)) - f^2*(m - 1)/(2*e*(p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x] - b*f*n/(2*c*(p + 1))* Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 1]
+Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1)) - f^2*(m - 1)/(2*e1*e2*(p + 1))* Int[(f*x)^(m - 2)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n, x] - b*f*n/(2*c*(p + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := -(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^ n/(2*d*f*(p + 1)) + (m + 2*p + 3)/(2*d*(p + 1))* Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x] - b*c*n/(2*f*(p + 1))* Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && Not[GtQ[m, 1]] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
+Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := -(f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(2*d1*d2*f*(p + 1)) + (m + 2*p + 3)/(2*d1*d2*(p + 1))* Int[(f*x)^ m*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^ n, x] - b*c*n/(2*f*(p + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && LtQ[p, -1] && Not[GtQ[m, 1]] && (IntegerQ[m] || EqQ[n, 1])
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^ n/(e*(m + 2*p + 1)) + f^2*(m - 1)/(c^2*(m + 2*p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x] - b*f*n/(c*(m + 2*p + 1))* Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
+Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(e1*e2*(m + 2*p + 1)) + f^2*(m - 1)/(c^2*(m + 2*p + 1))* Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^n, x] - b*f*n/(c*(m + 2*p + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol] := (f*x)^m* Simp[Sqrt[1 + c*x]* Sqrt[-1 + c*x]*(d + e*x^2)^p]*(a + b*ArcCosh[c*x])^(n + 1)/(b* c*(n + 1)) + f*m/(b*c*(n + 1))* Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Int[(f*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0]
+Int[(f_.*x_)^m_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol] := (f*x)^m* Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d1 + e1*x)^p]*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) + f*m/(b*c*(n + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Int[(f*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol] := (f*x)^m* Simp[Sqrt[1 + c*x]* Sqrt[-1 + c*x]*(d + e*x^2)^p]*(a + b*ArcCosh[c*x])^(n + 1)/(b* c*(n + 1)) + f*m/(b*c*(n + 1))* Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Int[(f*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x] - c*(m + 2*p + 1)/(b*f*(n + 1))* Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]
+Int[(f_.*x_)^m_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol] := (f*x)^m*Sqrt[1 + c*x]* Sqrt[-1 + c*x]*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) + f*m/(b*c*(n + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Int[(f*x)^(m - 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x] - c*(m + 2*p + 1)/(b*f*(n + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && LtQ[n, -1] && IGtQ[p + 1/2, 0] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]
+(* Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_ Symbol] := (f*x)^m*Simp[Sqrt[1+c*x]*Sqrt[-1+c*x]*(d+e*x^2)^p]*(a+b*ArcCosh[c*x] )^(n+1)/(b*c*(n+1)) - f*m/(b*c*(n+1))*Simp[(d+e*x^2)^p/((1+c*x)^p*(-1+c*x)^p)]* Int[(f*x)^(m-1)*(1+c*x)^(p+1/2)*(-1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x] )^(n+1),x] - c*(2*p+1)/(b*f*(n+1))*Simp[(d+e*x^2)^p/((1+c*x)^p*(-1+c*x)^p)]* Int[(f*x)^(m+1)*(1+c*x)^(p-1/2)*(-1+c*x)^(p-1/2)*(a+b*ArcCosh[c*x] )^(n+1),x] /; FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[c^2*d+e,0] && LtQ[n,-1] &&  NeQ[p,-1/2] && IntegerQ[2*p] && IGtQ[m,-3] *)
+(* Int[(f_.*x_)^m_.*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.* ArcCosh[c_.*x_])^n_,x_Symbol] := (f*x)^m*Sqrt[1+c*x]*Sqrt[-1+c*x]*(d1+e1*x)^p*(d2+e2*x)^p*(a+b* ArcCosh[c*x])^(n+1)/(b*c*(n+1)) - f*m/(b*c*(n+1))*Simp[(d1+e1*x)^p/(1+c*x)^p]*Simp[(d2+e2*x)^p/(-1+c* x)^p]* Int[(f*x)^(m-1)*(-1+c^2*x^2)^(p+1/2)*(a+b*ArcCosh[c*x])^(n+1),x] - c*(2*p+1)/(b*f*(n+1))*Simp[(d1+e1*x)^p/(1+c*x)^p]*Simp[(d2+e2*x)^p/( -1+c*x)^p]* Int[(f*x)^(m+1)*(-1+c^2*x^2)^(p-1/2)*(a+b*ArcCosh[c*x])^(n+1),x] /; FreeQ[{a,b,c,d1,e1,d2,e2,f,m,p},x] && EqQ[e1,c*d1] && EqQ[e2,-c*d2] &&  LtQ[n,-1] && ILtQ[p+1/2,0] && IGtQ[m,-3] *)
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCosh[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol] := f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])^n/(e*m) - b*f*n/(c*m)*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]/Sqrt[d + e*x^2]]* Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x] + f^2*(m - 1)/(c^2*m)* Int[(f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1]
+Int[(f_.*x_)^ m_*(a_. + b_.*ArcCosh[c_.*x_])^ n_./(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol] := f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n/(e1*e2*m) - b*f*n/(c*m)*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]* Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]* Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x] + f^2*(m - 1)/(c^2*m)* Int[(f*x)^(m - 2)*(a + b*ArcCosh[c*x])^ n/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && IGtQ[m, 1]
+Int[x_^m_*(a_. + b_.*ArcCosh[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol] := 1/c^(m + 1)*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]/Sqrt[d + e*x^2]]* Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
+Int[x_^m_*(a_. + b_.*ArcCosh[c_.*x_])^ n_./(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol] := 1/c^(m + 1)*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]* Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]* Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && IGtQ[n, 0] && IntegerQ[m]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCosh[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol] := (f*x)^(m + 1)/(f*(m + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]* (a + b*ArcCosh[c*x])* Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] + b*c*(f*x)^(m + 2)/(f^2*(m + 1)*(m + 2))* Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]/Sqrt[d + e*x^2]]* HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && Not[IntegerQ[m]]
+Int[(f_.*x_)^ m_*(a_. + b_.*ArcCosh[c_.*x_])/(Sqrt[d1_ + e1_.*x_]* Sqrt[d2_ + e2_.*x_]), x_Symbol] := (f*x)^(m + 1)/(f*(m + 1))* Simp[Sqrt[1 - c^2*x^2]/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])]* (a + b*ArcCosh[c*x])* Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] + b*c*(f*x)^(m + 2)/(f^2*(m + 1)*(m + 2))* Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]* Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]* HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && Not[IntegerQ[m]]
+Int[(f_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_/Sqrt[d_ + e_.*x_^2], x_Symbol] := (f*x)^m*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))* Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]/Sqrt[d + e*x^2]] - f*m/(b*c*(n + 1))* Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]/Sqrt[d + e*x^2]]* Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]
+Int[(f_.*x_)^ m_.*(a_. + b_.*ArcCosh[c_.*x_])^ n_/(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol] := (f*x)^m*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))* Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]* Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] - f*m/(b*c*(n + 1))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]* Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]* Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && LtQ[n, -1]
+Int[x_^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := 1/(b*c^(m + 1))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]* Subst[ Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
+Int[x_^m_.*(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := 1/(b*c^(m + 1))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]* Simp[(d2 + e2*x)^p/(-1 + c*x)^p]* Subst[ Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n/ Sqrt[d + e*x^2], (f*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && Not[IGtQ[(m + 1)/2, 0]] && (EqQ[m, -1] || EqQ[m, -2])
+Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^ n/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), (f*x)^ m*(d1 + e1*x)^(p + 1/2)*(d2 + e2*x)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[p + 1/2, 0] && Not[IGtQ[(m + 1)/2, 0]] && (EqQ[m, -1] || EqQ[m, -2])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol] := d*(f*x)^(m + 1)*(a + b*ArcCosh[c*x])/(f*(m + 1)) + e*(f*x)^(m + 3)*(a + b*ArcCosh[c*x])/(f^3*(m + 3)) - b*c/(f*(m + 1)*(m + 3))* Int[(f*x)^(m + 1)*(d*(m + 3) + e*(m + 1)*x^2)/(Sqrt[1 + c*x]* Sqrt[-1 + c*x]), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && NeQ[m, -1] && NeQ[m, -3]
+Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol] := (d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])/(2*e*(p + 1)) - b*c/(2*e*(p + 1))* Int[(d + e*x^2)^(p + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0])
+(* Int[x_^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^n_,x_Symbol]  := 1/(b*c^(m+2*p+1))*Subst[Int[x^n*Cosh[-a/b+x/b]^m*(c^2*d+e*Cosh[-a/b+ x/b]^2)^p*Sinh[-a/b+x/b],x],x,a+b*ArcCosh[c*x]] /; FreeQ[{a,b,c,d,e,n},x] && IGtQ[m,0] && IGtQ[p,0] *)
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Unintegrable[(f*x)^m*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
+Int[(f_.*x_)^m_.*(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Unintegrable[(f*x)^m*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, n, p}, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.5 u (a+b arccosh(c x))^n.m b/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.5 u (a+b arccosh(c x))^n.m
new file mode 100755
index 0000000..664faac
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.5 u (a+b arccosh(c x))^n.m	
@@ -0,0 +1,37 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.2.5 u (a+b arccosh(c x))^n *)
+Int[(a_. + b_.*ArcCosh[c_.*x_])^n_./(d_. + e_.*x_), x_Symbol] := Subst[Int[(a + b*x)^n*Sinh[x]/(c*d + e*Cosh[x]), x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (d + e*x)^(m + 1)*(a + b*ArcCosh[c*x])^n/(e*(m + 1)) - b*c*n/(e*(m + 1))* Int[(d + e*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[-1 + c*x]* Sqrt[1 + c*x]), x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
+Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := 1/c^(m + 1)* Subst[Int[(a + b*x)^n*Sinh[x]*(c*d + e*Cosh[x])^m, x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]
+Int[Px_*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol] := With[{u = IntHide[ExpandExpression[Px, x], x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Sqrt[1 - c^2*x^2]/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])* Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x]
+(* Int[Px_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol] := With[{u=IntHide[Px,x]}, Dist[(a+b*ArcCosh[c*x])^n,u,x] - b*c*n*Sqrt[1-c^2*x^2]/(Sqrt[-1+c*x]*Sqrt[1+c*x])*Int[ SimplifyIntegrand[u*(a+b*ArcCosh[c*x])^(n-1)/Sqrt[1-c^2*x^2],x],x]] /; FreeQ[{a,b,c},x] && PolyQ[Px,x] && IGtQ[n,0] *)
+Int[Px_*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol] := Int[ExpandIntegrand[Px*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, n}, x] && PolyQ[Px, x]
+Int[Px_*(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol] := With[{u = IntHide[Px*(d + e*x)^m, x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Sqrt[1 - c^2*x^2]/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])* Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Px, x]
+Int[(f_. + g_.*x_)^p_.*(d_ + e_.*x_)^m_*(a_. + b_.*ArcCosh[c_.*x_])^ n_, x_Symbol] := With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcCosh[c*x])^n, u, x] - b*c*n* Int[SimplifyIntegrand[ u*(a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]
+Int[(f_. + g_.*x_ + h_.*x_^2)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_/(d_ + e_.*x_)^2, x_Symbol] := With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcCosh[c*x])^n, u, x] - b*c*n* Int[SimplifyIntegrand[ u*(a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]
+Int[Px_*(d_ + e_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[Px*(d + e*x)^m*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && IGtQ[n, 0] && IntegerQ[m]
+Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_])^ n_., x_Symbol] := (-d)^IntPart[p]*(d + e*x^2)^ FracPart[p]/((-1 + c*x)^FracPart[p]*(1 + c*x)^FracPart[p])* Int[(f + g*x)^m*(-1 + c*x)^p*(1 + c*x)^p*(a + b*ArcCosh[c*x])^ n, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && IntegerQ[m]
+Int[Log[h_.*(f_. + g_.*x_)^m_.]*(d_ + e_.*x_^2)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (-d)^IntPart[p]*(d + e*x^2)^ FracPart[p]/((-1 + c*x)^FracPart[p]*(1 + c*x)^FracPart[p])* Int[ Log[h*(f + g*x)^m]*(-1 + c*x)^p*(1 + c*x)^p*(a + b*ArcCosh[c*x])^ n, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
+Int[(f_ + g_.*x_)^m_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol] := With[{u = IntHide[(f + g*x)^m*(d1 + e1*x)^p*(d2 + e2*x)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Int[Dist[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), u, x], x]] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && (GtQ[m, 3] || LtQ[m, -2*p - 1])
+Int[(f_ + g_.*x_)^m_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0] && (EqQ[n, 1] && GtQ[p, -1] || GtQ[p, 0] || EqQ[m, 1] || EqQ[m, 2] && LtQ[p, -2])
+Int[(f_ + g_.*x_)^m_*Sqrt[d1_ + e1_.*x_]* Sqrt[d2_ + e2_.*x_]*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (f + g*x)^ m*(d1*d2 + e1*e2*x^2)*(a + b*ArcCosh[c*x])^(n + 1)/(b*c* Sqrt[-d1*d2]*(n + 1)) - 1/(b*c*Sqrt[-d1*d2]*(n + 1))* Int[(d1*d2*g*m + 2*e1*e2*f*x + e1*e2*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && ILtQ[m, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^m_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[ Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n, (f + g*x)^ m*(d1 + e1*x)^(p - 1/2)*(d2 + e2*x)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^m_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (f + g*x)^ m*(d1 + e1*x)^(p + 1/2)*(d2 + e2*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n + 1)/(b*c* Sqrt[-d1*d2]*(n + 1)) - 1/(b*c*Sqrt[-d1*d2]*(n + 1))* Int[ ExpandIntegrand[(f + g*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), (d1*d2*g*m + e1*e2*f*(2*p + 1)*x + e1*e2*g*(m + 2*p + 1)*x^2)*(d1 + e1*x)^(p - 1/2)*(d2 + e2*x)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && ILtQ[m, 0] && IGtQ[p - 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^ m_.*(a_. + b_.*ArcCosh[c_.*x_])^ n_/(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol] := (f + g*x)^ m*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-d1*d2]*(n + 1)) - g*m/(b*c*Sqrt[-d1*d2]*(n + 1))* Int[(f + g*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && GtQ[d1, 0] && LtQ[d2, 0] && LtQ[n, -1]
+Int[(f_ + g_.*x_)^ m_.*(a_. + b_.*ArcCosh[c_.*x_])^ n_./(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol] := 1/(c^(m + 1)*Sqrt[-d1*d2])* Subst[Int[(a + b*x)^n*(c*f + g*Cosh[x])^m, x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && GtQ[d1, 0] && LtQ[d2, 0] && (GtQ[m, 0] || IGtQ[n, 0])
+Int[(f_ + g_.*x_)^m_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^ n/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), (f + g*x)^ m*(d1 + e1*x)^(p + 1/2)*(d2 + e2*x)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]
+Int[(f_ + g_.*x_)^m_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[p]/((-1 + c*x)^FracPart[p]*(1 + c*x)^FracPart[p])* Int[(f + g*x)^m*(-1 + c*x)^p*(1 + c*x)^p*(a + b*ArcCosh[c*x])^ n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && IntegerQ[p - 1/2] && Not[GtQ[d1, 0] && LtQ[d2, 0]]
+Int[Log[h_.*(f_. + g_.*x_)^m_.]*(a_. + b_.*ArcCosh[c_.*x_])^ n_./(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol] := Log[h*(f + g*x)^m]*(a + b*ArcCosh[c*x])^(n + 1)/(b*c* Sqrt[-d1*d2]*(n + 1)) - g*m/(b*c*Sqrt[-d1*d2]*(n + 1))* Int[(a + b*ArcCosh[c*x])^(n + 1)/(f + g*x), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, h, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]
+Int[Log[h_.*(f_. + g_.*x_)^m_.]*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := (-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[p]/((-1 + c*x)^FracPart[p]*(1 + c*x)^FracPart[p])* Int[ Log[h*(f + g*x)^m]*(-1 + c*x)^p*(1 + c*x)^p*(a + b*ArcCosh[c*x])^ n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, h, m, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2] && Not[GtQ[d1, 0] && LtQ[d2, 0]]
+Int[(d_ + e_.*x_)^m_*(f_ + g_.*x_)^m_*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x)^m*(f + g*x)^m, x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Int[Dist[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), u, x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m + 1/2, 0]
+Int[(d_ + e_.*x_)^m_.*(f_ + g_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_])^ n_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^ n, (d + e*x)^m*(f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && IntegerQ[m]
+Int[u_*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol] := With[{v = IntHide[u, x]}, Dist[a + b*ArcCosh[c*x], v, x] - b*c*Sqrt[1 - c^2*x^2]/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])* Int[SimplifyIntegrand[v/Sqrt[1 - c^2*x^2], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
+Int[Px_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol] := With[{u = ExpandIntegrand[ Px*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && PolyQ[Px, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2]
+Int[Px_.*(f_ + g_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^p_)^ m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := With[{u = ExpandIntegrand[ Px*(f + g*(d1 + e1*x)^p*(d2 + e2*x)^p)^m*(a + b*ArcCosh[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && PolyQ[Px, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[p + 1/2, 0] && IntegersQ[m, n]
+Int[RFx_*ArcCosh[c_.*x_]^n_., x_Symbol] := With[{u = ExpandIntegrand[ArcCosh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[c, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
+Int[RFx_*(a_ + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[RFx*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
+Int[RFx_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^p_*ArcCosh[c_.*x_]^n_., x_Symbol] := With[{u = ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p*ArcCosh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d1, e1, d2, e2}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2]
+Int[RFx_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_ + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Int[ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p, RFx*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2]
+Int[u_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol] := Unintegrable[u*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.6 Miscellaneous inverse hyperbolic cosine.m b/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.6 Miscellaneous inverse hyperbolic cosine.m
new file mode 100755
index 0000000..70178f8
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.6 Miscellaneous inverse hyperbolic cosine.m	
@@ -0,0 +1,29 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.2.6 Miscellaneous inverse hyperbolic cosine *)
+Int[(a_. + b_.*ArcCosh[c_ + d_.*x_])^n_., x_Symbol] := 1/d*Subst[Int[(a + b*ArcCosh[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, n}, x]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCosh[c_ + d_.*x_])^n_., x_Symbol] := 1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcCosh[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
+Int[(A_. + B_.*x_ + C_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_ + d_.*x_])^ n_., x_Symbol] := 1/d*Subst[Int[(-C/d^2 + C/d^2*x^2)^p*(a + b*ArcCosh[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, A, B, C, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[(e_. + f_.*x_)^m_.*(A_. + B_.*x_ + C_.*x_^2)^ p_.*(a_. + b_.*ArcCosh[c_ + d_.*x_])^n_., x_Symbol] := 1/d*Subst[ Int[((d*e - c*f)/d + f*x/d)^m*(-C/d^2 + C/d^2*x^2)^ p*(a + b*ArcCosh[x])^n, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[Sqrt[a_. + b_.*ArcCosh[1 + d_.*x_^2]], x_Symbol] := 2*Sqrt[a + b*ArcCosh[1 + d*x^2]]* Sinh[(1/2)*ArcCosh[1 + d*x^2]]^2/(d*x) - Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])* Sinh[(1/2)*ArcCosh[1 + d*x^2]]* Erfi[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[1 + d*x^2]]]/(d*x) + Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])* Sinh[(1/2)*ArcCosh[1 + d*x^2]]* Erf[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[1 + d*x^2]]]/(d*x) /; FreeQ[{a, b, d}, x]
+Int[Sqrt[a_. + b_.*ArcCosh[-1 + d_.*x_^2]], x_Symbol] := 2*Sqrt[a + b*ArcCosh[-1 + d*x^2]]* Cosh[(1/2)*ArcCosh[-1 + d*x^2]]^2/(d*x) - Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])* Cosh[(1/2)*ArcCosh[-1 + d*x^2]]* Erfi[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[-1 + d*x^2]]]/(d*x) - Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])* Cosh[(1/2)*ArcCosh[-1 + d*x^2]]* Erf[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[-1 + d*x^2]]]/(d*x) /; FreeQ[{a, b, d}, x]
+Int[(a_. + b_.*ArcCosh[c_ + d_.*x_^2])^n_, x_Symbol] := x*(a + b*ArcCosh[c + d*x^2])^n - 2*b* n*(2*c*d*x^2 + d^2*x^4)*(a + b*ArcCosh[c + d*x^2])^(n - 1)/(d*x* Sqrt[-1 + c + d*x^2]*Sqrt[1 + c + d*x^2]) + 4*b^2*n*(n - 1)*Int[(a + b*ArcCosh[c + d*x^2])^(n - 2), x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && GtQ[n, 1]
+Int[1/(a_. + b_.*ArcCosh[1 + d_.*x_^2]), x_Symbol] := x*Cosh[a/(2*b)]* CoshIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[d*x^2]) - x*Sinh[a/(2*b)]* SinhIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[d*x^2]) /; FreeQ[{a, b, d}, x]
+Int[1/(a_. + b_.*ArcCosh[-1 + d_.*x_^2]), x_Symbol] := -x*Sinh[a/(2*b)]* CoshIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[d*x^2]) + x*Cosh[a/(2*b)]* SinhIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[d*x^2]) /; FreeQ[{a, b, d}, x]
+Int[1/Sqrt[a_. + b_.*ArcCosh[1 + d_.*x_^2]], x_Symbol] := Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])* Sinh[ArcCosh[1 + d*x^2]/2]* Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x) + Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])* Sinh[ArcCosh[1 + d*x^2]/2]* Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x) /; FreeQ[{a, b, d}, x]
+Int[1/Sqrt[a_. + b_.*ArcCosh[-1 + d_.*x_^2]], x_Symbol] := Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])* Cosh[ArcCosh[-1 + d*x^2]/2]* Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x) - Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])* Cosh[ArcCosh[-1 + d*x^2]/2]* Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x) /; FreeQ[{a, b, d}, x]
+Int[1/(a_. + b_.*ArcCosh[1 + d_.*x_^2])^(3/2), x_Symbol] := -Sqrt[d*x^2]* Sqrt[2 + d*x^2]/(b*d*x*Sqrt[a + b*ArcCosh[1 + d*x^2]]) + Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])* Sinh[ArcCosh[1 + d*x^2]/2]* Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/Sqrt[2*b]]/(b^(3/2)*d*x) - Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])* Sinh[ArcCosh[1 + d*x^2]/2]* Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/Sqrt[2*b]]/(b^(3/2)*d*x) /; FreeQ[{a, b, d}, x]
+Int[1/(a_. + b_.*ArcCosh[-1 + d_.*x_^2])^(3/2), x_Symbol] := -Sqrt[d*x^2]* Sqrt[-2 + d*x^2]/(b*d*x*Sqrt[a + b*ArcCosh[-1 + d*x^2]]) + Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])* Cosh[ArcCosh[-1 + d*x^2]/2]* Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]]/(b^(3/2)*d*x) + Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])* Cosh[ArcCosh[-1 + d*x^2]/2]* Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]]/(b^(3/2)*d*x) /; FreeQ[{a, b, d}, x]
+Int[1/(a_. + b_.*ArcCosh[1 + d_.*x_^2])^2, x_Symbol] := -Sqrt[d*x^2]*Sqrt[2 + d*x^2]/(2*b*d*x*(a + b*ArcCosh[1 + d*x^2])) - x*Sinh[a/(2*b)]* CoshIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[d*x^2]) + x*Cosh[a/(2*b)]* SinhIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[d*x^2]) /; FreeQ[{a, b, d}, x]
+Int[1/(a_. + b_.*ArcCosh[-1 + d_.*x_^2])^2, x_Symbol] := -Sqrt[d*x^2]* Sqrt[-2 + d*x^2]/(2*b*d*x*(a + b*ArcCosh[-1 + d*x^2])) + x*Cosh[a/(2*b)]* CoshIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[d*x^2]) - x*Sinh[a/(2*b)]* SinhIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[d*x^2]) /; FreeQ[{a, b, d}, x]
+Int[(a_. + b_.*ArcCosh[c_ + d_.*x_^2])^n_, x_Symbol] := -x*(a + b*ArcCosh[c + d*x^2])^(n + 2)/(4*b^2*(n + 1)*(n + 2)) + (2*c*x^2 + d*x^4)*(a + b*ArcCosh[c + d*x^2])^(n + 1)/(2*b*(n + 1)*x* Sqrt[-1 + c + d*x^2]*Sqrt[1 + c + d*x^2]) + 1/(4*b^2*(n + 1)*(n + 2))* Int[(a + b*ArcCosh[c + d*x^2])^(n + 2), x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && LtQ[n, -1] && NeQ[n, -2]
+Int[ArcCosh[a_.*x_^p_]^n_./x_, x_Symbol] := 1/p*Subst[Int[x^n*Tanh[x], x], x, ArcCosh[a*x^p]] /; FreeQ[{a, p}, x] && IGtQ[n, 0]
+Int[u_.*ArcCosh[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol] := Int[u*ArcSech[a/c + b*x^n/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
+Int[ArcCosh[Sqrt[1 + b_.*x_^2]]^n_./Sqrt[1 + b_.*x_^2], x_Symbol] := Sqrt[-1 + Sqrt[1 + b*x^2]]*Sqrt[1 + Sqrt[1 + b*x^2]]/(b*x)* Subst[Int[ArcCosh[x]^n/(Sqrt[-1 + x]*Sqrt[1 + x]), x], x, Sqrt[1 + b*x^2]] /; FreeQ[{b, n}, x]
+Int[f_^(c_.*ArcCosh[a_. + b_.*x_]^n_.), x_Symbol] := 1/b*Subst[Int[f^(c*x^n)*Sinh[x], x], x, ArcCosh[a + b*x]] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
+Int[x_^m_.*f_^(c_.*ArcCosh[a_. + b_.*x_]^n_.), x_Symbol] := 1/b*Subst[Int[(-a/b + Cosh[x]/b)^m*f^(c*x^n)*Sinh[x], x], x, ArcCosh[a + b*x]] /; FreeQ[{a, b, c, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
+Int[ArcCosh[u_], x_Symbol] := x*ArcCosh[u] - Int[SimplifyIntegrand[x*D[u, x]/(Sqrt[-1 + u]*Sqrt[1 + u]), x], x] /; InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcCosh[u_]), x_Symbol] := (c + d*x)^(m + 1)*(a + b*ArcCosh[u])/(d*(m + 1)) - b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)* D[u, x]/(Sqrt[-1 + u]*Sqrt[1 + u]), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]
+Int[v_*(a_. + b_.*ArcCosh[u_]), x_Symbol] := With[{w = IntHide[v, x]}, Dist[(a + b*ArcCosh[u]), w, x] - b*Int[SimplifyIntegrand[w*D[u, x]/(Sqrt[-1 + u]*Sqrt[1 + u]), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_. /; FreeQ[{c, d, m}, x]]]
+Int[E^(n_.*ArcCosh[u_]), x_Symbol] := Int[(u + Sqrt[-1 + u]*Sqrt[1 + u])^n, x] /; IntegerQ[n] && PolyQ[u, x]
+Int[x_^m_.*E^(n_.*ArcCosh[u_]), x_Symbol] := Int[x^m*(u + Sqrt[-1 + u]*Sqrt[1 + u])^n, x] /; RationalQ[m] && IntegerQ[n] && PolyQ[u, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 (a+b arctanh(c x^n))^p.m b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 (a+b arctanh(c x^n))^p.m
new file mode 100755
index 0000000..cbc9744
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 (a+b arctanh(c x^n))^p.m	
@@ -0,0 +1,13 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.3.1 (a+b arctanh(c x^n))^p *)
+Int[(a_. + b_.*ArcTanh[c_.*x_^n_.])^p_., x_Symbol] := x*(a + b*ArcTanh[c*x^n])^p - b*c*n*p* Int[x^n*(a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n)), x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
+Int[(a_. + b_.*ArcCoth[c_.*x_^n_.])^p_., x_Symbol] := x*(a + b*ArcCoth[c*x^n])^p - b*c*n*p* Int[x^n*(a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n)), x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
+Int[(a_. + b_.*ArcTanh[c_.*x_^n_])^p_, x_Symbol] := Int[ExpandIntegrand[(a + b*Log[1 + c*x^n]/2 - b*Log[1 - c*x^n]/2)^ p, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_^n_])^p_, x_Symbol] := Int[ExpandIntegrand[(a + b*Log[1 + x^(-n)/c]/2 - b*Log[1 - x^(-n)/c]/2)^p, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_^n_])^p_, x_Symbol] := Int[(a + b*ArcCoth[x^(-n)/c])^p, x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && ILtQ[n, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_^n_])^p_, x_Symbol] := Int[(a + b*ArcTanh[x^(-n)/c])^p, x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && ILtQ[n, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_^n_])^p_, x_Symbol] := With[{k = Denominator[n]}, k*Subst[Int[x^(k - 1)*(a + b*ArcTanh[c*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && FractionQ[n]
+Int[(a_. + b_.*ArcCoth[c_.*x_^n_])^p_, x_Symbol] := With[{k = Denominator[n]}, k*Subst[Int[x^(k - 1)*(a + b*ArcCoth[c*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && FractionQ[n]
+Int[(a_. + b_.*ArcTanh[c_.*x_^n_.])^p_, x_Symbol] := Unintegrable[(a + b*ArcTanh[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p}, x]
+Int[(a_. + b_.*ArcCoth[c_.*x_^n_.])^p_, x_Symbol] := Unintegrable[(a + b*ArcCoth[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p}, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m
new file mode 100755
index 0000000..2d374bf
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m	
@@ -0,0 +1,198 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.3.1 u (a+b arctanh(c x^n))^p *)
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := x*(a + b*ArcTanh[c*x])^p - b*c*p*Int[x*(a + b*ArcTanh[c*x])^(p - 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := x*(a + b*ArcCoth[c*x])^p - b*c*p*Int[x*(a + b*ArcCoth[c*x])^(p - 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])/x_, x_Symbol] := a*Log[x] - b/2*PolyLog[2, -c*x] + b/2*PolyLog[2, c*x] /; FreeQ[{a, b, c}, x]
+Int[(a_. + b_.*ArcCoth[c_.*x_])/x_, x_Symbol] := a*Log[x] + b/2*PolyLog[2, -1/(c*x)] - b/2*PolyLog[2, 1/(c*x)] /; FreeQ[{a, b, c}, x]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_/x_, x_Symbol] := 2*(a + b*ArcTanh[c*x])^p*ArcTanh[1 - 2/(1 - c*x)] - 2*b*c*p* Int[(a + b*ArcTanh[c*x])^(p - 1)* ArcTanh[1 - 2/(1 - c*x)]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_/x_, x_Symbol] := 2*(a + b*ArcCoth[c*x])^p*ArcCoth[1 - 2/(1 - c*x)] - 2*b*c*p* Int[(a + b*ArcCoth[c*x])^(p - 1)* ArcCoth[1 - 2/(1 - c*x)]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := (d*x)^(m + 1)*(a + b*ArcTanh[c*x])^p/(d*(m + 1)) - b*c*p/(d*(m + 1))* Int[(d*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := (d*x)^(m + 1)*(a + b*ArcCoth[c*x])^p/(d*(m + 1)) - b*c*p/(d*(m + 1))* Int[(d*x)^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := -(a + b*ArcTanh[c*x])^p*Log[2/(1 + e*x/d)]/e + b*c*p/e* Int[(a + b*ArcTanh[c*x])^(p - 1)*Log[2/(1 + e*x/d)]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := -(a + b*ArcCoth[c*x])^p*Log[2/(1 + e*x/d)]/e + b*c*p/e* Int[(a + b*ArcCoth[c*x])^(p - 1)*Log[2/(1 + e*x/d)]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)]/e + b*c/e*Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x] + (a + b*ArcTanh[c*x])*Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b*c/e* Int[Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcCoth[c*x])*Log[2/(1 + c*x)]/e + b*c/e*Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x] + (a + b*ArcCoth[c*x])*Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b*c/e* Int[Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^2/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)]/e + b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)]/e + b^2*PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) + (a + b*ArcTanh[c*x])^2* Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b*(a + b*ArcTanh[c*x])* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b^2*PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^2/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcCoth[c*x])^2*Log[2/(1 + c*x)]/e + b*(a + b*ArcCoth[c*x])*PolyLog[2, 1 - 2/(1 + c*x)]/e + b^2*PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) + (a + b*ArcCoth[c*x])^2* Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b*(a + b*ArcCoth[c*x])* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b^2*PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^3/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcTanh[c*x])^3*Log[2/(1 + c*x)]/e + 3*b*(a + b*ArcTanh[c*x])^2*PolyLog[2, 1 - 2/(1 + c*x)]/(2*e) + 3*b^2*(a + b*ArcTanh[c*x])*PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) + 3*b^3*PolyLog[4, 1 - 2/(1 + c*x)]/(4*e) + (a + b*ArcTanh[c*x])^3* Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - 3*b*(a + b*ArcTanh[c*x])^2* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) - 3*b^2*(a + b*ArcTanh[c*x])* PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) - 3*b^3*PolyLog[4, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(4*e) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^3/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcCoth[c*x])^3*Log[2/(1 + c*x)]/e + 3*b*(a + b*ArcCoth[c*x])^2*PolyLog[2, 1 - 2/(1 + c*x)]/(2*e) + 3*b^2*(a + b*ArcCoth[c*x])*PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) + 3*b^3*PolyLog[4, 1 - 2/(1 + c*x)]/(4*e) + (a + b*ArcCoth[c*x])^3* Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - 3*b*(a + b*ArcCoth[c*x])^2* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) - 3*b^2*(a + b*ArcCoth[c*x])* PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) - 3*b^3*PolyLog[4, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(4*e) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := (d + e*x)^(q + 1)*(a + b*ArcTanh[c*x])/(e*(q + 1)) - b*c/(e*(q + 1))*Int[(d + e*x)^(q + 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := (d + e*x)^(q + 1)*(a + b*ArcCoth[c*x])/(e*(q + 1)) - b*c/(e*(q + 1))*Int[(d + e*x)^(q + 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol] := (d + e*x)^(q + 1)*(a + b*ArcTanh[c*x])^p/(e*(q + 1)) - b*c*p/(e*(q + 1))* Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol] := (d + e*x)^(q + 1)*(a + b*ArcCoth[c*x])^p/(e*(q + 1)) - b*c*p/(e*(q + 1))* Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
+Int[(f_.*x_)^m_.*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := f/e*Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x])^p, x] - d*f/e*Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x])^p/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && GtQ[m, 0]
+Int[(f_.*x_)^m_.*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := f/e*Int[(f*x)^(m - 1)*(a + b*ArcCoth[c*x])^p, x] - d*f/e*Int[(f*x)^(m - 1)*(a + b*ArcCoth[c*x])^p/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && GtQ[m, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(x_*(d_ + e_.*x_)), x_Symbol] := (a + b*ArcTanh[c*x])^p*Log[2 - 2/(1 + e*x/d)]/d - b*c*p/d* Int[(a + b*ArcTanh[c*x])^(p - 1)* Log[2 - 2/(1 + e*x/d)]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(x_*(d_ + e_.*x_)), x_Symbol] := (a + b*ArcCoth[c*x])^p*Log[2 - 2/(1 + e*x/d)]/d - b*c*p/d* Int[(a + b*ArcCoth[c*x])^(p - 1)* Log[2 - 2/(1 + e*x/d)]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := 1/d*Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x] - e/(d*f)*Int[(f*x)^(m + 1)*(a + b*ArcTanh[c*x])^p/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := 1/d*Int[(f*x)^m*(a + b*ArcCoth[c*x])^p, x] - e/(d*f)*Int[(f*x)^(m + 1)*(a + b*ArcCoth[c*x])^p/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcTanh[c*x]), u] - b*c*Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[ 2*m] && (IGtQ[m, 0] && IGtQ[q, 0] || ILtQ[m + q + 1, 0] && LtQ[m*q, 0])
+Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcCoth[c*x]), u] - b*c*Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[ 2*m] && (IGtQ[m, 0] && IGtQ[q, 0] || ILtQ[m + q + 1, 0] && LtQ[m*q, 0])
+Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcTanh[c*x])^p, u] - b*c*p*Int[ ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1), u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 - e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]
+Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcCoth[c*x])^p, u] - b*c*p*Int[ ExpandIntegrand[(a + b*ArcCoth[c*x])^(p - 1), u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 - e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := b*(d + e*x^2)^q/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcTanh[c*x])/(2*q + 1) + 2*d*q/(2*q + 1)* Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0]
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := b*(d + e*x^2)^q/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcCoth[c*x])/(2*q + 1) + 2*d*q/(2*q + 1)* Int[(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0]
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol] := b*p*(d + e*x^2)^ q*(a + b*ArcTanh[c*x])^(p - 1)/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p/(2*q + 1) + 2*d*q/(2*q + 1)* Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x] - b^2*d*p*(p - 1)/(2*q*(2*q + 1))* Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^(p - 2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && GtQ[p, 1]
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol] := b*p*(d + e*x^2)^ q*(a + b*ArcCoth[c*x])^(p - 1)/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^p/(2*q + 1) + 2*d*q/(2*q + 1)* Int[(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x])^p, x] - b^2*d*p*(p - 1)/(2*q*(2*q + 1))* Int[(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x])^(p - 2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && GtQ[p, 1]
+(* Int[(a_.+b_.*ArcTanh[c_.*x_])^p_./(d_+e_.*x_^2),x_Symbol] := 1/(c*d)*Subst[Int[(a+b*x)^p,x],x,ArcTanh[c*x]] /; FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e,0] *)
+(* Int[(a_.+b_.*ArcCoth[c_.*x_])^p_./(d_+e_.*x_^2),x_Symbol] := 1/(c*d)*Subst[Int[(a+b*x)^p,x],x,ArcCoth[c*x]] /; FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e,0] *)
+Int[1/((d_ + e_.*x_^2)*(a_. + b_.*ArcTanh[c_.*x_])), x_Symbol] := Log[RemoveContent[a + b*ArcTanh[c*x], x]]/(b*c*d) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
+Int[1/((d_ + e_.*x_^2)*(a_. + b_.*ArcCoth[c_.*x_])), x_Symbol] := Log[RemoveContent[a + b*ArcCoth[c*x], x]]/(b*c*d) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)) /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)) /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
+Int[(a_. + b_.*ArcTanh[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol] := -2*(a + b*ArcTanh[c*x])* ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*Sqrt[d]) - I*b*PolyLog[2, -I*Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*Sqrt[d]) + I*b*PolyLog[2, I*Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*Sqrt[d]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol] := -2*(a + b*ArcCoth[c*x])* ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*Sqrt[d]) - I*b*PolyLog[2, -I*Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*Sqrt[d]) + I*b*PolyLog[2, I*Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*Sqrt[d]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := 1/(c*Sqrt[d])*Subst[Int[(a + b*x)^p*Sech[x], x], x, ArcTanh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := -x*Sqrt[1 - 1/(c^2*x^2)]/Sqrt[d + e*x^2]* Subst[Int[(a + b*x)^p*Csch[x], x], x, ArcCoth[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcTanh[c*x])^p/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && Not[GtQ[d, 0]]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcCoth[c*x])^p/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && Not[GtQ[d, 0]]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := x*(a + b*ArcTanh[c*x])^p/(2*d*(d + e*x^2)) + (a + b*ArcTanh[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)) - b*c*p/2*Int[x*(a + b*ArcTanh[c*x])^(p - 1)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := x*(a + b*ArcCoth[c*x])^p/(2*d*(d + e*x^2)) + (a + b*ArcCoth[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)) - b*c*p/2*Int[x*(a + b*ArcCoth[c*x])^(p - 1)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])/(d_ + e_.*x_^2)^(3/2), x_Symbol] := -b/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcTanh[c*x])/(d*Sqrt[d + e*x^2]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])/(d_ + e_.*x_^2)^(3/2), x_Symbol] := -b/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcCoth[c*x])/(d*Sqrt[d + e*x^2]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := -b*(d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := -b*(d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_/(d_ + e_.*x_^2)^(3/2), x_Symbol] := -b*p*(a + b*ArcTanh[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcTanh[c*x])^p/(d*Sqrt[d + e*x^2]) + b^2*p*(p - 1)* Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2)^(3/2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_/(d_ + e_.*x_^2)^(3/2), x_Symbol] := -b*p*(a + b*ArcCoth[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcCoth[c*x])^p/(d*Sqrt[d + e*x^2]) + b^2*p*(p - 1)* Int[(a + b*ArcCoth[c*x])^(p - 2)/(d + e*x^2)^(3/2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol] := -b*p*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p - 1)/(4*c* d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x] + b^2*p*(p - 1)/(4*(q + 1)^2)* Int[(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol] := -b*p*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p - 1)/(4*c* d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p, x] + b^2*p*(p - 1)/(4*(q + 1)^2)* Int[(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p + 1)/(b*c* d*(p + 1)) + 2*c*(q + 1)/(b*(p + 1))* Int[x*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && LtQ[p, -1]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p + 1)/(b*c* d*(p + 1)) + 2*c*(q + 1)/(b*(p + 1))* Int[x*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && LtQ[p, -1]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := d^q/c* Subst[Int[(a + b*x)^p/Cosh[x]^(2*(q + 1)), x], x, ArcTanh[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := d^(q + 1/2)*Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(1 - c^2*x^2)^q*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && Not[IntegerQ[q] || GtQ[d, 0]]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := -(-d)^q/c* Subst[Int[(a + b*x)^p/Sinh[x]^(2*(q + 1)), x], x, ArcCoth[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && IntegerQ[q]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := -(-d)^(q + 1/2)*x*Sqrt[(c^2*x^2 - 1)/(c^2*x^2)]/Sqrt[d + e*x^2]* Subst[Int[(a + b*x)^p/Sinh[x]^(2*(q + 1)), x], x, ArcCoth[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && Not[IntegerQ[q]]
+Int[ArcTanh[c_.*x_]/(d_. + e_.*x_^2), x_Symbol] := 1/2*Int[Log[1 + c*x]/(d + e*x^2), x] - 1/2*Int[Log[1 - c*x]/(d + e*x^2), x] /; FreeQ[{c, d, e}, x]
+Int[ArcCoth[c_.*x_]/(d_. + e_.*x_^2), x_Symbol] := 1/2*Int[Log[1 + 1/(c*x)]/(d + e*x^2), x] - 1/2*Int[Log[1 - 1/(c*x)]/(d + e*x^2), x] /; FreeQ[{c, d, e}, x]
+Int[(a_ + b_.*ArcTanh[c_.*x_])/(d_. + e_.*x_^2), x_Symbol] := a*Int[1/(d + e*x^2), x] + b*Int[ArcTanh[c*x]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x]
+Int[(a_ + b_.*ArcCoth[c_.*x_])/(d_. + e_.*x_^2), x_Symbol] := a*Int[1/(d + e*x^2), x] + b*Int[ArcCoth[c*x]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x]
+Int[(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^q, x]}, Dist[a + b*ArcTanh[c*x], u, x] - b*c*Int[u/(1 - c^2*x^2), x]] /; FreeQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
+Int[(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^q, x]}, Dist[a + b*ArcCoth[c*x], u, x] - b*c*Int[u/(1 - c^2*x^2), x]] /; FreeQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, (d + e*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^p, (d + e*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := f^2/e*Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p, x] - d*f^2/e* Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := f^2/e*Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p, x] - d*f^2/e* Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/d*Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x] - e/(d*f^2)* Int[(f*x)^(m + 2)*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/d*Int[(f*x)^m*(a + b*ArcCoth[c*x])^p, x] - e/(d*f^2)* Int[(f*x)^(m + 2)*(a + b*ArcCoth[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
+Int[x_*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)) + 1/(c*d)*Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
+Int[x_*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcCoth[c*x])^(p + 1)/(b*e*(p + 1)) + 1/(c*d)*Int[(a + b*ArcCoth[c*x])^p/(1 - c*x), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
+Int[x_*(a_. + b_.*ArcTanh[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol] := x*(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)) - 1/(b*c*d*(p + 1))*Int[(a + b*ArcTanh[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Not[IGtQ[p, 0]] && NeQ[p, -1]
+Int[x_*(a_. + b_.*ArcCoth[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol] := -x*(a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)) - 1/(b*c*d*(p + 1))*Int[(a + b*ArcCoth[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Not[IGtQ[p, 0]] && NeQ[p, -1]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(x_*(d_ + e_.*x_^2)), x_Symbol] := (a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)) + 1/d*Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(x_*(d_ + e_.*x_^2)), x_Symbol] := (a + b*ArcCoth[c*x])^(p + 1)/(b*d*(p + 1)) + 1/d*Int[(a + b*ArcCoth[c*x])^p/(x*(1 + c*x)), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol] := (f*x)^m*(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)) - f*m/(b*c*d*(p + 1))* Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol] := (f*x)^m*(a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)) - f*m/(b*c*d*(p + 1))* Int[(f*x)^(m - 1)*(a + b*ArcCoth[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1]
+Int[x_^m_.*(a_. + b_.*ArcTanh[c_.*x_])/(d_ + e_.*x_^2), x_Symbol] := Int[ExpandIntegrand[(a + b*ArcTanh[c*x]), x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && Not[EqQ[m, 1] && NeQ[a, 0]]
+Int[x_^m_.*(a_. + b_.*ArcCoth[c_.*x_])/(d_ + e_.*x_^2), x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCoth[c*x]), x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && Not[EqQ[m, 1] && NeQ[a, 0]]
+Int[x_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p/(2*e*(q + 1)) + b*p/(2*c*(q + 1))* Int[(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]
+Int[x_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p/(2*e*(q + 1)) + b*p/(2*c*(q + 1))* Int[(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 1), x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]
+Int[x_*(a_. + b_.*ArcTanh[c_.*x_])^p_/(d_ + e_.*x_^2)^2, x_Symbol] := x*(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)*(d + e*x^2)) + (1 + c^2*x^2)*(a + b*ArcTanh[c*x])^(p + 2)/(b^2* e*(p + 1)*(p + 2)*(d + e*x^2)) + 4/(b^2*(p + 1)*(p + 2))* Int[x*(a + b*ArcTanh[c*x])^(p + 2)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1] && NeQ[p, -2]
+Int[x_*(a_. + b_.*ArcCoth[c_.*x_])^p_/(d_ + e_.*x_^2)^2, x_Symbol] := x*(a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)*(d + e*x^2)) + (1 + c^2*x^2)*(a + b*ArcCoth[c*x])^(p + 2)/(b^2* e*(p + 1)*(p + 2)*(d + e*x^2)) + 4/(b^2*(p + 1)*(p + 2))* Int[x*(a + b*ArcCoth[c*x])^(p + 2)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1] && NeQ[p, -2]
+Int[x_^2*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := -b*(d + e*x^2)^(q + 1)/(4*c^3*d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])/(2*c^2*d*(q + 1)) + 1/(2*c^2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -5/2]
+Int[x_^2*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := -b*(d + e*x^2)^(q + 1)/(4*c^3*d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])/(2*c^2*d*(q + 1)) + 1/(2*c^2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -5/2]
+Int[x_^2*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := -(a + b*ArcTanh[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)) + x*(a + b*ArcTanh[c*x])^p/(2*c^2*d*(d + e*x^2)) - b*p/(2*c)*Int[x*(a + b*ArcTanh[c*x])^(p - 1)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[x_^2*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := -(a + b*ArcCoth[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)) + x*(a + b*ArcCoth[c*x])^p/(2*c^2*d*(d + e*x^2)) - b*p/(2*c)*Int[x*(a + b*ArcCoth[c*x])^(p - 1)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := -b*(f*x)^m*(d + e*x^2)^(q + 1)/(c*d*m^2) + f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])/(c^2*d* m) - f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := -b*(f*x)^m*(d + e*x^2)^(q + 1)/(c*d*m^2) + f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])/(c^2*d* m) - f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol] := -b*p*(f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p - 1)/(c*d*m^2) + f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^ p/(c^2*d*m) + b^2*p*(p - 1)/m^2* Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 2), x] - f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol] := -b*p*(f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p - 1)/(c*d*m^2) + f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^ p/(c^2*d*m) + b^2*p*(p - 1)/m^2* Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 2), x] - f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol] := (f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p + 1)/(b*c* d*(p + 1)) - f*m/(b*c*(p + 1))* Int[(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol] := (f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p + 1)/(b*c* d*(p + 1)) - f*m/(b*c*(p + 1))* Int[(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^ p/(d*(m + 1)) - b*c*p/(m + 1)* Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^ p/(d*f*(m + 1)) - b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
+Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := (f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])/(f*(m + 2)) - b*c*d/(f*(m + 2))*Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x] + d/(m + 2)*Int[(f*x)^m*(a + b*ArcTanh[c*x])/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && NeQ[m, -2]
+Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := (f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x])/(f*(m + 2)) - b*c*d/(f*(m + 2))*Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x] + d/(m + 2)*Int[(f*x)^m*(a + b*ArcCoth[c*x])/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && NeQ[m, -2]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := d*Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x] - c^2*d/f^2* Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || EqQ[p, 1] && IntegerQ[q])
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := d*Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x])^p, x] - c^2*d/f^2* Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || EqQ[p, 1] && IntegerQ[q])
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := -f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])^p/(c^2*d*m) + b*f*p/(c*m)* Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x])^(p - 1)/Sqrt[d + e*x^2], x] + f^2*(m - 1)/(c^2*m)* Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && GtQ[m, 1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := -f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x])^p/(c^2*d*m) + b*f*p/(c*m)* Int[(f*x)^(m - 1)*(a + b*ArcCoth[c*x])^(p - 1)/Sqrt[d + e*x^2], x] + f^2*(m - 1)/(c^2*m)* Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && GtQ[m, 1]
+Int[(a_. + b_.*ArcTanh[c_.*x_])/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := -2/Sqrt[d]*(a + b*ArcTanh[c*x])* ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]] + b/Sqrt[d]*PolyLog[2, -Sqrt[1 - c*x]/Sqrt[1 + c*x]] - b/Sqrt[d]*PolyLog[2, Sqrt[1 - c*x]/Sqrt[1 + c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := -2/Sqrt[d]*(a + b*ArcCoth[c*x])* ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]] + b/Sqrt[d]*PolyLog[2, -Sqrt[1 - c*x]/Sqrt[1 + c*x]] - b/Sqrt[d]*PolyLog[2, Sqrt[1 - c*x]/Sqrt[1 + c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := 1/Sqrt[d]*Subst[Int[(a + b*x)^p*Csch[x], x], x, ArcTanh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := -c*x*Sqrt[1 - 1/(c^2*x^2)]/Sqrt[d + e*x^2]* Subst[Int[(a + b*x)^p*Sech[x], x], x, ArcCoth[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcTanh[c*x])^p/(x*Sqrt[1 - c^2*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && Not[GtQ[d, 0]]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcCoth[c*x])^p/(x*Sqrt[1 - c^2*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && Not[GtQ[d, 0]]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(x_^2*Sqrt[d_ + e_.*x_^2]), x_Symbol] := -Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])^p/(d*x) + b*c*p*Int[(a + b*ArcTanh[c*x])^(p - 1)/(x*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(x_^2*Sqrt[d_ + e_.*x_^2]), x_Symbol] := -Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x])^p/(d*x) + b*c*p*Int[(a + b*ArcCoth[c*x])^(p - 1)/(x*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := (f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])^p/(d*f*(m + 1)) - b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1)/Sqrt[d + e*x^2], x] + c^2*(m + 2)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(a + b*ArcTanh[c*x])^p/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := (f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x])^p/(d*f*(m + 1)) - b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1)/Sqrt[d + e*x^2], x] + c^2*(m + 2)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(a + b*ArcCoth[c*x])^p/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
+Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := 1/e*Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x] - d/e*Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
+Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := 1/e*Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p, x] - d/e*Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^p, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
+Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := 1/d*Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x] - e/d*Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
+Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := 1/d*Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p, x] - e/d*Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^p, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p + 1)/(b*c* d*(p + 1)) - m/(b*c*(p + 1))* Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x] + c*(m + 2*q + 2)/(b*(p + 1))* Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && LtQ[q, -1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := x^m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p + 1)/(b*c* d*(p + 1)) - m/(b*c*(p + 1))* Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p + 1), x] + c*(m + 2*q + 2)/(b*(p + 1))* Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && LtQ[q, -1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := d^q/c^(m + 1)* Subst[Int[(a + b*x)^p*Sinh[x]^m/Cosh[x]^(m + 2*(q + 1)), x], x, ArcTanh[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q] || GtQ[d, 0])
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := d^(q + 1/2)*Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[x^m*(1 - c^2*x^2)^q*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && Not[IntegerQ[q] || GtQ[d, 0]]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := -(-d)^q/c^(m + 1)* Subst[Int[(a + b*x)^p*Cosh[x]^m/Sinh[x]^(m + 2*(q + 1)), x], x, ArcCoth[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && IntegerQ[q]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := -(-d)^(q + 1/2)*x* Sqrt[(c^2*x^2 - 1)/(c^2*x^2)]/(c^m*Sqrt[d + e*x^2])* Subst[Int[(a + b*x)^p*Cosh[x]^m/Sinh[x]^(m + 2*(q + 1)), x], x, ArcCoth[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && Not[IntegerQ[q]]
+Int[x_*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])/(2*e*(q + 1)) - b*c/(2*e*(q + 1))*Int[(d + e*x^2)^(q + 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
+Int[x_*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])/(2*e*(q + 1)) - b*c/(2*e*(q + 1))*Int[(d + e*x^2)^(q + 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
+Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcTanh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ( IGtQ[q, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[q, 0] && GtQ[m + 2*q + 3, 0]] || ILtQ[(m + 2*q + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]] )
+Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcCoth[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ( IGtQ[q, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[q, 0] && GtQ[m + 2*q + 3, 0]] || ILtQ[(m + 2*q + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]] )
+Int[x_*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcTanh[c*x])^p/(1 - Rt[-e/d, 2]*x)^2, x] - 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcTanh[c*x])^p/(1 + Rt[-e/d, 2]*x)^2, x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0]
+Int[x_*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcCoth[c*x])^p/(1 - Rt[-e/d, 2]*x)^2, x] - 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcCoth[c*x])^p/(1 + Rt[-e/d, 2]*x)^2, x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := With[{u = ExpandIntegrand[(a + b*ArcTanh[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (GtQ[q, 0] || IntegerQ[m])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := With[{u = ExpandIntegrand[(a + b*ArcCoth[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (GtQ[q, 0] || IntegerQ[m])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*ArcTanh[c_.*x_]), x_Symbol] := a*Int[(f*x)^m*(d + e*x^2)^q, x] + b*Int[(f*x)^m*(d + e*x^2)^q*ArcTanh[c*x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*ArcCoth[c_.*x_]), x_Symbol] := a*Int[(f*x)^m*(d + e*x^2)^q, x] + b*Int[(f*x)^m*(d + e*x^2)^q*ArcCoth[c*x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x]
+Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p/(d + e*x^2), (f + g*x)^ m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && IGtQ[m, 0]
+Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^p/(d + e*x^2), (f + g*x)^ m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && IGtQ[m, 0]
+Int[ArcTanh[u_]*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/2*Int[Log[1 + u]*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x] - 1/2*Int[Log[1 - u]*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
+Int[ArcCoth[u_]*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/2*Int[ Log[SimplifyIntegrand[1 + 1/u, x]]*(a + b*ArcCoth[c*x])^ p/(d + e*x^2), x] - 1/2* Int[Log[SimplifyIntegrand[1 - 1/u, x]]*(a + b*ArcCoth[c*x])^ p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
+Int[ArcTanh[u_]*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/2*Int[Log[1 + u]*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x] - 1/2*Int[Log[1 - u]*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
+Int[ArcCoth[u_]*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/2*Int[ Log[SimplifyIntegrand[1 + 1/u, x]]*(a + b*ArcCoth[c*x])^ p/(d + e*x^2), x] - 1/2* Int[Log[SimplifyIntegrand[1 - 1/u, x]]*(a + b*ArcCoth[c*x])^ p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_.*Log[f_ + g_.*x_]/(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcTanh[c*x])^(p + 1)*Log[f + g*x]/(b*c*d*(p + 1)) - g/(b*c*d*(p + 1))* Int[(a + b*ArcTanh[c*x])^(p + 1)/(f + g*x), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[c^2*f^2 - g^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_.*Log[f_ + g_.*x_]/(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcCoth[c*x])^(p + 1)*Log[f + g*x]/(b*c*d*(p + 1)) - g/(b*c*d*(p + 1))* Int[(a + b*ArcCoth[c*x])^(p + 1)/(f + g*x), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[c^2*f^2 - g^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcTanh[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) - b*p/2* Int[(a + b*ArcTanh[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcCoth[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) - b*p/2* Int[(a + b*ArcCoth[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol] := -(a + b*ArcTanh[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) + b*p/2* Int[(a + b*ArcTanh[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol] := -(a + b*ArcCoth[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) + b*p/2* Int[(a + b*ArcCoth[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol] := -(a + b*ArcTanh[c*x])^p*PolyLog[k + 1, u]/(2*c*d) + b*p/2* Int[(a + b*ArcTanh[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol] := -(a + b*ArcCoth[c*x])^p*PolyLog[k + 1, u]/(2*c*d) + b*p/2* Int[(a + b*ArcCoth[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcTanh[c*x])^p*PolyLog[k + 1, u]/(2*c*d) - b*p/2* Int[(a + b*ArcTanh[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcCoth[c*x])^p*PolyLog[k + 1, u]/(2*c*d) - b*p/2* Int[(a + b*ArcCoth[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
+Int[1/((d_ + e_.*x_^2)*(a_. + b_.*ArcCoth[c_.*x_])*(a_. + b_.*ArcTanh[c_.*x_])), x_Symbol] := (-Log[a + b*ArcCoth[c*x]] + Log[a + b*ArcTanh[c*x]])/(b^2*c* d*(ArcCoth[c*x] - ArcTanh[c*x])) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^ m_.*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcCoth[c*x])^(m + 1)*(a + b*ArcTanh[c*x])^ p/(b*c*d*(m + 1)) - p/(m + 1)* Int[(a + b*ArcCoth[c*x])^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1)/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGeQ[m, p]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^ m_.*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcTanh[c*x])^(m + 1)*(a + b*ArcCoth[c*x])^ p/(b*c*d*(m + 1)) - p/(m + 1)* Int[(a + b*ArcTanh[c*x])^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1)/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[m, p]
+Int[ArcTanh[a_.*x_]/(c_ + d_.*x_^n_.), x_Symbol] := 1/2*Int[Log[1 + a*x]/(c + d*x^n), x] - 1/2*Int[Log[1 - a*x]/(c + d*x^n), x] /; FreeQ[{a, c, d}, x] && IntegerQ[n] && Not[EqQ[n, 2] && EqQ[a^2*c + d, 0]]
+Int[ArcCoth[a_.*x_]/(c_ + d_.*x_^n_.), x_Symbol] := 1/2*Int[Log[1 + 1/(a*x)]/(c + d*x^n), x] - 1/2*Int[Log[1 - 1/(a*x)]/(c + d*x^n), x] /; FreeQ[{a, c, d}, x] && IntegerQ[n] && Not[EqQ[n, 2] && EqQ[a^2*c + d, 0]]
+Int[Log[d_.*x_^m_.]*ArcTanh[c_.*x_^n_.]/x_, x_Symbol] := 1/2*Int[Log[d*x^m]*Log[1 + c*x^n]/x, x] - 1/2*Int[Log[d*x^m]*Log[1 - c*x^n]/x, x] /; FreeQ[{c, d, m, n}, x]
+Int[Log[d_.*x_^m_.]*ArcCoth[c_.*x_^n_.]/x_, x_Symbol] := 1/2*Int[Log[d*x^m]*Log[1 + 1/(c*x^n)]/x, x] - 1/2*Int[Log[d*x^m]*Log[1 - 1/(c*x^n)]/x, x] /; FreeQ[{c, d, m, n}, x]
+Int[Log[d_.*x_^m_.]*(a_ + b_.*ArcTanh[c_.*x_^n_.])/x_, x_Symbol] := a*Int[Log[d*x^m]/x, x] + b*Int[(Log[d*x^m]*ArcTanh[c*x^n])/x, x] /; FreeQ[{a, b, c, d, m, n}, x]
+Int[Log[d_.*x_^m_.]*(a_ + b_.*ArcCoth[c_.*x_^n_.])/x_, x_Symbol] := a*Int[Log[d*x^m]/x, x] + b*Int[(Log[d*x^m]*ArcCoth[c*x^n])/x, x] /; FreeQ[{a, b, c, d, m, n}, x]
+Int[(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := x*(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x]) - 2*e*g*Int[x^2*(a + b*ArcTanh[c*x])/(f + g*x^2), x] - b*c*Int[x*(d + e*Log[f + g*x^2])/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x]
+Int[(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := x*(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x]) - 2*e*g*Int[x^2*(a + b*ArcCoth[c*x])/(f + g*x^2), x] - b*c*Int[x*(d + e*Log[f + g*x^2])/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x]
+Int[Log[f_. + g_.*x_^2]*ArcTanh[c_.*x_]/x_, x_Symbol] := (Log[f + g*x^2] - Log[1 - c*x] - Log[1 + c*x])* Int[ArcTanh[c*x]/x, x] - 1/2*Int[Log[1 - c*x]^2/x, x] + 1/2*Int[Log[1 + c*x]^2/x, x] /; FreeQ[{c, f, g}, x] && EqQ[c^2*f + g, 0]
+Int[Log[f_. + g_.*x_^2]*ArcCoth[c_.*x_]/x_, x_Symbol] := (Log[f + g*x^2] - Log[-c^2*x^2] - Log[1 - 1/(c*x)] - Log[1 + 1/(c*x)])*Int[ArcCoth[c*x]/x, x] + Int[Log[-c^2*x^2]*ArcCoth[c*x]/x, x] - 1/2*Int[Log[1 - 1/(c*x)]^2/x, x] + 1/2*Int[Log[1 + 1/(c*x)]^2/x, x] /; FreeQ[{c, f, g}, x] && EqQ[c^2*f + g, 0]
+Int[Log[f_. + g_.*x_^2]*(a_ + b_.*ArcTanh[c_.*x_])/x_, x_Symbol] := a*Int[Log[f + g*x^2]/x, x] + b*Int[Log[f + g*x^2]*ArcTanh[c*x]/x, x] /; FreeQ[{a, b, c, f, g}, x]
+Int[Log[f_. + g_.*x_^2]*(a_ + b_.*ArcCoth[c_.*x_])/x_, x_Symbol] := a*Int[Log[f + g*x^2]/x, x] + b*Int[Log[f + g*x^2]*ArcCoth[c*x]/x, x] /; FreeQ[{a, b, c, f, g}, x]
+Int[(d_ + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_])/x_, x_Symbol] := d*Int[(a + b*ArcTanh[c*x])/x, x] + e*Int[Log[f + g*x^2]*(a + b*ArcTanh[c*x])/x, x] /; FreeQ[{a, b, c, d, e, f, g}, x]
+Int[(d_ + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_])/x_, x_Symbol] := d*Int[(a + b*ArcCoth[c*x])/x, x] + e*Int[Log[f + g*x^2]*(a + b*ArcCoth[c*x])/x, x] /; FreeQ[{a, b, c, d, e, f, g}, x]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := x^(m + 1)*(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x])/(m + 1) - 2*e*g/(m + 1)* Int[x^(m + 2)*(a + b*ArcTanh[c*x])/(f + g*x^2), x] - b*c/(m + 1)* Int[x^(m + 1)*(d + e*Log[f + g*x^2])/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m/2, 0]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := x^(m + 1)*(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x])/(m + 1) - 2*e*g/(m + 1)* Int[x^(m + 2)*(a + b*ArcCoth[c*x])/(f + g*x^2), x] - b*c/(m + 1)* Int[x^(m + 1)*(d + e*Log[f + g*x^2])/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m/2, 0]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcTanh[c*x], u, x] - b*c*Int[ExpandIntegrand[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcCoth[c*x], u, x] - b*c*Int[ExpandIntegrand[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(a + b*ArcTanh[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - 2*e*g*Int[ExpandIntegrand[x*u/(f + g*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(a + b*ArcCoth[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - 2*e*g*Int[ExpandIntegrand[x*u/(f + g*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]
+Int[x_*(d_. + e_.*Log[f_ + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_])^2, x_Symbol] := (f + g*x^2)*(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x])^2/(2*g) - e*x^2*(a + b*ArcTanh[c*x])^2/2 + b/c*Int[(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x]), x] + b*c*e*Int[x^2*(a + b*ArcTanh[c*x])/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*f + g, 0]
+Int[x_*(d_. + e_.*Log[f_ + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_])^2, x_Symbol] := (f + g*x^2)*(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x])^2/(2*g) - e*x^2*(a + b*ArcCoth[c*x])^2/2 + b/c*Int[(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x]), x] + b*c*e*Int[x^2*(a + b*ArcCoth[c*x])/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*f + g, 0]
+Int[u_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := Unintegrable[u*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || MatchQ[u, (d_. + e_.*x)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x)^q_. /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[u, (d_. + e_.*x^2)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x^2)^q_. /; FreeQ[{d, e, f, m, q}, x]])
+Int[u_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := Unintegrable[u*(a + b*ArcCoth[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || MatchQ[u, (d_. + e_.*x)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x)^q_. /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[u, (d_. + e_.*x^2)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x^2)^q_. /; FreeQ[{d, e, f, m, q}, x]])
+Int[ArcTanh[c_.*x_^n_], x_Symbol] := x*ArcTanh[c*x^n] - c*n*Int[x^n/(1 - c^2*x^(2*n)), x] /; FreeQ[{c, n}, x]
+Int[ArcCoth[c_.*x_^n_], x_Symbol] := x*ArcCoth[c*x^n] - c*n*Int[x^n/(1 - c^2*x^(2*n)), x] /; FreeQ[{c, n}, x]
+Int[(a_. + b_.*ArcTanh[c_.*x_^n_])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*Log[1 + c*x^n]/2 - b*Log[1 - c*x^n]/2)^ p, x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && IntegerQ[n]
+Int[(a_. + b_.*ArcCoth[c_.*x_^n_])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*Log[1 + x^(-n)/c]/2 - b*Log[1 - x^(-n)/c]/2)^p, x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && IntegerQ[n]
+Int[(a_. + b_.*ArcTanh[c_.*x_^n_])^p_./x_, x_Symbol] := 1/n*Subst[Int[(a + b*ArcTanh[c*x])^p/x, x], x, x^n] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_^n_])^p_./x_, x_Symbol] := 1/n*Subst[Int[(a + b*ArcCoth[c*x])^p/x, x], x, x^n] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcTanh[c_.*x_^n_]), x_Symbol] := (d*x)^(m + 1)*(a + b*ArcTanh[c*x^n])/(d*(m + 1)) - b*c*n/(d*(m + 1))* Int[x^(n - 1)*(d*x)^(m + 1)/(1 - c^2*x^(2*n)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcCoth[c_.*x_^n_]), x_Symbol] := (d*x)^(m + 1)*(a + b*ArcCoth[c*x^n])/(d*(m + 1)) - b*c*n/(d*(m + 1))* Int[x^(n - 1)*(d*x)^(m + 1)/(1 - c^2*x^(2*n)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcTanh[c_.*x_^n_])^p_., x_Symbol] := Int[ExpandIntegrand[(d*x)^ m*(a + b*Log[1 + c*x^n]/2 - b*Log[1 - c*x^n]/2)^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] && IntegerQ[m] && IntegerQ[n]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcCoth[c_.*x_^n_])^p_., x_Symbol] := Int[ExpandIntegrand[(d*x)^ m*(a + b*Log[1 + x^(-n)/c]/2 - b*Log[1 - x^(-n)/c]/2)^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] && IntegerQ[m] && IntegerQ[n]
+Int[u_.*(a_. + b_.*ArcTanh[c_.*x_^n_])^p_., x_Symbol] := Unintegrable[u*(a + b*ArcTanh[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p}, x] && (EqQ[u, 1] || MatchQ[u, (d_.*x)^m_. /; FreeQ[{d, m}, x]]) 
+Int[u_.*(a_. + b_.*ArcCoth[c_.*x_^n_])^p_., x_Symbol] := Unintegrable[u*(a + b*ArcCoth[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p}, x] && (EqQ[u, 1] || MatchQ[u, (d_.*x)^m_. /; FreeQ[{d, m}, x]]) 
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.2 (d x)^m (a+b arctanh(c x^n))^p.m b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.2 (d x)^m (a+b arctanh(c x^n))^p.m
new file mode 100755
index 0000000..2b71784
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.2 (d x)^m (a+b arctanh(c x^n))^p.m	
@@ -0,0 +1,27 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.3.2 (d x)^m (a+b arctanh(c x^n))^p *)
+Int[(a_. + b_.*ArcTanh[c_.*x_])/x_, x_Symbol] := a*Log[x] - b/2*PolyLog[2, -c*x] + b/2*PolyLog[2, c*x] /; FreeQ[{a, b, c}, x]
+Int[(a_. + b_.*ArcCoth[c_.*x_])/x_, x_Symbol] := a*Log[x] + b/2*PolyLog[2, -1/(c*x)] - b/2*PolyLog[2, 1/(c*x)] /; FreeQ[{a, b, c}, x]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_/x_, x_Symbol] := 2*(a + b*ArcTanh[c*x])^p*ArcTanh[1 - 2/(1 - c*x)] - 2*b*c*p* Int[(a + b*ArcTanh[c*x])^(p - 1)* ArcTanh[1 - 2/(1 - c*x)]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_/x_, x_Symbol] := 2*(a + b*ArcCoth[c*x])^p*ArcCoth[1 - 2/(1 - c*x)] - 2*b*c*p* Int[(a + b*ArcCoth[c*x])^(p - 1)* ArcCoth[1 - 2/(1 - c*x)]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
+Int[(a_. + b_.*ArcTanh[c_.*x_^n_])^p_./x_, x_Symbol] := 1/n*Subst[Int[(a + b*ArcTanh[c*x])^p/x, x], x, x^n] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_^n_])^p_./x_, x_Symbol] := 1/n*Subst[Int[(a + b*ArcCoth[c*x])^p/x, x], x, x^n] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]
+Int[x_^m_.*(a_. + b_.*ArcTanh[c_.*x_^n_.])^p_., x_Symbol] := x^(m + 1)*(a + b*ArcTanh[c*x^n])^p/(m + 1) - b*c*n*p/(m + 1)* Int[x^(m + n)*(a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n)), x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || EqQ[n, 1] && IntegerQ[m]) && NeQ[m, -1]
+Int[x_^m_.*(a_. + b_.*ArcCoth[c_.*x_^n_.])^p_., x_Symbol] := x^(m + 1)*(a + b*ArcCoth[c*x^n])^p/(m + 1) - b*c*n*p/(m + 1)* Int[x^(m + n)*(a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n)), x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || EqQ[n, 1] && IntegerQ[m]) && NeQ[m, -1]
+Int[x_^m_.*(a_. + b_.*ArcTanh[c_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcTanh[c*x])^p, x], x, x^n] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simplify[(m + 1)/n]]
+Int[x_^m_.*(a_. + b_.*ArcCoth[c_.*x_^n_])^p_., x_Symbol] := 1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcCoth[c*x])^p, x], x, x^n] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simplify[(m + 1)/n]]
+Int[x_^m_.*(a_. + b_.*ArcTanh[c_.*x_^n_])^p_, x_Symbol] := Int[ExpandIntegrand[ x^m*(a + b*Log[1 + c*x^n]/2 - b*Log[1 - c*x^n]/2)^p, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0] && IntegerQ[m]
+Int[x_^m_.*(a_. + b_.*ArcCoth[c_.*x_^n_])^p_, x_Symbol] := Int[ExpandIntegrand[ x^m*(a + b*Log[1 + x^(-n)/c]/2 - b*Log[1 - x^(-n)/c]/2)^p, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0] && IntegerQ[m]
+Int[x_^m_.*(a_. + b_.*ArcTanh[c_.*x_^n_])^p_, x_Symbol] := With[{k = Denominator[m]}, k*Subst[Int[x^(k*(m + 1) - 1)*(a + b*ArcTanh[c*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0] && FractionQ[m]
+Int[x_^m_.*(a_. + b_.*ArcCoth[c_.*x_^n_])^p_, x_Symbol] := With[{k = Denominator[m]}, k*Subst[Int[x^(k*(m + 1) - 1)*(a + b*ArcCoth[c*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0] && FractionQ[m]
+Int[x_^m_.*(a_. + b_.*ArcTanh[c_.*x_^n_])^p_, x_Symbol] := Int[x^m*(a + b*ArcCoth[x^(-n)/c])^p, x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p, 1] && ILtQ[n, 0]
+Int[x_^m_.*(a_. + b_.*ArcCoth[c_.*x_^n_])^p_, x_Symbol] := Int[x^m*(a + b*ArcTanh[x^(-n)/c])^p, x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p, 1] && ILtQ[n, 0]
+Int[x_^m_.*(a_. + b_.*ArcTanh[c_.*x_^n_])^p_, x_Symbol] := With[{k = Denominator[n]}, k*Subst[Int[x^(k*(m + 1) - 1)*(a + b*ArcTanh[c*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c, m}, x] && IGtQ[p, 1] && FractionQ[n]
+Int[x_^m_.*(a_. + b_.*ArcCoth[c_.*x_^n_])^p_, x_Symbol] := With[{k = Denominator[n]}, k*Subst[Int[x^(k*(m + 1) - 1)*(a + b*ArcCoth[c*x^(k*n)])^p, x], x, x^(1/k)]] /; FreeQ[{a, b, c, m}, x] && IGtQ[p, 1] && FractionQ[n]
+Int[(d_*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_^n_.]), x_Symbol] := (d*x)^(m + 1)*(a + b*ArcTanh[c*x^n])/(d*(m + 1)) - b*c*n/(d^n*(m + 1))*Int[(d*x)^(m + n)/(1 - c^2*x^(2*n)), x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[n] && NeQ[m, -1]
+Int[(d_*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_^n_.]), x_Symbol] := (d*x)^(m + 1)*(a + b*ArcCoth[c*x^n])/(d*(m + 1)) - b*c*n/(d^n*(m + 1))*Int[(d*x)^(m + n)/(1 - c^2*x^(2*n)), x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[n] && NeQ[m, -1]
+Int[(d_*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_^n_.])^p_., x_Symbol] := d^IntPart[m]*(d*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*ArcTanh[c*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || RationalQ[m, n])
+Int[(d_*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_^n_.])^p_., x_Symbol] := d^IntPart[m]*(d*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*ArcCoth[c*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || RationalQ[m, n])
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcTanh[c_.*x_^n_.])^p_., x_Symbol] := Unintegrable[(d*x)^m*(a + b*ArcTanh[c*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcCoth[c_.*x_^n_.])^p_., x_Symbol] := Unintegrable[(d*x)^m*(a + b*ArcCoth[c*x^n])^p, x] /; FreeQ[{a, b, c, d, m, n, p}, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.2 u (a+b arctanh(c+d x))^p.m b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.2 u (a+b arctanh(c+d x))^p.m
new file mode 100755
index 0000000..8b62506
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.2 u (a+b arctanh(c+d x))^p.m	
@@ -0,0 +1,23 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.3.2 u (a+b arctanh(c+d x))^p *)
+Int[(a_. + b_.*ArcTanh[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(a + b*ArcTanh[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*ArcCoth[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(a + b*ArcCoth[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*ArcTanh[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(a + b*ArcTanh[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]
+Int[(a_. + b_.*ArcCoth[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(a + b*ArcCoth[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcTanh[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(f*x/d)^m*(a + b*ArcTanh[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCoth[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(f*x/d)^m*(a + b*ArcCoth[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_*(a_. + b_.*ArcTanh[c_ + d_.*x_])^p_., x_Symbol] := (e + f*x)^(m + 1)*(a + b*ArcTanh[c + d*x])^p/(f*(m + 1)) - b*d*p/(f*(m + 1))* Int[(e + f*x)^(m + 1)*(a + b*ArcTanh[c + d*x])^(p - 1)/(1 - (c + d*x)^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[m, -1]
+Int[(e_. + f_.*x_)^m_*(a_. + b_.*ArcCoth[c_ + d_.*x_])^p_., x_Symbol] := (e + f*x)^(m + 1)*(a + b*ArcCoth[c + d*x])^p/(f*(m + 1)) - b*d*p/(f*(m + 1))* Int[(e + f*x)^(m + 1)*(a + b*ArcCoth[c + d*x])^(p - 1)/(1 - (c + d*x)^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[m, -1]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcTanh[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcTanh[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCoth[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcCoth[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcTanh[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(e + f*x)^m*(a + b*ArcTanh[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && Not[IGtQ[p, 0]]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCoth[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && Not[IGtQ[p, 0]]
+Int[ArcTanh[c_ + d_.*x_]/(e_ + f_.*x_^n_.), x_Symbol] := 1/2*Int[Log[1 + c + d*x]/(e + f*x^n), x] - 1/2*Int[Log[1 - c - d*x]/(e + f*x^n), x] /; FreeQ[{c, d, e, f}, x] && RationalQ[n]
+Int[ArcCoth[c_ + d_.*x_]/(e_ + f_.*x_^n_.), x_Symbol] := 1/2*Int[Log[(1 + c + d*x)/(c + d*x)]/(e + f*x^n), x] - 1/2*Int[Log[(-1 + c + d*x)/(c + d*x)]/(e + f*x^n), x] /; FreeQ[{c, d, e, f}, x] && RationalQ[n]
+Int[ArcTanh[c_ + d_.*x_]/(e_ + f_.*x_^n_), x_Symbol] := Unintegrable[ArcTanh[c + d*x]/(e + f*x^n), x] /; FreeQ[{c, d, e, f, n}, x] && Not[RationalQ[n]]
+Int[ArcCoth[c_ + d_.*x_]/(e_ + f_.*x_^n_), x_Symbol] := Unintegrable[ArcCoth[c + d*x]/(e + f*x^n), x] /; FreeQ[{c, d, e, f, n}, x] && Not[RationalQ[n]]
+Int[(A_. + B_.*x_ + C_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_ + d_.*x_])^ p_., x_Symbol] := 1/d*Subst[Int[(-C/d^2 + C/d^2*x^2)^q*(a + b*ArcTanh[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, A, B, C, p, q}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[(A_. + B_.*x_ + C_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_ + d_.*x_])^ p_., x_Symbol] := 1/d*Subst[Int[(C/d^2 + C/d^2*x^2)^q*(a + b*ArcCoth[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, A, B, C, p, q}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[(e_. + f_.*x_)^m_.*(A_. + B_.*x_ + C_.*x_^2)^ q_.*(a_. + b_.*ArcTanh[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[ Int[((d*e - c*f)/d + f*x/d)^m*(-C/d^2 + C/d^2*x^2)^ q*(a + b*ArcTanh[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[(e_. + f_.*x_)^m_.*(A_. + B_.*x_ + C_.*x_^2)^ q_.*(a_. + b_.*ArcCoth[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[ Int[((d*e - c*f)/d + f*x/d)^m*(-C/d^2 + C/d^2*x^2)^ q*(a + b*ArcCoth[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 (d+e x)^m (a+b arctanh(c x^n))^p.m b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 (d+e x)^m (a+b arctanh(c x^n))^p.m
new file mode 100755
index 0000000..b2cfe36
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 (d+e x)^m (a+b arctanh(c x^n))^p.m	
@@ -0,0 +1,25 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.3.3 (d+e x)^m (a+b arctanh(c x^n))^p *)
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := -(a + b*ArcTanh[c*x])^p*Log[2/(1 + e*x/d)]/e + b*c*p/e* Int[(a + b*ArcTanh[c*x])^(p - 1)*Log[2/(1 + e*x/d)]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := -(a + b*ArcCoth[c*x])^p*Log[2/(1 + e*x/d)]/e + b*c*p/e* Int[(a + b*ArcCoth[c*x])^(p - 1)*Log[2/(1 + e*x/d)]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)]/e + b*c/e*Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x] + (a + b*ArcTanh[c*x])*Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b*c/e* Int[Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcCoth[c*x])*Log[2/(1 + c*x)]/e + b*c/e*Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x] + (a + b*ArcCoth[c*x])*Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b*c/e* Int[Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^2/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)]/e + b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)]/e + b^2*PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) + (a + b*ArcTanh[c*x])^2* Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b*(a + b*ArcTanh[c*x])* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b^2*PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^2/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcCoth[c*x])^2*Log[2/(1 + c*x)]/e + b*(a + b*ArcCoth[c*x])*PolyLog[2, 1 - 2/(1 + c*x)]/e + b^2*PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) + (a + b*ArcCoth[c*x])^2* Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b*(a + b*ArcCoth[c*x])* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b^2*PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^3/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcTanh[c*x])^3*Log[2/(1 + c*x)]/e + 3*b*(a + b*ArcTanh[c*x])^2*PolyLog[2, 1 - 2/(1 + c*x)]/(2*e) + 3*b^2*(a + b*ArcTanh[c*x])*PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) + 3*b^3*PolyLog[4, 1 - 2/(1 + c*x)]/(4*e) + (a + b*ArcTanh[c*x])^3* Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - 3*b*(a + b*ArcTanh[c*x])^2* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) - 3*b^2*(a + b*ArcTanh[c*x])* PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) - 3*b^3*PolyLog[4, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(4*e) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^3/(d_ + e_.*x_), x_Symbol] := -(a + b*ArcCoth[c*x])^3*Log[2/(1 + c*x)]/e + 3*b*(a + b*ArcCoth[c*x])^2*PolyLog[2, 1 - 2/(1 + c*x)]/(2*e) + 3*b^2*(a + b*ArcCoth[c*x])*PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) + 3*b^3*PolyLog[4, 1 - 2/(1 + c*x)]/(4*e) + (a + b*ArcCoth[c*x])^3* Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - 3*b*(a + b*ArcCoth[c*x])^2* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) - 3*b^2*(a + b*ArcCoth[c*x])* PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) - 3*b^3*PolyLog[4, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(4*e) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := (d + e*x)^(q + 1)*(a + b*ArcTanh[c*x])/(e*(q + 1)) - b*c/(e*(q + 1))*Int[(d + e*x)^(q + 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := (d + e*x)^(q + 1)*(a + b*ArcCoth[c*x])/(e*(q + 1)) - b*c/(e*(q + 1))*Int[(d + e*x)^(q + 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol] := (d + e*x)^(q + 1)*(a + b*ArcTanh[c*x])^p/(e*(q + 1)) - b*c*p/(e*(q + 1))* Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
+Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol] := (d + e*x)^(q + 1)*(a + b*ArcCoth[c*x])^p/(e*(q + 1)) - b*c*p/(e*(q + 1))* Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
+Int[(a_. + b_.*ArcTanh[c_.*x_^n_])/(d_. + e_.*x_), x_Symbol] := Log[d + e*x]*(a + b*ArcTanh[c*x^n])/e - b*c*n/e*Int[x^(n - 1)*Log[d + e*x]/(1 - c^2*x^(2*n)), x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[n]
+Int[(a_. + b_.*ArcCoth[c_.*x_^n_])/(d_. + e_.*x_), x_Symbol] := Log[d + e*x]*(a + b*ArcCoth[c*x^n])/e - b*c*n/e*Int[x^(n - 1)*Log[d + e*x]/(1 - c^2*x^(2*n)), x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[n]
+Int[(a_. + b_.*ArcTanh[c_.*x_^n_])/(d_ + e_.*x_), x_Symbol] := With[{k = Denominator[n]}, k*Subst[Int[x^(k - 1)*(a + b*ArcTanh[c*x^(k*n)])/(d + e*x^k), x], x, x^(1/k)]] /; FreeQ[{a, b, c, d, e}, x] && FractionQ[n]
+Int[(a_. + b_.*ArcCoth[c_.*x_^n_])/(d_ + e_.*x_), x_Symbol] := With[{k = Denominator[n]}, k*Subst[Int[x^(k - 1)*(a + b*ArcCoth[c*x^(k*n)])/(d + e*x^k), x], x, x^(1/k)]] /; FreeQ[{a, b, c, d, e}, x] && FractionQ[n]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcTanh[c_.*x_^n_]), x_Symbol] := (d + e*x)^(m + 1)*(a + b*ArcTanh[c*x^n])/(e*(m + 1)) - b*c*n/(e*(m + 1))* Int[x^(n - 1)*(d + e*x)^(m + 1)/(1 - c^2*x^(2*n)), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCoth[c_.*x_^n_]), x_Symbol] := (d + e*x)^(m + 1)*(a + b*ArcCoth[c*x^n])/(e*(m + 1)) - b*c*n/(e*(m + 1))* Int[x^(n - 1)*(d + e*x)^(m + 1)/(1 - c^2*x^(2*n)), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
+Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcTanh[c_.*x_^n_])^p_, x_Symbol] := Int[ExpandIntegrand[(a + b*ArcTanh[c*x^n])^p, (d + e*x)^m, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 1] && IGtQ[m, 0]
+Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcCoth[c_.*x_^n_])^p_, x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCoth[c*x^n])^p, (d + e*x)^m, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 1] && IGtQ[m, 0]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcTanh[c_.*x_^n_])^p_., x_Symbol] := Unintegrable[(d + e*x)^m*(a + b*ArcTanh[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCoth[c_.*x_^n_])^p_., x_Symbol] := Unintegrable[(d + e*x)^m*(a + b*ArcCoth[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m
new file mode 100755
index 0000000..cd2b5f7
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m	
@@ -0,0 +1,90 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.3.3 Exponentials of inverse hyperbolic tangent *)
+Int[E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := Int[((1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/2)* Sqrt[1 - a^2*x^2])), x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2]
+Int[x_^m_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := Int[x^ m*((1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/2)* Sqrt[1 - a^2*x^2])), x] /; FreeQ[{a, m}, x] && IntegerQ[(n - 1)/2]
+Int[E^(n_*ArcTanh[a_.*x_]), x_Symbol] := Int[(1 + a*x)^(n/2)/(1 - a*x)^(n/2), x] /; FreeQ[{a, n}, x] && Not[IntegerQ[(n - 1)/2]]
+Int[x_^m_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := Int[x^m*(1 + a*x)^(n/2)/(1 - a*x)^(n/2), x] /; FreeQ[{a, m, n}, x] && Not[IntegerQ[(n - 1)/2]]
+Int[(c_ + d_.*x_)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^n*Int[(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
+Int[(e_. + f_.*x_)^m_.*(c_ + d_.*x_)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^n*Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1, 0]) && IntegerQ[2*p]
+Int[u_.*(c_ + d_.*x_)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[u*(1 + d*x/c)^p*(1 + a*x)^(n/2)/(1 - a*x)^(n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] || GtQ[c, 0])
+Int[u_.*(c_ + d_.*x_)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := Int[u*(c + d*x)^p*(1 + a*x)^(n/2)/(1 - a*x)^(n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && Not[IntegerQ[p] || GtQ[c, 0]]
+Int[u_.*(c_ + d_./x_)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := d^p*Int[u*(1 + c*x/d)^p*E^(n*ArcTanh[a*x])/x^p, x] /; FreeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]
+Int[u_.*(c_ + d_./x_)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := (-1)^(n/2)*c^p* Int[u*(1 + d/(c*x))^p*(1 + 1/(a*x))^(n/2)/(1 - 1/(a*x))^(n/2), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[p]] && IntegerQ[n/2] && GtQ[c, 0]
+Int[u_.*(c_ + d_./x_)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := Int[u*(c + d/x)^p*(1 + a*x)^(n/2)/(1 - a*x)^(n/2), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[p]] && IntegerQ[n/2] && Not[GtQ[c, 0]]
+Int[u_.*(c_ + d_./x_)^p_*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := x^p*(c + d/x)^p/(1 + c*x/d)^p* Int[u*(1 + c*x/d)^p*E^(n*ArcTanh[a*x])/x^p, x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[p]]
+Int[E^(n_*ArcTanh[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol] := (n - a*x)*E^(n*ArcTanh[a*x])/(a*c*(n^2 - 1)*Sqrt[c + d*x^2]) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n]]
+Int[(c_ + d_.*x_^2)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := (n + 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)* E^(n*ArcTanh[a*x])/(a*c*(n^2 - 4*(p + 1)^2)) - 2*(p + 1)*(2*p + 3)/(c*(n^2 - 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p, -1] && Not[IntegerQ[n]] && NeQ[n^2 - 4*(p + 1)^2, 0] && IntegerQ[2*p]
+Int[E^(n_.*ArcTanh[a_.*x_])/(c_ + d_.*x_^2), x_Symbol] := E^(n*ArcTanh[a*x])/(a*c*n) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]]
+Int[(c_ + d_.*x_^2)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && IntegerQ[p] && IGtQ[(n + 1)/2, 0] && Not[IntegerQ[p - n/2]]
+Int[(c_ + d_.*x_^2)^p_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[(1 - a^2*x^2)^(p + n/2)/(1 - a*x)^n, x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && IntegerQ[p] && ILtQ[(n - 1)/2, 0] && Not[IntegerQ[p - n/2]]
+Int[(c_ + d_.*x_^2)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
+Int[(c_ + d_.*x_^2)^p_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := c^(n/2)*Int[(c + d*x^2)^(p - n/2)*(1 + a*x)^n, x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && IGtQ[n/2, 0]
+Int[(c_ + d_.*x_^2)^p_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := 1/c^(n/2)*Int[(c + d*x^2)^(p + n/2)/(1 - a*x)^n, x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && ILtQ[n/2, 0]
+Int[(c_ + d_.*x_^2)^p_*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^IntPart[p]*(c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPart[p]* Int[(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]]
+Int[x_*E^(n_*ArcTanh[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol] := (1 - a*n*x)*E^(n*ArcTanh[a*x])/(d*(n^2 - 1)*Sqrt[c + d*x^2]) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n]]
+Int[x_*(c_ + d_.*x_^2)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := (c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x])/(2*d*(p + 1)) - a*c*n/(2*d*(p + 1))*Int[(c + d*x^2)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p, -1] && Not[IntegerQ[n]] && IntegerQ[2*p]
+(* Int[x_*(c_+d_.*x_^2)^p_*E^(n_*ArcTanh[a_.*x_]),x_Symbol] := -(2*(p+1)+a*n*x)*(c+d*x^2)^(p+1)*E^(n*ArcTanh[a*x])/(d*(n^2-4*(p+1)^ 2)) - n*(2*p+3)/(a*c*(n^2-4*(p+1)^2))*Int[(c+d*x^2)^(p+1)*E^(n*ArcTanh[a* x]),x] /; FreeQ[{a,c,d,n},x] && EqQ[a^2*c+d,0] && LeQ[p,-1] &&  NeQ[n^2-4*(p+1)^2,0] && Not[IntegerQ[n]] *)
+Int[x_^2*(c_ + d_.*x_^2)^p_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := (1 - a*n*x)*(c + d*x^2)^(p + 1)* E^(n*ArcTanh[a*x])/(a*d*n*(n^2 - 1)) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && EqQ[n^2 + 2*(p + 1), 0] && Not[IntegerQ[n]]
+Int[x_^2*(c_ + d_.*x_^2)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := -(n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)* E^(n*ArcTanh[a*x])/(a*d*(n^2 - 4*(p + 1)^2)) + (n^2 + 2*(p + 1))/(d*(n^2 - 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p, -1] && Not[IntegerQ[n]] && NeQ[n^2 - 4*(p + 1)^2, 0] && IntegerQ[2*p]
+Int[x_^m_.*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[x^m*(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0]) && IGtQ[(n + 1)/2, 0] && Not[IntegerQ[p - n/2]]
+Int[x_^m_.*(c_ + d_.*x_^2)^p_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[x^m*(1 - a^2*x^2)^(p + n/2)/(1 - a*x)^n, x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0]) && ILtQ[(n - 1)/2, 0] && Not[IntegerQ[p - n/2]]
+Int[x_^m_.*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
+Int[x_^m_.*(c_ + d_.*x_^2)^p_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := c^(n/2)*Int[x^m*(c + d*x^2)^(p - n/2)*(1 + a*x)^n, x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && IGtQ[n/2, 0]
+Int[x_^m_.*(c_ + d_.*x_^2)^p_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := 1/c^(n/2)*Int[x^m*(c + d*x^2)^(p + n/2)/(1 - a*x)^n, x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && ILtQ[n/2, 0]
+Int[x_^m_.*(c_ + d_.*x_^2)^p_*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^IntPart[p]*(c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPart[p]* Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && Not[IntegerQ[n/2]]
+Int[u_*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[u*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
+Int[u_*(c_ + d_.*x_^2)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := c^IntPart[p]*(c + d*x^2)^ FracPart[p]/((1 - a*x)^FracPart[p]*(1 + a*x)^FracPart[p])* Int[u*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && IntegerQ[n/2]
+Int[u_*(c_ + d_.*x_^2)^p_*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^IntPart[p]*(c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPart[p]* Int[u*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && Not[IntegerQ[n/2]]
+Int[u_.*(c_ + d_./x_^2)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := d^p*Int[u/x^(2*p)*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]
+Int[u_.*(c_ + d_./x_^2)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[u*(1 - 1/(a*x))^p*(1 + 1/(a*x))^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[p]] && IntegerQ[n/2] && GtQ[c, 0]
+Int[u_.*(c_ + d_./x_^2)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := x^(2*p)*(c + d/x^2)^p/((1 - a*x)^p*(1 + a*x)^p)* Int[u/x^(2*p)*(1 - a*x)^p*(1 + a*x)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[p]] && IntegerQ[n/2] && Not[GtQ[c, 0]]
+Int[u_.*(c_ + d_./x_^2)^p_*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := x^(2*p)*(c + d/x^2)^p/(1 + c*x^2/d)^p* Int[u/x^(2*p)*(1 + c*x^2/d)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[p]] && Not[IntegerQ[n/2]]
+Int[E^(n_.*ArcTanh[c_.*(a_ + b_.*x_)]), x_Symbol] := Int[(1 + a*c + b*c*x)^(n/2)/(1 - a*c - b*c*x)^(n/2), x] /; FreeQ[{a, b, c, n}, x]
+Int[x_^m_*E^(n_*ArcTanh[c_.*(a_ + b_.*x_)]), x_Symbol] := 4/(n*b^(m + 1)*c^(m + 1))* Subst[ Int[x^(2/n)*(-1 - a*c + (1 - a*c)*x^(2/n))^ m/(1 + x^(2/n))^(m + 2), x], x, (1 + c*(a + b*x))^(n/2)/(1 - c*(a + b*x))^(n/2)] /; FreeQ[{a, b, c}, x] && ILtQ[m, 0] && LtQ[-1, n, 1]
+Int[(d_. + e_.*x_)^m_.*E^(n_.*ArcTanh[c_.*(a_ + b_.*x_)]), x_Symbol] := Int[(d + e*x)^m*(1 + a*c + b*c*x)^(n/2)/(1 - a*c - b*c*x)^(n/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcTanh[a_ + b_.*x_]), x_Symbol] := (c/(1 - a^2))^p* Int[u*(1 - a - b*x)^(p - n/2)*(1 + a + b*x)^(p + n/2), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e (1 - a^2), 0] && (IntegerQ[p] || GtQ[c/(1 - a^2), 0])
+Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcTanh[a_ + b_.*x_]), x_Symbol] := (c + d*x + e*x^2)^p/(1 - a^2 - 2*a*b*x - b^2*x^2)^p* Int[u*(1 - a^2 - 2*a*b*x - b^2*x^2)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e (1 - a^2), 0] && Not[IntegerQ[p] || GtQ[c/(1 - a^2), 0]]
+Int[u_.*E^(n_.*ArcTanh[c_./(a_. + b_.*x_)]), x_Symbol] := Int[u*E^(n*ArcCoth[a/c + b*x/c]), x] /; FreeQ[{a, b, c, n}, x]
+Int[u_.*E^(n_*ArcCoth[a_.*x_]), x_Symbol] := (-1)^(n/2)*Int[u*E^(n*ArcTanh[a*x]), x] /; FreeQ[a, x] && IntegerQ[n/2]
+Int[E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -Subst[ Int[(1 + x/a)^((n + 1)/2)/(x^2*(1 - x/a)^((n - 1)/2)* Sqrt[1 - x^2/a^2]), x], x, 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2]
+Int[x_^m_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -Subst[ Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/a)^((n - 1)/2)* Sqrt[1 - x^2/a^2]), x], x, 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2] && IntegerQ[m]
+Int[E^(n_*ArcCoth[a_.*x_]), x_Symbol] := -Subst[Int[(1 + x/a)^(n/2)/(x^2*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && Not[IntegerQ[n]]
+Int[x_^m_.*E^(n_*ArcCoth[a_.*x_]), x_Symbol] := -Subst[Int[(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]
+Int[x_^m_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -x^m*(1/x)^m* Subst[Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/a)^((n - 1)/2)* Sqrt[1 - x^2/a^2]), x], x, 1/x] /; FreeQ[{a, m}, x] && IntegerQ[(n - 1)/2] && Not[IntegerQ[m]]
+Int[x_^m_*E^(n_*ArcCoth[a_.*x_]), x_Symbol] := -x^m*(1/x)^m* Subst[Int[(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, m, n}, x] && Not[IntegerQ[n]] && Not[IntegerQ[m]]
+Int[(c_ + d_.*x_)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := (1 + a*x)*(c + d*x)^p*E^(n*ArcCoth[a*x])/(a*(p + 1)) /; FreeQ[{a, c, d, n, p}, x] && EqQ[a*c + d, 0] && EqQ[p, n/2] && Not[IntegerQ[n/2]]
+(* Int[(c_+d_.*x_)^p_.*E^(n_.*ArcCoth[a_.*x_]),x_Symbol] := -(-a)^n*c^n*Subst[Int[(d+c*x)^(p-n)*(1-x^2/a^2)^(n/2)/x^(p+2),x],x, 1/x] /; FreeQ[{a,c,d},x] && EqQ[a*c+d,0] && IntegerQ[(n-1)/2] && IntegerQ[p] *)
+(* Int[x_^m_.*(c_+d_.*x_)^p_.*E^(n_.*ArcCoth[a_.*x_]),x_Symbol] := -(-a)^n*c^n*Subst[Int[(d+c*x)^(p-n)*(1-x^2/a^2)^(n/2)/x^(m+p+2),x], x,1/x] /; FreeQ[{a,c,d},x] && EqQ[a*c+d,0] && IntegerQ[(n-1)/2] && IntegerQ[m] &&  IntegerQ[p] *)
+(* Int[(c_+d_.*x_)^p_.*E^(n_.*ArcCoth[a_.*x_]),x_Symbol] := -(-a)^n*c^n*Sqrt[c+d*x]/(Sqrt[x]*Sqrt[d+c/x])*Subst[Int[(d+c*x)^(p- n)*(1-x^2/a^2)^(n/2)/x^(p+2),x],x,1/x] /; FreeQ[{a,c,d},x] && EqQ[a*c+d,0] && IntegerQ[(n-1)/2] &&  IntegerQ[p-1/2] *)
+(* Int[x_^m_.*(c_+d_.*x_)^p_.*E^(n_.*ArcCoth[a_.*x_]),x_Symbol] := -(-a)^n*c^n*Sqrt[c+d*x]/(Sqrt[x]*Sqrt[d+c/x])*Subst[Int[(d+c*x)^(p- n)*(1-x^2/a^2)^(n/2)/x^(m+p+2),x],x,1/x] /; FreeQ[{a,c,d},x] && EqQ[a*c+d,0] && IntegerQ[(n-1)/2] && IntegerQ[m] &&  IntegerQ[p-1/2] *)
+Int[u_.*(c_ + d_.*x_)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := d^p*Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c^2 - d^2, 0] && Not[IntegerQ[n/2]] && IntegerQ[p]
+Int[u_.*(c_ + d_.*x_)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := (c + d*x)^p/(x^p*(1 + c/(d*x))^p)* Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && Not[IntegerQ[n/2]] && Not[IntegerQ[p]]
+Int[(c_ + d_./x_)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^n*Subst[Int[(c + d*x)^(p - n)*(1 - x^2/a^2)^(n/2)/x^2, x], x, 1/x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1]) && IntegerQ[2*p]
+Int[x_^m_.*(c_ + d_./x_)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^n*Subst[Int[(c + d*x)^(p - n)*(1 - x^2/a^2)^(n/2)/x^(m + 2), x], x, 1/x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[ m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && IntegerQ[2*p]
+Int[(c_ + d_./x_)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^p*Subst[ Int[(1 + d*x/c)^p*(1 + x/a)^(n/2)/(x^2*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0])
+Int[x_^m_.*(c_ + d_./x_)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^p*Subst[ Int[(1 + d*x/c)^p*(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && IntegerQ[m]
+Int[x_^m_*(c_ + d_./x_)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^p*x^m*(1/x)^m* Subst[Int[(1 + d*x/c)^ p*(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegerQ[m]]
+Int[u_.*(c_ + d_./x_)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := (c + d/x)^p/(1 + d/(c*x))^p* Int[u*(1 + d/(c*x))^p*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[n/2]] && Not[IntegerQ[p] || GtQ[c, 0]]
+Int[E^(n_.*ArcCoth[a_.*x_])/(c_ + d_.*x_^2), x_Symbol] := E^(n*ArcCoth[a*x])/(a*c*n) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]]
+Int[E^(n_*ArcCoth[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol] := (n - a*x)*E^(n*ArcCoth[a*x])/(a*c*(n^2 - 1)*Sqrt[c + d*x^2]) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n]]
+Int[(c_ + d_.*x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := (n + 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCoth[a*x])/(a*c*(n^2 - 4*(p + 1)^2)) - 2*(p + 1)*(2*p + 3)/(c*(n^2 - 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] || Not[IntegerQ[n]])
+Int[x_*E^(n_*ArcCoth[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol] := -(1 - a*n*x)*E^(n*ArcCoth[a*x])/(a^2*c*(n^2 - 1)*Sqrt[c + d*x^2]) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n]]
+Int[x_*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := (2*(p + 1) + a*n*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCoth[a*x])/(a^2*c*(n^2 - 4*(p + 1)^2)) - n*(2*p + 3)/(a*c*(n^2 - 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && LeQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] || Not[IntegerQ[n]])
+Int[x_^2*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -(n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCoth[a*x])/(a^3*c*n^2*(n^2 - 1)) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && EqQ[n^2 + 2*(p + 1), 0] && NeQ[n^2, 1]
+Int[x_^2*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := (n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCoth[a*x])/(a^3*c*(n^2 - 4*(p + 1)^2)) - (n^2 + 2*(p + 1))/(a^2*c*(n^2 - 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && LeQ[p, -1] && NeQ[n^2 + 2*(p + 1), 0] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] || Not[IntegerQ[n]])
+Int[x_^m_.*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -(-c)^p/a^(m + 1)* Subst[Int[E^(n*x)*Coth[x]^(m + 2*(p + 1))/Cosh[x]^(2*(p + 1)), x], x, ArcCoth[a*x]] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && IntegerQ[m] && LeQ[3, m, -2 (p + 1)] && IntegerQ[p]
+Int[u_.*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := d^p*Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && IntegerQ[p]
+Int[u_.*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := (c + d*x^2)^p/(x^(2*p)*(1 - 1/(a^2*x^2))^p)* Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && Not[IntegerQ[p]]
+Int[u_.*(c_ + d_./x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := c^p/a^(2*p)* Int[u/x^(2*p)*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]
+Int[(c_ + d_./x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^p*Subst[Int[(1 - x/a)^(p - n/2)*(1 + x/a)^(p + n/2)/x^2, x], x, 1/x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegersQ[2*p, p + n/2]]
+Int[x_^m_.*(c_ + d_./x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^p*Subst[ Int[(1 - x/a)^(p - n/2)*(1 + x/a)^(p + n/2)/x^(m + 2), x], x, 1/x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegersQ[2*p, p + n/2]] && IntegerQ[m]
+Int[x_^m_*(c_ + d_./x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^p*x^m*(1/x)^m* Subst[Int[(1 - x/a)^(p - n/2)*(1 + x/a)^(p + n/2)/x^(m + 2), x], x, 1/x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegersQ[2*p, p + n/2]] && Not[IntegerQ[m]]
+Int[u_.*(c_ + d_./x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := c^IntPart[p]*(c + d/x^2)^FracPart[p]/(1 - 1/(a^2*x^2))^FracPart[p]* Int[u*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[n/2]] && Not[IntegerQ[p] || GtQ[c, 0]]
+Int[u_.*E^(n_*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol] := (-1)^(n/2)*Int[u*E^(n*ArcTanh[c*(a + b*x)]), x] /; FreeQ[{a, b, c}, x] && IntegerQ[n/2]
+Int[E^(n_.*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol] := (c*(a + b*x))^(n/ 2)*(1 + 1/(c*(a + b*x)))^(n/2)/(1 + a*c + b*c*x)^(n/2)* Int[(1 + a*c + b*c*x)^(n/2)/(-1 + a*c + b*c*x)^(n/2), x] /; FreeQ[{a, b, c, n}, x] && Not[IntegerQ[n/2]]
+Int[x_^m_*E^(n_*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol] := -4/(n*b^(m + 1)*c^(m + 1))* Subst[ Int[x^(2/n)*(1 + a*c + (1 - a*c)*x^(2/n))^ m/(-1 + x^(2/n))^(m + 2), x], x, (1 + 1/(c*(a + b*x)))^(n/2)/(1 - 1/(c*(a + b*x)))^(n/2)] /; FreeQ[{a, b, c}, x] && ILtQ[m, 0] && LtQ[-1, n, 1]
+Int[(d_. + e_.*x_)^m_.*E^(n_.*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol] := (c*(a + b*x))^(n/ 2)*(1 + 1/(c*(a + b*x)))^(n/2)/(1 + a*c + b*c*x)^(n/2)* Int[(d + e*x)^m*(1 + a*c + b*c*x)^(n/2)/(-1 + a*c + b*c*x)^(n/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && Not[IntegerQ[n/2]]
+Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcCoth[a_ + b_.*x_]), x_Symbol] := (c/(1 - a^2))^ p*((a + b*x)/(1 + a + b*x))^(n/2)*((1 + a + b*x)/(a + b*x))^(n/ 2)*((1 - a - b*x)^(n/2)/(-1 + a + b*x)^(n/2))* Int[u*(1 - a - b*x)^(p - n/2)*(1 + a + b*x)^(p + n/2), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && Not[IntegerQ[n/2]] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e (1 - a^2), 0] && (IntegerQ[p] || GtQ[c/(1 - a^2), 0])
+Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcCoth[a_ + b_.*x_]), x_Symbol] := (c + d*x + e*x^2)^p/(1 - a^2 - 2*a*b*x - b^2*x^2)^p* Int[u*(1 - a^2 - 2*a*b*x - b^2*x^2)^p*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && Not[IntegerQ[n/2]] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e (1 - a^2), 0] && Not[IntegerQ[p] || GtQ[c/(1 - a^2), 0]]
+Int[u_.*E^(n_.*ArcCoth[c_./(a_. + b_.*x_)]), x_Symbol] := Int[u*E^(n*ArcTanh[a/c + b*x/c]), x] /; FreeQ[{a, b, c, n}, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m
new file mode 100755
index 0000000..6beeb7b
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m	
@@ -0,0 +1,75 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.3.4 Miscellaneous inverse hyperbolic tangent *)
+Int[ArcTanh[a_ + b_.*x_^n_], x_Symbol] := x*ArcTanh[a + b*x^n] - b*n*Int[x^n/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x] /; FreeQ[{a, b, n}, x]
+Int[ArcCoth[a_ + b_.*x_^n_], x_Symbol] := x*ArcCoth[a + b*x^n] - b*n*Int[x^n/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x] /; FreeQ[{a, b, n}, x]
+Int[ArcTanh[a_. + b_.*x_^n_.]/x_, x_Symbol] := 1/2*Int[Log[1 + a + b*x^n]/x, x] - 1/2*Int[Log[1 - a - b*x^n]/x, x] /; FreeQ[{a, b, n}, x]
+Int[ArcCoth[a_. + b_.*x_^n_.]/x_, x_Symbol] := 1/2*Int[Log[1 + 1/(a + b*x^n)]/x, x] - 1/2*Int[Log[1 - 1/(a + b*x^n)]/x, x] /; FreeQ[{a, b, n}, x]
+Int[x_^m_.*ArcTanh[a_ + b_.*x_^n_], x_Symbol] := x^(m + 1)*ArcTanh[a + b*x^n]/(m + 1) - b*n/(m + 1)* Int[x^(m + n)/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x] /; FreeQ[{a, b}, x] && RationalQ[m, n] && NeQ[m, -1] && NeQ[m + 1, n]
+Int[x_^m_.*ArcCoth[a_ + b_.*x_^n_], x_Symbol] := x^(m + 1)*ArcCoth[a + b*x^n]/(m + 1) - b*n/(m + 1)* Int[x^(m + n)/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x] /; FreeQ[{a, b}, x] && RationalQ[m, n] && NeQ[m, -1] && NeQ[m + 1, n]
+Int[ArcTanh[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol] := 1/2*Int[Log[1 + a + b*f^(c + d*x)], x] - 1/2*Int[Log[1 - a - b*f^(c + d*x)], x] /; FreeQ[{a, b, c, d, f}, x] 
+Int[ArcCoth[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol] := 1/2*Int[Log[1 + 1/(a + b*f^(c + d*x))], x] - 1/2*Int[Log[1 - 1/(a + b*f^(c + d*x))], x] /; FreeQ[{a, b, c, d, f}, x] 
+Int[x_^m_.*ArcTanh[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol] := 1/2*Int[x^m*Log[1 + a + b*f^(c + d*x)], x] - 1/2*Int[x^m*Log[1 - a - b*f^(c + d*x)], x] /; FreeQ[{a, b, c, d, f}, x] && IGtQ[m, 0]
+Int[x_^m_.*ArcCoth[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol] := 1/2*Int[x^m*Log[1 + 1/(a + b*f^(c + d*x))], x] - 1/2*Int[x^m*Log[1 - 1/(a + b*f^(c + d*x))], x] /; FreeQ[{a, b, c, d, f}, x] && IGtQ[m, 0]
+Int[u_.*ArcTanh[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol] := Int[u*ArcCoth[a/c + b*x^n/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
+Int[u_.*ArcCoth[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol] := Int[u*ArcTanh[a/c + b*x^n/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
+Int[ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol] := x*ArcTanh[(c*x)/Sqrt[a + b*x^2]] - c*Int[x/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
+Int[ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol] := x*ArcCoth[(c*x)/Sqrt[a + b*x^2]] - c*Int[x/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
+Int[ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]]/x_, x_Symbol] := ArcTanh[c*x/Sqrt[a + b*x^2]]*Log[x] - c*Int[Log[x]/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
+Int[ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]]/x_, x_Symbol] := ArcCoth[c*x/Sqrt[a + b*x^2]]*Log[x] - c*Int[Log[x]/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
+Int[(d_.*x_)^m_.*ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol] := (d*x)^(m + 1)*ArcTanh[(c*x)/Sqrt[a + b*x^2]]/(d*(m + 1)) - c/(d*(m + 1))*Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b, c^2] && NeQ[m, -1]
+Int[(d_.*x_)^m_.*ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol] := (d*x)^(m + 1)*ArcCoth[(c*x)/Sqrt[a + b*x^2]]/(d*(m + 1)) - c/(d*(m + 1))*Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b, c^2] && NeQ[m, -1]
+Int[1/(Sqrt[a_. + b_.*x_^2]*ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]]), x_Symbol] := 1/c*Log[ArcTanh[c*x/Sqrt[a + b*x^2]]] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
+Int[1/(Sqrt[a_. + b_.*x_^2]*ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]]), x_Symbol] := -1/c*Log[ArcCoth[c*x/Sqrt[a + b*x^2]]] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
+Int[ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[a_. + b_.*x_^2], x_Symbol] := ArcTanh[c*x/Sqrt[a + b*x^2]]^(m + 1)/(c*(m + 1)) /; FreeQ[{a, b, c, m}, x] && EqQ[b, c^2] && NeQ[m, -1]
+Int[ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[a_. + b_.*x_^2], x_Symbol] := -ArcCoth[c*x/Sqrt[a + b*x^2]]^(m + 1)/(c*(m + 1)) /; FreeQ[{a, b, c, m}, x] && EqQ[b, c^2] && NeQ[m, -1]
+Int[ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[d_. + e_.*x_^2], x_Symbol] := Sqrt[a + b*x^2]/Sqrt[d + e*x^2]* Int[ArcTanh[c*x/Sqrt[a + b*x^2]]^m/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b, c^2] && EqQ[b*d - a*e, 0]
+Int[ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[d_. + e_.*x_^2], x_Symbol] := Sqrt[a + b*x^2]/Sqrt[d + e*x^2]* Int[ArcCoth[c*x/Sqrt[a + b*x^2]]^m/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b, c^2] && EqQ[b*d - a*e, 0]
+If[TrueQ[$LoadShowSteps], Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, ShowStep["", "Int[f[x,ArcTanh[a+b*x]]/(1-(a+b*x)^2),x]", "Subst[Int[f[-a/b+Tanh[x]/b,x],x],x,ArcTanh[a+b*x]]/b", Hold[ (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[Int[ SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*Sech[x]^(2*(n + 1)), x], x], x, tmp]]] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcTanh] && EqQ[Discriminant[v, x]*tmp[[1]]^2 - D[v, x]^2, 0]] /; SimplifyFlag && QuadraticQ[v, x] && ILtQ[n, 0] && PosQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]], Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[ Int[SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*Sech[x]^(2*(n + 1)), x], x], x, tmp] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcTanh] && EqQ[Discriminant[v, x]*tmp[[1]]^2 - D[v, x]^2, 0]] /; QuadraticQ[v, x] && ILtQ[n, 0] && PosQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]]]
+If[TrueQ[$LoadShowSteps], Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, ShowStep["", "Int[f[x,ArcCoth[a+b*x]]/(1-(a+b*x)^2),x]", "Subst[Int[f[-a/b+Coth[x]/b,x],x],x,ArcCoth[a+b*x]]/b", Hold[ (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[Int[ SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*(-Csch[x]^2)^(n + 1), x], x], x, tmp]]] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcCoth] && EqQ[Discriminant[v, x]*tmp[[1]]^2 - D[v, x]^2, 0]] /; SimplifyFlag && QuadraticQ[v, x] && ILtQ[n, 0] && PosQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]], Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[ Int[SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*(-Csch[x]^2)^(n + 1), x], x], x, tmp] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcCoth] && EqQ[Discriminant[v, x]*tmp[[1]]^2 - D[v, x]^2, 0]] /; QuadraticQ[v, x] && ILtQ[n, 0] && PosQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]]]
+Int[ArcTanh[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Tanh[a + b*x]] + b*Int[x/(c - d + c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]
+Int[ArcCoth[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Tanh[a + b*x]] + b*Int[x/(c - d + c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]
+Int[ArcTanh[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Coth[a + b*x]] + b*Int[x/(c - d - c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]
+Int[ArcCoth[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Coth[a + b*x]] + b*Int[x/(c - d - c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]
+Int[ArcTanh[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Tanh[a + b*x]] + b*(1 - c - d)* Int[x*E^(2*a + 2*b*x)/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x] - b*(1 + c + d)* Int[x*E^(2*a + 2*b*x)/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]
+Int[ArcCoth[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Tanh[a + b*x]] + b*(1 - c - d)* Int[x*E^(2*a + 2*b*x)/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x] - b*(1 + c + d)* Int[x*E^(2*a + 2*b*x)/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]
+Int[ArcTanh[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Coth[a + b*x]] + b*(1 + c + d)* Int[x*E^(2*a + 2*b*x)/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x] - b*(1 - c - d)* Int[x*E^(2*a + 2*b*x)/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]
+Int[ArcCoth[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Coth[a + b*x]] + b*(1 + c + d)* Int[x*E^(2*a + 2*b*x)/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x] - b*(1 - c - d)* Int[x*E^(2*a + 2*b*x)/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Tanh[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d + c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Tanh[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d + c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Coth[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Coth[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Tanh[a + b*x]]/(f*(m + 1)) + b*(1 - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x] - b*(1 + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Tanh[a + b*x]]/(f*(m + 1)) + b*(1 - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x] - b*(1 + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Coth[a + b*x]]/(f*(m + 1)) + b*(1 + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x] - b*(1 - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Coth[a + b*x]]/(f*(m + 1)) + b*(1 + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x] - b*(1 - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]
+Int[ArcTanh[Tan[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[Tan[a + b*x]] - b*Int[x*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b}, x]
+Int[ArcCoth[Tan[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[Tan[a + b*x]] - b*Int[x*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b}, x]
+Int[ArcTanh[Cot[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[Cot[a + b*x]] - b*Int[x*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b}, x]
+Int[ArcCoth[Cot[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[Cot[a + b*x]] - b*Int[x*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b}, x]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[Tan[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[Tan[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[Cot[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[Cot[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
+Int[ArcTanh[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Tan[a + b*x]] + I*b*Int[x/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c + I*d)^2, 1]
+Int[ArcCoth[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Tan[a + b*x]] + I*b*Int[x/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c + I*d)^2, 1]
+Int[ArcTanh[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Cot[a + b*x]] + I*b*Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, 1]
+Int[ArcCoth[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Cot[a + b*x]] + I*b*Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, 1]
+Int[ArcTanh[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Tan[a + b*x]] + I*b*(1 - c + I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x] - I*b*(1 + c - I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, 1]
+Int[ArcCoth[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Tan[a + b*x]] + I*b*(1 - c + I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x] - I*b*(1 + c - I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, 1]
+Int[ArcTanh[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Cot[a + b*x]] - I*b*(1 - c - I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x] + I*b*(1 + c + I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - I*d)^2, 1]
+Int[ArcCoth[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Cot[a + b*x]] - I*b*(1 - c - I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x] + I*b*(1 + c + I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Tan[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c + I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Tan[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c + I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Cot[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Cot[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Tan[a + b*x]]/(f*(m + 1)) + I*b*(1 - c + I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x] - I*b*(1 + c - I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c + I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Tan[a + b*x]]/(f*(m + 1)) + I*b*(1 - c + I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x] - I*b*(1 + c - I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c + I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Cot[a + b*x]]/(f*(m + 1)) - I*b*(1 - c - I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x] + I*b*(1 + c + I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Cot[a + b*x]]/(f*(m + 1)) - I*b*(1 - c - I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x] + I*b*(1 + c + I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - I*d)^2, 1]
+Int[ArcTanh[u_], x_Symbol] := x*ArcTanh[u] - Int[SimplifyIntegrand[x*D[u, x]/(1 - u^2), x], x] /; InverseFunctionFreeQ[u, x]
+Int[ArcCoth[u_], x_Symbol] := x*ArcCoth[u] - Int[SimplifyIntegrand[x*D[u, x]/(1 - u^2), x], x] /; InverseFunctionFreeQ[u, x]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcTanh[u_]), x_Symbol] := (c + d*x)^(m + 1)*(a + b*ArcTanh[u])/(d*(m + 1)) - b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(1 - u^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && FalseQ[PowerVariableExpn[u, m + 1, x]]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcCoth[u_]), x_Symbol] := (c + d*x)^(m + 1)*(a + b*ArcCoth[u])/(d*(m + 1)) - b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(1 - u^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && FalseQ[PowerVariableExpn[u, m + 1, x]]
+Int[v_*(a_. + b_.*ArcTanh[u_]), x_Symbol] := With[{w = IntHide[v, x]}, Dist[(a + b*ArcTanh[u]), w, x] - b*Int[SimplifyIntegrand[w*D[u, x]/(1 - u^2), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_. /; FreeQ[{c, d, m}, x]]] && FalseQ[FunctionOfLinear[v*(a + b*ArcTanh[u]), x]]
+Int[v_*(a_. + b_.*ArcCoth[u_]), x_Symbol] := With[{w = IntHide[v, x]}, Dist[(a + b*ArcCoth[u]), w, x] - b*Int[SimplifyIntegrand[w*D[u, x]/(1 - u^2), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_. /; FreeQ[{c, d, m}, x]]] && FalseQ[FunctionOfLinear[v*(a + b*ArcCoth[u]), x]]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 u (a+b arctanh(c x))^p.m b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 u (a+b arctanh(c x))^p.m
new file mode 100755
index 0000000..613b774
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 u (a+b arctanh(c x))^p.m	
@@ -0,0 +1,166 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.3.4 u (a+b arctanh(c x))^p *)
+Int[(f_.*x_)^m_.*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := f/e*Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x])^p, x] - d*f/e*Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x])^p/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && GtQ[m, 0]
+Int[(f_.*x_)^m_.*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := f/e*Int[(f*x)^(m - 1)*(a + b*ArcCoth[c*x])^p, x] - d*f/e*Int[(f*x)^(m - 1)*(a + b*ArcCoth[c*x])^p/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && GtQ[m, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(x_*(d_ + e_.*x_)), x_Symbol] := (a + b*ArcTanh[c*x])^p*Log[2 - 2/(1 + e*x/d)]/d - b*c*p/d* Int[(a + b*ArcTanh[c*x])^(p - 1)* Log[2 - 2/(1 + e*x/d)]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(x_*(d_ + e_.*x_)), x_Symbol] := (a + b*ArcCoth[c*x])^p*Log[2 - 2/(1 + e*x/d)]/d - b*c*p/d* Int[(a + b*ArcCoth[c*x])^(p - 1)* Log[2 - 2/(1 + e*x/d)]/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := 1/d*Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x] - e/(d*f)*Int[(f*x)^(m + 1)*(a + b*ArcTanh[c*x])^p/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol] := 1/d*Int[(f*x)^m*(a + b*ArcCoth[c*x])^p, x] - e/(d*f)*Int[(f*x)^(m + 1)*(a + b*ArcCoth[c*x])^p/(d + e*x), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcTanh[c*x]), u] - b*c*Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[ 2*m] && (IGtQ[m, 0] && IGtQ[q, 0] || ILtQ[m + q + 1, 0] && LtQ[m*q, 0])
+Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcCoth[c*x]), u] - b*c*Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[ 2*m] && (IGtQ[m, 0] && IGtQ[q, 0] || ILtQ[m + q + 1, 0] && LtQ[m*q, 0])
+Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcTanh[c*x])^p, u] - b*c*p*Int[ ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1), u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 - e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]
+Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcCoth[c*x])^p, u] - b*c*p*Int[ ExpandIntegrand[(a + b*ArcCoth[c*x])^(p - 1), u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 - e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := b*(d + e*x^2)^q/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcTanh[c*x])/(2*q + 1) + 2*d*q/(2*q + 1)* Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0]
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := b*(d + e*x^2)^q/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcCoth[c*x])/(2*q + 1) + 2*d*q/(2*q + 1)* Int[(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0]
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol] := b*p*(d + e*x^2)^ q*(a + b*ArcTanh[c*x])^(p - 1)/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p/(2*q + 1) + 2*d*q/(2*q + 1)* Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x] - b^2*d*p*(p - 1)/(2*q*(2*q + 1))* Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^(p - 2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && GtQ[p, 1]
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol] := b*p*(d + e*x^2)^ q*(a + b*ArcCoth[c*x])^(p - 1)/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^p/(2*q + 1) + 2*d*q/(2*q + 1)* Int[(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x])^p, x] - b^2*d*p*(p - 1)/(2*q*(2*q + 1))* Int[(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x])^(p - 2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && GtQ[p, 1]
+(* Int[(a_.+b_.*ArcTanh[c_.*x_])^p_./(d_+e_.*x_^2),x_Symbol] := 1/(c*d)*Subst[Int[(a+b*x)^p,x],x,ArcTanh[c*x]] /; FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e,0] *)
+(* Int[(a_.+b_.*ArcCoth[c_.*x_])^p_./(d_+e_.*x_^2),x_Symbol] := 1/(c*d)*Subst[Int[(a+b*x)^p,x],x,ArcCoth[c*x]] /; FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e,0] *)
+Int[1/((d_ + e_.*x_^2)*(a_. + b_.*ArcTanh[c_.*x_])), x_Symbol] := Log[RemoveContent[a + b*ArcTanh[c*x], x]]/(b*c*d) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
+Int[1/((d_ + e_.*x_^2)*(a_. + b_.*ArcCoth[c_.*x_])), x_Symbol] := Log[RemoveContent[a + b*ArcCoth[c*x], x]]/(b*c*d) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)) /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)) /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
+Int[(a_. + b_.*ArcTanh[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol] := -2*(a + b*ArcTanh[c*x])* ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*Sqrt[d]) - I*b*PolyLog[2, -I*Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*Sqrt[d]) + I*b*PolyLog[2, I*Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*Sqrt[d]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol] := -2*(a + b*ArcCoth[c*x])* ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*Sqrt[d]) - I*b*PolyLog[2, -I*Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*Sqrt[d]) + I*b*PolyLog[2, I*Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*Sqrt[d]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := 1/(c*Sqrt[d])*Subst[Int[(a + b*x)^p*Sech[x], x], x, ArcTanh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := -x*Sqrt[1 - 1/(c^2*x^2)]/Sqrt[d + e*x^2]* Subst[Int[(a + b*x)^p*Csch[x], x], x, ArcCoth[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcTanh[c*x])^p/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && Not[GtQ[d, 0]]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcCoth[c*x])^p/Sqrt[1 - c^2*x^2], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && Not[GtQ[d, 0]]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := x*(a + b*ArcTanh[c*x])^p/(2*d*(d + e*x^2)) + (a + b*ArcTanh[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)) - b*c*p/2*Int[x*(a + b*ArcTanh[c*x])^(p - 1)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := x*(a + b*ArcCoth[c*x])^p/(2*d*(d + e*x^2)) + (a + b*ArcCoth[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)) - b*c*p/2*Int[x*(a + b*ArcCoth[c*x])^(p - 1)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])/(d_ + e_.*x_^2)^(3/2), x_Symbol] := -b/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcTanh[c*x])/(d*Sqrt[d + e*x^2]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])/(d_ + e_.*x_^2)^(3/2), x_Symbol] := -b/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcCoth[c*x])/(d*Sqrt[d + e*x^2]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := -b*(d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := -b*(d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_/(d_ + e_.*x_^2)^(3/2), x_Symbol] := -b*p*(a + b*ArcTanh[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcTanh[c*x])^p/(d*Sqrt[d + e*x^2]) + b^2*p*(p - 1)* Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2)^(3/2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_/(d_ + e_.*x_^2)^(3/2), x_Symbol] := -b*p*(a + b*ArcCoth[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcCoth[c*x])^p/(d*Sqrt[d + e*x^2]) + b^2*p*(p - 1)* Int[(a + b*ArcCoth[c*x])^(p - 2)/(d + e*x^2)^(3/2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol] := -b*p*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p - 1)/(4*c* d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x] + b^2*p*(p - 1)/(4*(q + 1)^2)* Int[(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol] := -b*p*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p - 1)/(4*c* d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p, x] + b^2*p*(p - 1)/(4*(q + 1)^2)* Int[(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p + 1)/(b*c* d*(p + 1)) + 2*c*(q + 1)/(b*(p + 1))* Int[x*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && LtQ[p, -1]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p + 1)/(b*c* d*(p + 1)) + 2*c*(q + 1)/(b*(p + 1))* Int[x*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && LtQ[p, -1]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := d^q/c* Subst[Int[(a + b*x)^p/Cosh[x]^(2*(q + 1)), x], x, ArcTanh[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := d^(q + 1/2)*Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(1 - c^2*x^2)^q*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && Not[IntegerQ[q] || GtQ[d, 0]]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := -(-d)^q/c* Subst[Int[(a + b*x)^p/Sinh[x]^(2*(q + 1)), x], x, ArcCoth[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && IntegerQ[q]
+Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := -(-d)^(q + 1/2)*x*Sqrt[(c^2*x^2 - 1)/(c^2*x^2)]/Sqrt[d + e*x^2]* Subst[Int[(a + b*x)^p/Sinh[x]^(2*(q + 1)), x], x, ArcCoth[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && Not[IntegerQ[q]]
+Int[ArcTanh[c_.*x_]/(d_. + e_.*x_^2), x_Symbol] := 1/2*Int[Log[1 + c*x]/(d + e*x^2), x] - 1/2*Int[Log[1 - c*x]/(d + e*x^2), x] /; FreeQ[{c, d, e}, x]
+Int[ArcCoth[c_.*x_]/(d_. + e_.*x_^2), x_Symbol] := 1/2*Int[Log[1 + 1/(c*x)]/(d + e*x^2), x] - 1/2*Int[Log[1 - 1/(c*x)]/(d + e*x^2), x] /; FreeQ[{c, d, e}, x]
+Int[(a_ + b_.*ArcTanh[c_.*x_])/(d_. + e_.*x_^2), x_Symbol] := a*Int[1/(d + e*x^2), x] + b*Int[ArcTanh[c*x]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x]
+Int[(a_ + b_.*ArcCoth[c_.*x_])/(d_. + e_.*x_^2), x_Symbol] := a*Int[1/(d + e*x^2), x] + b*Int[ArcCoth[c*x]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x]
+Int[(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^q, x]}, Dist[a + b*ArcTanh[c*x], u, x] - b*c*Int[u/(1 - c^2*x^2), x]] /; FreeQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
+Int[(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^q, x]}, Dist[a + b*ArcCoth[c*x], u, x] - b*c*Int[u/(1 - c^2*x^2), x]] /; FreeQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, (d + e*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]
+Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^p, (d + e*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := f^2/e*Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p, x] - d*f^2/e* Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := f^2/e*Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p, x] - d*f^2/e* Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/d*Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x] - e/(d*f^2)* Int[(f*x)^(m + 2)*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/d*Int[(f*x)^m*(a + b*ArcCoth[c*x])^p, x] - e/(d*f^2)* Int[(f*x)^(m + 2)*(a + b*ArcCoth[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
+Int[x_*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)) + 1/(c*d)*Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
+Int[x_*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcCoth[c*x])^(p + 1)/(b*e*(p + 1)) + 1/(c*d)*Int[(a + b*ArcCoth[c*x])^p/(1 - c*x), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
+Int[x_*(a_. + b_.*ArcTanh[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol] := x*(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)) - 1/(b*c*d*(p + 1))*Int[(a + b*ArcTanh[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Not[IGtQ[p, 0]] && NeQ[p, -1]
+Int[x_*(a_. + b_.*ArcCoth[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol] := -x*(a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)) - 1/(b*c*d*(p + 1))*Int[(a + b*ArcCoth[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Not[IGtQ[p, 0]] && NeQ[p, -1]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(x_*(d_ + e_.*x_^2)), x_Symbol] := (a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)) + 1/d*Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(x_*(d_ + e_.*x_^2)), x_Symbol] := (a + b*ArcCoth[c*x])^(p + 1)/(b*d*(p + 1)) + 1/d*Int[(a + b*ArcCoth[c*x])^p/(x*(1 + c*x)), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol] := (f*x)^m*(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)) - f*m/(b*c*d*(p + 1))* Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol] := (f*x)^m*(a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)) - f*m/(b*c*d*(p + 1))* Int[(f*x)^(m - 1)*(a + b*ArcCoth[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1]
+Int[x_^m_.*(a_. + b_.*ArcTanh[c_.*x_])/(d_ + e_.*x_^2), x_Symbol] := Int[ExpandIntegrand[(a + b*ArcTanh[c*x]), x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && Not[EqQ[m, 1] && NeQ[a, 0]]
+Int[x_^m_.*(a_. + b_.*ArcCoth[c_.*x_])/(d_ + e_.*x_^2), x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCoth[c*x]), x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && Not[EqQ[m, 1] && NeQ[a, 0]]
+Int[x_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p/(2*e*(q + 1)) + b*p/(2*c*(q + 1))* Int[(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]
+Int[x_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p/(2*e*(q + 1)) + b*p/(2*c*(q + 1))* Int[(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 1), x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]
+Int[x_*(a_. + b_.*ArcTanh[c_.*x_])^p_/(d_ + e_.*x_^2)^2, x_Symbol] := x*(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)*(d + e*x^2)) + (1 + c^2*x^2)*(a + b*ArcTanh[c*x])^(p + 2)/(b^2* e*(p + 1)*(p + 2)*(d + e*x^2)) + 4/(b^2*(p + 1)*(p + 2))* Int[x*(a + b*ArcTanh[c*x])^(p + 2)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1] && NeQ[p, -2]
+Int[x_*(a_. + b_.*ArcCoth[c_.*x_])^p_/(d_ + e_.*x_^2)^2, x_Symbol] := x*(a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)*(d + e*x^2)) + (1 + c^2*x^2)*(a + b*ArcCoth[c*x])^(p + 2)/(b^2* e*(p + 1)*(p + 2)*(d + e*x^2)) + 4/(b^2*(p + 1)*(p + 2))* Int[x*(a + b*ArcCoth[c*x])^(p + 2)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1] && NeQ[p, -2]
+Int[x_^2*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := -b*(d + e*x^2)^(q + 1)/(4*c^3*d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])/(2*c^2*d*(q + 1)) + 1/(2*c^2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -5/2]
+Int[x_^2*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := -b*(d + e*x^2)^(q + 1)/(4*c^3*d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])/(2*c^2*d*(q + 1)) + 1/(2*c^2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -5/2]
+Int[x_^2*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := -(a + b*ArcTanh[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)) + x*(a + b*ArcTanh[c*x])^p/(2*c^2*d*(d + e*x^2)) - b*p/(2*c)*Int[x*(a + b*ArcTanh[c*x])^(p - 1)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[x_^2*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := -(a + b*ArcCoth[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)) + x*(a + b*ArcCoth[c*x])^p/(2*c^2*d*(d + e*x^2)) - b*p/(2*c)*Int[x*(a + b*ArcCoth[c*x])^(p - 1)/(d + e*x^2)^2, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := -b*(f*x)^m*(d + e*x^2)^(q + 1)/(c*d*m^2) + f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])/(c^2*d* m) - f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := -b*(f*x)^m*(d + e*x^2)^(q + 1)/(c*d*m^2) + f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])/(c^2*d* m) - f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol] := -b*p*(f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p - 1)/(c*d*m^2) + f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^ p/(c^2*d*m) + b^2*p*(p - 1)/m^2* Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 2), x] - f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol] := -b*p*(f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p - 1)/(c*d*m^2) + f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^ p/(c^2*d*m) + b^2*p*(p - 1)/m^2* Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 2), x] - f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol] := (f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p + 1)/(b*c* d*(p + 1)) - f*m/(b*c*(p + 1))* Int[(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol] := (f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p + 1)/(b*c* d*(p + 1)) - f*m/(b*c*(p + 1))* Int[(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^ p/(d*(m + 1)) - b*c*p/(m + 1)* Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := (f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^ p/(d*f*(m + 1)) - b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
+Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := (f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])/(f*(m + 2)) - b*c*d/(f*(m + 2))*Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x] + d/(m + 2)*Int[(f*x)^m*(a + b*ArcTanh[c*x])/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && NeQ[m, -2]
+Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := (f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x])/(f*(m + 2)) - b*c*d/(f*(m + 2))*Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x] + d/(m + 2)*Int[(f*x)^m*(a + b*ArcCoth[c*x])/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && NeQ[m, -2]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := d*Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x] - c^2*d/f^2* Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || EqQ[p, 1] && IntegerQ[q])
+Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := d*Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x])^p, x] - c^2*d/f^2* Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || EqQ[p, 1] && IntegerQ[q])
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := -f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])^p/(c^2*d*m) + b*f*p/(c*m)* Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x])^(p - 1)/Sqrt[d + e*x^2], x] + f^2*(m - 1)/(c^2*m)* Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && GtQ[m, 1]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := -f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x])^p/(c^2*d*m) + b*f*p/(c*m)* Int[(f*x)^(m - 1)*(a + b*ArcCoth[c*x])^(p - 1)/Sqrt[d + e*x^2], x] + f^2*(m - 1)/(c^2*m)* Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && GtQ[m, 1]
+Int[(a_. + b_.*ArcTanh[c_.*x_])/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := -2/Sqrt[d]*(a + b*ArcTanh[c*x])* ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]] + b/Sqrt[d]*PolyLog[2, -Sqrt[1 - c*x]/Sqrt[1 + c*x]] - b/Sqrt[d]*PolyLog[2, Sqrt[1 - c*x]/Sqrt[1 + c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := -2/Sqrt[d]*(a + b*ArcCoth[c*x])* ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]] + b/Sqrt[d]*PolyLog[2, -Sqrt[1 - c*x]/Sqrt[1 + c*x]] - b/Sqrt[d]*PolyLog[2, Sqrt[1 - c*x]/Sqrt[1 + c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := 1/Sqrt[d]*Subst[Int[(a + b*x)^p*Csch[x], x], x, ArcTanh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := -c*x*Sqrt[1 - 1/(c^2*x^2)]/Sqrt[d + e*x^2]* Subst[Int[(a + b*x)^p*Sech[x], x], x, ArcCoth[c*x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcTanh[c*x])^p/(x*Sqrt[1 - c^2*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && Not[GtQ[d, 0]]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol] := Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcCoth[c*x])^p/(x*Sqrt[1 - c^2*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && Not[GtQ[d, 0]]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(x_^2*Sqrt[d_ + e_.*x_^2]), x_Symbol] := -Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])^p/(d*x) + b*c*p*Int[(a + b*ArcTanh[c*x])^(p - 1)/(x*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(x_^2*Sqrt[d_ + e_.*x_^2]), x_Symbol] := -Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x])^p/(d*x) + b*c*p*Int[(a + b*ArcCoth[c*x])^(p - 1)/(x*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := (f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])^p/(d*f*(m + 1)) - b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1)/Sqrt[d + e*x^2], x] + c^2*(m + 2)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(a + b*ArcTanh[c*x])^p/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
+Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol] := (f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x])^p/(d*f*(m + 1)) - b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1)/Sqrt[d + e*x^2], x] + c^2*(m + 2)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(a + b*ArcCoth[c*x])^p/Sqrt[d + e*x^2], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
+Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := 1/e*Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x] - d/e*Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
+Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := 1/e*Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p, x] - d/e*Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^p, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
+Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := 1/d*Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x] - e/d*Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
+Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := 1/d*Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p, x] - e/d*Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^p, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p + 1)/(b*c* d*(p + 1)) - m/(b*c*(p + 1))* Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x] + c*(m + 2*q + 2)/(b*(p + 1))* Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && LtQ[q, -1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := x^m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p + 1)/(b*c* d*(p + 1)) - m/(b*c*(p + 1))* Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p + 1), x] + c*(m + 2*q + 2)/(b*(p + 1))* Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p + 1), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && LtQ[q, -1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := d^q/c^(m + 1)* Subst[Int[(a + b*x)^p*Sinh[x]^m/Cosh[x]^(m + 2*(q + 1)), x], x, ArcTanh[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q] || GtQ[d, 0])
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := d^(q + 1/2)*Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[x^m*(1 - c^2*x^2)^q*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && Not[IntegerQ[q] || GtQ[d, 0]]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := -(-d)^q/c^(m + 1)* Subst[Int[(a + b*x)^p*Cosh[x]^m/Sinh[x]^(m + 2*(q + 1)), x], x, ArcCoth[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && IntegerQ[q]
+Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := -(-d)^(q + 1/2)*x* Sqrt[(c^2*x^2 - 1)/(c^2*x^2)]/(c^m*Sqrt[d + e*x^2])* Subst[Int[(a + b*x)^p*Cosh[x]^m/Sinh[x]^(m + 2*(q + 1)), x], x, ArcCoth[c*x]] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && Not[IntegerQ[q]]
+Int[x_*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])/(2*e*(q + 1)) - b*c/(2*e*(q + 1))*Int[(d + e*x^2)^(q + 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
+Int[x_*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := (d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])/(2*e*(q + 1)) - b*c/(2*e*(q + 1))*Int[(d + e*x^2)^(q + 1)/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
+Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcTanh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ( IGtQ[q, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[q, 0] && GtQ[m + 2*q + 3, 0]] || ILtQ[(m + 2*q + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]] )
+Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcCoth[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ( IGtQ[q, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[q, 0] && GtQ[m + 2*q + 3, 0]] || ILtQ[(m + 2*q + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]] )
+Int[x_*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcTanh[c*x])^p/(1 - Rt[-e/d, 2]*x)^2, x] - 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcTanh[c*x])^p/(1 + Rt[-e/d, 2]*x)^2, x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0]
+Int[x_*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol] := 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcCoth[c*x])^p/(1 - Rt[-e/d, 2]*x)^2, x] - 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcCoth[c*x])^p/(1 + Rt[-e/d, 2]*x)^2, x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := With[{u = ExpandIntegrand[(a + b*ArcTanh[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (GtQ[q, 0] || IntegerQ[m])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := With[{u = ExpandIntegrand[(a + b*ArcCoth[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (GtQ[q, 0] || IntegerQ[m])
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*ArcTanh[c_.*x_]), x_Symbol] := a*Int[(f*x)^m*(d + e*x^2)^q, x] + b*Int[(f*x)^m*(d + e*x^2)^q*ArcTanh[c*x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x]
+Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*ArcCoth[c_.*x_]), x_Symbol] := a*Int[(f*x)^m*(d + e*x^2)^q, x] + b*Int[(f*x)^m*(d + e*x^2)^q*ArcCoth[c*x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x]
+Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p/(d + e*x^2), (f + g*x)^ m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && IGtQ[m, 0]
+Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^p/(d + e*x^2), (f + g*x)^ m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && IGtQ[m, 0]
+Int[ArcTanh[u_]*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/2*Int[Log[1 + u]*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x] - 1/2*Int[Log[1 - u]*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
+Int[ArcCoth[u_]*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/2*Int[ Log[SimplifyIntegrand[1 + 1/u, x]]*(a + b*ArcCoth[c*x])^ p/(d + e*x^2), x] - 1/2* Int[Log[SimplifyIntegrand[1 - 1/u, x]]*(a + b*ArcCoth[c*x])^ p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
+Int[ArcTanh[u_]*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/2*Int[Log[1 + u]*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x] - 1/2*Int[Log[1 - u]*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
+Int[ArcCoth[u_]*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := 1/2*Int[ Log[SimplifyIntegrand[1 + 1/u, x]]*(a + b*ArcCoth[c*x])^ p/(d + e*x^2), x] - 1/2* Int[Log[SimplifyIntegrand[1 - 1/u, x]]*(a + b*ArcCoth[c*x])^ p/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_.*Log[f_ + g_.*x_]/(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcTanh[c*x])^(p + 1)*Log[f + g*x]/(b*c*d*(p + 1)) - g/(b*c*d*(p + 1))* Int[(a + b*ArcTanh[c*x])^(p + 1)/(f + g*x), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[c^2*f^2 - g^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_.*Log[f_ + g_.*x_]/(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcCoth[c*x])^(p + 1)*Log[f + g*x]/(b*c*d*(p + 1)) - g/(b*c*d*(p + 1))* Int[(a + b*ArcCoth[c*x])^(p + 1)/(f + g*x), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[c^2*f^2 - g^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcTanh[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) - b*p/2* Int[(a + b*ArcTanh[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcCoth[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) - b*p/2* Int[(a + b*ArcCoth[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol] := -(a + b*ArcTanh[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) + b*p/2* Int[(a + b*ArcTanh[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol] := -(a + b*ArcCoth[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) + b*p/2* Int[(a + b*ArcCoth[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol] := -(a + b*ArcTanh[c*x])^p*PolyLog[k + 1, u]/(2*c*d) + b*p/2* Int[(a + b*ArcTanh[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol] := -(a + b*ArcCoth[c*x])^p*PolyLog[k + 1, u]/(2*c*d) + b*p/2* Int[(a + b*ArcCoth[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcTanh[c*x])^p*PolyLog[k + 1, u]/(2*c*d) - b*p/2* Int[(a + b*ArcTanh[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcCoth[c*x])^p*PolyLog[k + 1, u]/(2*c*d) - b*p/2* Int[(a + b*ArcCoth[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
+Int[1/((d_ + e_.*x_^2)*(a_. + b_.*ArcCoth[c_.*x_])*(a_. + b_.*ArcTanh[c_.*x_])), x_Symbol] := (-Log[a + b*ArcCoth[c*x]] + Log[a + b*ArcTanh[c*x]])/(b^2*c* d*(ArcCoth[c*x] - ArcTanh[c*x])) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
+Int[(a_. + b_.*ArcCoth[c_.*x_])^ m_.*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcCoth[c*x])^(m + 1)*(a + b*ArcTanh[c*x])^ p/(b*c*d*(m + 1)) - p/(m + 1)* Int[(a + b*ArcCoth[c*x])^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1)/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGeQ[m, p]
+Int[(a_. + b_.*ArcTanh[c_.*x_])^ m_.*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol] := (a + b*ArcTanh[c*x])^(m + 1)*(a + b*ArcCoth[c*x])^ p/(b*c*d*(m + 1)) - p/(m + 1)* Int[(a + b*ArcTanh[c*x])^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1)/(d + e*x^2), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[m, p]
+Int[ArcTanh[a_.*x_]/(c_ + d_.*x_^n_.), x_Symbol] := 1/2*Int[Log[1 + a*x]/(c + d*x^n), x] - 1/2*Int[Log[1 - a*x]/(c + d*x^n), x] /; FreeQ[{a, c, d}, x] && IntegerQ[n] && Not[EqQ[n, 2] && EqQ[a^2*c + d, 0]]
+Int[ArcCoth[a_.*x_]/(c_ + d_.*x_^n_.), x_Symbol] := 1/2*Int[Log[1 + 1/(a*x)]/(c + d*x^n), x] - 1/2*Int[Log[1 - 1/(a*x)]/(c + d*x^n), x] /; FreeQ[{a, c, d}, x] && IntegerQ[n] && Not[EqQ[n, 2] && EqQ[a^2*c + d, 0]]
+Int[Log[d_.*x_^m_.]*ArcTanh[c_.*x_^n_.]/x_, x_Symbol] := 1/2*Int[Log[d*x^m]*Log[1 + c*x^n]/x, x] - 1/2*Int[Log[d*x^m]*Log[1 - c*x^n]/x, x] /; FreeQ[{c, d, m, n}, x]
+Int[Log[d_.*x_^m_.]*ArcCoth[c_.*x_^n_.]/x_, x_Symbol] := 1/2*Int[Log[d*x^m]*Log[1 + 1/(c*x^n)]/x, x] - 1/2*Int[Log[d*x^m]*Log[1 - 1/(c*x^n)]/x, x] /; FreeQ[{c, d, m, n}, x]
+Int[Log[d_.*x_^m_.]*(a_ + b_.*ArcTanh[c_.*x_^n_.])/x_, x_Symbol] := a*Int[Log[d*x^m]/x, x] + b*Int[(Log[d*x^m]*ArcTanh[c*x^n])/x, x] /; FreeQ[{a, b, c, d, m, n}, x]
+Int[Log[d_.*x_^m_.]*(a_ + b_.*ArcCoth[c_.*x_^n_.])/x_, x_Symbol] := a*Int[Log[d*x^m]/x, x] + b*Int[(Log[d*x^m]*ArcCoth[c*x^n])/x, x] /; FreeQ[{a, b, c, d, m, n}, x]
+Int[(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := x*(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x]) - 2*e*g*Int[x^2*(a + b*ArcTanh[c*x])/(f + g*x^2), x] - b*c*Int[x*(d + e*Log[f + g*x^2])/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x]
+Int[(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := x*(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x]) - 2*e*g*Int[x^2*(a + b*ArcCoth[c*x])/(f + g*x^2), x] - b*c*Int[x*(d + e*Log[f + g*x^2])/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x]
+Int[Log[f_. + g_.*x_^2]*ArcTanh[c_.*x_]/x_, x_Symbol] := (Log[f + g*x^2] - Log[1 - c*x] - Log[1 + c*x])* Int[ArcTanh[c*x]/x, x] - 1/2*Int[Log[1 - c*x]^2/x, x] + 1/2*Int[Log[1 + c*x]^2/x, x] /; FreeQ[{c, f, g}, x] && EqQ[c^2*f + g, 0]
+Int[Log[f_. + g_.*x_^2]*ArcCoth[c_.*x_]/x_, x_Symbol] := (Log[f + g*x^2] - Log[-c^2*x^2] - Log[1 - 1/(c*x)] - Log[1 + 1/(c*x)])*Int[ArcCoth[c*x]/x, x] + Int[Log[-c^2*x^2]*ArcCoth[c*x]/x, x] - 1/2*Int[Log[1 - 1/(c*x)]^2/x, x] + 1/2*Int[Log[1 + 1/(c*x)]^2/x, x] /; FreeQ[{c, f, g}, x] && EqQ[c^2*f + g, 0]
+Int[Log[f_. + g_.*x_^2]*(a_ + b_.*ArcTanh[c_.*x_])/x_, x_Symbol] := a*Int[Log[f + g*x^2]/x, x] + b*Int[Log[f + g*x^2]*ArcTanh[c*x]/x, x] /; FreeQ[{a, b, c, f, g}, x]
+Int[Log[f_. + g_.*x_^2]*(a_ + b_.*ArcCoth[c_.*x_])/x_, x_Symbol] := a*Int[Log[f + g*x^2]/x, x] + b*Int[Log[f + g*x^2]*ArcCoth[c*x]/x, x] /; FreeQ[{a, b, c, f, g}, x]
+Int[(d_ + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_])/x_, x_Symbol] := d*Int[(a + b*ArcTanh[c*x])/x, x] + e*Int[Log[f + g*x^2]*(a + b*ArcTanh[c*x])/x, x] /; FreeQ[{a, b, c, d, e, f, g}, x]
+Int[(d_ + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_])/x_, x_Symbol] := d*Int[(a + b*ArcCoth[c*x])/x, x] + e*Int[Log[f + g*x^2]*(a + b*ArcCoth[c*x])/x, x] /; FreeQ[{a, b, c, d, e, f, g}, x]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := x^(m + 1)*(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x])/(m + 1) - 2*e*g/(m + 1)* Int[x^(m + 2)*(a + b*ArcTanh[c*x])/(f + g*x^2), x] - b*c/(m + 1)* Int[x^(m + 1)*(d + e*Log[f + g*x^2])/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m/2, 0]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := x^(m + 1)*(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x])/(m + 1) - 2*e*g/(m + 1)* Int[x^(m + 2)*(a + b*ArcCoth[c*x])/(f + g*x^2), x] - b*c/(m + 1)* Int[x^(m + 1)*(d + e*Log[f + g*x^2])/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m/2, 0]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcTanh[c*x], u, x] - b*c*Int[ExpandIntegrand[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcCoth[c*x], u, x] - b*c*Int[ExpandIntegrand[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(a + b*ArcTanh[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - 2*e*g*Int[ExpandIntegrand[x*u/(f + g*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]
+Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol] := With[{u = IntHide[x^m*(a + b*ArcCoth[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - 2*e*g*Int[ExpandIntegrand[x*u/(f + g*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]
+Int[x_*(d_. + e_.*Log[f_ + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_])^2, x_Symbol] := (f + g*x^2)*(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x])^2/(2*g) - e*x^2*(a + b*ArcTanh[c*x])^2/2 + b/c*Int[(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x]), x] + b*c*e*Int[x^2*(a + b*ArcTanh[c*x])/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*f + g, 0]
+Int[x_*(d_. + e_.*Log[f_ + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_])^2, x_Symbol] := (f + g*x^2)*(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x])^2/(2*g) - e*x^2*(a + b*ArcCoth[c*x])^2/2 + b/c*Int[(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x]), x] + b*c*e*Int[x^2*(a + b*ArcCoth[c*x])/(1 - c^2*x^2), x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*f + g, 0]
+Int[u_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol] := Unintegrable[u*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || MatchQ[u, (d_. + e_.*x)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x)^q_. /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[u, (d_. + e_.*x^2)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x^2)^q_. /; FreeQ[{d, e, f, m, q}, x]])
+Int[u_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol] := Unintegrable[u*(a + b*ArcCoth[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || MatchQ[u, (d_. + e_.*x)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x)^q_. /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[u, (d_. + e_.*x^2)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x^2)^q_. /; FreeQ[{d, e, f, m, q}, x]])
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.5 u (a+b arctanh(c+d x))^p.m b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.5 u (a+b arctanh(c+d x))^p.m
new file mode 100755
index 0000000..67fbe56
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.5 u (a+b arctanh(c+d x))^p.m	
@@ -0,0 +1,23 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.3.5 u (a+b arctanh(c+d x))^p *)
+Int[(a_. + b_.*ArcTanh[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(a + b*ArcTanh[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*ArcCoth[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(a + b*ArcCoth[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*ArcTanh[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(a + b*ArcTanh[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]
+Int[(a_. + b_.*ArcCoth[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(a + b*ArcCoth[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcTanh[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(f*x/d)^m*(a + b*ArcTanh[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCoth[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(f*x/d)^m*(a + b*ArcCoth[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_*(a_. + b_.*ArcTanh[c_ + d_.*x_])^p_., x_Symbol] := (e + f*x)^(m + 1)*(a + b*ArcTanh[c + d*x])^p/(f*(m + 1)) - b*d*p/(f*(m + 1))* Int[(e + f*x)^(m + 1)*(a + b*ArcTanh[c + d*x])^(p - 1)/(1 - (c + d*x)^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[m, -1]
+Int[(e_. + f_.*x_)^m_*(a_. + b_.*ArcCoth[c_ + d_.*x_])^p_., x_Symbol] := (e + f*x)^(m + 1)*(a + b*ArcCoth[c + d*x])^p/(f*(m + 1)) - b*d*p/(f*(m + 1))* Int[(e + f*x)^(m + 1)*(a + b*ArcCoth[c + d*x])^(p - 1)/(1 - (c + d*x)^2), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[m, -1]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcTanh[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcTanh[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCoth[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcCoth[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcTanh[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(e + f*x)^m*(a + b*ArcTanh[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && Not[IGtQ[p, 0]]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCoth[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && Not[IGtQ[p, 0]]
+Int[ArcTanh[c_ + d_.*x_]/(e_ + f_.*x_^n_.), x_Symbol] := 1/2*Int[Log[1 + c + d*x]/(e + f*x^n), x] - 1/2*Int[Log[1 - c - d*x]/(e + f*x^n), x] /; FreeQ[{c, d, e, f}, x] && RationalQ[n]
+Int[ArcCoth[c_ + d_.*x_]/(e_ + f_.*x_^n_.), x_Symbol] := 1/2*Int[Log[(1 + c + d*x)/(c + d*x)]/(e + f*x^n), x] - 1/2*Int[Log[(-1 + c + d*x)/(c + d*x)]/(e + f*x^n), x] /; FreeQ[{c, d, e, f}, x] && RationalQ[n]
+Int[ArcTanh[c_ + d_.*x_]/(e_ + f_.*x_^n_), x_Symbol] := Unintegrable[ArcTanh[c + d*x]/(e + f*x^n), x] /; FreeQ[{c, d, e, f, n}, x] && Not[RationalQ[n]]
+Int[ArcCoth[c_ + d_.*x_]/(e_ + f_.*x_^n_), x_Symbol] := Unintegrable[ArcCoth[c + d*x]/(e + f*x^n), x] /; FreeQ[{c, d, e, f, n}, x] && Not[RationalQ[n]]
+Int[(A_. + B_.*x_ + C_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_ + d_.*x_])^ p_., x_Symbol] := 1/d*Subst[Int[(-C/d^2 + C/d^2*x^2)^q*(a + b*ArcTanh[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, A, B, C, p, q}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[(A_. + B_.*x_ + C_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_ + d_.*x_])^ p_., x_Symbol] := 1/d*Subst[Int[(C/d^2 + C/d^2*x^2)^q*(a + b*ArcCoth[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, A, B, C, p, q}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[(e_. + f_.*x_)^m_.*(A_. + B_.*x_ + C_.*x_^2)^ q_.*(a_. + b_.*ArcTanh[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[ Int[((d*e - c*f)/d + f*x/d)^m*(-C/d^2 + C/d^2*x^2)^ q*(a + b*ArcTanh[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
+Int[(e_. + f_.*x_)^m_.*(A_. + B_.*x_ + C_.*x_^2)^ q_.*(a_. + b_.*ArcCoth[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[ Int[((d*e - c*f)/d + f*x/d)^m*(-C/d^2 + C/d^2*x^2)^ q*(a + b*ArcCoth[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.6 Exponentials of inverse hyperbolic tangent.m b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.6 Exponentials of inverse hyperbolic tangent.m
new file mode 100755
index 0000000..c1736b3
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.6 Exponentials of inverse hyperbolic tangent.m	
@@ -0,0 +1,90 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.3.6 Exponentials of inverse hyperbolic tangent *)
+Int[E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := Int[((1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/2)* Sqrt[1 - a^2*x^2])), x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2]
+Int[x_^m_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := Int[x^ m*((1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/2)* Sqrt[1 - a^2*x^2])), x] /; FreeQ[{a, m}, x] && IntegerQ[(n - 1)/2]
+Int[E^(n_*ArcTanh[a_.*x_]), x_Symbol] := Int[(1 + a*x)^(n/2)/(1 - a*x)^(n/2), x] /; FreeQ[{a, n}, x] && Not[IntegerQ[(n - 1)/2]]
+Int[x_^m_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := Int[x^m*(1 + a*x)^(n/2)/(1 - a*x)^(n/2), x] /; FreeQ[{a, m, n}, x] && Not[IntegerQ[(n - 1)/2]]
+Int[(c_ + d_.*x_)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^n*Int[(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
+Int[(e_. + f_.*x_)^m_.*(c_ + d_.*x_)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^n*Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1, 0]) && IntegerQ[2*p]
+Int[u_.*(c_ + d_.*x_)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[u*(1 + d*x/c)^p*(1 + a*x)^(n/2)/(1 - a*x)^(n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] || GtQ[c, 0])
+Int[u_.*(c_ + d_.*x_)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := Int[u*(c + d*x)^p*(1 + a*x)^(n/2)/(1 - a*x)^(n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && Not[IntegerQ[p] || GtQ[c, 0]]
+Int[u_.*(c_ + d_./x_)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := d^p*Int[u*(1 + c*x/d)^p*E^(n*ArcTanh[a*x])/x^p, x] /; FreeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]
+Int[u_.*(c_ + d_./x_)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := (-1)^(n/2)*c^p* Int[u*(1 + d/(c*x))^p*(1 + 1/(a*x))^(n/2)/(1 - 1/(a*x))^(n/2), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[p]] && IntegerQ[n/2] && GtQ[c, 0]
+Int[u_.*(c_ + d_./x_)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := Int[u*(c + d/x)^p*(1 + a*x)^(n/2)/(1 - a*x)^(n/2), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[p]] && IntegerQ[n/2] && Not[GtQ[c, 0]]
+Int[u_.*(c_ + d_./x_)^p_*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := x^p*(c + d/x)^p/(1 + c*x/d)^p* Int[u*(1 + c*x/d)^p*E^(n*ArcTanh[a*x])/x^p, x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[p]]
+Int[E^(n_*ArcTanh[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol] := (n - a*x)*E^(n*ArcTanh[a*x])/(a*c*(n^2 - 1)*Sqrt[c + d*x^2]) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n]]
+Int[(c_ + d_.*x_^2)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := (n + 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)* E^(n*ArcTanh[a*x])/(a*c*(n^2 - 4*(p + 1)^2)) - 2*(p + 1)*(2*p + 3)/(c*(n^2 - 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p, -1] && Not[IntegerQ[n]] && NeQ[n^2 - 4*(p + 1)^2, 0] && IntegerQ[2*p]
+Int[E^(n_.*ArcTanh[a_.*x_])/(c_ + d_.*x_^2), x_Symbol] := E^(n*ArcTanh[a*x])/(a*c*n) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]]
+Int[(c_ + d_.*x_^2)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && IntegerQ[p] && IGtQ[(n + 1)/2, 0] && Not[IntegerQ[p - n/2]]
+Int[(c_ + d_.*x_^2)^p_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[(1 - a^2*x^2)^(p + n/2)/(1 - a*x)^n, x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && IntegerQ[p] && ILtQ[(n - 1)/2, 0] && Not[IntegerQ[p - n/2]]
+Int[(c_ + d_.*x_^2)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
+Int[(c_ + d_.*x_^2)^p_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := c^(n/2)*Int[(c + d*x^2)^(p - n/2)*(1 + a*x)^n, x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && IGtQ[n/2, 0]
+Int[(c_ + d_.*x_^2)^p_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := 1/c^(n/2)*Int[(c + d*x^2)^(p + n/2)/(1 - a*x)^n, x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && ILtQ[n/2, 0]
+Int[(c_ + d_.*x_^2)^p_*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^IntPart[p]*(c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPart[p]* Int[(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]]
+Int[x_*E^(n_*ArcTanh[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol] := (1 - a*n*x)*E^(n*ArcTanh[a*x])/(d*(n^2 - 1)*Sqrt[c + d*x^2]) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n]]
+Int[x_*(c_ + d_.*x_^2)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := (c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x])/(2*d*(p + 1)) - a*c*n/(2*d*(p + 1))*Int[(c + d*x^2)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p, -1] && Not[IntegerQ[n]] && IntegerQ[2*p]
+(* Int[x_*(c_+d_.*x_^2)^p_*E^(n_*ArcTanh[a_.*x_]),x_Symbol] := -(2*(p+1)+a*n*x)*(c+d*x^2)^(p+1)*E^(n*ArcTanh[a*x])/(d*(n^2-4*(p+1)^ 2)) - n*(2*p+3)/(a*c*(n^2-4*(p+1)^2))*Int[(c+d*x^2)^(p+1)*E^(n*ArcTanh[a* x]),x] /; FreeQ[{a,c,d,n},x] && EqQ[a^2*c+d,0] && LeQ[p,-1] &&  NeQ[n^2-4*(p+1)^2,0] && Not[IntegerQ[n]] *)
+Int[x_^2*(c_ + d_.*x_^2)^p_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := (1 - a*n*x)*(c + d*x^2)^(p + 1)* E^(n*ArcTanh[a*x])/(a*d*n*(n^2 - 1)) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && EqQ[n^2 + 2*(p + 1), 0] && Not[IntegerQ[n]]
+Int[x_^2*(c_ + d_.*x_^2)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := -(n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)* E^(n*ArcTanh[a*x])/(a*d*(n^2 - 4*(p + 1)^2)) + (n^2 + 2*(p + 1))/(d*(n^2 - 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p, -1] && Not[IntegerQ[n]] && NeQ[n^2 - 4*(p + 1)^2, 0] && IntegerQ[2*p]
+Int[x_^m_.*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[x^m*(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0]) && IGtQ[(n + 1)/2, 0] && Not[IntegerQ[p - n/2]]
+Int[x_^m_.*(c_ + d_.*x_^2)^p_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[x^m*(1 - a^2*x^2)^(p + n/2)/(1 - a*x)^n, x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0]) && ILtQ[(n - 1)/2, 0] && Not[IntegerQ[p - n/2]]
+Int[x_^m_.*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
+Int[x_^m_.*(c_ + d_.*x_^2)^p_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := c^(n/2)*Int[x^m*(c + d*x^2)^(p - n/2)*(1 + a*x)^n, x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && IGtQ[n/2, 0]
+Int[x_^m_.*(c_ + d_.*x_^2)^p_.*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := 1/c^(n/2)*Int[x^m*(c + d*x^2)^(p + n/2)/(1 - a*x)^n, x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && ILtQ[n/2, 0]
+Int[x_^m_.*(c_ + d_.*x_^2)^p_*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^IntPart[p]*(c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPart[p]* Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && Not[IntegerQ[n/2]]
+Int[u_*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[u*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
+Int[u_*(c_ + d_.*x_^2)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := c^IntPart[p]*(c + d*x^2)^ FracPart[p]/((1 - a*x)^FracPart[p]*(1 + a*x)^FracPart[p])* Int[u*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && IntegerQ[n/2]
+Int[u_*(c_ + d_.*x_^2)^p_*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := c^IntPart[p]*(c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPart[p]* Int[u*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && Not[IntegerQ[n/2]]
+Int[u_.*(c_ + d_./x_^2)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := d^p*Int[u/x^(2*p)*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]
+Int[u_.*(c_ + d_./x_^2)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := c^p*Int[u*(1 - 1/(a*x))^p*(1 + 1/(a*x))^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[p]] && IntegerQ[n/2] && GtQ[c, 0]
+Int[u_.*(c_ + d_./x_^2)^p_*E^(n_*ArcTanh[a_.*x_]), x_Symbol] := x^(2*p)*(c + d/x^2)^p/((1 - a*x)^p*(1 + a*x)^p)* Int[u/x^(2*p)*(1 - a*x)^p*(1 + a*x)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[p]] && IntegerQ[n/2] && Not[GtQ[c, 0]]
+Int[u_.*(c_ + d_./x_^2)^p_*E^(n_.*ArcTanh[a_.*x_]), x_Symbol] := x^(2*p)*(c + d/x^2)^p/(1 + c*x^2/d)^p* Int[u/x^(2*p)*(1 + c*x^2/d)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[p]] && Not[IntegerQ[n/2]]
+Int[E^(n_.*ArcTanh[c_.*(a_ + b_.*x_)]), x_Symbol] := Int[(1 + a*c + b*c*x)^(n/2)/(1 - a*c - b*c*x)^(n/2), x] /; FreeQ[{a, b, c, n}, x]
+Int[x_^m_*E^(n_*ArcTanh[c_.*(a_ + b_.*x_)]), x_Symbol] := 4/(n*b^(m + 1)*c^(m + 1))* Subst[ Int[x^(2/n)*(-1 - a*c + (1 - a*c)*x^(2/n))^ m/(1 + x^(2/n))^(m + 2), x], x, (1 + c*(a + b*x))^(n/2)/(1 - c*(a + b*x))^(n/2)] /; FreeQ[{a, b, c}, x] && ILtQ[m, 0] && LtQ[-1, n, 1]
+Int[(d_. + e_.*x_)^m_.*E^(n_.*ArcTanh[c_.*(a_ + b_.*x_)]), x_Symbol] := Int[(d + e*x)^m*(1 + a*c + b*c*x)^(n/2)/(1 - a*c - b*c*x)^(n/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcTanh[a_ + b_.*x_]), x_Symbol] := (c/(1 - a^2))^p* Int[u*(1 - a - b*x)^(p - n/2)*(1 + a + b*x)^(p + n/2), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e (1 - a^2), 0] && (IntegerQ[p] || GtQ[c/(1 - a^2), 0])
+Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcTanh[a_ + b_.*x_]), x_Symbol] := (c + d*x + e*x^2)^p/(1 - a^2 - 2*a*b*x - b^2*x^2)^p* Int[u*(1 - a^2 - 2*a*b*x - b^2*x^2)^p*E^(n*ArcTanh[a*x]), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e (1 - a^2), 0] && Not[IntegerQ[p] || GtQ[c/(1 - a^2), 0]]
+Int[u_.*E^(n_.*ArcTanh[c_./(a_. + b_.*x_)]), x_Symbol] := Int[u*E^(n*ArcCoth[a/c + b*x/c]), x] /; FreeQ[{a, b, c, n}, x]
+Int[u_.*E^(n_*ArcCoth[a_.*x_]), x_Symbol] := (-1)^(n/2)*Int[u*E^(n*ArcTanh[a*x]), x] /; FreeQ[a, x] && IntegerQ[n/2]
+Int[E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -Subst[ Int[(1 + x/a)^((n + 1)/2)/(x^2*(1 - x/a)^((n - 1)/2)* Sqrt[1 - x^2/a^2]), x], x, 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2]
+Int[x_^m_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -Subst[ Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/a)^((n - 1)/2)* Sqrt[1 - x^2/a^2]), x], x, 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2] && IntegerQ[m]
+Int[E^(n_*ArcCoth[a_.*x_]), x_Symbol] := -Subst[Int[(1 + x/a)^(n/2)/(x^2*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && Not[IntegerQ[n]]
+Int[x_^m_.*E^(n_*ArcCoth[a_.*x_]), x_Symbol] := -Subst[Int[(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]
+Int[x_^m_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -x^m*(1/x)^m* Subst[Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/a)^((n - 1)/2)* Sqrt[1 - x^2/a^2]), x], x, 1/x] /; FreeQ[{a, m}, x] && IntegerQ[(n - 1)/2] && Not[IntegerQ[m]]
+Int[x_^m_*E^(n_*ArcCoth[a_.*x_]), x_Symbol] := -x^m*(1/x)^m* Subst[Int[(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, m, n}, x] && Not[IntegerQ[n]] && Not[IntegerQ[m]]
+Int[(c_ + d_.*x_)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := (1 + a*x)*(c + d*x)^p*E^(n*ArcCoth[a*x])/(a*(p + 1)) /; FreeQ[{a, c, d, n, p}, x] && EqQ[a*c + d, 0] && EqQ[p, n/2] && Not[IntegerQ[n/2]]
+(* Int[(c_+d_.*x_)^p_.*E^(n_.*ArcCoth[a_.*x_]),x_Symbol] := -(-a)^n*c^n*Subst[Int[(d+c*x)^(p-n)*(1-x^2/a^2)^(n/2)/x^(p+2),x],x, 1/x] /; FreeQ[{a,c,d},x] && EqQ[a*c+d,0] && IntegerQ[(n-1)/2] && IntegerQ[p] *)
+(* Int[x_^m_.*(c_+d_.*x_)^p_.*E^(n_.*ArcCoth[a_.*x_]),x_Symbol] := -(-a)^n*c^n*Subst[Int[(d+c*x)^(p-n)*(1-x^2/a^2)^(n/2)/x^(m+p+2),x], x,1/x] /; FreeQ[{a,c,d},x] && EqQ[a*c+d,0] && IntegerQ[(n-1)/2] && IntegerQ[m] &&  IntegerQ[p] *)
+(* Int[(c_+d_.*x_)^p_.*E^(n_.*ArcCoth[a_.*x_]),x_Symbol] := -(-a)^n*c^n*Sqrt[c+d*x]/(Sqrt[x]*Sqrt[d+c/x])*Subst[Int[(d+c*x)^(p- n)*(1-x^2/a^2)^(n/2)/x^(p+2),x],x,1/x] /; FreeQ[{a,c,d},x] && EqQ[a*c+d,0] && IntegerQ[(n-1)/2] &&  IntegerQ[p-1/2] *)
+(* Int[x_^m_.*(c_+d_.*x_)^p_.*E^(n_.*ArcCoth[a_.*x_]),x_Symbol] := -(-a)^n*c^n*Sqrt[c+d*x]/(Sqrt[x]*Sqrt[d+c/x])*Subst[Int[(d+c*x)^(p- n)*(1-x^2/a^2)^(n/2)/x^(m+p+2),x],x,1/x] /; FreeQ[{a,c,d},x] && EqQ[a*c+d,0] && IntegerQ[(n-1)/2] && IntegerQ[m] &&  IntegerQ[p-1/2] *)
+Int[u_.*(c_ + d_.*x_)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := d^p*Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c^2 - d^2, 0] && Not[IntegerQ[n/2]] && IntegerQ[p]
+Int[u_.*(c_ + d_.*x_)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := (c + d*x)^p/(x^p*(1 + c/(d*x))^p)* Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && Not[IntegerQ[n/2]] && Not[IntegerQ[p]]
+Int[(c_ + d_./x_)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^n*Subst[Int[(c + d*x)^(p - n)*(1 - x^2/a^2)^(n/2)/x^2, x], x, 1/x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1]) && IntegerQ[2*p]
+Int[x_^m_.*(c_ + d_./x_)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^n*Subst[Int[(c + d*x)^(p - n)*(1 - x^2/a^2)^(n/2)/x^(m + 2), x], x, 1/x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[ m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && IntegerQ[2*p]
+Int[(c_ + d_./x_)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^p*Subst[ Int[(1 + d*x/c)^p*(1 + x/a)^(n/2)/(x^2*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0])
+Int[x_^m_.*(c_ + d_./x_)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^p*Subst[ Int[(1 + d*x/c)^p*(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && IntegerQ[m]
+Int[x_^m_*(c_ + d_./x_)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^p*x^m*(1/x)^m* Subst[Int[(1 + d*x/c)^ p*(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegerQ[m]]
+Int[u_.*(c_ + d_./x_)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := (c + d/x)^p/(1 + d/(c*x))^p* Int[u*(1 + d/(c*x))^p*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[n/2]] && Not[IntegerQ[p] || GtQ[c, 0]]
+Int[E^(n_.*ArcCoth[a_.*x_])/(c_ + d_.*x_^2), x_Symbol] := E^(n*ArcCoth[a*x])/(a*c*n) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]]
+Int[E^(n_*ArcCoth[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol] := (n - a*x)*E^(n*ArcCoth[a*x])/(a*c*(n^2 - 1)*Sqrt[c + d*x^2]) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n]]
+Int[(c_ + d_.*x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := (n + 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCoth[a*x])/(a*c*(n^2 - 4*(p + 1)^2)) - 2*(p + 1)*(2*p + 3)/(c*(n^2 - 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] || Not[IntegerQ[n]])
+Int[x_*E^(n_*ArcCoth[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol] := -(1 - a*n*x)*E^(n*ArcCoth[a*x])/(a^2*c*(n^2 - 1)*Sqrt[c + d*x^2]) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n]]
+Int[x_*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := (2*(p + 1) + a*n*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCoth[a*x])/(a^2*c*(n^2 - 4*(p + 1)^2)) - n*(2*p + 3)/(a*c*(n^2 - 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && LeQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] || Not[IntegerQ[n]])
+Int[x_^2*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -(n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCoth[a*x])/(a^3*c*n^2*(n^2 - 1)) /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && EqQ[n^2 + 2*(p + 1), 0] && NeQ[n^2, 1]
+Int[x_^2*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := (n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCoth[a*x])/(a^3*c*(n^2 - 4*(p + 1)^2)) - (n^2 + 2*(p + 1))/(a^2*c*(n^2 - 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && LeQ[p, -1] && NeQ[n^2 + 2*(p + 1), 0] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] || Not[IntegerQ[n]])
+Int[x_^m_.*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -(-c)^p/a^(m + 1)* Subst[Int[E^(n*x)*Coth[x]^(m + 2*(p + 1))/Cosh[x]^(2*(p + 1)), x], x, ArcCoth[a*x]] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && IntegerQ[m] && LeQ[3, m, -2 (p + 1)] && IntegerQ[p]
+Int[u_.*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := d^p*Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && IntegerQ[p]
+Int[u_.*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := (c + d*x^2)^p/(x^(2*p)*(1 - 1/(a^2*x^2))^p)* Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && Not[IntegerQ[p]]
+Int[u_.*(c_ + d_./x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := c^p/a^(2*p)* Int[u/x^(2*p)*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]
+Int[(c_ + d_./x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^p*Subst[Int[(1 - x/a)^(p - n/2)*(1 + x/a)^(p + n/2)/x^2, x], x, 1/x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegersQ[2*p, p + n/2]]
+Int[x_^m_.*(c_ + d_./x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^p*Subst[ Int[(1 - x/a)^(p - n/2)*(1 + x/a)^(p + n/2)/x^(m + 2), x], x, 1/x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegersQ[2*p, p + n/2]] && IntegerQ[m]
+Int[x_^m_*(c_ + d_./x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := -c^p*x^m*(1/x)^m* Subst[Int[(1 - x/a)^(p - n/2)*(1 + x/a)^(p + n/2)/x^(m + 2), x], x, 1/x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegersQ[2*p, p + n/2]] && Not[IntegerQ[m]]
+Int[u_.*(c_ + d_./x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol] := c^IntPart[p]*(c + d/x^2)^FracPart[p]/(1 - 1/(a^2*x^2))^FracPart[p]* Int[u*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[n/2]] && Not[IntegerQ[p] || GtQ[c, 0]]
+Int[u_.*E^(n_*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol] := (-1)^(n/2)*Int[u*E^(n*ArcTanh[c*(a + b*x)]), x] /; FreeQ[{a, b, c}, x] && IntegerQ[n/2]
+Int[E^(n_.*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol] := (c*(a + b*x))^(n/ 2)*(1 + 1/(c*(a + b*x)))^(n/2)/(1 + a*c + b*c*x)^(n/2)* Int[(1 + a*c + b*c*x)^(n/2)/(-1 + a*c + b*c*x)^(n/2), x] /; FreeQ[{a, b, c, n}, x] && Not[IntegerQ[n/2]]
+Int[x_^m_*E^(n_*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol] := -4/(n*b^(m + 1)*c^(m + 1))* Subst[ Int[x^(2/n)*(1 + a*c + (1 - a*c)*x^(2/n))^ m/(-1 + x^(2/n))^(m + 2), x], x, (1 + 1/(c*(a + b*x)))^(n/2)/(1 - 1/(c*(a + b*x)))^(n/2)] /; FreeQ[{a, b, c}, x] && ILtQ[m, 0] && LtQ[-1, n, 1]
+Int[(d_. + e_.*x_)^m_.*E^(n_.*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol] := (c*(a + b*x))^(n/ 2)*(1 + 1/(c*(a + b*x)))^(n/2)/(1 + a*c + b*c*x)^(n/2)* Int[(d + e*x)^m*(1 + a*c + b*c*x)^(n/2)/(-1 + a*c + b*c*x)^(n/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && Not[IntegerQ[n/2]]
+Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcCoth[a_ + b_.*x_]), x_Symbol] := (c/(1 - a^2))^ p*((a + b*x)/(1 + a + b*x))^(n/2)*((1 + a + b*x)/(a + b*x))^(n/ 2)*((1 - a - b*x)^(n/2)/(-1 + a + b*x)^(n/2))* Int[u*(1 - a - b*x)^(p - n/2)*(1 + a + b*x)^(p + n/2), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && Not[IntegerQ[n/2]] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e (1 - a^2), 0] && (IntegerQ[p] || GtQ[c/(1 - a^2), 0])
+Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcCoth[a_ + b_.*x_]), x_Symbol] := (c + d*x + e*x^2)^p/(1 - a^2 - 2*a*b*x - b^2*x^2)^p* Int[u*(1 - a^2 - 2*a*b*x - b^2*x^2)^p*E^(n*ArcCoth[a*x]), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && Not[IntegerQ[n/2]] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e (1 - a^2), 0] && Not[IntegerQ[p] || GtQ[c/(1 - a^2), 0]]
+Int[u_.*E^(n_.*ArcCoth[c_./(a_. + b_.*x_)]), x_Symbol] := Int[u*E^(n*ArcTanh[a/c + b*x/c]), x] /; FreeQ[{a, b, c, n}, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.7 Miscellaneous inverse hyperbolic tangent.m b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.7 Miscellaneous inverse hyperbolic tangent.m
new file mode 100755
index 0000000..e2d6051
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.7 Miscellaneous inverse hyperbolic tangent.m	
@@ -0,0 +1,75 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.3.7 Miscellaneous inverse hyperbolic tangent *)
+Int[ArcTanh[a_ + b_.*x_^n_], x_Symbol] := x*ArcTanh[a + b*x^n] - b*n*Int[x^n/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x] /; FreeQ[{a, b, n}, x]
+Int[ArcCoth[a_ + b_.*x_^n_], x_Symbol] := x*ArcCoth[a + b*x^n] - b*n*Int[x^n/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x] /; FreeQ[{a, b, n}, x]
+Int[ArcTanh[a_. + b_.*x_^n_.]/x_, x_Symbol] := 1/2*Int[Log[1 + a + b*x^n]/x, x] - 1/2*Int[Log[1 - a - b*x^n]/x, x] /; FreeQ[{a, b, n}, x]
+Int[ArcCoth[a_. + b_.*x_^n_.]/x_, x_Symbol] := 1/2*Int[Log[1 + 1/(a + b*x^n)]/x, x] - 1/2*Int[Log[1 - 1/(a + b*x^n)]/x, x] /; FreeQ[{a, b, n}, x]
+Int[x_^m_.*ArcTanh[a_ + b_.*x_^n_], x_Symbol] := x^(m + 1)*ArcTanh[a + b*x^n]/(m + 1) - b*n/(m + 1)* Int[x^(m + n)/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x] /; FreeQ[{a, b}, x] && RationalQ[m, n] && NeQ[m, -1] && NeQ[m + 1, n]
+Int[x_^m_.*ArcCoth[a_ + b_.*x_^n_], x_Symbol] := x^(m + 1)*ArcCoth[a + b*x^n]/(m + 1) - b*n/(m + 1)* Int[x^(m + n)/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x] /; FreeQ[{a, b}, x] && RationalQ[m, n] && NeQ[m, -1] && NeQ[m + 1, n]
+Int[ArcTanh[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol] := 1/2*Int[Log[1 + a + b*f^(c + d*x)], x] - 1/2*Int[Log[1 - a - b*f^(c + d*x)], x] /; FreeQ[{a, b, c, d, f}, x] 
+Int[ArcCoth[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol] := 1/2*Int[Log[1 + 1/(a + b*f^(c + d*x))], x] - 1/2*Int[Log[1 - 1/(a + b*f^(c + d*x))], x] /; FreeQ[{a, b, c, d, f}, x] 
+Int[x_^m_.*ArcTanh[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol] := 1/2*Int[x^m*Log[1 + a + b*f^(c + d*x)], x] - 1/2*Int[x^m*Log[1 - a - b*f^(c + d*x)], x] /; FreeQ[{a, b, c, d, f}, x] && IGtQ[m, 0]
+Int[x_^m_.*ArcCoth[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol] := 1/2*Int[x^m*Log[1 + 1/(a + b*f^(c + d*x))], x] - 1/2*Int[x^m*Log[1 - 1/(a + b*f^(c + d*x))], x] /; FreeQ[{a, b, c, d, f}, x] && IGtQ[m, 0]
+Int[u_.*ArcTanh[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol] := Int[u*ArcCoth[a/c + b*x^n/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
+Int[u_.*ArcCoth[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol] := Int[u*ArcTanh[a/c + b*x^n/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
+Int[ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol] := x*ArcTanh[(c*x)/Sqrt[a + b*x^2]] - c*Int[x/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
+Int[ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol] := x*ArcCoth[(c*x)/Sqrt[a + b*x^2]] - c*Int[x/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
+Int[ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]]/x_, x_Symbol] := ArcTanh[c*x/Sqrt[a + b*x^2]]*Log[x] - c*Int[Log[x]/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
+Int[ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]]/x_, x_Symbol] := ArcCoth[c*x/Sqrt[a + b*x^2]]*Log[x] - c*Int[Log[x]/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
+Int[(d_.*x_)^m_.*ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol] := (d*x)^(m + 1)*ArcTanh[(c*x)/Sqrt[a + b*x^2]]/(d*(m + 1)) - c/(d*(m + 1))*Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b, c^2] && NeQ[m, -1]
+Int[(d_.*x_)^m_.*ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol] := (d*x)^(m + 1)*ArcCoth[(c*x)/Sqrt[a + b*x^2]]/(d*(m + 1)) - c/(d*(m + 1))*Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b, c^2] && NeQ[m, -1]
+Int[1/(Sqrt[a_. + b_.*x_^2]*ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]]), x_Symbol] := 1/c*Log[ArcTanh[c*x/Sqrt[a + b*x^2]]] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
+Int[1/(Sqrt[a_. + b_.*x_^2]*ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]]), x_Symbol] := -1/c*Log[ArcCoth[c*x/Sqrt[a + b*x^2]]] /; FreeQ[{a, b, c}, x] && EqQ[b, c^2]
+Int[ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[a_. + b_.*x_^2], x_Symbol] := ArcTanh[c*x/Sqrt[a + b*x^2]]^(m + 1)/(c*(m + 1)) /; FreeQ[{a, b, c, m}, x] && EqQ[b, c^2] && NeQ[m, -1]
+Int[ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[a_. + b_.*x_^2], x_Symbol] := -ArcCoth[c*x/Sqrt[a + b*x^2]]^(m + 1)/(c*(m + 1)) /; FreeQ[{a, b, c, m}, x] && EqQ[b, c^2] && NeQ[m, -1]
+Int[ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[d_. + e_.*x_^2], x_Symbol] := Sqrt[a + b*x^2]/Sqrt[d + e*x^2]* Int[ArcTanh[c*x/Sqrt[a + b*x^2]]^m/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b, c^2] && EqQ[b*d - a*e, 0]
+Int[ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[d_. + e_.*x_^2], x_Symbol] := Sqrt[a + b*x^2]/Sqrt[d + e*x^2]* Int[ArcCoth[c*x/Sqrt[a + b*x^2]]^m/Sqrt[a + b*x^2], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b, c^2] && EqQ[b*d - a*e, 0]
+If[TrueQ[$LoadShowSteps], Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, ShowStep["", "Int[f[x,ArcTanh[a+b*x]]/(1-(a+b*x)^2),x]", "Subst[Int[f[-a/b+Tanh[x]/b,x],x],x,ArcTanh[a+b*x]]/b", Hold[ (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[Int[ SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*Sech[x]^(2*(n + 1)), x], x], x, tmp]]] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcTanh] && EqQ[Discriminant[v, x]*tmp[[1]]^2 - D[v, x]^2, 0]] /; SimplifyFlag && QuadraticQ[v, x] && ILtQ[n, 0] && PosQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]], Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[ Int[SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*Sech[x]^(2*(n + 1)), x], x], x, tmp] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcTanh] && EqQ[Discriminant[v, x]*tmp[[1]]^2 - D[v, x]^2, 0]] /; QuadraticQ[v, x] && ILtQ[n, 0] && PosQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]]]
+If[TrueQ[$LoadShowSteps], Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, ShowStep["", "Int[f[x,ArcCoth[a+b*x]]/(1-(a+b*x)^2),x]", "Subst[Int[f[-a/b+Coth[x]/b,x],x],x,ArcCoth[a+b*x]]/b", Hold[ (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[Int[ SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*(-Csch[x]^2)^(n + 1), x], x], x, tmp]]] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcCoth] && EqQ[Discriminant[v, x]*tmp[[1]]^2 - D[v, x]^2, 0]] /; SimplifyFlag && QuadraticQ[v, x] && ILtQ[n, 0] && PosQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]], Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[ Int[SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*(-Csch[x]^2)^(n + 1), x], x], x, tmp] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcCoth] && EqQ[Discriminant[v, x]*tmp[[1]]^2 - D[v, x]^2, 0]] /; QuadraticQ[v, x] && ILtQ[n, 0] && PosQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]]]
+Int[ArcTanh[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Tanh[a + b*x]] + b*Int[x/(c - d + c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]
+Int[ArcCoth[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Tanh[a + b*x]] + b*Int[x/(c - d + c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]
+Int[ArcTanh[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Coth[a + b*x]] + b*Int[x/(c - d - c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]
+Int[ArcCoth[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Coth[a + b*x]] + b*Int[x/(c - d - c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]
+Int[ArcTanh[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Tanh[a + b*x]] + b*(1 - c - d)* Int[x*E^(2*a + 2*b*x)/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x] - b*(1 + c + d)* Int[x*E^(2*a + 2*b*x)/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]
+Int[ArcCoth[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Tanh[a + b*x]] + b*(1 - c - d)* Int[x*E^(2*a + 2*b*x)/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x] - b*(1 + c + d)* Int[x*E^(2*a + 2*b*x)/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]
+Int[ArcTanh[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Coth[a + b*x]] + b*(1 + c + d)* Int[x*E^(2*a + 2*b*x)/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x] - b*(1 - c - d)* Int[x*E^(2*a + 2*b*x)/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]
+Int[ArcCoth[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Coth[a + b*x]] + b*(1 + c + d)* Int[x*E^(2*a + 2*b*x)/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x] - b*(1 - c - d)* Int[x*E^(2*a + 2*b*x)/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Tanh[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d + c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Tanh[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d + c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Coth[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Coth[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Tanh[a + b*x]]/(f*(m + 1)) + b*(1 - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x] - b*(1 + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Tanh[a + b*x]]/(f*(m + 1)) + b*(1 - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x] - b*(1 + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Coth[a + b*x]]/(f*(m + 1)) + b*(1 + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x] - b*(1 - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Coth[a + b*x]]/(f*(m + 1)) + b*(1 + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x] - b*(1 - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]
+Int[ArcTanh[Tan[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[Tan[a + b*x]] - b*Int[x*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b}, x]
+Int[ArcCoth[Tan[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[Tan[a + b*x]] - b*Int[x*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b}, x]
+Int[ArcTanh[Cot[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[Cot[a + b*x]] - b*Int[x*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b}, x]
+Int[ArcCoth[Cot[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[Cot[a + b*x]] - b*Int[x*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b}, x]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[Tan[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[Tan[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[Cot[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[Cot[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]
+Int[ArcTanh[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Tan[a + b*x]] + I*b*Int[x/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c + I*d)^2, 1]
+Int[ArcCoth[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Tan[a + b*x]] + I*b*Int[x/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c + I*d)^2, 1]
+Int[ArcTanh[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Cot[a + b*x]] + I*b*Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, 1]
+Int[ArcCoth[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Cot[a + b*x]] + I*b*Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, 1]
+Int[ArcTanh[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Tan[a + b*x]] + I*b*(1 - c + I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x] - I*b*(1 + c - I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, 1]
+Int[ArcCoth[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Tan[a + b*x]] + I*b*(1 - c + I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x] - I*b*(1 + c - I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, 1]
+Int[ArcTanh[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := x*ArcTanh[c + d*Cot[a + b*x]] - I*b*(1 - c - I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x] + I*b*(1 + c + I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - I*d)^2, 1]
+Int[ArcCoth[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := x*ArcCoth[c + d*Cot[a + b*x]] - I*b*(1 - c - I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x] + I*b*(1 + c + I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[(c - I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Tan[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c + I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Tan[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c + I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Cot[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Cot[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Tan[a + b*x]]/(f*(m + 1)) + I*b*(1 - c + I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x] - I*b*(1 + c - I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c + I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Tan[a + b*x]]/(f*(m + 1)) + I*b*(1 - c + I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x] - I*b*(1 + c - I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c + I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcTanh[c + d*Cot[a + b*x]]/(f*(m + 1)) - I*b*(1 - c - I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x] + I*b*(1 + c + I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - I*d)^2, 1]
+Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol] := (e + f*x)^(m + 1)*ArcCoth[c + d*Cot[a + b*x]]/(f*(m + 1)) - I*b*(1 - c - I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x] + I*b*(1 + c + I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - I*d)^2, 1]
+Int[ArcTanh[u_], x_Symbol] := x*ArcTanh[u] - Int[SimplifyIntegrand[x*D[u, x]/(1 - u^2), x], x] /; InverseFunctionFreeQ[u, x]
+Int[ArcCoth[u_], x_Symbol] := x*ArcCoth[u] - Int[SimplifyIntegrand[x*D[u, x]/(1 - u^2), x], x] /; InverseFunctionFreeQ[u, x]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcTanh[u_]), x_Symbol] := (c + d*x)^(m + 1)*(a + b*ArcTanh[u])/(d*(m + 1)) - b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(1 - u^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && FalseQ[PowerVariableExpn[u, m + 1, x]]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcCoth[u_]), x_Symbol] := (c + d*x)^(m + 1)*(a + b*ArcCoth[u])/(d*(m + 1)) - b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(1 - u^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && FalseQ[PowerVariableExpn[u, m + 1, x]]
+Int[v_*(a_. + b_.*ArcTanh[u_]), x_Symbol] := With[{w = IntHide[v, x]}, Dist[(a + b*ArcTanh[u]), w, x] - b*Int[SimplifyIntegrand[w*D[u, x]/(1 - u^2), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_. /; FreeQ[{c, d, m}, x]]] && FalseQ[FunctionOfLinear[v*(a + b*ArcTanh[u]), x]]
+Int[v_*(a_. + b_.*ArcCoth[u_]), x_Symbol] := With[{w = IntHide[v, x]}, Dist[(a + b*ArcCoth[u]), w, x] - b*Int[SimplifyIntegrand[w*D[u, x]/(1 - u^2), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_. /; FreeQ[{c, d, m}, x]]] && FalseQ[FunctionOfLinear[v*(a + b*ArcCoth[u]), x]]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m b/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m
new file mode 100755
index 0000000..be23281
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m	
@@ -0,0 +1,39 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.5.1 u (a+b arcsech(c x))^n *)
+Int[ArcSech[c_.*x_], x_Symbol] := x*ArcSech[c*x] + Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)]*Int[1/Sqrt[1 - c^2*x^2], x] /; FreeQ[c, x]
+Int[ArcCsch[c_.*x_], x_Symbol] := x*ArcCsch[c*x] + 1/c*Int[1/(x*Sqrt[1 + 1/(c^2*x^2)]), x] /; FreeQ[c, x]
+Int[(a_. + b_.*ArcSech[c_.*x_])^n_, x_Symbol] := -1/c*Subst[Int[(a + b*x)^n*Sech[x]*Tanh[x], x], x, ArcSech[c*x]] /; FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]
+Int[(a_. + b_.*ArcCsch[c_.*x_])^n_, x_Symbol] := -1/c*Subst[Int[(a + b*x)^n*Csch[x]*Coth[x], x], x, ArcCsch[c*x]] /; FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]
+Int[(a_. + b_.*ArcSech[c_.*x_])/x_, x_Symbol] := -Subst[Int[(a + b*ArcCosh[x/c])/x, x], x, 1/x] /; FreeQ[{a, b, c}, x]
+Int[(a_. + b_.*ArcCsch[c_.*x_])/x_, x_Symbol] := -Subst[Int[(a + b*ArcSinh[x/c])/x, x], x, 1/x] /; FreeQ[{a, b, c}, x]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcSech[c_.*x_]), x_Symbol] := (d*x)^(m + 1)*(a + b*ArcSech[c*x])/(d*(m + 1)) + b*Sqrt[1 + c*x]/(m + 1)*Sqrt[1/(1 + c*x)]* Int[(d*x)^m/(Sqrt[1 - c*x]*Sqrt[1 + c*x]), x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
+Int[(d_.*x_)^m_.*(a_. + b_.*ArcCsch[c_.*x_]), x_Symbol] := (d*x)^(m + 1)*(a + b*ArcCsch[c*x])/(d*(m + 1)) + b*d/(c*(m + 1))*Int[(d*x)^(m - 1)/Sqrt[1 + 1/(c^2*x^2)], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
+Int[x_^m_.*(a_. + b_.*ArcSech[c_.*x_])^n_, x_Symbol] := -1/c^(m + 1)* Subst[Int[(a + b*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, ArcSech[c*x]] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n, 0] || LtQ[m, -1])
+Int[x_^m_.*(a_. + b_.*ArcCsch[c_.*x_])^n_, x_Symbol] := -1/c^(m + 1)* Subst[Int[(a + b*x)^n*Csch[x]^(m + 1)*Coth[x], x], x, ArcCsch[c*x]] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n, 0] || LtQ[m, -1])
+Int[(a_. + b_.*ArcSech[c_.*x_])/(d_. + e_.*x_), x_Symbol] := (a + b*ArcSech[c*x])* Log[1 + (e - Sqrt[-c^2*d^2 + e^2])/(c*d*E^ArcSech[c*x])]/e + (a + b*ArcSech[c*x])* Log[1 + (e + Sqrt[-c^2*d^2 + e^2])/(c*d*E^ArcSech[c*x])]/e - (a + b*ArcSech[c*x])*Log[1 + 1/E^(2*ArcSech[c*x])]/e + b/e* Int[(Sqrt[(1 - c*x)/(1 + c*x)]* Log[1 + (e - Sqrt[-c^2*d^2 + e^2])/(c*d* E^ArcSech[c*x])])/(x*(1 - c*x)), x] + b/e* Int[(Sqrt[(1 - c*x)/(1 + c*x)]* Log[1 + (e + Sqrt[-c^2*d^2 + e^2])/(c*d* E^ArcSech[c*x])])/(x*(1 - c*x)), x] - b/e* Int[(Sqrt[(1 - c*x)/(1 + c*x)]* Log[1 + 1/E^(2*ArcSech[c*x])])/(x*(1 - c*x)), x] /; FreeQ[{a, b, c, d, e}, x]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcSech[c_.*x_]), x_Symbol] := (d + e*x)^(m + 1)*(a + b*ArcSech[c*x])/(e*(m + 1)) + b*Sqrt[1 + c*x]/(e*(m + 1))*Sqrt[1/(1 + c*x)]* Int[(d + e*x)^(m + 1)/(x*Sqrt[1 - c^2*x^2]), x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
+Int[(a_. + b_.*ArcCsch[c_.*x_])/(d_. + e_.*x_), x_Symbol] := (a + b*ArcCsch[c*x])* Log[1 - (e - Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x]/(c*d)]/e + (a + b*ArcCsch[c*x])* Log[1 - (e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x]/(c*d)]/e - (a + b*ArcCsch[c*x])*Log[1 - E^(2*ArcCsch[c*x])]/e + b/(c*e)* Int[Log[1 - (e - Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x]/(c*d)]/(x^2* Sqrt[1 + 1/(c^2*x^2)]), x] + b/(c*e)* Int[Log[1 - (e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x]/(c*d)]/(x^2* Sqrt[1 + 1/(c^2*x^2)]), x] - b/(c*e)* Int[Log[1 - E^(2*ArcCsch[c*x])]/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x] /; FreeQ[{a, b, c, d, e}, x]
+Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCsch[c_.*x_]), x_Symbol] := (d + e*x)^(m + 1)*(a + b*ArcCsch[c*x])/(e*(m + 1)) + b/(c*e*(m + 1))* Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
+Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSech[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[(a + b*ArcSech[c*x]), u, x] + b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)]* Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
+Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsch[c_.*x_]), x_Symbol] := With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[(a + b*ArcCsch[c*x]), u, x] - b*c*x/Sqrt[-c^2*x^2]* Int[SimplifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
+Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSech[c_.*x_])^n_., x_Symbol] := -Subst[Int[(e + d*x^2)^p*(a + b*ArcCosh[x/c])^n/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[p]
+Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsch[c_.*x_])^n_., x_Symbol] := -Subst[Int[(e + d*x^2)^p*(a + b*ArcSinh[x/c])^n/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[p]
+Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSech[c_.*x_])^n_., x_Symbol] := -Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcCosh[x/c])^n/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
+Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsch[c_.*x_])^n_., x_Symbol] := -Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcSinh[x/c])^n/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[e - c^2*d, 0] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
+Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSech[c_.*x_])^n_., x_Symbol] := -Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcCosh[x/c])^n/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]
+Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsch[c_.*x_])^n_., x_Symbol] := -Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcSinh[x/c])^n/x^(2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[e - c^2*d, 0] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]
+Int[x_*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSech[c_.*x_]), x_Symbol] := (d + e*x^2)^(p + 1)*(a + b*ArcSech[c*x])/(2*e*(p + 1)) + b*Sqrt[1 + c*x]/(2*e*(p + 1))*Sqrt[1/(1 + c*x)]* Int[(d + e*x^2)^(p + 1)/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
+Int[x_*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsch[c_.*x_]), x_Symbol] := (d + e*x^2)^(p + 1)*(a + b*ArcCsch[c*x])/(2*e*(p + 1)) - b*c*x/(2*e*(p + 1)*Sqrt[-c^2*x^2])* Int[(d + e*x^2)^(p + 1)/(x*Sqrt[-1 - c^2*x^2]), x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
+Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSech[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[(a + b*ArcSech[c*x]), u, x] + b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)]* Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ( IGtQ[p, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]] || ILtQ[(m + 2*p + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]])
+Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsch[c_.*x_]), x_Symbol] := With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[(a + b*ArcCsch[c*x]), u, x] - b*c*x/Sqrt[-c^2*x^2]* Int[SimplifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ( IGtQ[p, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]] || ILtQ[(m + 2*p + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]] )
+Int[x_^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSech[c_.*x_])^n_., x_Symbol] := -Subst[ Int[(e + d*x^2)^p*(a + b*ArcCosh[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegersQ[m, p]
+Int[x_^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsch[c_.*x_])^n_., x_Symbol] := -Subst[ Int[(e + d*x^2)^p*(a + b*ArcSinh[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegersQ[m, p]
+Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSech[c_.*x_])^n_., x_Symbol] := -Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcCosh[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
+Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsch[c_.*x_])^n_., x_Symbol] := -Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcSinh[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[e - c^2*d, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]
+Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSech[c_.*x_])^n_., x_Symbol] := -Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcCosh[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]
+Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsch[c_.*x_])^n_., x_Symbol] := -Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcSinh[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[e - c^2*d, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]
+Int[u_*(a_. + b_.*ArcSech[c_.*x_]), x_Symbol] := With[{v = IntHide[u, x]}, Dist[(a + b*ArcSech[c*x]), v, x] + b* Sqrt[1 - c^2*x^2]/(c*x*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])* Int[SimplifyIntegrand[v/(x*Sqrt[1 - c^2*x^2]), x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
+Int[u_*(a_. + b_.*ArcCsch[c_.*x_]), x_Symbol] := With[{v = IntHide[u, x]}, Dist[(a + b*ArcCsch[c*x]), v, x] + b/c* Int[SimplifyIntegrand[v/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
+Int[u_.*(a_. + b_.*ArcSech[c_.*x_])^n_., x_Symbol] := Unintegrable[u*(a + b*ArcSech[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
+Int[u_.*(a_. + b_.*ArcCsch[c_.*x_])^n_., x_Symbol] := Unintegrable[u*(a + b*ArcCsch[c*x])^n, x] /; FreeQ[{a, b, c, n}, x]
diff --git a/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m b/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m
new file mode 100755
index 0000000..1571087
--- /dev/null
+++ b/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m	
@@ -0,0 +1,41 @@
+
+(* ::Subsection::Closed:: *)
+(* 7.5.2 Miscellaneous inverse hyperbolic secant *)
+Int[ArcSech[c_ + d_.*x_], x_Symbol] := (c + d*x)*ArcSech[c + d*x]/d + Int[Sqrt[(1 - c - d*x)/(1 + c + d*x)]/(1 - c - d*x), x] /; FreeQ[{c, d}, x]
+Int[ArcCsch[c_ + d_.*x_], x_Symbol] := (c + d*x)*ArcCsch[c + d*x]/d + Int[1/((c + d*x)*Sqrt[1 + 1/(c + d*x)^2]), x] /; FreeQ[{c, d}, x]
+Int[(a_. + b_.*ArcSech[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(a + b*ArcSech[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*ArcCsch[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(a + b*ArcCsch[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]
+Int[(a_. + b_.*ArcSech[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(a + b*ArcSech[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]
+Int[(a_. + b_.*ArcCsch[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(a + b*ArcCsch[c + d*x])^p, x] /; FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSech[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(f*x/d)^m*(a + b*ArcSech[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsch[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[(f*x/d)^m*(a + b*ArcCsch[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
+(* Int[x_^m_.*ArcSech[a_+b_.*x_],x_Symbol] := -((-a)^(m+1)-b^(m+1)*x^(m+1))*ArcSech[a+b*x]/(b^(m+1)*(m+1)) + 1/(b^(m+1)*(m+1))*Subst[Int[((-a*x)^(m+1)-(1-a*x)^(m+1))/(x^(m+1)* Sqrt[-1+x]*Sqrt[1+x]),x],x,1/(a+b*x)] /; FreeQ[{a,b},x] && IntegerQ[m] && NeQ[m,-1] *)
+(* Int[x_^m_.*ArcCsch[a_+b_.*x_],x_Symbol] := -((-a)^(m+1)-b^(m+1)*x^(m+1))*ArcCsch[a+b*x]/(b^(m+1)*(m+1)) + 1/(b^(m+1)*(m+1))*Subst[Int[((-a*x)^(m+1)-(1-a*x)^(m+1))/(x^(m+1)* Sqrt[1+x^2]),x],x,1/(a+b*x)] /; FreeQ[{a,b},x] && IntegerQ[m] && NeQ[m,-1] *)
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSech[c_ + d_.*x_])^p_., x_Symbol] := -1/d^(m + 1)* Subst[Int[(a + b*x)^p*Sech[x]*Tanh[x]*(d*e - c*f + f*Sech[x])^m, x], x, ArcSech[c + d*x]] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsch[c_ + d_.*x_])^p_., x_Symbol] := -1/d^(m + 1)* Subst[Int[(a + b*x)^p*Csch[x]*Coth[x]*(d*e - c*f + f*Csch[x])^m, x], x, ArcCsch[c + d*x]] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSech[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcSech[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsch[c_ + d_.*x_])^p_., x_Symbol] := 1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcCsch[x])^p, x], x, c + d*x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSech[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(e + f*x)^m*(a + b*ArcSech[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && Not[IGtQ[p, 0]]
+Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsch[c_ + d_.*x_])^p_, x_Symbol] := Unintegrable[(e + f*x)^m*(a + b*ArcCsch[c + d*x])^p, x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && Not[IGtQ[p, 0]]
+Int[u_.*ArcSech[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol] := Int[u*ArcCosh[a/c + b*x^n/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
+Int[u_.*ArcCsch[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol] := Int[u*ArcSinh[a/c + b*x^n/c]^m, x] /; FreeQ[{a, b, c, n, m}, x]
+Int[E^ArcSech[a_.*x_], x_Symbol] := x*E^ArcSech[a*x] + Log[x]/a + 1/a*Int[1/(x*(1 - a*x))*Sqrt[(1 - a*x)/(1 + a*x)], x] /; FreeQ[a, x]
+Int[E^ArcSech[a_.*x_^p_], x_Symbol] := x*E^ArcSech[a*x^p] + p/a*Int[1/x^p, x] + p*Sqrt[1 + a*x^p]/a*Sqrt[1/(1 + a*x^p)]* Int[1/(x^p*Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x] /; FreeQ[{a, p}, x]
+Int[E^ArcCsch[a_.*x_^p_.], x_Symbol] := 1/a*Int[1/x^p, x] + Int[Sqrt[1 + 1/(a^2*x^(2*p))], x] /; FreeQ[{a, p}, x]
+Int[E^(n_.*ArcSech[u_]), x_Symbol] := Int[(1/u + Sqrt[(1 - u)/(1 + u)] + 1/u*Sqrt[(1 - u)/(1 + u)])^n, x] /; IntegerQ[n]
+Int[E^(n_.*ArcCsch[u_]), x_Symbol] := Int[(1/u + Sqrt[1 + 1/u^2])^n, x] /; IntegerQ[n]
+Int[E^ArcSech[a_.*x_^p_.]/x_, x_Symbol] := -1/(a*p*x^p) + Sqrt[1 + a*x^p]/a*Sqrt[1/(1 + a*x^p)]* Int[Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]/x^(p + 1), x] /; FreeQ[{a, p}, x]
+Int[x_^m_.*E^ArcSech[a_.*x_^p_.], x_Symbol] := x^(m + 1)*E^ArcSech[a*x^p]/(m + 1) + p/(a*(m + 1))*Int[x^(m - p), x] + p*Sqrt[1 + a*x^p]/(a*(m + 1))*Sqrt[1/(1 + a*x^p)]* Int[x^(m - p)/(Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x] /; FreeQ[{a, m, p}, x] && NeQ[m, -1]
+Int[x_^m_.*E^ArcCsch[a_.*x_^p_.], x_Symbol] := 1/a*Int[x^(m - p), x] + Int[x^m*Sqrt[1 + 1/(a^2*x^(2*p))], x] /; FreeQ[{a, m, p}, x]
+Int[x_^m_.*E^(n_.*ArcSech[u_]), x_Symbol] := Int[x^m*(1/u + Sqrt[(1 - u)/(1 + u)] + 1/u*Sqrt[(1 - u)/(1 + u)])^ n, x] /; FreeQ[m, x] && IntegerQ[n]
+Int[x_^m_.*E^(n_.*ArcCsch[u_]), x_Symbol] := Int[x^m*(1/u + Sqrt[1 + 1/u^2])^n, x] /; FreeQ[m, x] && IntegerQ[n]
+Int[E^(ArcSech[c_.*x_])/(a_ + b_.*x_^2), x_Symbol] := 1/(a*c)*Int[Sqrt[1/(1 + c*x)]/(x*Sqrt[1 - c*x]), x] + 1/c*Int[1/(x*(a + b*x^2)), x] /; FreeQ[{a, b, c}, x] && EqQ[b + a*c^2, 0]
+Int[E^(ArcCsch[c_.*x_])/(a_ + b_.*x_^2), x_Symbol] := 1/(a*c^2)*Int[1/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x] + 1/c*Int[1/(x*(a + b*x^2)), x] /; FreeQ[{a, b, c}, x] && EqQ[b - a*c^2, 0]
+Int[(d_.*x_)^m_.*E^(ArcSech[c_.*x_])/(a_ + b_.*x_^2), x_Symbol] := d/(a*c)*Int[(d*x)^(m - 1)*Sqrt[1/(1 + c*x)]/Sqrt[1 - c*x], x] + d/c*Int[(d*x)^(m - 1)/(a + b*x^2), x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b + a*c^2, 0]
+Int[(d_.*x_)^m_.*E^(ArcCsch[c_.*x_])/(a_ + b_.*x_^2), x_Symbol] := d^2/(a*c^2)*Int[(d*x)^(m - 2)/Sqrt[1 + 1/(c^2*x^2)], x] + d/c*Int[(d*x)^(m - 1)/(a + b*x^2), x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b - a*c^2, 0]
+Int[ArcSech[u_], x_Symbol] := x*ArcSech[u] + Sqrt[1 - u^2]/(u*Sqrt[-1 + 1/u]*Sqrt[1 + 1/u])* Int[SimplifyIntegrand[x*D[u, x]/(u*Sqrt[1 - u^2]), x], x] /; InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]
+Int[ArcCsch[u_], x_Symbol] := x*ArcCsch[u] - u/Sqrt[-u^2]* Int[SimplifyIntegrand[x*D[u, x]/(u*Sqrt[-1 - u^2]), x], x] /; InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcSech[u_]), x_Symbol] := (c + d*x)^(m + 1)*(a + b*ArcSech[u])/(d*(m + 1)) + b*Sqrt[1 - u^2]/(d*(m + 1)*u*Sqrt[-1 + 1/u]*Sqrt[1 + 1/u])* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(u*Sqrt[1 - u^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]
+Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcCsch[u_]), x_Symbol] := (c + d*x)^(m + 1)*(a + b*ArcCsch[u])/(d*(m + 1)) - b*u/(d*(m + 1)*Sqrt[-u^2])* Int[SimplifyIntegrand[(c + d*x)^(m + 1)* D[u, x]/(u*Sqrt[-1 - u^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]
+Int[v_*(a_. + b_.*ArcSech[u_]), x_Symbol] := With[{w = IntHide[v, x]}, Dist[(a + b*ArcSech[u]), w, x] + b*Sqrt[1 - u^2]/(u*Sqrt[-1 + 1/u]*Sqrt[1 + 1/u])* Int[SimplifyIntegrand[w*D[u, x]/(u*Sqrt[1 - u^2]), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_. /; FreeQ[{c, d, m}, x]]]
+Int[v_*(a_. + b_.*ArcCsch[u_]), x_Symbol] := With[{w = IntHide[v, x]}, Dist[(a + b*ArcCsch[u]), w, x] - b*u/Sqrt[-u^2]* Int[SimplifyIntegrand[w*D[u, x]/(u*Sqrt[-1 - u^2]), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_. /; FreeQ[{c, d, m}, x]]]
diff --git a/IntegrationRules/8 Special functions/8.1 Error functions.m b/IntegrationRules/8 Special functions/8.1 Error functions.m
new file mode 100755
index 0000000..e5daf8a
--- /dev/null
+++ b/IntegrationRules/8 Special functions/8.1 Error functions.m	
@@ -0,0 +1,72 @@
+
+(* ::Subsection::Closed:: *)
+(* 8.1 Error functions *)
+Int[Erf[a_. + b_.*x_], x_Symbol] := (a + b*x)*Erf[a + b*x]/b + 1/(b*Sqrt[Pi]*E^(a + b*x)^2) /; FreeQ[{a, b}, x]
+Int[Erfc[a_. + b_.*x_], x_Symbol] := (a + b*x)*Erfc[a + b*x]/b - 1/(b*Sqrt[Pi]*E^(a + b*x)^2) /; FreeQ[{a, b}, x]
+Int[Erfi[a_. + b_.*x_], x_Symbol] := (a + b*x)*Erfi[a + b*x]/b - E^(a + b*x)^2/(b*Sqrt[Pi]) /; FreeQ[{a, b}, x]
+Int[Erf[a_. + b_.*x_]^2, x_Symbol] := (a + b*x)*Erf[a + b*x]^2/b - 4/Sqrt[Pi]*Int[(a + b*x)*Erf[a + b*x]/E^(a + b*x)^2, x] /; FreeQ[{a, b}, x]
+Int[Erfc[a_. + b_.*x_]^2, x_Symbol] := (a + b*x)*Erfc[a + b*x]^2/b + 4/Sqrt[Pi]*Int[(a + b*x)*Erfc[a + b*x]/E^(a + b*x)^2, x] /; FreeQ[{a, b}, x]
+Int[Erfi[a_. + b_.*x_]^2, x_Symbol] := (a + b*x)*Erfi[a + b*x]^2/b - 4/Sqrt[Pi]*Int[(a + b*x)*E^(a + b*x)^2*Erfi[a + b*x], x] /; FreeQ[{a, b}, x]
+Int[Erf[a_. + b_.*x_]^n_, x_Symbol] := Unintegrable[Erf[a + b*x]^n, x] /; FreeQ[{a, b, n}, x] && NeQ[n, 1] && NeQ[n, 2]
+Int[Erfc[a_. + b_.*x_]^n_, x_Symbol] := Unintegrable[Erfc[a + b*x]^n, x] /; FreeQ[{a, b, n}, x] && NeQ[n, 1] && NeQ[n, 2]
+Int[Erfi[a_. + b_.*x_]^n_, x_Symbol] := Unintegrable[Erfi[a + b*x]^n, x] /; FreeQ[{a, b, n}, x] && NeQ[n, 1] && NeQ[n, 2]
+Int[Erf[b_.*x_]/x_, x_Symbol] := 2*b*x/Sqrt[Pi]* HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, -b^2*x^2] /; FreeQ[b, x]
+Int[Erfc[b_.*x_]/x_, x_Symbol] := Log[x] - Int[Erf[b*x]/x, x] /; FreeQ[b, x]
+Int[Erfi[b_.*x_]/x_, x_Symbol] := 2*b*x/Sqrt[Pi]*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, b^2*x^2] /; FreeQ[b, x]
+Int[(c_. + d_.*x_)^m_.*Erf[a_. + b_.*x_], x_Symbol] := (c + d*x)^(m + 1)*Erf[a + b*x]/(d*(m + 1)) - 2*b/(Sqrt[Pi]*d*(m + 1))* Int[(c + d*x)^(m + 1)/E^(a + b*x)^2, x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
+Int[(c_. + d_.*x_)^m_.*Erfc[a_. + b_.*x_], x_Symbol] := (c + d*x)^(m + 1)*Erfc[a + b*x]/(d*(m + 1)) + 2*b/(Sqrt[Pi]*d*(m + 1))* Int[(c + d*x)^(m + 1)/E^(a + b*x)^2, x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
+Int[(c_. + d_.*x_)^m_.*Erfi[a_. + b_.*x_], x_Symbol] := (c + d*x)^(m + 1)*Erfi[a + b*x]/(d*(m + 1)) - 2*b/(Sqrt[Pi]*d*(m + 1))* Int[(c + d*x)^(m + 1)*E^(a + b*x)^2, x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
+Int[x_^m_.*Erf[b_.*x_]^2, x_Symbol] := x^(m + 1)*Erf[b*x]^2/(m + 1) - 4*b/(Sqrt[Pi]*(m + 1))*Int[x^(m + 1)*E^(-b^2*x^2)*Erf[b*x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])
+Int[x_^m_.*Erfc[b_.*x_]^2, x_Symbol] := x^(m + 1)*Erfc[b*x]^2/(m + 1) + 4*b/(Sqrt[Pi]*(m + 1))*Int[x^(m + 1)*E^(-b^2*x^2)*Erfc[b*x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])
+Int[x_^m_.*Erfi[b_.*x_]^2, x_Symbol] := x^(m + 1)*Erfi[b*x]^2/(m + 1) - 4*b/(Sqrt[Pi]*(m + 1))*Int[x^(m + 1)*E^(b^2*x^2)*Erfi[b*x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])
+Int[(c_. + d_.*x_)^m_.*Erf[a_ + b_.*x_]^2, x_Symbol] := 1/b^(m + 1)* Subst[Int[ExpandIntegrand[Erf[x]^2, (b*c - a*d + d*x)^m, x], x], x, a + b*x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*Erfc[a_ + b_.*x_]^2, x_Symbol] := 1/b^(m + 1)* Subst[Int[ExpandIntegrand[Erfc[x]^2, (b*c - a*d + d*x)^m, x], x], x, a + b*x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*Erfi[a_ + b_.*x_]^2, x_Symbol] := 1/b^(m + 1)* Subst[Int[ExpandIntegrand[Erfi[x]^2, (b*c - a*d + d*x)^m, x], x], x, a + b*x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*Erf[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[(c + d*x)^m*Erf[a + b*x]^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
+Int[(c_. + d_.*x_)^m_.*Erfc[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[(c + d*x)^m*Erfc[a + b*x]^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
+Int[(c_. + d_.*x_)^m_.*Erfi[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[(c + d*x)^m*Erfi[a + b*x]^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
+Int[E^(c_. + d_.*x_^2)*Erf[b_.*x_]^n_., x_Symbol] := E^c*Sqrt[Pi]/(2*b)*Subst[Int[x^n, x], x, Erf[b*x]] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]
+Int[E^(c_. + d_.*x_^2)*Erfc[b_.*x_]^n_., x_Symbol] := -E^c*Sqrt[Pi]/(2*b)*Subst[Int[x^n, x], x, Erfc[b*x]] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]
+Int[E^(c_. + d_.*x_^2)*Erfi[b_.*x_]^n_., x_Symbol] := E^c*Sqrt[Pi]/(2*b)*Subst[Int[x^n, x], x, Erfi[b*x]] /; FreeQ[{b, c, d, n}, x] && EqQ[d, b^2]
+Int[E^(c_. + d_.*x_^2)*Erf[b_.*x_], x_Symbol] := b*E^c*x^2/Sqrt[Pi]*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]
+Int[E^(c_. + d_.*x_^2)*Erfc[b_.*x_], x_Symbol] := Int[E^(c + d*x^2), x] - Int[E^(c + d*x^2)*Erf[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]
+Int[E^(c_. + d_.*x_^2)*Erfi[b_.*x_], x_Symbol] := b*E^c*x^2/Sqrt[Pi]*HypergeometricPFQ[{1, 1}, {3/2, 2}, -b^2*x^2] /; FreeQ[{b, c, d}, x] && EqQ[d, -b^2]
+Int[E^(c_. + d_.*x_^2)*Erf[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[E^(c + d*x^2)*Erf[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
+Int[E^(c_. + d_.*x_^2)*Erfc[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[E^(c + d*x^2)*Erfc[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
+Int[E^(c_. + d_.*x_^2)*Erfi[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[E^(c + d*x^2)*Erfi[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
+Int[x_*E^(c_. + d_.*x_^2)*Erf[a_. + b_.*x_], x_Symbol] := E^(c + d*x^2)*Erf[a + b*x]/(2*d) - b/(d*Sqrt[Pi])*Int[E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x] /; FreeQ[{a, b, c, d}, x]
+Int[x_*E^(c_. + d_.*x_^2)*Erfc[a_. + b_.*x_], x_Symbol] := E^(c + d*x^2)*Erfc[a + b*x]/(2*d) + b/(d*Sqrt[Pi])*Int[E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x] /; FreeQ[{a, b, c, d}, x]
+Int[x_*E^(c_. + d_.*x_^2)*Erfi[a_. + b_.*x_], x_Symbol] := E^(c + d*x^2)*Erfi[a + b*x]/(2*d) - b/(d*Sqrt[Pi])*Int[E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x] /; FreeQ[{a, b, c, d}, x]
+Int[x_^m_*E^(c_. + d_.*x_^2)*Erf[a_. + b_.*x_], x_Symbol] := x^(m - 1)*E^(c + d*x^2)*Erf[a + b*x]/(2*d) - b/(d*Sqrt[Pi])* Int[x^(m - 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x] - (m - 1)/(2*d)*Int[x^(m - 2)*E^(c + d*x^2)*Erf[a + b*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1]
+Int[x_^m_*E^(c_. + d_.*x_^2)*Erfc[a_. + b_.*x_], x_Symbol] := x^(m - 1)*E^(c + d*x^2)*Erfc[a + b*x]/(2*d) + b/(d*Sqrt[Pi])* Int[x^(m - 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x] - (m - 1)/(2*d)*Int[x^(m - 2)*E^(c + d*x^2)*Erfc[a + b*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1]
+Int[x_^m_*E^(c_. + d_.*x_^2)*Erfi[a_. + b_.*x_], x_Symbol] := x^(m - 1)*E^(c + d*x^2)*Erfi[a + b*x]/(2*d) - b/(d*Sqrt[Pi])* Int[x^(m - 1)*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x] - (m - 1)/(2*d)*Int[x^(m - 2)*E^(c + d*x^2)*Erfi[a + b*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1]
+Int[E^(c_. + d_.*x_^2)*Erf[b_.*x_]/x_, x_Symbol] := 2*b*E^c*x/Sqrt[Pi]* HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, b^2*x^2] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]
+Int[E^(c_. + d_.*x_^2)*Erfc[b_.*x_]/x_, x_Symbol] := Int[E^(c + d*x^2)/x, x] - Int[E^(c + d*x^2)*Erf[b*x]/x, x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]
+Int[E^(c_. + d_.*x_^2)*Erfi[b_.*x_]/x_, x_Symbol] := 2*b*E^c*x/Sqrt[Pi]* HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, -b^2*x^2] /; FreeQ[{b, c, d}, x] && EqQ[d, -b^2]
+Int[x_^m_*E^(c_. + d_.*x_^2)*Erf[a_. + b_.*x_], x_Symbol] := x^(m + 1)*E^(c + d*x^2)*Erf[a + b*x]/(m + 1) - 2*b/((m + 1)*Sqrt[Pi])* Int[x^(m + 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x] - 2*d/(m + 1)*Int[x^(m + 2)*E^(c + d*x^2)*Erf[a + b*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -1]
+Int[x_^m_*E^(c_. + d_.*x_^2)*Erfc[a_. + b_.*x_], x_Symbol] := x^(m + 1)*E^(c + d*x^2)*Erfc[a + b*x]/(m + 1) + 2*b/((m + 1)*Sqrt[Pi])* Int[x^(m + 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x] - 2*d/(m + 1)*Int[x^(m + 2)*E^(c + d*x^2)*Erfc[a + b*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -1]
+Int[x_^m_*E^(c_. + d_.*x_^2)*Erfi[a_. + b_.*x_], x_Symbol] := x^(m + 1)*E^(c + d*x^2)*Erfi[a + b*x]/(m + 1) - 2*b/((m + 1)*Sqrt[Pi])* Int[x^(m + 1)*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x] - 2*d/(m + 1)*Int[x^(m + 2)*E^(c + d*x^2)*Erfi[a + b*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -1]
+Int[(e_.*x_)^m_.*E^(c_. + d_.*x_^2)*Erf[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[(e*x)^m*E^(c + d*x^2)*Erf[a + b*x]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[(e_.*x_)^m_.*E^(c_. + d_.*x_^2)*Erfc[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[(e*x)^m*E^(c + d*x^2)*Erfc[a + b*x]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[(e_.*x_)^m_.*E^(c_. + d_.*x_^2)*Erfi[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[(e*x)^m*E^(c + d*x^2)*Erfi[a + b*x]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[Erf[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := x*Erf[d*(a + b*Log[c*x^n])] - 2*b*d*n/(Sqrt[Pi])*Int[1/E^(d*(a + b*Log[c*x^n]))^2, x] /; FreeQ[{a, b, c, d, n}, x]
+Int[Erfc[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := x*Erfc[d*(a + b*Log[c*x^n])] + 2*b*d*n/(Sqrt[Pi])*Int[1/E^(d*(a + b*Log[c*x^n]))^2, x] /; FreeQ[{a, b, c, d, n}, x]
+Int[Erfi[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := x*Erfi[d*(a + b*Log[c*x^n])] - 2*b*d*n/(Sqrt[Pi])*Int[E^(d*(a + b*Log[c*x^n]))^2, x] /; FreeQ[{a, b, c, d, n}, x]
+Int[F_[d_.*(a_. + b_.*Log[c_.*x_^n_.])]/x_, x_Symbol] := 1/n*Subst[F[d*(a + b*x)], x, Log[c*x^n]] /; FreeQ[{a, b, c, d, n}, x] && MemberQ[{Erf, Erfc, Erfi}, F]
+Int[(e_.*x_)^m_.*Erf[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := (e*x)^(m + 1)*Erf[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - 2*b*d*n/(Sqrt[Pi]*(m + 1))* Int[(e*x)^m/E^(d*(a + b*Log[c*x^n]))^2, x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
+Int[(e_.*x_)^m_.*Erfc[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := (e*x)^(m + 1)*Erfc[d*(a + b*Log[c*x^n])]/(e*(m + 1)) + 2*b*d*n/(Sqrt[Pi]*(m + 1))* Int[(e*x)^m/E^(d*(a + b*Log[c*x^n]))^2, x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
+Int[(e_.*x_)^m_.*Erfi[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := (e*x)^(m + 1)*Erfi[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - 2*b*d*n/(Sqrt[Pi]*(m + 1))* Int[(e*x)^m*E^(d*(a + b*Log[c*x^n]))^2, x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
+Int[Sin[c_. + d_.*x_^2]*Erf[b_.*x_], x_Symbol] := I/2*Int[E^(-I*c - I*d*x^2)*Erf[b*x], x] - I/2*Int[E^(I*c + I*d*x^2)*Erf[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -b^4]
+Int[Sin[c_. + d_.*x_^2]*Erfc[b_.*x_], x_Symbol] := I/2*Int[E^(-I*c - I*d*x^2)*Erfc[b*x], x] - I/2*Int[E^(I*c + I*d*x^2)*Erfc[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -b^4]
+Int[Sin[c_. + d_.*x_^2]*Erfi[b_.*x_], x_Symbol] := I/2*Int[E^(-I*c - I*d*x^2)*Erfi[b*x], x] - I/2*Int[E^(I*c + I*d*x^2)*Erfi[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -b^4]
+Int[Cos[c_. + d_.*x_^2]*Erf[b_.*x_], x_Symbol] := 1/2*Int[E^(-I*c - I*d*x^2)*Erf[b*x], x] + 1/2*Int[E^(I*c + I*d*x^2)*Erf[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -b^4]
+Int[Cos[c_. + d_.*x_^2]*Erfc[b_.*x_], x_Symbol] := 1/2*Int[E^(-I*c - I*d*x^2)*Erfc[b*x], x] + 1/2*Int[E^(I*c + I*d*x^2)*Erfc[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -b^4]
+Int[Cos[c_. + d_.*x_^2]*Erfi[b_.*x_], x_Symbol] := 1/2*Int[E^(-I*c - I*d*x^2)*Erfi[b*x], x] + 1/2*Int[E^(I*c + I*d*x^2)*Erfi[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -b^4]
+Int[Sinh[c_. + d_.*x_^2]*Erf[b_.*x_], x_Symbol] := 1/2*Int[E^(c + d*x^2)*Erf[b*x], x] - 1/2*Int[E^(-c - d*x^2)*Erf[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, b^4]
+Int[Sinh[c_. + d_.*x_^2]*Erfc[b_.*x_], x_Symbol] := 1/2*Int[E^(c + d*x^2)*Erfc[b*x], x] - 1/2*Int[E^(-c - d*x^2)*Erfc[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, b^4]
+Int[Sinh[c_. + d_.*x_^2]*Erfi[b_.*x_], x_Symbol] := 1/2*Int[E^(c + d*x^2)*Erfi[b*x], x] - 1/2*Int[E^(-c - d*x^2)*Erfi[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, b^4]
+Int[Cosh[c_. + d_.*x_^2]*Erf[b_.*x_], x_Symbol] := 1/2*Int[E^(c + d*x^2)*Erf[b*x], x] + 1/2*Int[E^(-c - d*x^2)*Erf[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, b^4]
+Int[Cosh[c_. + d_.*x_^2]*Erfc[b_.*x_], x_Symbol] := 1/2*Int[E^(c + d*x^2)*Erfc[b*x], x] + 1/2*Int[E^(-c - d*x^2)*Erfc[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, b^4]
+Int[Cosh[c_. + d_.*x_^2]*Erfi[b_.*x_], x_Symbol] := 1/2*Int[E^(c + d*x^2)*Erfi[b*x], x] + 1/2*Int[E^(-c - d*x^2)*Erfi[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, b^4]
+Int[F_[f_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])], x_Symbol] := 1/e*Subst[Int[F[f*(a + b*Log[c*x^n])], x], x, d + e*x] /; FreeQ[{a, b, c, d, e, f, n}, x] && MemberQ[{Erf, Erfc, Erfi, FresnelS, FresnelC, ExpIntegralEi, SinIntegral, CosIntegral, SinhIntegral, CoshIntegral}, F]
+Int[(g_ + h_. x_)^m_.*F_[f_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])], x_Symbol] := 1/e*Subst[Int[(g*x/d)^m*F[f*(a + b*Log[c*x^n])], x], x, d + e*x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && EqQ[e*f - d*g, 0] && MemberQ[{Erf, Erfc, Erfi, FresnelS, FresnelC, ExpIntegralEi, SinIntegral, CosIntegral, SinhIntegral, CoshIntegral}, F]
diff --git a/IntegrationRules/8 Special functions/8.10 Bessel functions.m b/IntegrationRules/8 Special functions/8.10 Bessel functions.m
new file mode 100755
index 0000000..22c3eb6
--- /dev/null
+++ b/IntegrationRules/8 Special functions/8.10 Bessel functions.m	
@@ -0,0 +1,7 @@
+
+(* ::Subsection::Closed:: *)
+(* 8.10 Bessel functions *)
+Int[BesselJ[1, a_. + b_.*x_], x_Symbol] := -BesselJ[0, a + b*x]/b /; FreeQ[{a, b}, x]
+Int[BesselJ[n_, a_. + b_.*x_], x_Symbol] := -2*BesselJ[n - 1, a + b*x]/b + Int[BesselJ[n - 2, a + b*x], x] /; FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0]
+(* Int[BesselJ[n_,a_.+b_.*x_],x_Symbol] := (-1)^n*Int[BesselJ[-n,a+b*x],x] /; FreeQ[{a,b},x] && ILtQ[n,0] *)
+Int[BesselJ[n_, a_. + b_.*x_], x_Symbol] := (a + b*x)^(n + 1)* HypergeometricPFQ[{(n + 1)/2}, {(n + 3)/2, n + 1}, -1/4*(a + b*x)^2]/(2^n*b*Gamma[n + 2]) /; FreeQ[{a, b, n}, x]
diff --git a/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m b/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m
new file mode 100755
index 0000000..ef156fb
--- /dev/null
+++ b/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m	
@@ -0,0 +1,58 @@
+
+(* ::Subsection::Closed:: *)
+(* 8.2 Fresnel integral functions *)
+Int[FresnelS[a_. + b_.*x_], x_Symbol] := (a + b*x)*FresnelS[a + b*x]/b + Cos[Pi/2*(a + b*x)^2]/(b*Pi) /; FreeQ[{a, b}, x]
+Int[FresnelC[a_. + b_.*x_], x_Symbol] := (a + b*x)*FresnelC[a + b*x]/b - Sin[Pi/2*(a + b*x)^2]/(b*Pi) /; FreeQ[{a, b}, x]
+Int[FresnelS[a_. + b_.*x_]^2, x_Symbol] := (a + b*x)*FresnelS[a + b*x]^2/b - 2*Int[(a + b*x)*Sin[Pi/2*(a + b*x)^2]*FresnelS[a + b*x], x] /; FreeQ[{a, b}, x]
+Int[FresnelC[a_. + b_.*x_]^2, x_Symbol] := (a + b*x)*FresnelC[a + b*x]^2/b - 2*Int[(a + b*x)*Cos[Pi/2*(a + b*x)^2]*FresnelC[a + b*x], x] /; FreeQ[{a, b}, x]
+Int[FresnelS[a_. + b_.*x_]^n_, x_Symbol] := Unintegrable[FresnelS[a + b*x]^n, x] /; FreeQ[{a, b, n}, x] && NeQ[n, 1] && NeQ[n, 2]
+Int[FresnelC[a_. + b_.*x_]^n_, x_Symbol] := Unintegrable[FresnelC[a + b*x]^n, x] /; FreeQ[{a, b, n}, x] && NeQ[n, 1] && NeQ[n, 2]
+Int[FresnelS[b_.*x_]/x_, x_Symbol] := (1 + I)/4*Int[Erf[Sqrt[Pi]/2*(1 + I)*b*x]/x, x] + (1 - I)/4* Int[Erf[Sqrt[Pi]/2*(1 - I)*b*x]/x, x] /; FreeQ[b, x]
+Int[FresnelC[b_.*x_]/x_, x_Symbol] := (1 - I)/4*Int[Erf[Sqrt[Pi]/2*(1 + I)*b*x]/x, x] + (1 + I)/4* Int[Erf[Sqrt[Pi]/2*(1 - I)*b*x]/x, x] /; FreeQ[b, x]
+Int[(d_.*x_)^m_.*FresnelS[b_.*x_], x_Symbol] := (d*x)^(m + 1)*FresnelS[b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(d*x)^(m + 1)*Sin[Pi/2*b^2*x^2], x] /; FreeQ[{b, d, m}, x] && NeQ[m, -1]
+Int[(d_.*x_)^m_.*FresnelC[b_.*x_], x_Symbol] := (d*x)^(m + 1)*FresnelC[b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(d*x)^(m + 1)*Cos[Pi/2*b^2*x^2], x] /; FreeQ[{b, d, m}, x] && NeQ[m, -1]
+Int[(c_. + d_.*x_)^m_.*FresnelS[a_. + b_.*x_], x_Symbol] := (c + d*x)^(m + 1)*FresnelS[a + b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(c + d*x)^(m + 1)*Sin[Pi/2*(a + b*x)^2], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*FresnelC[a_. + b_.*x_], x_Symbol] := (c + d*x)^(m + 1)*FresnelC[a + b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(c + d*x)^(m + 1)*Cos[Pi/2*(a + b*x)^2], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
+Int[x_^m_.*FresnelS[b_.*x_]^2, x_Symbol] := x^(m + 1)*FresnelS[b*x]^2/(m + 1) - 2*b/(m + 1)*Int[x^(m + 1)*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x] /; FreeQ[b, x] && IntegerQ[m] && NeQ[m, -1]
+Int[x_^m_.*FresnelC[b_.*x_]^2, x_Symbol] := x^(m + 1)*FresnelC[b*x]^2/(m + 1) - 2*b/(m + 1)*Int[x^(m + 1)*Cos[Pi/2*b^2*x^2]*FresnelC[b*x], x] /; FreeQ[b, x] && IntegerQ[m] && NeQ[m, -1]
+Int[(c_. + d_.*x_)^m_.*FresnelS[a_ + b_.*x_]^2, x_Symbol] := 1/b^(m + 1)* Subst[Int[ExpandIntegrand[FresnelS[x]^2, (b*c - a*d + d*x)^m, x], x], x, a + b*x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*FresnelC[a_ + b_.*x_]^2, x_Symbol] := 1/b^(m + 1)* Subst[Int[ExpandIntegrand[FresnelC[x]^2, (b*c - a*d + d*x)^m, x], x], x, a + b*x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*FresnelS[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[(c + d*x)^m*FresnelS[a + b*x]^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
+Int[(c_. + d_.*x_)^m_.*FresnelC[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[(c + d*x)^m*FresnelC[a + b*x]^n, x] /; FreeQ[{a, b, c, d, m, n}, x]
+Int[E^(c_. + d_.*x_^2)*FresnelS[b_.*x_], x_Symbol] := (1 + I)/4* Int[E^(c + d*x^2)*Erf[Sqrt[Pi]/2*(1 + I)*b*x], x] + (1 - I)/4* Int[E^(c + d*x^2)*Erf[Sqrt[Pi]/2*(1 - I)*b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -Pi^2/4*b^4]
+Int[E^(c_. + d_.*x_^2)*FresnelC[b_.*x_], x_Symbol] := (1 - I)/4* Int[E^(c + d*x^2)*Erf[Sqrt[Pi]/2*(1 + I)*b*x], x] + (1 + I)/4* Int[E^(c + d*x^2)*Erf[Sqrt[Pi]/2*(1 - I)*b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, -Pi^2/4*b^4]
+Int[E^(c_. + d_.*x_^2)*FresnelS[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[E^(c + d*x^2)*FresnelS[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
+Int[E^(c_. + d_.*x_^2)*FresnelC[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[E^(c + d*x^2)*FresnelC[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
+Int[Sin[d_.*x_^2]*FresnelS[b_.*x_]^n_., x_Symbol] := Pi*b/(2*d)*Subst[Int[x^n, x], x, FresnelS[b*x]] /; FreeQ[{b, d, n}, x] && EqQ[d^2, Pi^2/4*b^4]
+Int[Cos[d_.*x_^2]*FresnelC[b_.*x_]^n_., x_Symbol] := Pi*b/(2*d)*Subst[Int[x^n, x], x, FresnelC[b*x]] /; FreeQ[{b, d, n}, x] && EqQ[d^2, Pi^2/4*b^4]
+Int[Sin[c_ + d_.*x_^2]*FresnelS[b_.*x_], x_Symbol] := Sin[c]*Int[Cos[d*x^2]*FresnelS[b*x], x] + Cos[c]*Int[Sin[d*x^2]*FresnelS[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, Pi^2/4*b^4]
+Int[Cos[c_ + d_.*x_^2]*FresnelC[b_.*x_], x_Symbol] := Cos[c]*Int[Cos[d*x^2]*FresnelC[b*x], x] - Sin[c]*Int[Sin[d*x^2]*FresnelC[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, Pi^2/4*b^4]
+Int[Sin[c_. + d_.*x_^2]*FresnelS[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[Sin[c + d*x^2]*FresnelS[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
+Int[Cos[c_. + d_.*x_^2]*FresnelC[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[Cos[c + d*x^2]*FresnelC[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
+Int[Cos[d_.*x_^2]*FresnelS[b_.*x_], x_Symbol] := FresnelC[b*x]*FresnelS[b*x]/(2*b) - 1/8*I*b*x^2* HypergeometricPFQ[{1, 1}, {3/2, 2}, -1/2*I*b^2*Pi*x^2] + 1/8*I*b*x^2* HypergeometricPFQ[{1, 1}, {3/2, 2}, 1/2*I*b^2*Pi*x^2] /; FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4]
+Int[Sin[d_.*x_^2]*FresnelC[b_.*x_], x_Symbol] := b*Pi*FresnelC[b*x]*FresnelS[b*x]/(4*d) + 1/8*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -I*d*x^2] - 1/8*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, I*d*x^2] /; FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4]
+Int[Cos[c_ + d_.*x_^2]*FresnelS[b_.*x_], x_Symbol] := Cos[c]*Int[Cos[d*x^2]*FresnelS[b*x], x] - Sin[c]*Int[Sin[d*x^2]*FresnelS[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, Pi^2/4*b^4]
+Int[Sin[c_ + d_.*x_^2]*FresnelC[b_.*x_], x_Symbol] := Sin[c]*Int[Cos[d*x^2]*FresnelC[b*x], x] + Cos[c]*Int[Sin[d*x^2]*FresnelC[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, Pi^2/4*b^4]
+Int[Cos[c_. + d_.*x_^2]*FresnelS[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[Cos[c + d*x^2]*FresnelS[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
+Int[Sin[c_. + d_.*x_^2]*FresnelC[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[Sin[c + d*x^2]*FresnelC[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
+Int[x_*Sin[d_.*x_^2]*FresnelS[b_.*x_], x_Symbol] := -Cos[d*x^2]*FresnelS[b*x]/(2*d) + 1/(2*b*Pi)*Int[Sin[2*d*x^2], x] /; FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4]
+Int[x_*Cos[d_.*x_^2]*FresnelC[b_.*x_], x_Symbol] := Sin[d*x^2]*FresnelC[b*x]/(2*d) - b/(4*d)*Int[Sin[2*d*x^2], x] /; FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4]
+Int[x_^m_*Sin[d_.*x_^2]*FresnelS[b_.*x_], x_Symbol] := -x^(m - 1)*Cos[d*x^2]*FresnelS[b*x]/(2*d) + 1/(2*b*Pi)*Int[x^(m - 1)*Sin[2*d*x^2], x] + (m - 1)/(2*d)*Int[x^(m - 2)*Cos[d*x^2]*FresnelS[b*x], x] /; FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && IGtQ[m, 1]
+Int[x_^m_*Cos[d_.*x_^2]*FresnelC[b_.*x_], x_Symbol] := x^(m - 1)*Sin[d*x^2]*FresnelC[b*x]/(2*d) - b/(4*d)*Int[x^(m - 1)*Sin[2*d*x^2], x] - (m - 1)/(2*d)*Int[x^(m - 2)*Sin[d*x^2]*FresnelC[b*x], x] /; FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && IGtQ[m, 1]
+Int[x_^m_*Sin[d_.*x_^2]*FresnelS[b_.*x_], x_Symbol] := x^(m + 1)*Sin[d*x^2]*FresnelS[b*x]/(m + 1) - d*x^(m + 2)/(Pi*b*(m + 1)*(m + 2)) + d/(Pi*b*(m + 1))*Int[x^(m + 1)*Cos[2*d*x^2], x] - 2*d/(m + 1)*Int[x^(m + 2)*Cos[d*x^2]*FresnelS[b*x], x] /; FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && ILtQ[m, -2]
+Int[x_^m_*Cos[d_.*x_^2]*FresnelC[b_.*x_], x_Symbol] := x^(m + 1)*Cos[d*x^2]*FresnelC[b*x]/(m + 1) - b*x^(m + 2)/(2*(m + 1)*(m + 2)) - b/(2*(m + 1))*Int[x^(m + 1)*Cos[2*d*x^2], x] + 2*d/(m + 1)*Int[x^(m + 2)*Sin[d*x^2]*FresnelC[b*x], x] /; FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && ILtQ[m, -2]
+Int[(e_.*x_)^m_.*Sin[c_. + d_.*x_^2]*FresnelS[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[(e*x)^m*Sin[c + d*x^2]*FresnelS[a + b*x]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[(e_.*x_)^m_.*Cos[c_. + d_.*x_^2]*FresnelC[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[(e*x)^m*Cos[c + d*x^2]*FresnelC[a + b*x]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[x_*Cos[d_.*x_^2]*FresnelS[b_.*x_], x_Symbol] := Sin[d*x^2]*FresnelS[b*x]/(2*d) - 1/(Pi*b)*Int[Sin[d*x^2]^2, x] /; FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4]
+Int[x_*Sin[d_.*x_^2]*FresnelC[b_.*x_], x_Symbol] := -Cos[d*x^2]*FresnelC[b*x]/(2*d) + b/(2*d)*Int[Cos[d*x^2]^2, x] /; FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4]
+Int[x_^m_*Cos[d_.*x_^2]*FresnelS[b_.*x_], x_Symbol] := x^(m - 1)*Sin[d*x^2]*FresnelS[b*x]/(2*d) - 1/(Pi*b)*Int[x^(m - 1)*Sin[d*x^2]^2, x] - (m - 1)/(2*d)*Int[x^(m - 2)*Sin[d*x^2]*FresnelS[b*x], x] /; FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && IGtQ[m, 1]
+Int[x_^m_*Sin[d_.*x_^2]*FresnelC[b_.*x_], x_Symbol] := -x^(m - 1)*Cos[d*x^2]*FresnelC[b*x]/(2*d) + b/(2*d)*Int[x^(m - 1)*Cos[d*x^2]^2, x] + (m - 1)/(2*d)*Int[x^(m - 2)*Cos[d*x^2]*FresnelC[b*x], x] /; FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && IGtQ[m, 1]
+Int[x_^m_*Cos[d_.*x_^2]*FresnelS[b_.*x_], x_Symbol] := x^(m + 1)*Cos[d*x^2]*FresnelS[b*x]/(m + 1) - d/(Pi*b*(m + 1))*Int[x^(m + 1)*Sin[2*d*x^2], x] + 2*d/(m + 1)*Int[x^(m + 2)*Sin[d*x^2]*FresnelS[b*x], x] /; FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && ILtQ[m, -1]
+Int[x_^m_*Sin[d_.*x_^2]*FresnelC[b_.*x_], x_Symbol] := x^(m + 1)*Sin[d*x^2]*FresnelC[b*x]/(m + 1) - b/(2*(m + 1))*Int[x^(m + 1)*Sin[2*d*x^2], x] - 2*d/(m + 1)*Int[x^(m + 2)*Cos[d*x^2]*FresnelC[b*x], x] /; FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && ILtQ[m, -1]
+Int[(e_.*x_)^m_.*Cos[c_. + d_.*x_^2]*FresnelS[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[(e*x)^m*Cos[c + d*x^2]*FresnelS[a + b*x]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[(e_.*x_)^m_.*Sin[c_. + d_.*x_^2]*FresnelC[a_. + b_.*x_]^n_., x_Symbol] := Unintegrable[(e*x)^m*Sin[c + d*x^2]*FresnelC[a + b*x]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
+Int[FresnelS[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := x*FresnelS[d*(a + b*Log[c*x^n])] - b*d*n*Int[Sin[Pi/2*(d*(a + b*Log[c*x^n]))^2], x] /; FreeQ[{a, b, c, d, n}, x]
+Int[FresnelC[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := x*FresnelC[d*(a + b*Log[c*x^n])] - b*d*n*Int[Cos[Pi/2*(d*(a + b*Log[c*x^n]))^2], x] /; FreeQ[{a, b, c, d, n}, x]
+Int[F_[d_.*(a_. + b_.*Log[c_.*x_^n_.])]/x_, x_Symbol] := 1/n*Subst[F[d*(a + b*x)], x, Log[c*x^n]] /; FreeQ[{a, b, c, d, n}, x] && MemberQ[{FresnelS, FresnelC}, F]
+Int[(e_.*x_)^m_.*FresnelS[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := (e*x)^(m + 1)*FresnelS[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - b*d*n/(m + 1)* Int[(e*x)^m*Sin[Pi/2*(d*(a + b*Log[c*x^n]))^2], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
+Int[(e_.*x_)^m_.*FresnelC[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := (e*x)^(m + 1)*FresnelC[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - b*d*n/(m + 1)* Int[(e*x)^m*Cos[Pi/2*(d*(a + b*Log[c*x^n]))^2], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
diff --git a/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m b/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m
new file mode 100755
index 0000000..1e594c9
--- /dev/null
+++ b/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m	
@@ -0,0 +1,30 @@
+
+(* ::Subsection::Closed:: *)
+(* 8.3 Exponential integral functions *)
+Int[ExpIntegralE[n_, a_. + b_.*x_], x_Symbol] := -ExpIntegralE[n + 1, a + b*x]/b /; FreeQ[{a, b, n}, x]
+Int[x_^m_.*ExpIntegralE[n_, b_.*x_], x_Symbol] := -x^m*ExpIntegralE[n + 1, b*x]/b + m/b*Int[x^(m - 1)*ExpIntegralE[n + 1, b*x], x] /; FreeQ[b, x] && EqQ[m + n, 0] && IGtQ[m, 0]
+Int[ExpIntegralE[1, b_.*x_]/x_, x_Symbol] := b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -b*x] - EulerGamma*Log[x] - 1/2*Log[b*x]^2 /; FreeQ[b, x]
+Int[x_^m_*ExpIntegralE[n_, b_.*x_], x_Symbol] := x^(m + 1)*ExpIntegralE[n, b*x]/(m + 1) + b/(m + 1)*Int[x^(m + 1)*ExpIntegralE[n - 1, b*x], x] /; FreeQ[b, x] && EqQ[m + n, 0] && ILtQ[m, -1]
+Int[(d_.*x_)^m_*ExpIntegralE[n_, b_.*x_], x_Symbol] := (d*x)^m*Gamma[m + 1]*Log[x]/(b*(b*x)^m) - (d*x)^(m + 1)* HypergeometricPFQ[{m + 1, m + 1}, {m + 2, m + 2}, -b* x]/(d*(m + 1)^2) /; FreeQ[{b, d, m, n}, x] && EqQ[m + n, 0] && Not[IntegerQ[m]]
+Int[(d_.*x_)^m_.*ExpIntegralE[n_, b_.*x_], x_Symbol] := (d*x)^(m + 1)*ExpIntegralE[n, b*x]/(d*(m + n)) - (d*x)^(m + 1)* ExpIntegralE[-m, b*x]/(d*(m + n)) /; FreeQ[{b, d, m, n}, x] && NeQ[m + n, 0]
+Int[(c_. + d_.*x_)^m_.*ExpIntegralE[n_, a_ + b_.*x_], x_Symbol] := -(c + d*x)^m*ExpIntegralE[n + 1, a + b*x]/b + d*m/b*Int[(c + d*x)^(m - 1)*ExpIntegralE[n + 1, a + b*x], x] /; FreeQ[{a, b, c, d, m, n}, x] && (IGtQ[m, 0] || ILtQ[n, 0] || GtQ[m, 0] && LtQ[n, -1])
+Int[(c_. + d_.*x_)^m_.*ExpIntegralE[n_, a_ + b_.*x_], x_Symbol] := (c + d*x)^(m + 1)*ExpIntegralE[n, a + b*x]/(d*(m + 1)) + b/(d*(m + 1))* Int[(c + d*x)^(m + 1)*ExpIntegralE[n - 1, a + b*x], x] /; FreeQ[{a, b, c, d, m, n}, x] && (IGtQ[n, 0] || LtQ[m, -1] && GtQ[n, 0]) && NeQ[m, -1]
+Int[(c_. + d_.*x_)^m_.*ExpIntegralE[n_, a_ + b_.*x_], x_Symbol] := Unintegrable[(c + d*x)^m*ExpIntegralE[n, a + b*x], x] /; FreeQ[{a, b, c, d, m, n}, x]
+Int[ExpIntegralEi[a_. + b_.*x_], x_Symbol] := (a + b*x)*ExpIntegralEi[a + b*x]/b - E^(a + b*x)/b /; FreeQ[{a, b}, x]
+Int[ExpIntegralEi[b_.*x_]/x_, x_Symbol] := Log[x]*(ExpIntegralEi[b*x] + ExpIntegralE[1, -b*x]) - Int[ExpIntegralE[1, -b*x]/x, x] /; FreeQ[b, x]
+Int[ExpIntegralEi[a_. + b_.*x_]/(c_. + d_.*x_), x_Symbol] := Unintegrable[ExpIntegralEi[a + b*x]/(c + d*x), x] /; FreeQ[{a, b, c, d}, x]
+Int[(c_. + d_.*x_)^m_.*ExpIntegralEi[a_. + b_.*x_], x_Symbol] := (c + d*x)^(m + 1)*ExpIntegralEi[a + b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(c + d*x)^(m + 1)*E^(a + b*x)/(a + b*x), x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
+Int[ExpIntegralEi[a_. + b_.*x_]^2, x_Symbol] := (a + b*x)*ExpIntegralEi[a + b*x]^2/b - 2*Int[E^(a + b*x)*ExpIntegralEi[a + b*x], x] /; FreeQ[{a, b}, x]
+Int[x_^m_.*ExpIntegralEi[b_.*x_]^2, x_Symbol] := x^(m + 1)*ExpIntegralEi[b*x]^2/(m + 1) - 2/(m + 1)*Int[x^m*E^(b*x)*ExpIntegralEi[b*x], x] /; FreeQ[b, x] && IGtQ[m, 0]
+Int[x_^m_.*ExpIntegralEi[a_ + b_.*x_]^2, x_Symbol] := x^(m + 1)*ExpIntegralEi[a + b*x]^2/(m + 1) + a*x^m*ExpIntegralEi[a + b*x]^2/(b*(m + 1)) - 2/(m + 1)*Int[x^m*E^(a + b*x)*ExpIntegralEi[a + b*x], x] - a*m/(b*(m + 1))*Int[x^(m - 1)*ExpIntegralEi[a + b*x]^2, x] /; FreeQ[{a, b}, x] && IGtQ[m, 0]
+(* Int[x_^m_.*ExpIntegralEi[a_+b_.*x_]^2,x_Symbol] := b*x^(m+2)*ExpIntegralEi[a+b*x]^2/(a*(m+1)) + x^(m+1)*ExpIntegralEi[a+b*x]^2/(m+1) - 2*b/(a*(m+1))*Int[x^(m+1)*E^(a+b*x)*ExpIntegralEi[a+b*x],x] - b*(m+2)/(a*(m+1))*Int[x^(m+1)*ExpIntegralEi[a+b*x]^2,x] /; FreeQ[{a,b},x] && ILtQ[m,-2] *)
+Int[E^(a_. + b_.*x_)*ExpIntegralEi[c_. + d_.*x_], x_Symbol] := E^(a + b*x)*ExpIntegralEi[c + d*x]/b - d/b*Int[E^(a + c + (b + d)*x)/(c + d*x), x] /; FreeQ[{a, b, c, d}, x]
+Int[x_^m_.*E^(a_. + b_.*x_)*ExpIntegralEi[c_. + d_.*x_], x_Symbol] := x^m*E^(a + b*x)*ExpIntegralEi[c + d*x]/b - d/b*Int[x^m*E^(a + c + (b + d)*x)/(c + d*x), x] - m/b*Int[x^(m - 1)*E^(a + b*x)*ExpIntegralEi[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
+Int[x_^m_*E^(a_. + b_.*x_)*ExpIntegralEi[c_. + d_.*x_], x_Symbol] := x^(m + 1)*E^(a + b*x)*ExpIntegralEi[c + d*x]/(m + 1) - d/(m + 1)*Int[x^(m + 1)*E^(a + c + (b + d)*x)/(c + d*x), x] - b/(m + 1)*Int[x^(m + 1)*E^(a + b*x)*ExpIntegralEi[c + d*x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -1]
+Int[ExpIntegralEi[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := x*ExpIntegralEi[d*(a + b*Log[c*x^n])] - b*n*E^(a*d)*Int[(c*x^n)^(b*d)/(a + b*Log[c*x^n]), x] /; FreeQ[{a, b, c, d, n}, x]
+Int[ExpIntegralEi[d_.*(a_. + b_.*Log[c_.*x_^n_.])]/x_, x_Symbol] := 1/n*Subst[ExpIntegralEi[d*(a + b*x)], x, Log[c*x^n]] /; FreeQ[{a, b, c, d, n}, x]
+Int[(e_.*x_)^m_.*ExpIntegralEi[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := (e*x)^(m + 1)*ExpIntegralEi[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - b*n*E^(a*d)*(c*x^n)^(b*d)/((m + 1)*(e*x)^(b*d*n))* Int[(e*x)^(m + b*d*n)/(a + b*Log[c*x^n]), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
+Int[LogIntegral[a_. + b_.*x_], x_Symbol] := (a + b*x)*LogIntegral[a + b*x]/b - ExpIntegralEi[2*Log[a + b*x]]/b /; FreeQ[{a, b}, x]
+Int[LogIntegral[b_.*x_]/x_, x_Symbol] := -b*x + Log[b*x]*LogIntegral[b*x] /; FreeQ[b, x]
+Int[LogIntegral[a_. + b_.*x_]/(c_. + d_.*x_), x_Symbol] := Unintegrable[LogIntegral[a + b*x]/(c + d*x), x] /; FreeQ[{a, b, c, d}, x]
+Int[(c_. + d_.*x_)^m_.*LogIntegral[a_. + b_.*x_], x_Symbol] := (c + d*x)^(m + 1)*LogIntegral[a + b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(c + d*x)^(m + 1)/Log[a + b*x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
diff --git a/IntegrationRules/8 Special functions/8.4 Trig integral functions.m b/IntegrationRules/8 Special functions/8.4 Trig integral functions.m
new file mode 100755
index 0000000..0531280
--- /dev/null
+++ b/IntegrationRules/8 Special functions/8.4 Trig integral functions.m	
@@ -0,0 +1,34 @@
+
+(* ::Subsection::Closed:: *)
+(* 8.4 Trig integral functions *)
+Int[SinIntegral[a_. + b_.*x_], x_Symbol] := (a + b*x)*SinIntegral[a + b*x]/b + Cos[a + b*x]/b /; FreeQ[{a, b}, x]
+Int[CosIntegral[a_. + b_.*x_], x_Symbol] := (a + b*x)*CosIntegral[a + b*x]/b - Sin[a + b*x]/b /; FreeQ[{a, b}, x]
+Int[SinIntegral[b_.*x_]/x_, x_Symbol] := 1/2*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -I*b*x] + 1/2*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, I*b*x] /; FreeQ[b, x]
+Int[CosIntegral[b_.*x_]/x_, x_Symbol] := -1/2*I*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -I*b*x] + 1/2*I*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, I*b*x] + EulerGamma*Log[x] + 1/2*Log[b*x]^2 /; FreeQ[b, x]
+Int[(c_. + d_.*x_)^m_.*SinIntegral[a_. + b_.*x_], x_Symbol] := (c + d*x)^(m + 1)*SinIntegral[a + b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(c + d*x)^(m + 1)*Sin[a + b*x]/(a + b*x), x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
+Int[(c_. + d_.*x_)^m_.*CosIntegral[a_. + b_.*x_], x_Symbol] := (c + d*x)^(m + 1)*CosIntegral[a + b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(c + d*x)^(m + 1)*Cos[a + b*x]/(a + b*x), x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
+Int[SinIntegral[a_. + b_.*x_]^2, x_Symbol] := (a + b*x)*SinIntegral[a + b*x]^2/b - 2*Int[Sin[a + b*x]*SinIntegral[a + b*x], x] /; FreeQ[{a, b}, x]
+Int[CosIntegral[a_. + b_.*x_]^2, x_Symbol] := (a + b*x)*CosIntegral[a + b*x]^2/b - 2*Int[Cos[a + b*x]*CosIntegral[a + b*x], x] /; FreeQ[{a, b}, x]
+Int[x_^m_.*SinIntegral[b_.*x_]^2, x_Symbol] := x^(m + 1)*SinIntegral[b*x]^2/(m + 1) - 2/(m + 1)*Int[x^m*Sin[b*x]*SinIntegral[b*x], x] /; FreeQ[b, x] && IGtQ[m, 0]
+Int[x_^m_.*CosIntegral[b_.*x_]^2, x_Symbol] := x^(m + 1)*CosIntegral[b*x]^2/(m + 1) - 2/(m + 1)*Int[x^m*Cos[b*x]*CosIntegral[b*x], x] /; FreeQ[b, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*SinIntegral[a_ + b_.*x_]^2, x_Symbol] := (a + b*x)*(c + d*x)^m*SinIntegral[a + b*x]^2/(b*(m + 1)) - 2/(m + 1)*Int[(c + d*x)^m*Sin[a + b*x]*SinIntegral[a + b*x], x] + (b*c - a*d)*m/(b*(m + 1))* Int[(c + d*x)^(m - 1)*SinIntegral[a + b*x]^2, x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*CosIntegral[a_ + b_.*x_]^2, x_Symbol] := (a + b*x)*(c + d*x)^m*CosIntegral[a + b*x]^2/(b*(m + 1)) - 2/(m + 1)*Int[(c + d*x)^m*Cos[a + b*x]*CosIntegral[a + b*x], x] + (b*c - a*d)*m/(b*(m + 1))* Int[(c + d*x)^(m - 1)*CosIntegral[a + b*x]^2, x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
+(* Int[x_^m_.*SinIntegral[a_+b_.*x_]^2,x_Symbol] := b*x^(m+2)*SinIntegral[a+b*x]^2/(a*(m+1)) + x^(m+1)*SinIntegral[a+b*x]^2/(m+1) - 2*b/(a*(m+1))*Int[x^(m+1)*Sin[a+b*x]*SinIntegral[a+b*x],x] - b*(m+2)/(a*(m+1))*Int[x^(m+1)*SinIntegral[a+b*x]^2,x] /; FreeQ[{a,b},x] && ILtQ[m,-2] *)
+(* Int[x_^m_.*CosIntegral[a_+b_.*x_]^2,x_Symbol] := b*x^(m+2)*CosIntegral[a+b*x]^2/(a*(m+1)) + x^(m+1)*CosIntegral[a+b*x]^2/(m+1) - 2*b/(a*(m+1))*Int[x^(m+1)*Cos[a+b*x]*CosIntegral[a+b*x],x] - b*(m+2)/(a*(m+1))*Int[x^(m+1)*CosIntegral[a+b*x]^2,x] /; FreeQ[{a,b},x] && ILtQ[m,-2] *)
+Int[Sin[a_. + b_.*x_]*SinIntegral[c_. + d_.*x_], x_Symbol] := -Cos[a + b*x]*SinIntegral[c + d*x]/b + d/b*Int[Cos[a + b*x]*Sin[c + d*x]/(c + d*x), x] /; FreeQ[{a, b, c, d}, x]
+Int[Cos[a_. + b_.*x_]*CosIntegral[c_. + d_.*x_], x_Symbol] := Sin[a + b*x]*CosIntegral[c + d*x]/b - d/b*Int[Sin[a + b*x]*Cos[c + d*x]/(c + d*x), x] /; FreeQ[{a, b, c, d}, x]
+Int[(e_. + f_.*x_)^m_.*Sin[a_. + b_.*x_]*SinIntegral[c_. + d_.*x_], x_Symbol] := -(e + f*x)^m*Cos[a + b*x]*SinIntegral[c + d*x]/b + d/b*Int[(e + f*x)^m*Cos[a + b*x]*Sin[c + d*x]/(c + d*x), x] + f*m/b* Int[(e + f*x)^(m - 1)*Cos[a + b*x]*SinIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.*Cos[a_. + b_.*x_]*CosIntegral[c_. + d_.*x_], x_Symbol] := (e + f*x)^m*Sin[a + b*x]*CosIntegral[c + d*x]/b - d/b*Int[(e + f*x)^m*Sin[a + b*x]*Cos[c + d*x]/(c + d*x), x] - f*m/b* Int[(e + f*x)^(m - 1)*Sin[a + b*x]*CosIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_*Sin[a_. + b_.*x_]*SinIntegral[c_. + d_.*x_], x_Symbol] := (e + f*x)^(m + 1)*Sin[a + b*x]*SinIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sin[a + b*x]*Sin[c + d*x]/(c + d*x), x] - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cos[a + b*x]*SinIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
+Int[(e_. + f_.*x_)^m_.*Cos[a_. + b_.*x_]*CosIntegral[c_. + d_.*x_], x_Symbol] := (e + f*x)^(m + 1)*Cos[a + b*x]*CosIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cos[a + b*x]*Cos[c + d*x]/(c + d*x), x] + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sin[a + b*x]*CosIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
+Int[Cos[a_. + b_.*x_]*SinIntegral[c_. + d_.*x_], x_Symbol] := Sin[a + b*x]*SinIntegral[c + d*x]/b - d/b*Int[Sin[a + b*x]*Sin[c + d*x]/(c + d*x), x] /; FreeQ[{a, b, c, d}, x]
+Int[Sin[a_. + b_.*x_]*CosIntegral[c_. + d_.*x_], x_Symbol] := -Cos[a + b*x]*CosIntegral[c + d*x]/b + d/b*Int[Cos[a + b*x]*Cos[c + d*x]/(c + d*x), x] /; FreeQ[{a, b, c, d}, x]
+Int[(e_. + f_.*x_)^m_.*Cos[a_. + b_.*x_]*SinIntegral[c_. + d_.*x_], x_Symbol] := (e + f*x)^m*Sin[a + b*x]*SinIntegral[c + d*x]/b - d/b*Int[(e + f*x)^m*Sin[a + b*x]*Sin[c + d*x]/(c + d*x), x] - f*m/b* Int[(e + f*x)^(m - 1)*Sin[a + b*x]*SinIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.*Sin[a_. + b_.*x_]*CosIntegral[c_. + d_.*x_], x_Symbol] := -(e + f*x)^m*Cos[a + b*x]*CosIntegral[c + d*x]/b + d/b*Int[(e + f*x)^m*Cos[a + b*x]*Cos[c + d*x]/(c + d*x), x] + f*m/b* Int[(e + f*x)^(m - 1)*Cos[a + b*x]*CosIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.*Cos[a_. + b_.*x_]*SinIntegral[c_. + d_.*x_], x_Symbol] := (e + f*x)^(m + 1)*Cos[a + b*x]*SinIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cos[a + b*x]*Sin[c + d*x]/(c + d*x), x] + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sin[a + b*x]*SinIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
+Int[(e_. + f_.*x_)^m_*Sin[a_. + b_.*x_]*CosIntegral[c_. + d_.*x_], x_Symbol] := (e + f*x)^(m + 1)*Sin[a + b*x]*CosIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sin[a + b*x]*Cos[c + d*x]/(c + d*x), x] - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cos[a + b*x]*CosIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
+Int[SinIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := x*SinIntegral[d*(a + b*Log[c*x^n])] - b*d*n*Int[Sin[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x] /; FreeQ[{a, b, c, d, n}, x]
+Int[CosIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := x*CosIntegral[d*(a + b*Log[c*x^n])] - b*d*n*Int[Cos[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x] /; FreeQ[{a, b, c, d, n}, x]
+Int[F_[d_.*(a_. + b_.*Log[c_.*x_^n_.])]/x_, x_Symbol] := 1/n*Subst[F[d*(a + b*x)], x, Log[c*x^n]] /; FreeQ[{a, b, c, d, n}, x] && MemberQ[{SinIntegral, CosIntegral}, x]
+Int[(e_.*x_)^m_.*SinIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := (e*x)^(m + 1)*SinIntegral[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - b*d*n/(m + 1)* Int[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
+Int[(e_.*x_)^m_.*CosIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := (e*x)^(m + 1)*CosIntegral[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - b*d*n/(m + 1)* Int[(e*x)^m*Cos[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
diff --git a/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m b/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m
new file mode 100755
index 0000000..32b2fa8
--- /dev/null
+++ b/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m	
@@ -0,0 +1,34 @@
+
+(* ::Subsection::Closed:: *)
+(* 8.5 Hyperbolic integral functions *)
+Int[SinhIntegral[a_. + b_.*x_], x_Symbol] := (a + b*x)*SinhIntegral[a + b*x]/b - Cosh[a + b*x]/b /; FreeQ[{a, b}, x]
+Int[CoshIntegral[a_. + b_.*x_], x_Symbol] := (a + b*x)*CoshIntegral[a + b*x]/b - Sinh[a + b*x]/b /; FreeQ[{a, b}, x]
+Int[SinhIntegral[b_.*x_]/x_, x_Symbol] := 1/2*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -b*x] + 1/2*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, b*x] /; FreeQ[b, x]
+Int[CoshIntegral[b_.*x_]/x_, x_Symbol] := -1/2*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -b*x] + 1/2*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, b*x] + EulerGamma*Log[x] + 1/2*Log[b*x]^2 /; FreeQ[b, x]
+Int[(c_. + d_.*x_)^m_.*SinhIntegral[a_. + b_.*x_], x_Symbol] := (c + d*x)^(m + 1)*SinhIntegral[a + b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(c + d*x)^(m + 1)*Sinh[a + b*x]/(a + b*x), x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
+Int[(c_. + d_.*x_)^m_.*CoshIntegral[a_. + b_.*x_], x_Symbol] := (c + d*x)^(m + 1)*CoshIntegral[a + b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(c + d*x)^(m + 1)*Cosh[a + b*x]/(a + b*x), x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
+Int[SinhIntegral[a_. + b_.*x_]^2, x_Symbol] := (a + b*x)*SinhIntegral[a + b*x]^2/b - 2*Int[Sinh[a + b*x]*SinhIntegral[a + b*x], x] /; FreeQ[{a, b}, x]
+Int[CoshIntegral[a_. + b_.*x_]^2, x_Symbol] := (a + b*x)*CoshIntegral[a + b*x]^2/b - 2*Int[Cosh[a + b*x]*CoshIntegral[a + b*x], x] /; FreeQ[{a, b}, x]
+Int[x_^m_.*SinhIntegral[b_.*x_]^2, x_Symbol] := x^(m + 1)*SinhIntegral[b*x]^2/(m + 1) - 2/(m + 1)*Int[x^m*Sinh[b*x]*SinhIntegral[b*x], x] /; FreeQ[b, x] && IGtQ[m, 0]
+Int[x_^m_.*CoshIntegral[b_.*x_]^2, x_Symbol] := x^(m + 1)*CoshIntegral[b*x]^2/(m + 1) - 2/(m + 1)*Int[x^m*Cosh[b*x]*CoshIntegral[b*x], x] /; FreeQ[b, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*SinhIntegral[a_ + b_.*x_]^2, x_Symbol] := (a + b*x)*(c + d*x)^m*SinhIntegral[a + b*x]^2/(b*(m + 1)) - 2/(m + 1)* Int[(c + d*x)^m*Sinh[a + b*x]*SinhIntegral[a + b*x], x] + (b*c - a*d)*m/(b*(m + 1))* Int[(c + d*x)^(m - 1)*SinhIntegral[a + b*x]^2, x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*CoshIntegral[a_ + b_.*x_]^2, x_Symbol] := (a + b*x)*(c + d*x)^m*CoshIntegral[a + b*x]^2/(b*(m + 1)) - 2/(m + 1)* Int[(c + d*x)^m*Cosh[a + b*x]*CoshIntegral[a + b*x], x] + (b*c - a*d)*m/(b*(m + 1))* Int[(c + d*x)^(m - 1)*CoshIntegral[a + b*x]^2, x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
+(* Int[x_^m_.*SinhIntegral[a_+b_.*x_]^2,x_Symbol] := b*x^(m+2)*SinhIntegral[a+b*x]^2/(a*(m+1)) + x^(m+1)*SinhIntegral[a+b*x]^2/(m+1) - 2*b/(a*(m+1))*Int[x^(m+1)*Sinh[a+b*x]*SinhIntegral[a+b*x],x] - b*(m+2)/(a*(m+1))*Int[x^(m+1)*SinhIntegral[a+b*x]^2,x] /; FreeQ[{a,b},x] && ILtQ[m,-2] *)
+(* Int[x_^m_.*CoshIntegral[a_+b_.*x_]^2,x_Symbol] := b*x^(m+2)*CoshIntegral[a+b*x]^2/(a*(m+1)) + x^(m+1)*CoshIntegral[a+b*x]^2/(m+1) - 2*b/(a*(m+1))*Int[x^(m+1)*Cosh[a+b*x]*CoshIntegral[a+b*x],x] - b*(m+2)/(a*(m+1))*Int[x^(m+1)*CoshIntegral[a+b*x]^2,x] /; FreeQ[{a,b},x] && ILtQ[m,-2] *)
+Int[Sinh[a_. + b_.*x_]*SinhIntegral[c_. + d_.*x_], x_Symbol] := Cosh[a + b*x]*SinhIntegral[c + d*x]/b - d/b*Int[Cosh[a + b*x]*Sinh[c + d*x]/(c + d*x), x] /; FreeQ[{a, b, c, d}, x]
+Int[Cosh[a_. + b_.*x_]*CoshIntegral[c_. + d_.*x_], x_Symbol] := Sinh[a + b*x]*CoshIntegral[c + d*x]/b - d/b*Int[Sinh[a + b*x]*Cosh[c + d*x]/(c + d*x), x] /; FreeQ[{a, b, c, d}, x]
+Int[(e_. + f_.*x_)^m_.*Sinh[a_. + b_.*x_]*SinhIntegral[c_. + d_.*x_], x_Symbol] := (e + f*x)^m*Cosh[a + b*x]*SinhIntegral[c + d*x]/b - d/b*Int[(e + f*x)^m*Cosh[a + b*x]*Sinh[c + d*x]/(c + d*x), x] - f*m/b* Int[(e + f*x)^(m - 1)*Cosh[a + b*x]*SinhIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.*Cosh[a_. + b_.*x_]*CoshIntegral[c_. + d_.*x_], x_Symbol] := (e + f*x)^m*Sinh[a + b*x]*CoshIntegral[c + d*x]/b - d/b*Int[(e + f*x)^m*Sinh[a + b*x]*Cosh[c + d*x]/(c + d*x), x] - f*m/b* Int[(e + f*x)^(m - 1)*Sinh[a + b*x]*CoshIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_*Sinh[a_. + b_.*x_]*SinhIntegral[c_. + d_.*x_], x_Symbol] := (e + f*x)^(m + 1)*Sinh[a + b*x]*SinhIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sinh[a + b*x]*Sinh[c + d*x]/(c + d*x), x] - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cosh[a + b*x]*SinhIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
+Int[(e_. + f_.*x_)^m_.*Cosh[a_. + b_.*x_]*CoshIntegral[c_. + d_.*x_], x_Symbol] := (e + f*x)^(m + 1)*Cosh[a + b*x]*CoshIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cosh[a + b*x]*Cosh[c + d*x]/(c + d*x), x] - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sinh[a + b*x]*CoshIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
+Int[Cosh[a_. + b_.*x_]*SinhIntegral[c_. + d_.*x_], x_Symbol] := Sinh[a + b*x]*SinhIntegral[c + d*x]/b - d/b*Int[Sinh[a + b*x]*Sinh[c + d*x]/(c + d*x), x] /; FreeQ[{a, b, c, d}, x]
+Int[Sinh[a_. + b_.*x_]*CoshIntegral[c_. + d_.*x_], x_Symbol] := Cosh[a + b*x]*CoshIntegral[c + d*x]/b - d/b*Int[Cosh[a + b*x]*Cosh[c + d*x]/(c + d*x), x] /; FreeQ[{a, b, c, d}, x]
+Int[(e_. + f_.*x_)^m_.*Cosh[a_. + b_.*x_]*SinhIntegral[c_. + d_.*x_], x_Symbol] := (e + f*x)^m*Sinh[a + b*x]*SinhIntegral[c + d*x]/b - d/b*Int[(e + f*x)^m*Sinh[a + b*x]*Sinh[c + d*x]/(c + d*x), x] - f*m/b* Int[(e + f*x)^(m - 1)*Sinh[a + b*x]*SinhIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.*Sinh[a_. + b_.*x_]*CoshIntegral[c_. + d_.*x_], x_Symbol] := (e + f*x)^m*Cosh[a + b*x]*CoshIntegral[c + d*x]/b - d/b*Int[(e + f*x)^m*Cosh[a + b*x]*Cosh[c + d*x]/(c + d*x), x] - f*m/b* Int[(e + f*x)^(m - 1)*Cosh[a + b*x]*CoshIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^m_.*Cosh[a_. + b_.*x_]*SinhIntegral[c_. + d_.*x_], x_Symbol] := (e + f*x)^(m + 1)*Cosh[a + b*x]* SinhIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cosh[a + b*x]*Sinh[c + d*x]/(c + d*x), x] - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sinh[a + b*x]*SinhIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
+Int[(e_. + f_.*x_)^m_*Sinh[a_. + b_.*x_]*CoshIntegral[c_. + d_.*x_], x_Symbol] := (e + f*x)^(m + 1)*Sinh[a + b*x]* CoshIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sinh[a + b*x]*Cosh[c + d*x]/(c + d*x), x] - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cosh[a + b*x]*CoshIntegral[c + d*x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
+Int[SinhIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := x*SinhIntegral[d*(a + b*Log[c*x^n])] - b*d*n*Int[Sinh[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x] /; FreeQ[{a, b, c, d, n}, x]
+Int[CoshIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := x*CoshIntegral[d*(a + b*Log[c*x^n])] - b*d*n*Int[Cosh[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x] /; FreeQ[{a, b, c, d, n}, x]
+Int[F_[d_.*(a_. + b_.*Log[c_.*x_^n_.])]/x_, x_Symbol] := 1/n*Subst[F[d*(a + b*x)], x, Log[c*x^n]] /; FreeQ[{a, b, c, d, n}, x] && MemberQ[{SinhIntegral, CoshIntegral}, x]
+Int[(e_.*x_)^m_.*SinhIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := (e*x)^(m + 1)*SinhIntegral[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - b*d*n/(m + 1)* Int[(e*x)^m*Sinh[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
+Int[(e_.*x_)^m_.*CoshIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := (e*x)^(m + 1)*CoshIntegral[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - b*d*n/(m + 1)* Int[(e*x)^m*Cosh[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
diff --git a/IntegrationRules/8 Special functions/8.6 Gamma functions.m b/IntegrationRules/8 Special functions/8.6 Gamma functions.m
new file mode 100755
index 0000000..390e846
--- /dev/null
+++ b/IntegrationRules/8 Special functions/8.6 Gamma functions.m	
@@ -0,0 +1,28 @@
+
+(* ::Subsection::Closed:: *)
+(* 8.6 Gamma functions *)
+Int[Gamma[n_, a_. + b_.*x_], x_Symbol] := (a + b*x)*Gamma[n, a + b*x]/b - Gamma[n + 1, a + b*x]/b /; FreeQ[{a, b, n}, x]
+Int[Gamma[0, b_.*x_]/x_, x_Symbol] := b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -b*x] - EulerGamma*Log[x] - 1/2*Log[b*x]^2 /; FreeQ[b, x]
+(* Int[Gamma[1,b_.*x_]/x_,x_Symbol] := Int[1/(x*E^(b*x)),x] /; FreeQ[b,x] *)
+Int[Gamma[n_, b_.*x_]/x_, x_Symbol] := -Gamma[n - 1, b*x] + (n - 1)*Int[Gamma[n - 1, b*x]/x, x] /; FreeQ[b, x] && IGtQ[n, 1]
+Int[Gamma[n_, b_.*x_]/x_, x_Symbol] := Gamma[n, b*x]/n + 1/n*Int[Gamma[n + 1, b*x]/x, x] /; FreeQ[b, x] && ILtQ[n, 0]
+Int[Gamma[n_, b_.*x_]/x_, x_Symbol] := Gamma[n]*Log[x] - (b*x)^n/n^2* HypergeometricPFQ[{n, n}, {1 + n, 1 + n}, -b*x] /; FreeQ[{b, n}, x] && Not[IntegerQ[n]]
+Int[(d_.*x_)^m_.*Gamma[n_, b_.*x_], x_Symbol] := (d*x)^(m + 1)*Gamma[n, b*x]/(d*(m + 1)) - (d*x)^m*Gamma[m + n + 1, b*x]/(b*(m + 1)*(b*x)^m) /; FreeQ[{b, d, m, n}, x] && NeQ[m, -1]
+Int[(c_ + d_.*x_)^m_.*Gamma[n_, a_ + b_.*x_], x_Symbol] := 1/b*Subst[Int[(d*x/b)^m*Gamma[n, x], x], x, a + b*x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0]
+Int[Gamma[n_, a_. + b_.*x_]/(c_. + d_.*x_), x_Symbol] := Int[(a + b*x)^(n - 1)/((c + d*x)*E^(a + b*x)), x] + (n - 1)* Int[Gamma[n - 1, a + b*x]/(c + d*x), x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 1]
+Int[(c_. + d_.*x_)^m_.*Gamma[n_, a_. + b_.*x_], x_Symbol] := Block[{$UseGamma = True}, (c + d*x)^(m + 1)*Gamma[n, a + b*x]/(d*(m + 1)) + b/(d*(m + 1))* Int[(c + d*x)^(m + 1)*(a + b*x)^(n - 1)/E^(a + b*x), x]] /; FreeQ[{a, b, c, d, m, n}, x] && (IGtQ[m, 0] || IGtQ[n, 0] || IntegersQ[m, n]) && NeQ[m, -1]
+Int[(c_. + d_.*x_)^m_.*Gamma[n_, a_. + b_.*x_], x_Symbol] := Unintegrable[(c + d*x)^m*Gamma[n, a + b*x], x] /; FreeQ[{a, b, c, d, m, n}, x]
+Int[LogGamma[a_. + b_.*x_], x_Symbol] := PolyGamma[-2, a + b*x]/b /; FreeQ[{a, b}, x]
+Int[(c_. + d_.*x_)^m_.*LogGamma[a_. + b_.*x_], x_Symbol] := (c + d*x)^m*PolyGamma[-2, a + b*x]/b - d*m/b*Int[(c + d*x)^(m - 1)*PolyGamma[-2, a + b*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*LogGamma[a_. + b_.*x_], x_Symbol] := Unintegrable[(c + d*x)^m*LogGamma[a + b*x], x] /; FreeQ[{a, b, c, d, m}, x]
+Int[PolyGamma[n_, a_. + b_.*x_], x_Symbol] := PolyGamma[n - 1, a + b*x]/b /; FreeQ[{a, b, n}, x]
+Int[(c_. + d_.*x_)^m_.*PolyGamma[n_, a_. + b_.*x_], x_Symbol] := (c + d*x)^m*PolyGamma[n - 1, a + b*x]/b - d*m/b*Int[(c + d*x)^(m - 1)*PolyGamma[n - 1, a + b*x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*PolyGamma[n_, a_. + b_.*x_], x_Symbol] := (c + d*x)^(m + 1)*PolyGamma[n, a + b*x]/(d*(m + 1)) - b/(d*(m + 1))* Int[(c + d*x)^(m + 1)*PolyGamma[n + 1, a + b*x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1]
+Int[(c_. + d_.*x_)^m_.*PolyGamma[n_, a_. + b_.*x_], x_Symbol] := Unintegrable[(c + d*x)^m*PolyGamma[n, a + b*x], x] /; FreeQ[{a, b, c, d, m, n}, x]
+Int[Gamma[a_. + b_.*x_]^n_.*PolyGamma[0, a_. + b_.*x_], x_Symbol] := Gamma[a + b*x]^n/(b*n) /; FreeQ[{a, b, n}, x]
+Int[((a_. + b_.*x_)!)^n_.*PolyGamma[0, c_. + b_.*x_], x_Symbol] := ((a + b*x)!)^n/(b*n) /; FreeQ[{a, b, c, n}, x] && EqQ[c, a + 1]
+Int[Gamma[p_, d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := x*Gamma[p, d*(a + b*Log[c*x^n])] + b*d*n*E^(-a*d)* Int[(d*(a + b*Log[c*x^n]))^(p - 1)/(c*x^n)^(b*d), x] /; FreeQ[{a, b, c, d, n, p}, x]
+Int[Gamma[p_, d_.*(a_. + b_.*Log[c_.*x_^n_.])]/x_, x_Symbol] := 1/n*Subst[Gamma[p, d*(a + b*x)], x, Log[c*x^n]] /; FreeQ[{a, b, c, d, n, p}, x]
+Int[(e_.*x_)^m_.*Gamma[p_, d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol] := (e*x)^(m + 1)*Gamma[p, d*(a + b*Log[c*x^n])]/(e*(m + 1)) + b*d*n*E^(-a*d)*(e*x)^(b*d*n)/((m + 1)*(c*x^n)^(b*d))* Int[(e*x)^(m - b*d*n)*(d*(a + b*Log[c*x^n]))^(p - 1), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[m, -1]
+Int[Gamma[p_, f_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])], x_Symbol] := 1/e*Subst[Int[Gamma[p, f*(a + b*Log[c*x^n])], x], x, d + e*x] /; FreeQ[{a, b, c, d, e, f, n, p}, x]
+Int[(g_ + h_. x_)^m_.* Gamma[p_, f_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])], x_Symbol] := 1/e*Subst[Int[(g*x/d)^m*Gamma[p, f*(a + b*Log[c*x^n])], x], x, d + e*x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[e*g - d*h, 0]
diff --git a/IntegrationRules/8 Special functions/8.7 Zeta function.m b/IntegrationRules/8 Special functions/8.7 Zeta function.m
new file mode 100755
index 0000000..2062c39
--- /dev/null
+++ b/IntegrationRules/8 Special functions/8.7 Zeta function.m	
@@ -0,0 +1,8 @@
+
+(* ::Subsection::Closed:: *)
+(* 8.7 Zeta function *)
+Int[Zeta[2, a_. + b_.*x_], x_Symbol] := Int[PolyGamma[1, a + b*x], x] /; FreeQ[{a, b}, x]
+Int[Zeta[s_, a_. + b_.*x_], x_Symbol] := -Zeta[s - 1, a + b*x]/(b*(s - 1)) /; FreeQ[{a, b, s}, x] && NeQ[s, 1] && NeQ[s, 2]
+Int[(c_. + d_.*x_)^m_.*Zeta[2, a_. + b_.*x_], x_Symbol] := Int[(c + d*x)^m*PolyGamma[1, a + b*x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m]
+Int[(c_. + d_.*x_)^m_.*Zeta[s_, a_. + b_.*x_], x_Symbol] := -(c + d*x)^m*Zeta[s - 1, a + b*x]/(b*(s - 1)) + d*m/(b*(s - 1))*Int[(c + d*x)^(m - 1)*Zeta[s - 1, a + b*x], x] /; FreeQ[{a, b, c, d, s}, x] && NeQ[s, 1] && NeQ[s, 2] && GtQ[m, 0]
+Int[(c_. + d_.*x_)^m_.*Zeta[s_, a_. + b_.*x_], x_Symbol] := (c + d*x)^(m + 1)*Zeta[s, a + b*x]/(d*(m + 1)) + b*s/(d*(m + 1))*Int[(c + d*x)^(m + 1)*Zeta[s + 1, a + b*x], x] /; FreeQ[{a, b, c, d, s}, x] && NeQ[s, 1] && NeQ[s, 2] && LtQ[m, -1]
diff --git a/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m b/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m
new file mode 100755
index 0000000..0e495a9
--- /dev/null
+++ b/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m	
@@ -0,0 +1,31 @@
+
+(* ::Subsection::Closed:: *)
+(* 8.8 Polylogarithm function *)
+(* Int[PolyLog[2,a_.*(b_.*x_^p_.)^q_.],x_Symbol] := x*PolyLog[2,a*(b*x^p)^q] + p*q*Int[Log[1-a*(b*x^p)^q],x] /; FreeQ[{a,b,p,q},x] *)
+Int[PolyLog[n_, a_.*(b_.*x_^p_.)^q_.], x_Symbol] := x*PolyLog[n, a*(b*x^p)^q] - p*q*Int[PolyLog[n - 1, a*(b*x^p)^q], x] /; FreeQ[{a, b, p, q}, x] && GtQ[n, 0]
+Int[PolyLog[n_, a_.*(b_.*x_^p_.)^q_.], x_Symbol] := x*PolyLog[n + 1, a*(b*x^p)^q]/(p*q) - 1/(p*q)*Int[PolyLog[n + 1, a*(b*x^p)^q], x] /; FreeQ[{a, b, p, q}, x] && LtQ[n, -1]
+Int[PolyLog[n_, a_.*(b_.*x_^p_.)^q_.], x_Symbol] := Unintegrable[PolyLog[n, a*(b*x^p)^q], x] /; FreeQ[{a, b, n, p, q}, x]
+Int[PolyLog[n_, c_.*(a_. + b_.*x_)^p_.]/(d_. + e_.*x_), x_Symbol] := PolyLog[n + 1, c*(a + b*x)^p]/(e*p) /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]
+Int[PolyLog[n_, a_.*(b_.*x_^p_.)^q_.]/x_, x_Symbol] := PolyLog[n + 1, a*(b*x^p)^q]/(p*q) /; FreeQ[{a, b, n, p, q}, x]
+Int[(d_.*x_)^m_.*PolyLog[n_, a_.*(b_.*x_^p_.)^q_.], x_Symbol] := (d*x)^(m + 1)*PolyLog[n, a*(b*x^p)^q]/(d*(m + 1)) - p*q/(m + 1)*Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x] /; FreeQ[{a, b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]
+Int[(d_.*x_)^m_.*PolyLog[n_, a_.*(b_.*x_^p_.)^q_.], x_Symbol] := (d*x)^(m + 1)*PolyLog[n + 1, a*(b*x^p)^q]/(d*p*q) - (m + 1)/(p*q)*Int[(d*x)^m*PolyLog[n + 1, a*(b*x^p)^q], x] /; FreeQ[{a, b, d, m, p, q}, x] && NeQ[m, -1] && LtQ[n, -1]
+Int[(d_.*x_)^m_.*PolyLog[n_, a_.*(b_.*x_^p_.)^q_.], x_Symbol] := Unintegrable[(d*x)^m*PolyLog[n, a*(b*x^p)^q], x] /; FreeQ[{a, b, d, m, n, p, q}, x]
+Int[Log[c_.*x_^m_.]^r_.*PolyLog[n_, a_.*(b_.*x_^p_.)^q_.]/x_, x_Symbol] := Log[c*x^m]^r*PolyLog[n + 1, a*(b*x^p)^q]/(p*q) - m*r/(p*q)* Int[Log[c*x^m]^(r - 1)*PolyLog[n + 1, a*(b*x^p)^q]/x, x] /; FreeQ[{a, b, c, m, n, q, r}, x] && GtQ[r, 0]
+Int[PolyLog[n_, c_.*(a_. + b_.*x_)^p_.], x_Symbol] := x*PolyLog[n, c*(a + b*x)^p] - p*Int[PolyLog[n - 1, c*(a + b*x)^p], x] + a*p*Int[PolyLog[n - 1, c*(a + b*x)^p]/(a + b*x), x] /; FreeQ[{a, b, c, p}, x] && GtQ[n, 0]
+Int[PolyLog[2, c_.*(a_. + b_.*x_)]/(d_. + e_.*x_), x_Symbol] := Log[1 - a*c - b*c*x]*PolyLog[2, c*(a + b*x)]/e + b/e*Int[Log[1 - a*c - b*c*x]^2/(a + b*x), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*(b*d - a*e) + e, 0]
+Int[PolyLog[2, c_.*(a_. + b_.*x_)]/(d_. + e_.*x_), x_Symbol] := Log[d + e*x]*PolyLog[2, c*(a + b*x)]/e + b/e*Int[Log[d + e*x]*Log[1 - a*c - b*c*x]/(a + b*x), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c*(b*d - a*e) + e, 0]
+Int[(d_. + e_.*x_)^m_.*PolyLog[2, c_.*(a_. + b_.*x_)], x_Symbol] := (d + e*x)^(m + 1)*PolyLog[2, c*(a + b*x)]/(e*(m + 1)) + b/(e*(m + 1))* Int[(d + e*x)^(m + 1)*Log[1 - a*c - b*c*x]/(a + b*x), x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
+(* Int[(d_.+e_.*x_)^m_.*PolyLog[n_,c_.*(a_.+b_.*x_)^p_.],x_Symbol] := (d+e*x)^(m+1)*PolyLog[n,c*(a+b*x)^p]/(e*(m+1)) - b*p/(e*(m+1))*Int[(d+e*x)^(m+1)*PolyLog[n-1,c*(a+b*x)^p]/(a+b*x),x]  /; FreeQ[{a,b,c,d,e,m,p},x] && GtQ[n,0] && IGtQ[m,0] *)
+Int[x_^m_.*PolyLog[n_, c_.*(a_. + b_.*x_)^p_.], x_Symbol] := -(a^(m + 1) - b^(m + 1)*x^(m + 1))* PolyLog[n, c*(a + b*x)^p]/((m + 1)*b^(m + 1)) + p/((m + 1)*b^m)* Int[ExpandIntegrand[ PolyLog[n - 1, c*(a + b*x)^p], (a^(m + 1) - b^(m + 1)*x^(m + 1))/(a + b*x), x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[n, 0] && IntegerQ[m] && NeQ[m, -1]
+Int[(g_. + h_.*Log[f_.*(d_. + e_.*x_)^n_.])* PolyLog[2, c_.*(a_. + b_.*x_)], x_Symbol] := x*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)] + b*Int[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x]* ExpandIntegrand[x/(a + b*x), x], x] - e*h*n* Int[PolyLog[2, c*(a + b*x)]*ExpandIntegrand[x/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n}, x]
+Int[Log[1 + e_.*x_]*PolyLog[2, c_.*x_]/x_, x_Symbol] := -PolyLog[2, c*x]^2/2 /; FreeQ[{c, e}, x] && EqQ[c + e, 0]
+Int[(g_ + h_.*Log[1 + e_.*x_])*PolyLog[2, c_.*x_]/x_, x_Symbol] := g*Int[PolyLog[2, c*x]/x, x] + h*Int[(Log[1 + e*x]*PolyLog[2, c*x])/x, x] /; FreeQ[{c, e, g, h}, x] && EqQ[c + e, 0]
+Int[x_^m_.*(g_. + h_.*Log[f_.*(d_. + e_.*x_)^n_.])* PolyLog[2, c_.*(a_. + b_.*x_)], x_Symbol] := x^(m + 1)*(g + h*Log[f*(d + e*x)^n])* PolyLog[2, c*(a + b*x)]/(m + 1) + b/(m + 1)* Int[ExpandIntegrand[(g + h*Log[f*(d + e*x)^n])* Log[1 - a*c - b*c*x], x^(m + 1)/(a + b*x), x], x] - e*h*n/(m + 1)* Int[ExpandIntegrand[PolyLog[2, c*(a + b*x)], x^(m + 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n}, x] && IntegerQ[m] && NeQ[m, -1]
+Int[Px_*(g_. + h_.*Log[f_.*(d_. + e_.*x_)^n_.])* PolyLog[2, c_.*(a_. + b_.*x_)], x_Symbol] := With[{u = IntHide[Px, x]}, u*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)] + b*Int[ ExpandIntegrand[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x], u/(a + b*x), x], x] - e*h*n* Int[ExpandIntegrand[PolyLog[2, c*(a + b*x)], u/(d + e*x), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h, n}, x] && PolyQ[Px, x]
+Int[x_^m_*Px_*(g_. + h_.*Log[1 + e_.*x_])*PolyLog[2, c_.*x_], x_Symbol] := Coeff[Px, x, -m - 1]* Int[(g + h*Log[1 + e*x])*PolyLog[2, c*x]/x, x] + Int[ x^m*(Px - Coeff[Px, x, -m - 1]*x^(-m - 1))*(g + h*Log[1 + e*x])* PolyLog[2, c*x], x] /; FreeQ[{c, e, g, h}, x] && PolyQ[Px, x] && ILtQ[m, 0] && EqQ[c + e, 0] && NeQ[Coeff[Px, x, -m - 1], 0]
+Int[x_^m_.*Px_*(g_. + h_.*Log[f_.*(d_. + e_.*x_)^n_.])* PolyLog[2, c_.*(a_. + b_.*x_)], x_Symbol] := With[{u = IntHide[x^m*Px, x]}, u*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)] + b*Int[ ExpandIntegrand[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x], u/(a + b*x), x], x] - e*h*n* Int[ExpandIntegrand[PolyLog[2, c*(a + b*x)], u/(d + e*x), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h, n}, x] && PolyQ[Px, x] && IntegerQ[m]
+Int[x_^m_*Px_.*(g_. + h_.*Log[f_.*(d_. + e_.*x_)^n_.])* PolyLog[2, c_.*(a_. + b_.*x_)], x_Symbol] := Unintegrable[ x^m*Px*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && PolyQ[Px, x]
+Int[PolyLog[n_, d_.*(F_^(c_.*(a_. + b_.*x_)))^p_.], x_Symbol] := PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F]) /; FreeQ[{F, a, b, c, d, n, p}, x]
+Int[(e_. + f_.*x_)^m_.*PolyLog[n_, d_.*(F_^(c_.*(a_. + b_.*x_)))^p_.], x_Symbol] := (e + f*x)^m* PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F]) - f*m/(b*c*p*Log[F])* Int[(e + f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m, 0]
+Int[u_*PolyLog[n_, v_], x_Symbol] := With[{w = DerivativeDivides[v, u*v, x]}, w*PolyLog[n + 1, v] /; Not[FalseQ[w]]] /; FreeQ[n, x]
+Int[u_*Log[w_]*PolyLog[n_, v_], x_Symbol] := With[{z = DerivativeDivides[v, u*v, x]}, z*Log[w]*PolyLog[n + 1, v] - Int[SimplifyIntegrand[z*D[w, x]*PolyLog[n + 1, v]/w, x], x] /; Not[FalseQ[z]]] /; FreeQ[n, x] && InverseFunctionFreeQ[w, x]
diff --git a/IntegrationRules/8 Special functions/8.9 Product logarithm function.m b/IntegrationRules/8 Special functions/8.9 Product logarithm function.m
new file mode 100755
index 0000000..b1f9bef
--- /dev/null
+++ b/IntegrationRules/8 Special functions/8.9 Product logarithm function.m	
@@ -0,0 +1,47 @@
+
+(* ::Subsection::Closed:: *)
+(* 8.9 Product logarithm function *)
+Int[(c_.*ProductLog[a_. + b_.*x_])^p_, x_Symbol] := (a + b*x)*(c*ProductLog[a + b*x])^p/(b*(p + 1)) + p/(c*(p + 1))* Int[(c*ProductLog[a + b*x])^(p + 1)/(1 + ProductLog[a + b*x]), x] /; FreeQ[{a, b, c}, x] && LtQ[p, -1]
+Int[(c_.*ProductLog[a_. + b_.*x_])^p_., x_Symbol] := (a + b*x)*(c*ProductLog[a + b*x])^p/b - p*Int[(c*ProductLog[a + b*x])^p/(1 + ProductLog[a + b*x]), x] /; FreeQ[{a, b, c}, x] && Not[LtQ[p, -1]]
+Int[(e_. + f_.*x_)^m_.*(c_.*ProductLog[a_ + b_.*x_])^p_., x_Symbol] := 1/b^(m + 1)* Subst[Int[ ExpandIntegrand[(c*ProductLog[x])^p, (b*e - a*f + f*x)^m, x], x], x, a + b*x] /; FreeQ[{a, b, c, e, f, p}, x] && IGtQ[m, 0]
+Int[(c_.*ProductLog[a_.*x_^n_])^p_., x_Symbol] := x*(c*ProductLog[a*x^n])^p - n*p*Int[(c*ProductLog[a*x^n])^p/(1 + ProductLog[a*x^n]), x] /; FreeQ[{a, c, n, p}, x] && (EqQ[n*(p - 1), -1] || IntegerQ[p - 1/2] && EqQ[n*(p - 1/2), -1])
+Int[(c_.*ProductLog[a_.*x_^n_])^p_., x_Symbol] := x*(c*ProductLog[a*x^n])^p/(n*p + 1) + n*p/(c*(n*p + 1))* Int[(c*ProductLog[a*x^n])^(p + 1)/(1 + ProductLog[a*x^n]), x] /; FreeQ[{a, c, n}, x] && (IntegerQ[p] && EqQ[n*(p + 1), -1] || IntegerQ[p - 1/2] && EqQ[n*(p + 1/2), -1])
+Int[(c_.*ProductLog[a_.*x_^n_])^p_., x_Symbol] := -Subst[Int[(c*ProductLog[a*x^(-n)])^p/x^2, x], x, 1/x] /; FreeQ[{a, c, p}, x] && ILtQ[n, 0]
+Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^p_., x_Symbol] := x^(m + 1)*(c*ProductLog[a*x^n])^p/(m + 1) - n*p/(m + 1)* Int[x^m*(c*ProductLog[a*x^n])^p/(1 + ProductLog[a*x^n]), x] /; FreeQ[{a, c, m, n, p}, x] && NeQ[m, -1] && (IntegerQ[p - 1/2] && IGtQ[2*Simplify[p + (m + 1)/n], 0] || Not[IntegerQ[p - 1/2]] && IGtQ[Simplify[p + (m + 1)/n] + 1, 0])
+Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^p_., x_Symbol] := x^(m + 1)*(c*ProductLog[a*x^n])^p/(m + n*p + 1) + n*p/(c*(m + n*p + 1))* Int[x^m*(c*ProductLog[a*x^n])^(p + 1)/(1 + ProductLog[a*x^n]), x] /; FreeQ[{a, c, m, n, p}, x] && (EqQ[m, -1] || IntegerQ[p - 1/2] && ILtQ[Simplify[p + (m + 1)/n] - 1/2, 0] || Not[IntegerQ[p - 1/2]] && ILtQ[Simplify[p + (m + 1)/n], 0])
+Int[x_^m_.*(c_.*ProductLog[a_.*x_])^p_., x_Symbol] := Int[x^m*(c*ProductLog[a*x])^p/(1 + ProductLog[a*x]), x] + 1/c* Int[x^m*(c*ProductLog[a*x])^(p + 1)/(1 + ProductLog[a*x]), x] /; FreeQ[{a, c, m}, x]
+Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_])^p_., x_Symbol] := -Subst[Int[(c*ProductLog[a*x^(-n)])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, c, p}, x] && ILtQ[n, 0] && IntegerQ[m] && NeQ[m, -1]
+Int[1/(d_ + d_.*ProductLog[a_. + b_.*x_]), x_Symbol] := (a + b*x)/(b*d*ProductLog[a + b*x]) /; FreeQ[{a, b, d}, x]
+Int[ProductLog[a_. + b_.*x_]/(d_ + d_.*ProductLog[a_. + b_.*x_]), x_Symbol] := d*x - Int[1/(d + d*ProductLog[a + b*x]), x] /; FreeQ[{a, b, d}, x]
+Int[(c_.*ProductLog[a_. + b_.*x_])^ p_/(d_ + d_.*ProductLog[a_. + b_.*x_]), x_Symbol] := c*(a + b*x)*(c*ProductLog[a + b*x])^(p - 1)/(b*d) - c*p* Int[(c*ProductLog[a + b*x])^(p - 1)/(d + d*ProductLog[a + b*x]), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0]
+Int[1/(ProductLog[a_. + b_.*x_]*(d_ + d_.*ProductLog[a_. + b_.*x_])), x_Symbol] := ExpIntegralEi[ProductLog[a + b*x]]/(b*d) /; FreeQ[{a, b, d}, x]
+Int[1/(Sqrt[ c_.*ProductLog[a_. + b_.*x_]]*(d_ + d_.*ProductLog[a_. + b_.*x_])), x_Symbol] := Rt[Pi*c, 2]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Rt[c, 2]]/(b*c*d) /; FreeQ[{a, b, c, d}, x] && PosQ[c]
+Int[1/(Sqrt[ c_.*ProductLog[a_. + b_.*x_]]*(d_ + d_.*ProductLog[a_. + b_.*x_])), x_Symbol] := Rt[-Pi*c, 2]*Erf[Sqrt[c*ProductLog[a + b*x]]/Rt[-c, 2]]/(b*c*d) /; FreeQ[{a, b, c, d}, x] && NegQ[c]
+Int[(c_.*ProductLog[a_. + b_.*x_])^ p_/(d_ + d_.*ProductLog[a_. + b_.*x_]), x_Symbol] := (a + b*x)*(c*ProductLog[a + b*x])^p/(b*d*(p + 1)) - 1/(c*(p + 1))* Int[(c*ProductLog[a + b*x])^(p + 1)/(d + d*ProductLog[a + b*x]), x] /; FreeQ[{a, b, c, d}, x] && LtQ[p, -1]
+Int[(c_.*ProductLog[a_. + b_.*x_])^ p_./(d_ + d_.*ProductLog[a_. + b_.*x_]), x_Symbol] := Gamma[ p + 1, -ProductLog[a + b*x]]*(c*ProductLog[a + b*x])^ p/(b*d*(-ProductLog[a + b*x])^p) /; FreeQ[{a, b, c, d, p}, x]
+Int[(e_. + f_.*x_)^m_./(d_ + d_.*ProductLog[a_ + b_.*x_]), x_Symbol] := 1/b^(m + 1)* Subst[Int[ ExpandIntegrand[1/(d + d*ProductLog[x]), (b*e - a*f + f*x)^m, x], x], x, a + b*x] /; FreeQ[{a, b, d, e, f}, x] && IGtQ[m, 0]
+Int[(e_. + f_.*x_)^ m_.*(c_.*ProductLog[a_ + b_.*x_])^ p_./(d_ + d_.*ProductLog[a_ + b_.*x_]), x_Symbol] := 1/b^(m + 1)* Subst[Int[ ExpandIntegrand[(c*ProductLog[x])^ p/(d + d*ProductLog[x]), (b*e - a*f + f*x)^m, x], x], x, a + b*x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IGtQ[m, 0]
+Int[1/(d_ + d_.*ProductLog[a_.*x_^n_]), x_Symbol] := -Subst[Int[1/(x^2*(d + d*ProductLog[a*x^(-n)])), x], x, 1/x] /; FreeQ[{a, d}, x] && ILtQ[n, 0]
+Int[(c_.*ProductLog[a_.*x_^n_.])^ p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol] := c*x*(c*ProductLog[a*x^n])^(p - 1)/d /; FreeQ[{a, c, d, n, p}, x] && EqQ[n*(p - 1), -1]
+Int[ProductLog[a_.*x_^n_.]^p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol] := a^p*ExpIntegralEi[-p*ProductLog[a*x^n]]/(d*n) /; FreeQ[{a, d}, x] && IntegerQ[p] && EqQ[n*p, -1]
+Int[(c_.*ProductLog[a_.*x_^n_.])^p_/(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol] := Rt[Pi*c*n, 2]/(d*n*a^(1/n)*c^(1/n))* Erfi[Sqrt[c*ProductLog[a*x^n]]/Rt[c*n, 2]] /; FreeQ[{a, c, d}, x] && IntegerQ[1/n] && EqQ[p, 1/2 - 1/n] && PosQ[c*n]
+Int[(c_.*ProductLog[a_.*x_^n_.])^p_/(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol] := Rt[-Pi*c*n, 2]/(d*n*a^(1/n)*c^(1/n))* Erf[Sqrt[c*ProductLog[a*x^n]]/Rt[-c*n, 2]] /; FreeQ[{a, c, d}, x] && IntegerQ[1/n] && EqQ[p, 1/2 - 1/n] && NegQ[c*n]
+Int[(c_.*ProductLog[a_.*x_^n_.])^ p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol] := c*x*(c*ProductLog[a*x^n])^(p - 1)/d - c*(n*(p - 1) + 1)* Int[(c*ProductLog[a*x^n])^(p - 1)/(d + d*ProductLog[a*x^n]), x] /; FreeQ[{a, c, d}, x] && GtQ[n, 0] && GtQ[n*(p - 1) + 1, 0]
+Int[(c_.*ProductLog[a_.*x_^n_.])^ p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol] := x*(c*ProductLog[a*x^n])^p/(d*(n*p + 1)) - 1/(c*(n*p + 1))* Int[(c*ProductLog[a*x^n])^(p + 1)/(d + d*ProductLog[a*x^n]), x] /; FreeQ[{a, c, d}, x] && GtQ[n, 0] && LtQ[n*p + 1, 0]
+Int[(c_.*ProductLog[a_.*x_^n_])^p_./(d_ + d_.*ProductLog[a_.*x_^n_]), x_Symbol] := -Subst[ Int[(c*ProductLog[a*x^(-n)])^p/(x^2*(d + d*ProductLog[a*x^(-n)])), x], x, 1/x] /; FreeQ[{a, c, d, p}, x] && ILtQ[n, 0]
+Int[x_^m_./(d_ + d_.*ProductLog[a_.*x_]), x_Symbol] := x^(m + 1)/(d*(m + 1)*ProductLog[a*x]) - m/(m + 1)*Int[x^m/(ProductLog[a*x]*(d + d*ProductLog[a*x])), x] /; FreeQ[{a, d}, x] && GtQ[m, 0]
+Int[1/(x_*(d_ + d_.*ProductLog[a_.*x_])), x_Symbol] := Log[ProductLog[a*x]]/d /; FreeQ[{a, d}, x]
+Int[x_^m_./(d_ + d_.*ProductLog[a_.*x_]), x_Symbol] := x^(m + 1)/(d*(m + 1)) - Int[x^m*ProductLog[a*x]/(d + d*ProductLog[a*x]), x] /; FreeQ[{a, d}, x] && LtQ[m, -1]
+Int[x_^m_./(d_ + d_.*ProductLog[a_.*x_]), x_Symbol] := x^m*Gamma[m + 1, -(m + 1)*ProductLog[a*x]]/ (a*d*(m + 1)* E^(m*ProductLog[a*x])*(-(m + 1)*ProductLog[a*x])^m) /; FreeQ[{a, d, m}, x] && Not[IntegerQ[m]]
+Int[1/(x_*(d_ + d_.*ProductLog[a_.*x_^n_.])), x_Symbol] := Log[ProductLog[a*x^n]]/(d*n) /; FreeQ[{a, d, n}, x]
+Int[x_^m_./(d_ + d_.*ProductLog[a_.*x_^n_]), x_Symbol] := -Subst[Int[1/(x^(m + 2)*(d + d*ProductLog[a*x^(-n)])), x], x, 1/x] /; FreeQ[{a, d}, x] && IntegerQ[m] && ILtQ[n, 0] && NeQ[m, -1]
+Int[(c_.*ProductLog[a_.*x_^n_.])^ p_./(x_*(d_ + d_.*ProductLog[a_.*x_^n_.])), x_Symbol] := (c*ProductLog[a*x^n])^p/(d*n*p) /; FreeQ[{a, c, d, n, p}, x]
+Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^ p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol] := c*x^(m + 1)*(c*ProductLog[a*x^n])^(p - 1)/(d*(m + 1)) /; FreeQ[{a, c, d, m, n, p}, x] && NeQ[m, -1] && EqQ[m + n*(p - 1), -1]
+Int[x_^m_.* ProductLog[a_.*x_^n_.]^p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol] := a^p*ExpIntegralEi[-p*ProductLog[a*x^n]]/(d*n) /; FreeQ[{a, d, m, n}, x] && IntegerQ[p] && EqQ[m + n*p, -1]
+Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^ p_/(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol] := a^(p - 1/2)*c^(p - 1/2)*Rt[Pi*c/(p - 1/2), 2]* Erf[Sqrt[c*ProductLog[a*x^n]]/Rt[c/(p - 1/2), 2]]/(d*n) /; FreeQ[{a, c, d, m, n}, x] && NeQ[m, -1] && IntegerQ[p - 1/2] && EqQ[m + n*(p - 1/2), -1] && PosQ[c/(p - 1/2)]
+Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^ p_/(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol] := a^(p - 1/2)*c^(p - 1/2)*Rt[-Pi*c/(p - 1/2), 2]* Erfi[Sqrt[c*ProductLog[a*x^n]]/Rt[-c/(p - 1/2), 2]]/(d*n) /; FreeQ[{a, c, d, m, n}, x] && NeQ[m, -1] && IntegerQ[p - 1/2] && EqQ[m + n*(p - 1/2), -1] && NegQ[c/(p - 1/2)]
+Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^ p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol] := c*x^(m + 1)*(c*ProductLog[a*x^n])^(p - 1)/(d*(m + 1)) - c*(m + n*(p - 1) + 1)/(m + 1)* Int[x^m*(c*ProductLog[a*x^n])^(p - 1)/(d + d*ProductLog[a*x^n]), x] /; FreeQ[{a, c, d, m, n, p}, x] && NeQ[m, -1] && GtQ[Simplify[p + (m + 1)/n], 1]
+Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^ p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol] := x^(m + 1)*(c*ProductLog[a*x^n])^p/(d*(m + n*p + 1)) - (m + 1)/(c*(m + n*p + 1))* Int[x^m*(c*ProductLog[a*x^n])^(p + 1)/(d + d*ProductLog[a*x^n]), x] /; FreeQ[{a, c, d, m, n, p}, x] && NeQ[m, -1] && LtQ[Simplify[p + (m + 1)/n], 0]
+Int[x_^m_.*(c_.*ProductLog[a_.*x_])^p_./(d_ + d_.*ProductLog[a_.*x_]), x_Symbol] := x^m*Gamma[ m + p + 1, -(m + 1)*ProductLog[a*x]]*(c*ProductLog[a*x])^p/ (a*d*(m + 1)* E^(m*ProductLog[a*x])*(-(m + 1)*ProductLog[a*x])^(m + p)) /; FreeQ[{a, c, d, m, p}, x] && NeQ[m, -1]
+Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^ p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol] := -Subst[ Int[(c*ProductLog[a*x^(-n)])^ p/(x^(m + 2)*(d + d*ProductLog[a*x^(-n)])), x], x, 1/x] /; FreeQ[{a, c, d, p}, x] && NeQ[m, -1] && IntegerQ[m] && LtQ[n, 0]
+Int[u_, x_Symbol] := Subst[ Int[SimplifyIntegrand[(x + 1)*E^x*SubstFor[ProductLog[x], u, x], x], x], x, ProductLog[x]] /; FunctionOfQ[ProductLog[x], u, x]
diff --git a/IntegrationRules/9 Miscellaneous/9.1 Derivative integration rules.m b/IntegrationRules/9 Miscellaneous/9.1 Derivative integration rules.m
new file mode 100755
index 0000000..881e789
--- /dev/null
+++ b/IntegrationRules/9 Miscellaneous/9.1 Derivative integration rules.m	
@@ -0,0 +1,160 @@
+(* ::Package:: *)
+
+(************************************************************************)
+(* This file was generated automatically by the Mathematica front end.  *)
+(* It contains Initialization cells from a Notebook file, which         *)
+(* typically will have the same name as this file except ending in      *)
+(* ".nb" instead of ".m".                                               *)
+(*                                                                      *)
+(* This file is intended to be loaded into the Mathematica kernel using *)
+(* the package loading commands Get or Needs.  Doing so is equivalent   *)
+(* to using the Evaluate Initialization Cells menu command in the front *)
+(* end.                                                                 *)
+(*                                                                      *)
+(* DO NOT EDIT THIS FILE.  This entire file is regenerated              *)
+(* automatically each time the parent Notebook file is saved in the     *)
+(* Mathematica front end.  Any changes you make to this file will be    *)
+(* overwritten.                                                         *)
+(************************************************************************)
+
+
+
+(* ::Code:: *)
+Int[Derivative[n_][f_][x_],x_Symbol] :=
+  Derivative[n-1][f][x] /;
+FreeQ[{f,n},x]
+
+
+(* ::Code:: *)
+Int[(c_.*F_^(a_.+b_.*x_))^p_.*Derivative[n_][f_][x_],x_Symbol] :=
+  (c*F^(a+b*x))^p*Derivative[n-1][f][x] - b*p*Log[F] \[Star] Int[(c*F^(a+b*x))^p*Derivative[n-1][f][x],x] /;
+FreeQ[{a,b,c,f,F,p},x] && IGtQ[n,0]
+
+
+(* ::Code:: *)
+Int[(c_.*F_^(a_.+b_.*x_))^p_.*Derivative[n_][f_][x_],x_Symbol] :=
+  (c*F^(a+b*x))^p*Derivative[n][f][x]/(b*p*Log[F]) - 1/(b*p*Log[F]) \[Star] Int[(c*F^(a+b*x))^p*Derivative[n+1][f][x],x] /;
+FreeQ[{a,b,c,f,F,p},x] && ILtQ[n,0]
+
+
+(* ::Code:: *)
+Int[Sin[a_.+b_.*x_]*Derivative[n_][f_][x_],x_Symbol] :=
+  Sin[a+b*x]*Derivative[n-1][f][x] - b \[Star] Int[Cos[a+b*x]*Derivative[n-1][f][x],x] /;
+FreeQ[{a,b,f},x] && IGtQ[n,0]
+
+
+(* ::Code:: *)
+Int[Cos[a_.+b_.*x_]*Derivative[n_][f_][x_],x_Symbol] :=
+  Cos[a+b*x]*Derivative[n-1][f][x] + b \[Star] Int[Sin[a+b*x]*Derivative[n-1][f][x],x] /;
+FreeQ[{a,b,f},x] && IGtQ[n,0]
+
+
+(* ::Code:: *)
+Int[Sin[a_.+b_.*x_]*Derivative[n_][f_][x_],x_Symbol] :=
+  -Cos[a+b*x]*Derivative[n][f][x]/b + 1/b \[Star] Int[Cos[a+b*x]*Derivative[n+1][f][x],x] /;
+FreeQ[{a,b,f},x] && ILtQ[n,0]
+
+
+(* ::Code:: *)
+Int[Cos[a_.+b_.*x_]*Derivative[n_][f_][x_],x_Symbol] :=
+  Sin[a+b*x]*Derivative[n][f][x]/b - 1/b \[Star] Int[Sin[a+b*x]*Derivative[n+1][f][x],x] /;
+FreeQ[{a,b,f},x] && ILtQ[n,0]
+
+
+(* ::Code:: *)
+Int[u_*Derivative[n_][f_][x_],x_Symbol] :=
+  Subst[Int[SimplifyIntegrand[SubstFor[Derivative[n-1][f][x],u,x],x],x],x,Derivative[n-1][f][x]] /;
+FreeQ[{f,n},x] && FunctionOfQ[Derivative[n-1][f][x],u,x]
+
+
+(* ::Code:: *)
+Int[u_*(a_.*Derivative[1][f_][x_]*g_[x_]+a_.*f_[x_]*Derivative[1][g_][x_]),x_Symbol] :=
+  a \[Star] Subst[Int[SimplifyIntegrand[SubstFor[f[x]*g[x],u,x],x],x],x,f[x]*g[x]] /;
+FreeQ[{a,f,g},x] && FunctionOfQ[f[x]*g[x],u,x]
+
+
+(* ::Code:: *)
+Int[u_*(a_.*Derivative[m_][f_][x_]*g_[x_]+a_.*Derivative[m1_][f_][x_]*Derivative[1][g_][x_]),x_Symbol] :=
+  a \[Star] Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m-1][f][x]*g[x],u,x],x],x],x,Derivative[m-1][f][x]*g[x]] /;
+FreeQ[{a,f,g,m},x] && EqQ[m1,m-1] && FunctionOfQ[Derivative[m-1][f][x]*g[x],u,x]
+
+
+(* ::Code:: *)
+Int[u_*(a_.*Derivative[m_][f_][x_]*Derivative[n1_][g_][x_]+a_.*Derivative[m1_][f_][x_]*Derivative[n_][g_][x_]),x_Symbol] :=
+  a \[Star] Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m-1][f][x]*Derivative[n-1][g][x],u,x],x],x],x,Derivative[m-1][f][x]*Derivative[n-1][g][x]] /;
+FreeQ[{a,f,g,m,n},x] && EqQ[m1,m-1] && EqQ[n1,n-1] && FunctionOfQ[Derivative[m-1][f][x]*Derivative[n-1][g][x],u,x]
+
+
+(* ::Code:: *)
+Int[u_*f_[x_]^p_.*(a_.*Derivative[1][f_][x_]*g_[x_]+b_.*f_[x_]*Derivative[1][g_][x_]),x_Symbol] :=
+  b \[Star] Subst[Int[SimplifyIntegrand[SubstFor[f[x]^(p+1)*g[x],u,x],x],x],x,f[x]^(p+1)*g[x]] /;
+FreeQ[{a,b,f,g,p},x] && EqQ[a,b*(p+1)] && FunctionOfQ[f[x]^(p+1)*g[x],u,x]
+
+
+(* ::Code:: *)
+Int[u_*Derivative[m1_][f_][x_]^p_.*
+    (a_.*Derivative[m_][f_][x_]*g_[x_]+b_.*Derivative[m1_][f_][x_]*Derivative[1][g_][x_]),x_Symbol] :=
+  b \[Star] Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m-1][f][x]^(p+1)*g[x],u,x],x],x],x,
+    Derivative[m-1][f][x]^(p+1)*g[x]] /;
+FreeQ[{a,b,f,g,m,p},x] && EqQ[m1,m-1] && EqQ[a,b*(p+1)] && FunctionOfQ[Derivative[m-1][f][x]^(p+1)*g[x],u,x]
+
+
+(* ::Code:: *)
+Int[u_*g_[x_]^q_.*
+    (a_.*Derivative[m_][f_][x_]*g_[x_]+b_.*Derivative[m1_][f_][x_]*Derivative[1][g_][x_]),x_Symbol] :=
+  a \[Star] Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m-1][f][x]*g[x]^(q+1),u,x],x],x],x,
+    Derivative[m-1][f][x]*g[x]^(q+1)] /;
+FreeQ[{a,b,f,g,m,q},x] && EqQ[m1,m-1] && EqQ[a*(q+1),b] && FunctionOfQ[Derivative[m-1][f][x]*g[x]^(q+1),u,x]
+
+
+(* ::Code:: *)
+Int[u_*Derivative[m1_][f_][x_]^p_.*
+    (a_.*Derivative[m_][f_][x_]*Derivative[n1_][g_][x_]+b_.*Derivative[m1_][f_][x_]*Derivative[n_][g_][x_]),x_Symbol] :=
+  b \[Star] Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m-1][f][x]^(p+1)*Derivative[n-1][g][x],u,x],x],x],x,
+    Derivative[m-1][f][x]^(p+1)*Derivative[n-1][g][x]] /;
+FreeQ[{a,b,f,g,m,n,p},x] && EqQ[m1,m-1] && EqQ[n1,n-1] && EqQ[a,b*(p+1)] && 
+  FunctionOfQ[Derivative[m-1][f][x]^(p+1)*Derivative[n-1][g][x],u,x]
+
+
+(* ::Code:: *)
+Int[u_*f_[x_]^p_.*g_[x_]^q_.*(a_.*Derivative[1][f_][x_]*g_[x_]+b_.*f_[x_]*Derivative[1][g_][x_]),x_Symbol] :=
+  a/(p+1) \[Star] Subst[Int[SimplifyIntegrand[SubstFor[f[x]^(p+1)*g[x]^(q+1),u,x],x],x],x,f[x]^(p+1)*g[x]^(q+1)] /;
+FreeQ[{a,b,f,g,p,q},x] && EqQ[a*(q+1),b*(p+1)] && FunctionOfQ[f[x]^(p+1)*g[x]^(q+1),u,x]
+
+
+(* ::Code:: *)
+Int[u_*Derivative[m1_][f_][x_]^p_.*g_[x_]^q_.*
+    (a_.*Derivative[m_][f_][x_]*g_[x_]+b_.*Derivative[m1_][f_][x_]*Derivative[1][g_][x_]),x_Symbol] :=
+  a/(p+1) \[Star] Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m-1][f][x]^(p+1)*g[x]^(q+1),u,x],x],x],x,
+    Derivative[m-1][f][x]^(p+1)*g[x]^(q+1)] /;
+FreeQ[{a,b,f,g,m,p,q},x] && EqQ[m1,m-1] && EqQ[a*(q+1),b*(p+1)] && FunctionOfQ[Derivative[m-1][f][x]^(p+1)*g[x]^(q+1),u,x]
+
+
+(* ::Code:: *)
+Int[u_*Derivative[m1_][f_][x_]^p_.*Derivative[n1_][g_][x_]^q_.*
+    (a_.*Derivative[m_][f_][x_]*Derivative[n1_][g_][x_]+b_.*Derivative[m1_][f_][x_]*Derivative[n_][g_][x_]),x_Symbol] :=
+  a/(p+1) \[Star] Subst[Int[SimplifyIntegrand[SubstFor[Derivative[m-1][f][x]^(p+1)*Derivative[n-1][g][x]^(q+1),u,x],x],x],x,
+    Derivative[m-1][f][x]^(p+1)*Derivative[n-1][g][x]^(q+1)] /;
+FreeQ[{a,b,f,g,m,n,p,q},x] && EqQ[m1,m-1] && EqQ[n1,n-1] && EqQ[a*(q+1),b*(p+1)] && 
+  FunctionOfQ[Derivative[m-1][f][x]^(p+1)*Derivative[n-1][g][x]^(q+1),u,x]
+
+
+(* ::Code:: *)
+Int[f_'[x_]*g_[x_] + f_[x_]*g_'[x_],x_Symbol] :=
+  f[x]*g[x] /;
+FreeQ[{f,g},x]
+
+
+(* ::Code:: *)
+Int[(f_'[x_]*g_[x_] - f_[x_]*g_'[x_])/g_[x_]^2,x_Symbol] :=
+  f[x]/g[x] /;
+FreeQ[{f,g},x]
+
+
+(* ::Code:: *)
+Int[(f_'[x_]*g_[x_] - f_[x_]*g_'[x_])/(f_[x_]*g_[x_]),x_Symbol] :=
+  Log[f[x]/g[x]] /;
+FreeQ[{f,g},x]
+
+
+
diff --git a/Rubi.m b/Rubi.m
index c03ebe3..70aed3d 100644
--- a/Rubi.m
+++ b/Rubi.m
@@ -72,6 +72,9 @@
 $RubiVersion = StringJoin["Rubi ", Version /. List@@Get[FileNameJoin[{$RubiDir, "PacletInfo.m"}]]];
 Print["Loading " <> $RubiVersion <> " will take a minute or two. In the future this will take less than a second."];
 
+(* Elementary function rules *)
+$LoadElementaryFunctionRules = If[Not[ValueQ[Global`$LoadElementaryFunctionRules]], True, TrueQ[Global`$LoadElementaryFunctionRules]];
+
 (* Disable Steps *)
 (* $LoadShowSteps = If[Not[ValueQ[Global`$LoadShowSteps]], True, TrueQ[Global`$LoadShowSteps]]; *)
 $LoadShowSteps = False
@@ -186,6 +189,167 @@
 
 LoadRules[FileNameJoin[{"1 Algebraic functions", "1.4 Miscellaneous", "1.4.3 Miscellaneous algebraic functions"}]];
 
+If[$LoadElementaryFunctionRules===True,
+  LoadRules[FileNameJoin[{"2 Exponentials", "2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p"}]];
+  LoadRules[FileNameJoin[{"2 Exponentials", "2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p"}]];
+  LoadRules[FileNameJoin[{"2 Exponentials", "2.3 Miscellaneous exponentials"}]];
+
+  LoadRules[FileNameJoin[{"3 Logarithms", "3.1.1 (a+b log(c x^n))^p"}]];
+  LoadRules[FileNameJoin[{"3 Logarithms", "3.1.2 (d x)^m (a+b log(c x^n))^p"}]];
+  LoadRules[FileNameJoin[{"3 Logarithms", "3.1.3 (d+e x^r)^q (a+b log(c x^n))^p"}]];
+  LoadRules[FileNameJoin[{"3 Logarithms", "3.1.4 (f x)^m (d+e x^r)^q (a+b log(c x^n))^p"}]];
+  (*
+  LoadRules[FileNameJoin[{"3 Logarithms", "3.1.5 u (a+b log(c x^n))^p"}]];
+  LoadRules[FileNameJoin[{"3 Logarithms", "3.3 u (a+b log(c (d+e x)^n))^p"}]];
+  *)
+  LoadRules[FileNameJoin[{"3 Logarithms", "3.4 u (a+b log(c (d+e x^m)^n))^p"}]];
+  LoadRules[FileNameJoin[{"3 Logarithms", "3.2.1 (f+g x)^m (A+B log(e ((a+b x) over (c+d x))^n))^p"}]];
+  LoadRules[FileNameJoin[{"3 Logarithms", "3.2.2 (f+g x)^m (h+i x)^q (A+B log(e ((a+b x) over (c+d x))^n))^p"}]];
+  (*
+  LoadRules[FileNameJoin[{"3 Logarithms", "3.2.3 u log(e (f (a+b x)^p (c+d x)^q)^r)^s"}]];
+  *)
+  LoadRules[FileNameJoin[{"3 Logarithms", "3.5 Miscellaneous logarithms"}]];
+
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.0.1 (a sin)^m (b trg)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.0.2 (a trg)^m (b tan)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.0.3 (a csc)^m (b sec)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.1.1 (a+b sin)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.1.2 (g cos)^p (a+b sin)^m"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.1.3 (g tan)^p (a+b sin)^m"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.2.1 (a+b sin)^m (c+d sin)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin)"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2)"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2)"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.5 trig^m (a cos+b sin)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.6 (a+b cos+c sin)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.7 (d trig)^m (a+b (c sin)^n)^p"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.8 trig^m (a+b cos^p+c sin^q)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.10 (c+d x)^m (a+b sin)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.11 (e x)^m (a+b x^n)^p sin"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.12 (e x)^m (a+b sin(c+d x^n))^p"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.1 Sine", "4.1.13 (d+e x)^m sin(a+b x+c x^2)^n"}]];
+
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.3 Tangent", "4.3.1.1 (a+b tan)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.3 Tangent", "4.3.1.2 (d sec)^m (a+b tan)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.3 Tangent", "4.3.1.3 (d sin)^m (a+b tan)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.3 Tangent", "4.3.2.1 (a+b tan)^m (c+d tan)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.3 Tangent", "4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.3 Tangent", "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan)"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.3 Tangent", "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2)"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.3 Tangent", "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2)"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.3 Tangent", "4.3.7 (d trig)^m (a+b (c tan)^n)^p"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.3 Tangent", "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.3 Tangent", "4.3.10 (c+d x)^m (a+b tan)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.3 Tangent", "4.3.11 (e x)^m (a+b tan(c+d x^n))^p"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.3 Tangent", "4.3.12 (d+e x)^m tan(a+b x+c x^2)^n"}]];
+
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.5 Secant", "4.5.1.1 (a+b sec)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.5 Secant", "4.5.1.2 (d sec)^n (a+b sec)^m"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.5 Secant", "4.5.1.3 (d sin)^n (a+b sec)^m"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.5 Secant", "4.5.1.4 (d tan)^n (a+b sec)^m"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.5 Secant", "4.5.2.1 (a+b sec)^m (c+d sec)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.5 Secant", "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.5 Secant", "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec)"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.5 Secant", "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2)"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.5 Secant", "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2)"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.5 Secant", "4.5.7 (d trig)^m (a+b (c sec)^n)^p"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.5 Secant", "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.5 Secant", "4.5.10 (c+d x)^m (a+b sec)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.5 Secant", "4.5.11 (e x)^m (a+b sec(c+d x^n))^p"}]];
+
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.7 Miscellaneous", "4.7.1 Sine normalization rules"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.7 Miscellaneous", "4.7.2 Tangent normalization rules"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.7 Miscellaneous", "4.7.3 Secant normalization rules"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.7 Miscellaneous", "4.7.4 (c trig)^m (d trig)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.7 Miscellaneous", "4.7.5 Inert trig functions"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.7 Miscellaneous", "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.7 Miscellaneous", "4.7.7 F^(c (a+b x)) trig(d+e x)^n"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.7 Miscellaneous", "4.7.8 u trig(a+b log(c x^n))^p"}]];
+  LoadRules[FileNameJoin[{"4 Trig functions", "4.7 Miscellaneous", "4.7.9 Active trig functions"}]];
+
+  LoadRules[FileNameJoin[{"5 Inverse trig functions", "5.1 Inverse sine", "5.1.1 (a+b arcsin(c x))^n"}]];
+  LoadRules[FileNameJoin[{"5 Inverse trig functions", "5.1 Inverse sine", "5.1.2 (d x)^m (a+b arcsin(c x))^n"}]];
+  LoadRules[FileNameJoin[{"5 Inverse trig functions", "5.1 Inverse sine", "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n"}]];
+  LoadRules[FileNameJoin[{"5 Inverse trig functions", "5.1 Inverse sine", "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n"}]];
+  LoadRules[FileNameJoin[{"5 Inverse trig functions", "5.1 Inverse sine", "5.1.5 u (a+b arcsin(c x))^n"}]];
+  LoadRules[FileNameJoin[{"5 Inverse trig functions", "5.1 Inverse sine", "5.1.6 Miscellaneous inverse sine"}]];
+
+  LoadRules[FileNameJoin[{"5 Inverse trig functions", "5.3 Inverse tangent", "5.3.1 (a+b arctan(c x^n))^p"}]];
+  LoadRules[FileNameJoin[{"5 Inverse trig functions", "5.3 Inverse tangent", "5.3.2 (d x)^m (a+b arctan(c x^n))^p"}]];
+  LoadRules[FileNameJoin[{"5 Inverse trig functions", "5.3 Inverse tangent", "5.3.3 (d+e x)^m (a+b arctan(c x^n))^p"}]];
+  (*
+  LoadRules[FileNameJoin[{"5 Inverse trig functions", "5.3 Inverse tangent", "5.3.4 u (a+b arctan(c x))^p"}]];
+  *)
+  LoadRules[FileNameJoin[{"5 Inverse trig functions", "5.3 Inverse tangent", "5.3.5 u (a+b arctan(c+d x))^p"}]];
+  LoadRules[FileNameJoin[{"5 Inverse trig functions", "5.3 Inverse tangent", "5.3.6 Exponentials of inverse tangent"}]];
+  LoadRules[FileNameJoin[{"5 Inverse trig functions", "5.3 Inverse tangent", "5.3.7 Miscellaneous inverse tangent"}]];
+
+  LoadRules[FileNameJoin[{"5 Inverse trig functions", "5.5 Inverse secant", "5.5.1 u (a+b arcsec(c x))^n"}]];
+  LoadRules[FileNameJoin[{"5 Inverse trig functions", "5.5 Inverse secant", "5.5.2 Miscellaneous inverse secant"}]];
+
+  LoadRules[FileNameJoin[{"6 Hyperbolic functions", "6.1 Hyperbolic sine", "6.1.10 (c+d x)^m (a+b sinh)^n"}]];
+  LoadRules[FileNameJoin[{"6 Hyperbolic functions", "6.1 Hyperbolic sine", "6.1.11 (e x)^m (a+b x^n)^p sinh"}]];
+  LoadRules[FileNameJoin[{"6 Hyperbolic functions", "6.1 Hyperbolic sine", "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p"}]];
+  LoadRules[FileNameJoin[{"6 Hyperbolic functions", "6.1 Hyperbolic sine", "6.1.13 (d+e x)^m sinh(a+b x+c x^2)^n"}]];
+
+  LoadRules[FileNameJoin[{"6 Hyperbolic functions", "6.3 Hyperbolic tangent", "6.3.10 (c+d x)^m (a+b tanh)^n"}]];
+  LoadRules[FileNameJoin[{"6 Hyperbolic functions", "6.3 Hyperbolic tangent", "6.3.11 (e x)^m (a+b tanh(c+d x^n))^p"}]];
+  LoadRules[FileNameJoin[{"6 Hyperbolic functions", "6.3 Hyperbolic tangent", "6.3.12 (d+e x)^m tanh(a+b x+c x^2)^n"}]];
+
+  LoadRules[FileNameJoin[{"6 Hyperbolic functions", "6.5 Hyperbolic secant", "6.5.10 (c+d x)^m (a+b sech)^n"}]];
+  LoadRules[FileNameJoin[{"6 Hyperbolic functions", "6.5 Hyperbolic secant", "6.5.11 (e x)^m (a+b sech(c+d x^n))^p"}]];
+
+  LoadRules[FileNameJoin[{"6 Hyperbolic functions", "6.7 Miscellaneous", "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p"}]];
+  LoadRules[FileNameJoin[{"6 Hyperbolic functions", "6.7 Miscellaneous", "6.7.7 F^(c (a+b x)) hyper(d+e x)^n"}]];
+  LoadRules[FileNameJoin[{"6 Hyperbolic functions", "6.7 Miscellaneous", "6.7.8 u hyper(a+b log(c x^n))^p"}]];
+  LoadRules[FileNameJoin[{"6 Hyperbolic functions", "6.7 Miscellaneous", "6.7.9 Active hyperbolic functions"}]];
+
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.1 Inverse hyperbolic sine", "7.1.1 (a+b arcsinh(c x))^n"}]];
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.1 Inverse hyperbolic sine", "7.1.2 (d x)^m (a+b arcsinh(c x))^n"}]];
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.1 Inverse hyperbolic sine", "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n"}]];
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.1 Inverse hyperbolic sine", "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n"}]];
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.1 Inverse hyperbolic sine", "7.1.5 u (a+b arcsinh(c x))^n"}]];
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.1 Inverse hyperbolic sine", "7.1.6 Miscellaneous inverse hyperbolic sine"}]];
+
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.2 Inverse hyperbolic cosine", "7.2.1 (a+b arccosh(c x))^n"}]];
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.2 Inverse hyperbolic cosine", "7.2.2 (d x)^m (a+b arccosh(c x))^n"}]];
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.2 Inverse hyperbolic cosine", "7.2.3 (d+e x^2)^p (a+b arccosh(c x))^n"}]];
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.2 Inverse hyperbolic cosine", "7.2.4 (f x)^m (d+e x^2)^p (a+b arccosh(c x))^n"}]];
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.2 Inverse hyperbolic cosine", "7.2.5 u (a+b arccosh(c x))^n"}]];
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.2 Inverse hyperbolic cosine", "7.2.6 Miscellaneous inverse hyperbolic cosine"}]];
+
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.3 Inverse hyperbolic tangent", "7.3.1 (a+b arctanh(c x^n))^p"}]];
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.3 Inverse hyperbolic tangent", "7.3.2 (d x)^m (a+b arctanh(c x^n))^p"}]];
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.3 Inverse hyperbolic tangent", "7.3.3 (d+e x)^m (a+b arctanh(c x^n))^p"}]];
+  (*
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.3 Inverse hyperbolic tangent", "7.3.4 u (a+b arctanh(c x))^p"}]];
+  *)
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.3 Inverse hyperbolic tangent", "7.3.5 u (a+b arctanh(c+d x))^p"}]];
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.3 Inverse hyperbolic tangent", "7.3.6 Exponentials of inverse hyperbolic tangent"}]];
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.3 Inverse hyperbolic tangent", "7.3.7 Miscellaneous inverse hyperbolic tangent"}]];
+
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.5 Inverse hyperbolic secant", "7.5.1 u (a+b arcsech(c x))^n"}]];
+  LoadRules[FileNameJoin[{"7 Inverse hyperbolic functions", "7.5 Inverse hyperbolic secant", "7.5.2 Miscellaneous inverse hyperbolic secant"}]];
+
+  LoadRules[FileNameJoin[{"8 Special functions", "8.1 Error functions"}]];
+  LoadRules[FileNameJoin[{"8 Special functions", "8.2 Fresnel integral functions"}]];
+  LoadRules[FileNameJoin[{"8 Special functions", "8.3 Exponential integral functions"}]];
+  LoadRules[FileNameJoin[{"8 Special functions", "8.4 Trig integral functions"}]];
+  LoadRules[FileNameJoin[{"8 Special functions", "8.5 Hyperbolic integral functions"}]];
+  LoadRules[FileNameJoin[{"8 Special functions", "8.6 Gamma functions"}]];
+  LoadRules[FileNameJoin[{"8 Special functions", "8.7 Zeta function"}]];
+  (*
+  LoadRules[FileNameJoin[{"8 Special functions", "8.8 Polylogarithm function"}]];
+  *)
+  LoadRules[FileNameJoin[{"8 Special functions", "8.9 Product logarithm function"}]];
+  (*LoadRules[FileNameJoin[{"8 Special functions", "8.10 Bessel functions"}]]; *)
+
+  LoadRules[FileNameJoin[{"9 Miscellaneous", "9.1 Derivative integration rules"}]]
+];
+
 (* Required rules from Section 9 *)
 LoadRules[FileNameJoin[{"9 Miscellaneous", "9.1 Integrand simplification rules"}]];
 LoadRules[FileNameJoin[{"9 Miscellaneous", "9.2 Piecewise linear functions"}]];