# Maple integration test file: "4 Trig functions\4.3 Tangent\4.3.7 (d trig)^m (a+b (c tan)^n)^p.txt"

lst:=[

# Integrands of the form (b (c Tan[e+f x])^n)^p

# Integrands of the form (b (c Tan[e+f x])^n)^(p/2)

# Integrands of the form (b Tan[e+f x]^2)^(p/2)
[(b*tan(e+f*x)^2)^(5/2),x,4,-b^2*cot(e+f*x)*log(cos(e+f*x))*sqrt(b*tan(e+f*x)^2)/f-1/2*b^2*sqrt(b*tan(e+f*x)^2)*tan(e+f*x)/f+1/4*b^2*sqrt(b*tan(e+f*x)^2)*tan(e+f*x)^3/f],
[(b*tan(e+f*x)^2)^(3/2),x,3,b*cot(e+f*x)*log(cos(e+f*x))*sqrt(b*tan(e+f*x)^2)/f+1/2*b*sqrt(b*tan(e+f*x)^2)*tan(e+f*x)/f],
[(b*tan(e+f*x)^2)^(1/2),x,2,-cot(e+f*x)*log(cos(e+f*x))*sqrt(b*tan(e+f*x)^2)/f],
[1/(b*tan(e+f*x)^2)^(1/2),x,2,log(sin(e+f*x))*tan(e+f*x)/(f*sqrt(b*tan(e+f*x)^2))],
[1/(b*tan(e+f*x)^2)^(3/2),x,3,-1/2*cot(e+f*x)/(b*f*sqrt(b*tan(e+f*x)^2))-log(sin(e+f*x))*tan(e+f*x)/(b*f*sqrt(b*tan(e+f*x)^2))],
[1/(b*tan(e+f*x)^2)^(5/2),x,4,1/2*cot(e+f*x)/(b^2*f*sqrt(b*tan(e+f*x)^2))-1/4*cot(e+f*x)^3/(b^2*f*sqrt(b*tan(e+f*x)^2))+log(sin(e+f*x))*tan(e+f*x)/(b^2*f*sqrt(b*tan(e+f*x)^2))],

# Integrands of the form (b Tan[e+f x]^3)^(p/2)
[(b*tan(e+f*x)^3)^(5/2),x,16,-2*b^2*cot(e+f*x)*sqrt(b*tan(e+f*x)^3)/f-b^2*arctan(1-sqrt(2)*sqrt(tan(e+f*x)))*sqrt(b*tan(e+f*x)^3)/(f*sqrt(2)*tan(e+f*x)^(3/2))+b^2*arctan(1+sqrt(2)*sqrt(tan(e+f*x)))*sqrt(b*tan(e+f*x)^3)/(f*sqrt(2)*tan(e+f*x)^(3/2))-1/2*b^2*log(1-sqrt(2)*sqrt(tan(e+f*x))+tan(e+f*x))*sqrt(b*tan(e+f*x)^3)/(f*sqrt(2)*tan(e+f*x)^(3/2))+1/2*b^2*log(1+sqrt(2)*sqrt(tan(e+f*x))+tan(e+f*x))*sqrt(b*tan(e+f*x)^3)/(f*sqrt(2)*tan(e+f*x)^(3/2))+2/5*b^2*sqrt(b*tan(e+f*x)^3)*tan(e+f*x)/f-2/9*b^2*sqrt(b*tan(e+f*x)^3)*tan(e+f*x)^3/f+2/13*b^2*sqrt(b*tan(e+f*x)^3)*tan(e+f*x)^5/f],
[(b*tan(e+f*x)^3)^(3/2),x,14,-2/3*b*sqrt(b*tan(e+f*x)^3)/f-b*arctan(1-sqrt(2)*sqrt(tan(e+f*x)))*sqrt(b*tan(e+f*x)^3)/(f*sqrt(2)*tan(e+f*x)^(3/2))+b*arctan(1+sqrt(2)*sqrt(tan(e+f*x)))*sqrt(b*tan(e+f*x)^3)/(f*sqrt(2)*tan(e+f*x)^(3/2))+1/2*b*log(1-sqrt(2)*sqrt(tan(e+f*x))+tan(e+f*x))*sqrt(b*tan(e+f*x)^3)/(f*sqrt(2)*tan(e+f*x)^(3/2))-1/2*b*log(1+sqrt(2)*sqrt(tan(e+f*x))+tan(e+f*x))*sqrt(b*tan(e+f*x)^3)/(f*sqrt(2)*tan(e+f*x)^(3/2))+2/7*b*sqrt(b*tan(e+f*x)^3)*tan(e+f*x)^2/f],
[(b*tan(e+f*x)^3)^(1/2),x,13,2*cot(e+f*x)*sqrt(b*tan(e+f*x)^3)/f+arctan(1-sqrt(2)*sqrt(tan(e+f*x)))*sqrt(b*tan(e+f*x)^3)/(f*sqrt(2)*tan(e+f*x)^(3/2))-arctan(1+sqrt(2)*sqrt(tan(e+f*x)))*sqrt(b*tan(e+f*x)^3)/(f*sqrt(2)*tan(e+f*x)^(3/2))+1/2*log(1-sqrt(2)*sqrt(tan(e+f*x))+tan(e+f*x))*sqrt(b*tan(e+f*x)^3)/(f*sqrt(2)*tan(e+f*x)^(3/2))-1/2*log(1+sqrt(2)*sqrt(tan(e+f*x))+tan(e+f*x))*sqrt(b*tan(e+f*x)^3)/(f*sqrt(2)*tan(e+f*x)^(3/2))],
[1/(b*tan(e+f*x)^3)^(1/2),x,13,-2*tan(e+f*x)/(f*sqrt(b*tan(e+f*x)^3))+arctan(1-sqrt(2)*sqrt(tan(e+f*x)))*tan(e+f*x)^(3/2)/(f*sqrt(2)*sqrt(b*tan(e+f*x)^3))-arctan(1+sqrt(2)*sqrt(tan(e+f*x)))*tan(e+f*x)^(3/2)/(f*sqrt(2)*sqrt(b*tan(e+f*x)^3))-1/2*log(1-sqrt(2)*sqrt(tan(e+f*x))+tan(e+f*x))*tan(e+f*x)^(3/2)/(f*sqrt(2)*sqrt(b*tan(e+f*x)^3))+1/2*log(1+sqrt(2)*sqrt(tan(e+f*x))+tan(e+f*x))*tan(e+f*x)^(3/2)/(f*sqrt(2)*sqrt(b*tan(e+f*x)^3))],
[1/(b*tan(e+f*x)^3)^(3/2),x,14,2/3/(b*f*sqrt(b*tan(e+f*x)^3))-2/7*cot(e+f*x)^2/(b*f*sqrt(b*tan(e+f*x)^3))-arctan(1-sqrt(2)*sqrt(tan(e+f*x)))*tan(e+f*x)^(3/2)/(b*f*sqrt(2)*sqrt(b*tan(e+f*x)^3))+arctan(1+sqrt(2)*sqrt(tan(e+f*x)))*tan(e+f*x)^(3/2)/(b*f*sqrt(2)*sqrt(b*tan(e+f*x)^3))-1/2*log(1-sqrt(2)*sqrt(tan(e+f*x))+tan(e+f*x))*tan(e+f*x)^(3/2)/(b*f*sqrt(2)*sqrt(b*tan(e+f*x)^3))+1/2*log(1+sqrt(2)*sqrt(tan(e+f*x))+tan(e+f*x))*tan(e+f*x)^(3/2)/(b*f*sqrt(2)*sqrt(b*tan(e+f*x)^3))],
[1/(b*tan(e+f*x)^3)^(5/2),x,16,-2/5*cot(e+f*x)/(b^2*f*sqrt(b*tan(e+f*x)^3))+2/9*cot(e+f*x)^3/(b^2*f*sqrt(b*tan(e+f*x)^3))-2/13*cot(e+f*x)^5/(b^2*f*sqrt(b*tan(e+f*x)^3))+2*tan(e+f*x)/(b^2*f*sqrt(b*tan(e+f*x)^3))-arctan(1-sqrt(2)*sqrt(tan(e+f*x)))*tan(e+f*x)^(3/2)/(b^2*f*sqrt(2)*sqrt(b*tan(e+f*x)^3))+arctan(1+sqrt(2)*sqrt(tan(e+f*x)))*tan(e+f*x)^(3/2)/(b^2*f*sqrt(2)*sqrt(b*tan(e+f*x)^3))+1/2*log(1-sqrt(2)*sqrt(tan(e+f*x))+tan(e+f*x))*tan(e+f*x)^(3/2)/(b^2*f*sqrt(2)*sqrt(b*tan(e+f*x)^3))-1/2*log(1+sqrt(2)*sqrt(tan(e+f*x))+tan(e+f*x))*tan(e+f*x)^(3/2)/(b^2*f*sqrt(2)*sqrt(b*tan(e+f*x)^3))],

# Integrands of the form (b Tan[e+f x]^4)^(p/2)
[(b*tan(e+f*x)^4)^(5/2),x,7,b^2*cot(e+f*x)*sqrt(b*tan(e+f*x)^4)/f-b^2*x*cot(e+f*x)^2*sqrt(b*tan(e+f*x)^4)-1/3*b^2*sqrt(b*tan(e+f*x)^4)*tan(e+f*x)/f+1/5*b^2*sqrt(b*tan(e+f*x)^4)*tan(e+f*x)^3/f-1/7*b^2*sqrt(b*tan(e+f*x)^4)*tan(e+f*x)^5/f+1/9*b^2*sqrt(b*tan(e+f*x)^4)*tan(e+f*x)^7/f],
[(b*tan(e+f*x)^4)^(3/2),x,5,b*cot(e+f*x)*sqrt(b*tan(e+f*x)^4)/f-b*x*cot(e+f*x)^2*sqrt(b*tan(e+f*x)^4)-1/3*b*sqrt(b*tan(e+f*x)^4)*tan(e+f*x)/f+1/5*b*sqrt(b*tan(e+f*x)^4)*tan(e+f*x)^3/f],
[(b*tan(e+f*x)^4)^(1/2),x,3,cot(e+f*x)*sqrt(b*tan(e+f*x)^4)/f-x*cot(e+f*x)^2*sqrt(b*tan(e+f*x)^4)],
[1/(b*tan(e+f*x)^4)^(1/2),x,3,-tan(e+f*x)/(f*sqrt(b*tan(e+f*x)^4))-x*tan(e+f*x)^2/sqrt(b*tan(e+f*x)^4)],
[1/(b*tan(e+f*x)^4)^(3/2),x,5,1/3*cot(e+f*x)/(b*f*sqrt(b*tan(e+f*x)^4))-1/5*cot(e+f*x)^3/(b*f*sqrt(b*tan(e+f*x)^4))-tan(e+f*x)/(b*f*sqrt(b*tan(e+f*x)^4))-x*tan(e+f*x)^2/(b*sqrt(b*tan(e+f*x)^4))],
[1/(b*tan(e+f*x)^4)^(5/2),x,7,1/3*cot(e+f*x)/(b^2*f*sqrt(b*tan(e+f*x)^4))-1/5*cot(e+f*x)^3/(b^2*f*sqrt(b*tan(e+f*x)^4))+1/7*cot(e+f*x)^5/(b^2*f*sqrt(b*tan(e+f*x)^4))-1/9*cot(e+f*x)^7/(b^2*f*sqrt(b*tan(e+f*x)^4))-tan(e+f*x)/(b^2*f*sqrt(b*tan(e+f*x)^4))-x*tan(e+f*x)^2/(b^2*sqrt(b*tan(e+f*x)^4))],

# Integrands of the form (b (c Tan[e+f x])^n)^(p/2)
[(b*tan(e+f*x)^n)^(5/2),x,3,2*b^2*hypergeom([1,1/4*(2+5*n)],[1/4*(6+5*n)],-tan(e+f*x)^2)*sqrt(b*tan(e+f*x)^n)*tan(e+f*x)^(1+2*n)/(f*(2+5*n))],
[(b*tan(e+f*x)^n)^(3/2),x,3,2*b*hypergeom([1,1/4*(2+3*n)],[3/4*(2+n)],-tan(e+f*x)^2)*sqrt(b*tan(e+f*x)^n)*tan(e+f*x)^(1+n)/(f*(2+3*n))],
[(b*tan(e+f*x)^n)^(1/2),x,3,2*hypergeom([1,1/4*(2+n)],[1/4*(6+n)],-tan(e+f*x)^2)*sqrt(b*tan(e+f*x)^n)*tan(e+f*x)/(f*(2+n))],
[1/(b*tan(e+f*x)^n)^(1/2),x,3,2*hypergeom([1,1/4*(2-n)],[1/4*(6-n)],-tan(e+f*x)^2)*tan(e+f*x)/(f*(2-n)*sqrt(b*tan(e+f*x)^n))],
[1/(b*tan(e+f*x)^n)^(3/2),x,3,2*hypergeom([1,1/4*(2-3*n)],[3/4*(2-n)],-tan(e+f*x)^2)*tan(e+f*x)^(1-n)/(b*f*(2-3*n)*sqrt(b*tan(e+f*x)^n))],
[1/(b*tan(e+f*x)^n)^(5/2),x,3,2*hypergeom([1,1/4*(2-5*n)],[1/4*(6-5*n)],-tan(e+f*x)^2)*tan(e+f*x)^(1-2*n)/(b^2*f*(2-5*n)*sqrt(b*tan(e+f*x)^n))],

# Integrands of the form (b (c Tan[e+f x])^n)^p when p symbolic
[(b*tan(e+f*x)^n)^p,x,3,hypergeom([1,1/2*(1+n*p)],[1/2*(3+n*p)],-tan(e+f*x)^2)*tan(e+f*x)*(b*tan(e+f*x)^n)^p/(f*(1+n*p))],
[(b*tan(e+f*x)^2)^p,x,3,hypergeom([1,1/2*(1+2*p)],[1/2*(3+2*p)],-tan(e+f*x)^2)*tan(e+f*x)*(b*tan(e+f*x)^2)^p/(f*(1+2*p))],
[(b*tan(e+f*x)^3)^p,x,3,hypergeom([1,1/2*(1+3*p)],[3/2*(1+p)],-tan(e+f*x)^2)*tan(e+f*x)*(b*tan(e+f*x)^3)^p/(f*(1+3*p))],
[(b*tan(e+f*x)^4)^p,x,3,hypergeom([1,1/2*(1+4*p)],[1/2*(3+4*p)],-tan(e+f*x)^2)*tan(e+f*x)*(b*tan(e+f*x)^4)^p/(f*(1+4*p))],
[(b*tan(e+f*x)^n)^(1/n),x,2,-cot(e+f*x)*log(cos(e+f*x))*(b*tan(e+f*x)^n)^(1/n)/f],

# Integrands of the form (d Sin[e+f x])^m (a+b (c Tan[e+f x])^n)^p

# Integrands of the form (d Sin[e+f x])^m (a+b Tan[e+f x]^2)^p

# Integrands of the form Sin[e+f x]^m (a+b Tan[e+f x]^2)^p

# p>0
[sin(e+f*x)^5*(a+b*tan(e+f*x)^2),x,3,-(a-3*b)*cos(e+f*x)/f+1/3*(2*a-3*b)*cos(e+f*x)^3/f-1/5*(a-b)*cos(e+f*x)^5/f+b*sec(e+f*x)/f],
[sin(e+f*x)^3*(a+b*tan(e+f*x)^2),x,3,-(a-2*b)*cos(e+f*x)/f+1/3*(a-b)*cos(e+f*x)^3/f+b*sec(e+f*x)/f],
[sin(e+f*x)*(a+b*tan(e+f*x)^2),x,3,-(a-b)*cos(e+f*x)/f+b*sec(e+f*x)/f],
[csc(e+f*x)*(a+b*tan(e+f*x)^2),x,3,-a*arctanh(cos(e+f*x))/f+b*sec(e+f*x)/f],
[csc(e+f*x)^3*(a+b*tan(e+f*x)^2),x,4,-1/2*(a+2*b)*arctanh(cos(e+f*x))/f-1/2*a*cot(e+f*x)*csc(e+f*x)/f+b*sec(e+f*x)/f],
[csc(e+f*x)^5*(a+b*tan(e+f*x)^2),x,5,-3/8*(a+4*b)*arctanh(cos(e+f*x))/f-1/8*(5*a+4*b)*cot(e+f*x)*csc(e+f*x)/f-1/4*a*cot(e+f*x)^3*csc(e+f*x)/f+b*sec(e+f*x)/f],
[sin(e+f*x)^6*(a+b*tan(e+f*x)^2),x,6,5/16*(a-7*b)*x-1/16*(11*a-29*b)*cos(e+f*x)*sin(e+f*x)/f+1/24*(13*a-19*b)*cos(e+f*x)^3*sin(e+f*x)/f-1/6*(a-b)*cos(e+f*x)^5*sin(e+f*x)/f+b*tan(e+f*x)/f],
[sin(e+f*x)^4*(a+b*tan(e+f*x)^2),x,5,3/8*(a-5*b)*x-1/8*(5*a-9*b)*cos(e+f*x)*sin(e+f*x)/f+1/4*(a-b)*cos(e+f*x)^3*sin(e+f*x)/f+b*tan(e+f*x)/f],
[sin(e+f*x)^2*(a+b*tan(e+f*x)^2),x,4,1/2*(a-3*b)*x-1/2*(a-b)*cos(e+f*x)*sin(e+f*x)/f+b*tan(e+f*x)/f],
[a+b*tan(e+f*x)^2,x,3,a*x-b*x+b*tan(e+f*x)/f],
[csc(e+f*x)^2*(a+b*tan(e+f*x)^2),x,3,-a*cot(e+f*x)/f+b*tan(e+f*x)/f],
[csc(e+f*x)^4*(a+b*tan(e+f*x)^2),x,3,-(a+b)*cot(e+f*x)/f-1/3*a*cot(e+f*x)^3/f+b*tan(e+f*x)/f],
[csc(e+f*x)^6*(a+b*tan(e+f*x)^2),x,3,-(a+2*b)*cot(e+f*x)/f-1/3*(2*a+b)*cot(e+f*x)^3/f-1/5*a*cot(e+f*x)^5/f+b*tan(e+f*x)/f],
[sin(e+f*x)^5*(a+b*tan(e+f*x)^2)^2,x,3,-(a^2-6*a*b+6*b^2)*cos(e+f*x)/f+2/3*(a-2*b)*(a-b)*cos(e+f*x)^3/f-1/5*(a-b)^2*cos(e+f*x)^5/f+2*(a-2*b)*b*sec(e+f*x)/f+1/3*b^2*sec(e+f*x)^3/f],
[sin(e+f*x)^3*(a+b*tan(e+f*x)^2)^2,x,3,-(a-3*b)*(a-b)*cos(e+f*x)/f+1/3*(a-b)^2*cos(e+f*x)^3/f+(2*a-3*b)*b*sec(e+f*x)/f+1/3*b^2*sec(e+f*x)^3/f],
[sin(e+f*x)*(a+b*tan(e+f*x)^2)^2,x,3,-(a-b)^2*cos(e+f*x)/f+2*(a-b)*b*sec(e+f*x)/f+1/3*b^2*sec(e+f*x)^3/f],
[csc(e+f*x)*(a+b*tan(e+f*x)^2)^2,x,4,-a^2*arctanh(cos(e+f*x))/f+(2*a-b)*b*sec(e+f*x)/f+1/3*b^2*sec(e+f*x)^3/f],
[csc(e+f*x)^3*(a+b*tan(e+f*x)^2)^2,x,5,-1/2*a*(a+4*b)*arctanh(cos(e+f*x))/f+1/2*a*(a+4*b)*sec(e+f*x)/f-1/2*a^2*csc(e+f*x)^2*sec(e+f*x)/f+1/3*b^2*sec(e+f*x)^3/f],
[csc(e+f*x)^5*(a+b*tan(e+f*x)^2)^2,x,6,-1/8*(3*a^2+24*a*b+8*b^2)*arctanh(cos(e+f*x))/f-1/8*a*(a+8*b)*cot(e+f*x)*csc(e+f*x)/f+1/4*(a^2+8*a*b+4*b^2)*sec(e+f*x)/f-1/4*a^2*csc(e+f*x)^4*sec(e+f*x)/f+1/3*b^2*sec(e+f*x)^3/f],
[sin(e+f*x)^4*(a+b*tan(e+f*x)^2)^2,x,6,1/8*(3*a^2-30*a*b+35*b^2)*x-1/8*(a-9*b)*(a-b)*cos(e+f*x)*sin(e+f*x)/f-1/4*(a^2-10*a*b+13*b^2)*tan(e+f*x)/f+1/4*(a-b)^2*sin(e+f*x)^4*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],
[sin(e+f*x)^2*(a+b*tan(e+f*x)^2)^2,x,5,1/2*(a-5*b)*(a-b)*x-1/2*(a-5*b)*(a-b)*tan(e+f*x)/f+1/2*(a-b)^2*sin(e+f*x)^2*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],
[(a+b*tan(e+f*x)^2)^2,x,4,(a-b)^2*x+(2*a-b)*b*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],
[csc(e+f*x)^2*(a+b*tan(e+f*x)^2)^2,x,3,-a^2*cot(e+f*x)/f+2*a*b*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],
[csc(e+f*x)^4*(a+b*tan(e+f*x)^2)^2,x,3,-a*(a+2*b)*cot(e+f*x)/f-1/3*a^2*cot(e+f*x)^3/f+b*(2*a+b)*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],
[csc(e+f*x)^6*(a+b*tan(e+f*x)^2)^2,x,3,-(a^2+4*a*b+b^2)*cot(e+f*x)/f-2/3*a*(a+b)*cot(e+f*x)^3/f-1/5*a^2*cot(e+f*x)^5/f+2*b*(a+b)*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],

# p<0
[sin(e+f*x)^5/(a+b*tan(e+f*x)^2),x,4,-a^2*cos(e+f*x)/((a-b)^3*f)+1/3*(2*a-b)*cos(e+f*x)^3/((a-b)^2*f)-1/5*cos(e+f*x)^5/((a-b)*f)-a^2*arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/((a-b)^(7/2)*f)],
[sin(e+f*x)^3/(a+b*tan(e+f*x)^2),x,4,-a*cos(e+f*x)/((a-b)^2*f)+1/3*cos(e+f*x)^3/((a-b)*f)-a*arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/((a-b)^(5/2)*f)],
[sin(e+f*x)/(a+b*tan(e+f*x)^2),x,3,-cos(e+f*x)/((a-b)*f)-arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/((a-b)^(3/2)*f)],
[csc(e+f*x)/(a+b*tan(e+f*x)^2),x,4,-arctanh(cos(e+f*x))/(a*f)-arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/(a*f*sqrt(a-b))],
[csc(e+f*x)^3/(a+b*tan(e+f*x)^2),x,5,-1/2*(a-2*b)*arctanh(cos(e+f*x))/(a^2*f)-1/2*cot(e+f*x)*csc(e+f*x)/(a*f)-arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(a-b)*sqrt(b)/(a^2*f)],
[csc(e+f*x)^5/(a+b*tan(e+f*x)^2),x,6,-1/8*(3*a^2-12*a*b+8*b^2)*arctanh(cos(e+f*x))/(a^3*f)-1/8*(5*a-4*b)*cot(e+f*x)*csc(e+f*x)/(a^2*f)-1/4*cot(e+f*x)^3*csc(e+f*x)/(a*f)-(a-b)^(3/2)*arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/(a^3*f)],
[sin(e+f*x)^6/(a+b*tan(e+f*x)^2),x,7,1/16*(5*a^3+15*a^2*b-5*a*b^2+b^3)*x/(a-b)^4-1/16*(11*a^2-4*a*b+b^2)*cos(e+f*x)*sin(e+f*x)/((a-b)^3*f)+1/8*(3*a-b)*cos(e+f*x)^3*sin(e+f*x)/((a-b)^2*f)+1/6*cos(e+f*x)^3*sin(e+f*x)^3/((a-b)*f)-a^(5/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/((a-b)^4*f)],
[sin(e+f*x)^4/(a+b*tan(e+f*x)^2),x,6,1/8*(3*a^2+6*a*b-b^2)*x/(a-b)^3-1/8*(5*a-b)*cos(e+f*x)*sin(e+f*x)/((a-b)^2*f)+1/4*cos(e+f*x)^3*sin(e+f*x)/((a-b)*f)-a^(3/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/((a-b)^3*f)],
[sin(e+f*x)^2/(a+b*tan(e+f*x)^2),x,5,1/2*(a+b)*x/(a-b)^2-1/2*cos(e+f*x)*sin(e+f*x)/((a-b)*f)-arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(a)*sqrt(b)/((a-b)^2*f)],
[1/(a+b*tan(e+f*x)^2),x,3,x/(a-b)-arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/((a-b)*f*sqrt(a))],
[csc(e+f*x)^2/(a+b*tan(e+f*x)^2),x,3,-cot(e+f*x)/(a*f)-arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/(a^(3/2)*f)],
[csc(e+f*x)^4/(a+b*tan(e+f*x)^2),x,4,-(a-b)*cot(e+f*x)/(a^2*f)-1/3*cot(e+f*x)^3/(a*f)-(a-b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/(a^(5/2)*f)],
[csc(e+f*x)^6/(a+b*tan(e+f*x)^2),x,4,-(a-b)^2*cot(e+f*x)/(a^3*f)-1/3*(2*a-b)*cot(e+f*x)^3/(a^2*f)-1/5*cot(e+f*x)^5/(a*f)-(a-b)^2*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/(a^(7/2)*f)],
[sin(e+f*x)^5/(a+b*tan(e+f*x)^2)^2,x,6,-1/5*(5*a^2+10*a*b-b^2)*cos(e+f*x)/((a-b)^4*f)+1/15*(10*a-3*b)*cos(e+f*x)^3/((a-b)^3*f)-1/5*cos(e+f*x)^5/((a-b)*f*(a-b+b*sec(e+f*x)^2))-1/10*b*(5*a^2+2*b^2)*sec(e+f*x)/((a-b)^4*f*(a-b+b*sec(e+f*x)^2))-1/2*a*(3*a+4*b)*arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/((a-b)^(9/2)*f)],
[sin(e+f*x)^3/(a+b*tan(e+f*x)^2)^2,x,5,-(a+b)*cos(e+f*x)/((a-b)^3*f)+1/3*cos(e+f*x)^3/((a-b)^2*f)-1/2*a*b*sec(e+f*x)/((a-b)^3*f*(a-b+b*sec(e+f*x)^2))-1/2*(3*a+2*b)*arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/((a-b)^(7/2)*f)],
[sin(e+f*x)/(a+b*tan(e+f*x)^2)^2,x,4,-3/2*cos(e+f*x)/((a-b)^2*f)+1/2*cos(e+f*x)/((a-b)*f*(a-b+b*sec(e+f*x)^2))-3/2*arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/((a-b)^(5/2)*f)],
[csc(e+f*x)/(a+b*tan(e+f*x)^2)^2,x,5,-arctanh(cos(e+f*x))/(a^2*f)-1/2*b*sec(e+f*x)/(a*(a-b)*f*(a-b+b*sec(e+f*x)^2))-1/2*(3*a-2*b)*arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/(a^2*(a-b)^(3/2)*f)],
[csc(e+f*x)^3/(a+b*tan(e+f*x)^2)^2,x,6,-1/2*(a-4*b)*arctanh(cos(e+f*x))/(a^3*f)-1/2*cot(e+f*x)*csc(e+f*x)/(a*f*(a-b+b*sec(e+f*x)^2))-b*sec(e+f*x)/(a^2*f*(a-b+b*sec(e+f*x)^2))-1/2*(3*a-4*b)*arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/(a^3*f*sqrt(a-b))],
[csc(e+f*x)^5/(a+b*tan(e+f*x)^2)^2,x,7,-3/8*(a^2-8*a*b+8*b^2)*arctanh(cos(e+f*x))/(a^4*f)-1/8*(5*a-6*b)*cot(e+f*x)*csc(e+f*x)/(a^2*f*(a-b+b*sec(e+f*x)^2))-1/4*cot(e+f*x)^3*csc(e+f*x)/(a*f*(a-b+b*sec(e+f*x)^2))-3/8*(3*a-4*b)*b*sec(e+f*x)/(a^3*f*(a-b+b*sec(e+f*x)^2))-3/2*(a-2*b)*arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(a-b)*sqrt(b)/(a^4*f)],
[sin(e+f*x)^4/(a+b*tan(e+f*x)^2)^2,x,7,3/8*(a^2+6*a*b+b^2)*x/(a-b)^4-3/2*(a+b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(a)*sqrt(b)/((a-b)^4*f)-1/8*(5*a+b)*cos(e+f*x)*sin(e+f*x)/((a-b)^2*f*(a+b*tan(e+f*x)^2))+1/4*cos(e+f*x)^3*sin(e+f*x)/((a-b)*f*(a+b*tan(e+f*x)^2))-3/8*b*(3*a+b)*tan(e+f*x)/((a-b)^3*f*(a+b*tan(e+f*x)^2))],
[sin(e+f*x)^2/(a+b*tan(e+f*x)^2)^2,x,6,1/2*(a+3*b)*x/(a-b)^3-1/2*(3*a+b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/((a-b)^3*f*sqrt(a))-1/2*cos(e+f*x)*sin(e+f*x)/((a-b)*f*(a+b*tan(e+f*x)^2))-b*tan(e+f*x)/((a-b)^2*f*(a+b*tan(e+f*x)^2))],
[1/(a+b*tan(e+f*x)^2)^2,x,5,x/(a-b)^2-1/2*(3*a-b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/(a^(3/2)*(a-b)^2*f)-1/2*b*tan(e+f*x)/(a*(a-b)*f*(a+b*tan(e+f*x)^2))],
[csc(e+f*x)^2/(a+b*tan(e+f*x)^2)^2,x,4,-3/2*cot(e+f*x)/(a^2*f)-3/2*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/(a^(5/2)*f)+1/2*cot(e+f*x)/(a*f*(a+b*tan(e+f*x)^2))],
[csc(e+f*x)^4/(a+b*tan(e+f*x)^2)^2,x,5,-(a-2*b)*cot(e+f*x)/(a^3*f)-1/3*cot(e+f*x)^3/(a^2*f)-1/2*(3*a-5*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/(a^(7/2)*f)-1/2*(a-b)*b*tan(e+f*x)/(a^3*f*(a+b*tan(e+f*x)^2))],
[csc(e+f*x)^6/(a+b*tan(e+f*x)^2)^2,x,6,-1/5*(5*a^2-20*a*b+14*b^2)*cot(e+f*x)/(a^4*f)-1/15*(10*a-7*b)*cot(e+f*x)^3/(a^3*f)-1/2*(3*a-7*b)*(a-b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/(a^(9/2)*f)-1/5*cot(e+f*x)^5/(a*f*(a+b*tan(e+f*x)^2))-1/10*b*(5*a^2-10*a*b+7*b^2)*tan(e+f*x)/(a^4*f*(a+b*tan(e+f*x)^2))],
[sin(e+f*x)^5/(a+b*tan(e+f*x)^2)^3,x,7,-1/5*(5*a^2+20*a*b+2*b^2)*cos(e+f*x)/((a-b)^5*f)+1/15*(10*a-b)*cos(e+f*x)^3/((a-b)^4*f)-1/5*cos(e+f*x)^5/((a-b)*f*(a-b+b*sec(e+f*x)^2)^2)-1/20*b*(5*a^2+4*b^2)*sec(e+f*x)/((a-b)^4*f*(a-b+b*sec(e+f*x)^2)^2)-1/40*b*(35*a^2+40*a*b+24*b^2)*sec(e+f*x)/((a-b)^5*f*(a-b+b*sec(e+f*x)^2))-1/8*(15*a^2+40*a*b+8*b^2)*arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/((a-b)^(11/2)*f)],
[sin(e+f*x)^3/(a+b*tan(e+f*x)^2)^3,x,6,-(a+2*b)*cos(e+f*x)/((a-b)^4*f)+1/3*cos(e+f*x)^3/((a-b)^3*f)-1/4*a*b*sec(e+f*x)/((a-b)^3*f*(a-b+b*sec(e+f*x)^2)^2)-1/8*b*(7*a+4*b)*sec(e+f*x)/((a-b)^4*f*(a-b+b*sec(e+f*x)^2))-5/8*(3*a+4*b)*arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/((a-b)^(9/2)*f)],
[sin(e+f*x)/(a+b*tan(e+f*x)^2)^3,x,5,-15/8*cos(e+f*x)/((a-b)^3*f)+1/4*cos(e+f*x)/((a-b)*f*(a-b+b*sec(e+f*x)^2)^2)+5/8*cos(e+f*x)/((a-b)^2*f*(a-b+b*sec(e+f*x)^2))-15/8*arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/((a-b)^(7/2)*f)],
[csc(e+f*x)/(a+b*tan(e+f*x)^2)^3,x,6,-arctanh(cos(e+f*x))/(a^3*f)-1/4*b*sec(e+f*x)/(a*(a-b)*f*(a-b+b*sec(e+f*x)^2)^2)-1/8*(7*a-4*b)*b*sec(e+f*x)/(a^2*(a-b)^2*f*(a-b+b*sec(e+f*x)^2))-1/8*(15*a^2-20*a*b+8*b^2)*arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/(a^3*(a-b)^(5/2)*f)],
[csc(e+f*x)^3/(a+b*tan(e+f*x)^2)^3,x,7,-1/2*(a-6*b)*arctanh(cos(e+f*x))/(a^4*f)-1/2*cot(e+f*x)*csc(e+f*x)/(a*f*(a-b+b*sec(e+f*x)^2)^2)-3/4*b*sec(e+f*x)/(a^2*f*(a-b+b*sec(e+f*x)^2)^2)-1/8*(11*a-12*b)*b*sec(e+f*x)/(a^3*(a-b)*f*(a-b+b*sec(e+f*x)^2))-1/8*(15*a^2-40*a*b+24*b^2)*arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/(a^4*(a-b)^(3/2)*f)],
[csc(e+f*x)^5/(a+b*tan(e+f*x)^2)^3,x,8,-3/8*(a^2-12*a*b+16*b^2)*arctanh(cos(e+f*x))/(a^5*f)-1/8*(5*a-8*b)*cot(e+f*x)*csc(e+f*x)/(a^2*f*(a-b+b*sec(e+f*x)^2)^2)-1/4*cot(e+f*x)^3*csc(e+f*x)/(a*f*(a-b+b*sec(e+f*x)^2)^2)-1/8*(7*a-12*b)*b*sec(e+f*x)/(a^3*f*(a-b+b*sec(e+f*x)^2)^2)-3/2*(a-2*b)*b*sec(e+f*x)/(a^4*f*(a-b+b*sec(e+f*x)^2))-3/8*(5*a^2-20*a*b+16*b^2)*arctan(sec(e+f*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/(a^5*f*sqrt(a-b))],
[sin(e+f*x)^4/(a+b*tan(e+f*x)^2)^3,x,8,3/8*(a^2+10*a*b+5*b^2)*x/(a-b)^5-3/8*(5*a^2+10*a*b+b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/((a-b)^5*f*sqrt(a))-1/8*(5*a+3*b)*cos(e+f*x)*sin(e+f*x)/((a-b)^2*f*(a+b*tan(e+f*x)^2)^2)+1/4*cos(e+f*x)^3*sin(e+f*x)/((a-b)*f*(a+b*tan(e+f*x)^2)^2)-1/8*b*(7*a+5*b)*tan(e+f*x)/((a-b)^3*f*(a+b*tan(e+f*x)^2)^2)-3/2*b*(a+b)*tan(e+f*x)/((a-b)^4*f*(a+b*tan(e+f*x)^2))],
[sin(e+f*x)^2/(a+b*tan(e+f*x)^2)^3,x,7,1/2*(a+5*b)*x/(a-b)^4-1/8*(15*a^2+10*a*b-b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/(a^(3/2)*(a-b)^4*f)-1/2*cos(e+f*x)*sin(e+f*x)/((a-b)*f*(a+b*tan(e+f*x)^2)^2)-3/4*b*tan(e+f*x)/((a-b)^2*f*(a+b*tan(e+f*x)^2)^2)-1/8*b*(11*a+b)*tan(e+f*x)/(a*(a-b)^3*f*(a+b*tan(e+f*x)^2))],
[1/(a+b*tan(e+f*x)^2)^3,x,6,x/(a-b)^3-1/8*(15*a^2-10*a*b+3*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/(a^(5/2)*(a-b)^3*f)-1/4*b*tan(e+f*x)/(a*(a-b)*f*(a+b*tan(e+f*x)^2)^2)-1/8*(7*a-3*b)*b*tan(e+f*x)/(a^2*(a-b)^2*f*(a+b*tan(e+f*x)^2))],
[csc(e+f*x)^2/(a+b*tan(e+f*x)^2)^3,x,5,-15/8*cot(e+f*x)/(a^3*f)-15/8*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/(a^(7/2)*f)+1/4*cot(e+f*x)/(a*f*(a+b*tan(e+f*x)^2)^2)+5/8*cot(e+f*x)/(a^2*f*(a+b*tan(e+f*x)^2))],
[csc(e+f*x)^4/(a+b*tan(e+f*x)^2)^3,x,6,-(a-3*b)*cot(e+f*x)/(a^4*f)-1/3*cot(e+f*x)^3/(a^3*f)-5/8*(3*a-7*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/(a^(9/2)*f)-1/4*(a-b)*b*tan(e+f*x)/(a^3*f*(a+b*tan(e+f*x)^2)^2)-1/8*(7*a-11*b)*b*tan(e+f*x)/(a^4*f*(a+b*tan(e+f*x)^2))],
[csc(e+f*x)^6/(a+b*tan(e+f*x)^2)^3,x,7,-1/5*(5*a^2-30*a*b+27*b^2)*cot(e+f*x)/(a^5*f)-1/15*(10*a-9*b)*cot(e+f*x)^3/(a^4*f)-1/8*(15*a^2-70*a*b+63*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/(a^(11/2)*f)-1/5*cot(e+f*x)^5/(a*f*(a+b*tan(e+f*x)^2)^2)-1/20*b*(5*a^2-10*a*b+9*b^2)*tan(e+f*x)/(a^4*f*(a+b*tan(e+f*x)^2)^2)-1/40*b*(35*a^2-110*a*b+99*b^2)*tan(e+f*x)/(a^5*f*(a+b*tan(e+f*x)^2))],

# Integrands of the form Sin[e+f x]^m (a+b Tan[e+f x]^2)^(p/2)

# p>0
[sin(e+f*x)^5*sqrt(a+b*tan(e+f*x)^2),x,6,2/15*(5*a-4*b)*cos(e+f*x)^3*(a-b+b*sec(e+f*x)^2)^(3/2)/((a-b)^2*f)-1/5*cos(e+f*x)^5*(a-b+b*sec(e+f*x)^2)^(3/2)/((a-b)*f)+arctanh(sec(e+f*x)*sqrt(b)/sqrt(a-b+b*sec(e+f*x)^2))*sqrt(b)/f-cos(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/f],
[sin(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2),x,5,1/3*cos(e+f*x)^3*(a-b+b*sec(e+f*x)^2)^(3/2)/((a-b)*f)+arctanh(sec(e+f*x)*sqrt(b)/sqrt(a-b+b*sec(e+f*x)^2))*sqrt(b)/f-cos(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/f],
[sin(e+f*x)*sqrt(a+b*tan(e+f*x)^2),x,4,arctanh(sec(e+f*x)*sqrt(b)/sqrt(a-b+b*sec(e+f*x)^2))*sqrt(b)/f-cos(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/f],
[csc(e+f*x)*sqrt(a+b*tan(e+f*x)^2),x,6,-arctanh(sec(e+f*x)*sqrt(a)/sqrt(a-b+b*sec(e+f*x)^2))*sqrt(a)/f+arctanh(sec(e+f*x)*sqrt(b)/sqrt(a-b+b*sec(e+f*x)^2))*sqrt(b)/f],
[csc(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2),x,7,-1/2*(a+b)*arctanh(sec(e+f*x)*sqrt(a)/sqrt(a-b+b*sec(e+f*x)^2))/(f*sqrt(a))+arctanh(sec(e+f*x)*sqrt(b)/sqrt(a-b+b*sec(e+f*x)^2))*sqrt(b)/f-1/2*cot(e+f*x)*csc(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/f],
[csc(e+f*x)^5*sqrt(a+b*tan(e+f*x)^2),x,8,-1/8*(3*a^2+6*a*b-b^2)*arctanh(sec(e+f*x)*sqrt(a)/sqrt(a-b+b*sec(e+f*x)^2))/(a^(3/2)*f)+arctanh(sec(e+f*x)*sqrt(b)/sqrt(a-b+b*sec(e+f*x)^2))*sqrt(b)/f-1/8*(3*a+b)*cot(e+f*x)*csc(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/(a*f)-1/4*cot(e+f*x)*csc(e+f*x)^3*sqrt(a-b+b*sec(e+f*x)^2)/f],
[sin(e+f*x)^4*sqrt(a+b*tan(e+f*x)^2),x,8,1/8*(3*a^2-12*a*b+8*b^2)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(3/2)*f)+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(b)/f-1/8*(3*a-4*b)*cos(e+f*x)*sin(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/((a-b)*f)-1/4*cos(e+f*x)*sin(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2)/f],
[sin(e+f*x)^2*sqrt(a+b*tan(e+f*x)^2),x,7,1/2*(a-2*b)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(f*sqrt(a-b))+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(b)/f-1/2*cos(e+f*x)*sin(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/f],
[sqrt(a+b*tan(e+f*x)^2),x,6,arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(a-b)/f+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(b)/f],
[csc(e+f*x)^2*sqrt(a+b*tan(e+f*x)^2),x,4,arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(b)/f-cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/f],
[csc(e+f*x)^4*sqrt(a+b*tan(e+f*x)^2),x,5,arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(b)/f-cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/f-1/3*cot(e+f*x)^3*(a+b*tan(e+f*x)^2)^(3/2)/(a*f)],
[csc(e+f*x)^6*sqrt(a+b*tan(e+f*x)^2),x,6,arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(b)/f-cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/f-2/15*(5*a-b)*cot(e+f*x)^3*(a+b*tan(e+f*x)^2)^(3/2)/(a^2*f)-1/5*cot(e+f*x)^5*(a+b*tan(e+f*x)^2)^(3/2)/(a*f)],
[sin(e+f*x)^5*(a+b*tan(e+f*x)^2)^(3/2),x,7,-1/3*(3*a-7*b)*cos(e+f*x)*(a-b+b*sec(e+f*x)^2)^(3/2)/((a-b)*f)+2/3*cos(e+f*x)^3*(a-b+b*sec(e+f*x)^2)^(5/2)/((a-b)*f)-1/5*cos(e+f*x)^5*(a-b+b*sec(e+f*x)^2)^(5/2)/((a-b)*f)+1/2*(3*a-7*b)*arctanh(sec(e+f*x)*sqrt(b)/sqrt(a-b+b*sec(e+f*x)^2))*sqrt(b)/f+1/2*(3*a-7*b)*b*sec(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/((a-b)*f)],
[sin(e+f*x)^3*(a+b*tan(e+f*x)^2)^(3/2),x,6,-1/3*(3*a-5*b)*cos(e+f*x)*(a-b+b*sec(e+f*x)^2)^(3/2)/((a-b)*f)+1/3*cos(e+f*x)^3*(a-b+b*sec(e+f*x)^2)^(5/2)/((a-b)*f)+1/2*(3*a-5*b)*arctanh(sec(e+f*x)*sqrt(b)/sqrt(a-b+b*sec(e+f*x)^2))*sqrt(b)/f+1/2*(3*a-5*b)*b*sec(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/((a-b)*f)],
[sin(e+f*x)*(a+b*tan(e+f*x)^2)^(3/2),x,5,-cos(e+f*x)*(a-b+b*sec(e+f*x)^2)^(3/2)/f+3/2*(a-b)*arctanh(sec(e+f*x)*sqrt(b)/sqrt(a-b+b*sec(e+f*x)^2))*sqrt(b)/f+3/2*b*sec(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/f],
[csc(e+f*x)*(a+b*tan(e+f*x)^2)^(3/2),x,7,-a^(3/2)*arctanh(sec(e+f*x)*sqrt(a)/sqrt(a-b+b*sec(e+f*x)^2))/f+1/2*(3*a-b)*arctanh(sec(e+f*x)*sqrt(b)/sqrt(a-b+b*sec(e+f*x)^2))*sqrt(b)/f+1/2*b*sec(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/f],
[csc(e+f*x)^3*(a+b*tan(e+f*x)^2)^(3/2),x,8,-1/2*cot(e+f*x)*csc(e+f*x)*(a-b+b*sec(e+f*x)^2)^(3/2)/f-1/2*(a+3*b)*arctanh(sec(e+f*x)*sqrt(a)/sqrt(a-b+b*sec(e+f*x)^2))*sqrt(a)/f+1/2*(3*a+b)*arctanh(sec(e+f*x)*sqrt(b)/sqrt(a-b+b*sec(e+f*x)^2))*sqrt(b)/f+b*sec(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/f],
[csc(e+f*x)^5*(a+b*tan(e+f*x)^2)^(3/2),x,9,-1/4*cot(e+f*x)*csc(e+f*x)^3*(a-b+b*sec(e+f*x)^2)^(3/2)/f-3/8*(a^2+6*a*b+b^2)*arctanh(sec(e+f*x)*sqrt(a)/sqrt(a-b+b*sec(e+f*x)^2))/(f*sqrt(a))+3/2*(a+b)*arctanh(sec(e+f*x)*sqrt(b)/sqrt(a-b+b*sec(e+f*x)^2))*sqrt(b)/f+3/8*(a+3*b)*sec(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/f-3/8*(a+b)*csc(e+f*x)^2*sec(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/f],
[sin(e+f*x)^4*(a+b*tan(e+f*x)^2)^(3/2),x,9,3/8*(a^2-8*a*b+8*b^2)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(f*sqrt(a-b))+3/2*(a-2*b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(b)/f-3/8*(a-4*b)*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/f+3/8*(a-2*b)*sin(e+f*x)^2*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/f-1/4*cos(e+f*x)*sin(e+f*x)^3*(a+b*tan(e+f*x)^2)^(3/2)/f],
[sin(e+f*x)^2*(a+b*tan(e+f*x)^2)^(3/2),x,8,1/2*(a-4*b)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(a-b)/f+1/2*(3*a-4*b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(b)/f+b*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/f-1/2*cos(e+f*x)*sin(e+f*x)*(a+b*tan(e+f*x)^2)^(3/2)/f],
[(a+b*tan(e+f*x)^2)^(3/2),x,7,(a-b)^(3/2)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/f+1/2*(3*a-2*b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(b)/f+1/2*b*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/f],
[csc(e+f*x)^2*(a+b*tan(e+f*x)^2)^(3/2),x,5,3/2*a*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(b)/f+3/2*b*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/f-cot(e+f*x)*(a+b*tan(e+f*x)^2)^(3/2)/f],
[csc(e+f*x)^4*(a+b*tan(e+f*x)^2)^(3/2),x,6,1/2*(3*a+2*b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(b)/f+1/2*b*(3*a+2*b)*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/(a*f)-1/3*(3*a+2*b)*cot(e+f*x)*(a+b*tan(e+f*x)^2)^(3/2)/(a*f)-1/3*cot(e+f*x)^3*(a+b*tan(e+f*x)^2)^(5/2)/(a*f)],
[csc(e+f*x)^6*(a+b*tan(e+f*x)^2)^(3/2),x,7,1/2*(3*a+4*b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(b)/f+1/2*b*(3*a+4*b)*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/(a*f)-1/3*(3*a+4*b)*cot(e+f*x)*(a+b*tan(e+f*x)^2)^(3/2)/(a*f)-2/3*cot(e+f*x)^3*(a+b*tan(e+f*x)^2)^(5/2)/(a*f)-1/5*cot(e+f*x)^5*(a+b*tan(e+f*x)^2)^(5/2)/(a*f)],

# p<0
[sin(e+f*x)^5/sqrt(a+b*tan(e+f*x)^2),x,4,-1/15*(15*a^2-10*a*b+3*b^2)*cos(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/((a-b)^3*f)+2/15*(5*a-3*b)*cos(e+f*x)^3*sqrt(a-b+b*sec(e+f*x)^2)/((a-b)^2*f)-1/5*cos(e+f*x)^5*sqrt(a-b+b*sec(e+f*x)^2)/((a-b)*f)],
[sin(e+f*x)^3/sqrt(a+b*tan(e+f*x)^2),x,3,-1/3*(3*a-b)*cos(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/((a-b)^2*f)+1/3*cos(e+f*x)^3*sqrt(a-b+b*sec(e+f*x)^2)/((a-b)*f)],
[sin(e+f*x)/sqrt(a+b*tan(e+f*x)^2),x,2,-cos(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/((a-b)*f)],
[csc(e+f*x)/sqrt(a+b*tan(e+f*x)^2),x,3,-arctanh(sec(e+f*x)*sqrt(a)/sqrt(a-b+b*sec(e+f*x)^2))/(f*sqrt(a))],
[csc(e+f*x)^3/sqrt(a+b*tan(e+f*x)^2),x,5,-1/2*(a-b)*arctanh(sec(e+f*x)*sqrt(a)/sqrt(a-b+b*sec(e+f*x)^2))/(a^(3/2)*f)-1/2*cot(e+f*x)*csc(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/(a*f)],
[csc(e+f*x)^5/sqrt(a+b*tan(e+f*x)^2),x,6,-3/8*(a-b)^2*arctanh(sec(e+f*x)*sqrt(a)/sqrt(a-b+b*sec(e+f*x)^2))/(a^(5/2)*f)-1/8*(5*a-3*b)*cot(e+f*x)*csc(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/(a^2*f)-1/4*cot(e+f*x)^3*csc(e+f*x)*sqrt(a-b+b*sec(e+f*x)^2)/(a*f)],
[sin(e+f*x)^4/sqrt(a+b*tan(e+f*x)^2),x,6,3/8*a^2*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(5/2)*f)-1/8*(5*a-2*b)*cos(e+f*x)*sin(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/((a-b)^2*f)+1/4*cos(e+f*x)^3*sin(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/((a-b)*f)],
[sin(e+f*x)^2/sqrt(a+b*tan(e+f*x)^2),x,5,1/2*a*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(3/2)*f)-1/2*cos(e+f*x)*sin(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/((a-b)*f)],
[1/sqrt(a+b*tan(e+f*x)^2),x,3,arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(f*sqrt(a-b))],
[csc(e+f*x)^2/sqrt(a+b*tan(e+f*x)^2),x,2,-cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/(a*f)],
[csc(e+f*x)^4/sqrt(a+b*tan(e+f*x)^2),x,3,-1/3*(3*a-2*b)*cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/(a^2*f)-1/3*cot(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2)/(a*f)],
[csc(e+f*x)^6/sqrt(a+b*tan(e+f*x)^2),x,4,-1/15*(15*a^2-20*a*b+8*b^2)*cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/(a^3*f)-2/15*(5*a-2*b)*cot(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2)/(a^2*f)-1/5*cot(e+f*x)^5*sqrt(a+b*tan(e+f*x)^2)/(a*f)],
[sin(e+f*x)^5/(a+b*tan(e+f*x)^2)^(3/2),x,5,-1/15*(15*a^2+10*a*b-b^2)*cos(e+f*x)/((a-b)^3*f*sqrt(a-b+b*sec(e+f*x)^2))+2/15*(5*a-2*b)*cos(e+f*x)^3/((a-b)^2*f*sqrt(a-b+b*sec(e+f*x)^2))-1/5*cos(e+f*x)^5/((a-b)*f*sqrt(a-b+b*sec(e+f*x)^2))-2/15*b*(15*a^2+10*a*b-b^2)*sec(e+f*x)/((a-b)^4*f*sqrt(a-b+b*sec(e+f*x)^2))],
[sin(e+f*x)^3/(a+b*tan(e+f*x)^2)^(3/2),x,4,-1/3*(3*a+b)*cos(e+f*x)/((a-b)^2*f*sqrt(a-b+b*sec(e+f*x)^2))+1/3*cos(e+f*x)^3/((a-b)*f*sqrt(a-b+b*sec(e+f*x)^2))-2/3*b*(3*a+b)*sec(e+f*x)/((a-b)^3*f*sqrt(a-b+b*sec(e+f*x)^2))],
[sin(e+f*x)/(a+b*tan(e+f*x)^2)^(3/2),x,3,-cos(e+f*x)/((a-b)*f*sqrt(a-b+b*sec(e+f*x)^2))-2*b*sec(e+f*x)/((a-b)^2*f*sqrt(a-b+b*sec(e+f*x)^2))],
[csc(e+f*x)/(a+b*tan(e+f*x)^2)^(3/2),x,4,-arctanh(sec(e+f*x)*sqrt(a)/sqrt(a-b+b*sec(e+f*x)^2))/(a^(3/2)*f)-b*sec(e+f*x)/(a*(a-b)*f*sqrt(a-b+b*sec(e+f*x)^2))],
[csc(e+f*x)^3/(a+b*tan(e+f*x)^2)^(3/2),x,6,-1/2*(a-3*b)*arctanh(sec(e+f*x)*sqrt(a)/sqrt(a-b+b*sec(e+f*x)^2))/(a^(5/2)*f)-1/2*cot(e+f*x)*csc(e+f*x)/(a*f*sqrt(a-b+b*sec(e+f*x)^2))-3/2*b*sec(e+f*x)/(a^2*f*sqrt(a-b+b*sec(e+f*x)^2))],
[csc(e+f*x)^5/(a+b*tan(e+f*x)^2)^(3/2),x,7,-3/8*(a-5*b)*(a-b)*arctanh(sec(e+f*x)*sqrt(a)/sqrt(a-b+b*sec(e+f*x)^2))/(a^(7/2)*f)-5/8*(a-b)*cot(e+f*x)*csc(e+f*x)/(a^2*f*sqrt(a-b+b*sec(e+f*x)^2))-1/4*cot(e+f*x)^3*csc(e+f*x)/(a*f*sqrt(a-b+b*sec(e+f*x)^2))-1/8*(13*a-15*b)*b*sec(e+f*x)/(a^3*f*sqrt(a-b+b*sec(e+f*x)^2))],
[sin(e+f*x)^4/(a+b*tan(e+f*x)^2)^(3/2),x,7,3/8*a*(a+4*b)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(7/2)*f)-5/8*a*cos(e+f*x)*sin(e+f*x)/((a-b)^2*f*sqrt(a+b*tan(e+f*x)^2))+1/4*cos(e+f*x)^3*sin(e+f*x)/((a-b)*f*sqrt(a+b*tan(e+f*x)^2))-1/8*b*(13*a+2*b)*tan(e+f*x)/((a-b)^3*f*sqrt(a+b*tan(e+f*x)^2))],
[sin(e+f*x)^2/(a+b*tan(e+f*x)^2)^(3/2),x,6,1/2*(a+2*b)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(5/2)*f)-1/2*cos(e+f*x)*sin(e+f*x)/((a-b)*f*sqrt(a+b*tan(e+f*x)^2))-3/2*b*tan(e+f*x)/((a-b)^2*f*sqrt(a+b*tan(e+f*x)^2))],
[1/(a+b*tan(e+f*x)^2)^(3/2),x,4,arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(3/2)*f)-b*tan(e+f*x)/(a*(a-b)*f*sqrt(a+b*tan(e+f*x)^2))],
[csc(e+f*x)^2/(a+b*tan(e+f*x)^2)^(3/2),x,3,-cot(e+f*x)/(a*f*sqrt(a+b*tan(e+f*x)^2))-2*b*tan(e+f*x)/(a^2*f*sqrt(a+b*tan(e+f*x)^2))],
[csc(e+f*x)^4/(a+b*tan(e+f*x)^2)^(3/2),x,4,-1/3*(3*a-4*b)*cot(e+f*x)/(a^2*f*sqrt(a+b*tan(e+f*x)^2))-1/3*cot(e+f*x)^3/(a*f*sqrt(a+b*tan(e+f*x)^2))-2/3*(3*a-4*b)*b*tan(e+f*x)/(a^3*f*sqrt(a+b*tan(e+f*x)^2))],
[csc(e+f*x)^6/(a+b*tan(e+f*x)^2)^(3/2),x,5,-1/15*(15*a^2-40*a*b+24*b^2)*cot(e+f*x)/(a^3*f*sqrt(a+b*tan(e+f*x)^2))-2/15*(5*a-3*b)*cot(e+f*x)^3/(a^2*f*sqrt(a+b*tan(e+f*x)^2))-1/5*cot(e+f*x)^5/(a*f*sqrt(a+b*tan(e+f*x)^2))-2/15*b*(15*a^2-40*a*b+24*b^2)*tan(e+f*x)/(a^4*f*sqrt(a+b*tan(e+f*x)^2))],
[sin(e+f*x)^5/(a+b*tan(e+f*x)^2)^(5/2),x,6,-1/5*(5*a^2+10*a*b+b^2)*cos(e+f*x)/((a-b)^3*f*(a-b+b*sec(e+f*x)^2)^(3/2))+2/15*(5*a-b)*cos(e+f*x)^3/((a-b)^2*f*(a-b+b*sec(e+f*x)^2)^(3/2))-1/5*cos(e+f*x)^5/((a-b)*f*(a-b+b*sec(e+f*x)^2)^(3/2))-4/15*b*(5*a^2+10*a*b+b^2)*sec(e+f*x)/((a-b)^4*f*(a-b+b*sec(e+f*x)^2)^(3/2))-8/15*b*(5*a^2+10*a*b+b^2)*sec(e+f*x)/((a-b)^5*f*sqrt(a-b+b*sec(e+f*x)^2))],
[sin(e+f*x)^3/(a+b*tan(e+f*x)^2)^(5/2),x,5,-(a+b)*cos(e+f*x)/((a-b)^2*f*(a-b+b*sec(e+f*x)^2)^(3/2))+1/3*cos(e+f*x)^3/((a-b)*f*(a-b+b*sec(e+f*x)^2)^(3/2))-4/3*b*(a+b)*sec(e+f*x)/((a-b)^3*f*(a-b+b*sec(e+f*x)^2)^(3/2))-8/3*b*(a+b)*sec(e+f*x)/((a-b)^4*f*sqrt(a-b+b*sec(e+f*x)^2))],
[sin(e+f*x)/(a+b*tan(e+f*x)^2)^(5/2),x,4,-cos(e+f*x)/((a-b)*f*(a-b+b*sec(e+f*x)^2)^(3/2))-4/3*b*sec(e+f*x)/((a-b)^2*f*(a-b+b*sec(e+f*x)^2)^(3/2))-8/3*b*sec(e+f*x)/((a-b)^3*f*sqrt(a-b+b*sec(e+f*x)^2))],
[csc(e+f*x)/(a+b*tan(e+f*x)^2)^(5/2),x,6,-arctanh(sec(e+f*x)*sqrt(a)/sqrt(a-b+b*sec(e+f*x)^2))/(a^(5/2)*f)-1/3*b*sec(e+f*x)/(a*(a-b)*f*(a-b+b*sec(e+f*x)^2)^(3/2))-1/3*(5*a-3*b)*b*sec(e+f*x)/(a^2*(a-b)^2*f*sqrt(a-b+b*sec(e+f*x)^2))],
[csc(e+f*x)^3/(a+b*tan(e+f*x)^2)^(5/2),x,7,-1/2*(a-5*b)*arctanh(sec(e+f*x)*sqrt(a)/sqrt(a-b+b*sec(e+f*x)^2))/(a^(7/2)*f)-1/2*cot(e+f*x)*csc(e+f*x)/(a*f*(a-b+b*sec(e+f*x)^2)^(3/2))-5/6*b*sec(e+f*x)/(a^2*f*(a-b+b*sec(e+f*x)^2)^(3/2))-1/6*(13*a-15*b)*b*sec(e+f*x)/(a^3*(a-b)*f*sqrt(a-b+b*sec(e+f*x)^2))],
[csc(e+f*x)^5/(a+b*tan(e+f*x)^2)^(5/2),x,8,-1/8*(3*a^2-30*a*b+35*b^2)*arctanh(sec(e+f*x)*sqrt(a)/sqrt(a-b+b*sec(e+f*x)^2))/(a^(9/2)*f)-1/8*(5*a-7*b)*cot(e+f*x)*csc(e+f*x)/(a^2*f*(a-b+b*sec(e+f*x)^2)^(3/2))-1/4*cot(e+f*x)^3*csc(e+f*x)/(a*f*(a-b+b*sec(e+f*x)^2)^(3/2))-1/24*(23*a-35*b)*b*sec(e+f*x)/(a^3*f*(a-b+b*sec(e+f*x)^2)^(3/2))-5/24*(11*a-21*b)*b*sec(e+f*x)/(a^4*f*sqrt(a-b+b*sec(e+f*x)^2))],
[sin(e+f*x)^4/(a+b*tan(e+f*x)^2)^(5/2),x,8,1/8*(3*a^2+24*a*b+8*b^2)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(9/2)*f)-5/24*b*(11*a+10*b)*tan(e+f*x)/((a-b)^4*f*sqrt(a+b*tan(e+f*x)^2))-1/8*(5*a+2*b)*cos(e+f*x)*sin(e+f*x)/((a-b)^2*f*(a+b*tan(e+f*x)^2)^(3/2))+1/4*cos(e+f*x)^3*sin(e+f*x)/((a-b)*f*(a+b*tan(e+f*x)^2)^(3/2))-1/24*b*(23*a+12*b)*tan(e+f*x)/((a-b)^3*f*(a+b*tan(e+f*x)^2)^(3/2))],
[sin(e+f*x)^2/(a+b*tan(e+f*x)^2)^(5/2),x,7,1/2*(a+4*b)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(7/2)*f)-1/6*b*(13*a+2*b)*tan(e+f*x)/(a*(a-b)^3*f*sqrt(a+b*tan(e+f*x)^2))-1/2*cos(e+f*x)*sin(e+f*x)/((a-b)*f*(a+b*tan(e+f*x)^2)^(3/2))-5/6*b*tan(e+f*x)/((a-b)^2*f*(a+b*tan(e+f*x)^2)^(3/2))],
[1/(a+b*tan(e+f*x)^2)^(5/2),x,6,arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(5/2)*f)-1/3*(5*a-2*b)*b*tan(e+f*x)/(a^2*(a-b)^2*f*sqrt(a+b*tan(e+f*x)^2))-1/3*b*tan(e+f*x)/(a*(a-b)*f*(a+b*tan(e+f*x)^2)^(3/2))],
[csc(e+f*x)^2/(a+b*tan(e+f*x)^2)^(5/2),x,4,-8/3*b*tan(e+f*x)/(a^3*f*sqrt(a+b*tan(e+f*x)^2))-cot(e+f*x)/(a*f*(a+b*tan(e+f*x)^2)^(3/2))-4/3*b*tan(e+f*x)/(a^2*f*(a+b*tan(e+f*x)^2)^(3/2))],
[csc(e+f*x)^4/(a+b*tan(e+f*x)^2)^(5/2),x,5,-8/3*(a-2*b)*b*tan(e+f*x)/(a^4*f*sqrt(a+b*tan(e+f*x)^2))-(a-2*b)*cot(e+f*x)/(a^2*f*(a+b*tan(e+f*x)^2)^(3/2))-1/3*cot(e+f*x)^3/(a*f*(a+b*tan(e+f*x)^2)^(3/2))-4/3*(a-2*b)*b*tan(e+f*x)/(a^3*f*(a+b*tan(e+f*x)^2)^(3/2))],
[csc(e+f*x)^6/(a+b*tan(e+f*x)^2)^(5/2),x,6,-8/15*b*(5*a^2-20*a*b+16*b^2)*tan(e+f*x)/(a^5*f*sqrt(a+b*tan(e+f*x)^2))-1/5*(5*a^2-20*a*b+16*b^2)*cot(e+f*x)/(a^3*f*(a+b*tan(e+f*x)^2)^(3/2))-2/15*(5*a-4*b)*cot(e+f*x)^3/(a^2*f*(a+b*tan(e+f*x)^2)^(3/2))-1/5*cot(e+f*x)^5/(a*f*(a+b*tan(e+f*x)^2)^(3/2))-4/15*b*(5*a^2-20*a*b+16*b^2)*tan(e+f*x)/(a^4*f*(a+b*tan(e+f*x)^2)^(3/2))],

# Integrands of the form Sin[e+f x]^m (a+b Tan[e+f x]^2)^p when p symbolic
[(d*sin(e+f*x))^m*(b*tan(e+f*x)^2)^p,x,3,(cos(e+f*x)^2)^(1/2+p)*hypergeom([1/2*(1+2*p),1/2*(1+m+2*p)],[1/2*(3+m+2*p)],sin(e+f*x)^2)*(d*sin(e+f*x))^m*tan(e+f*x)*(b*tan(e+f*x)^2)^p/(f*(1+m+2*p))],
[(d*sin(e+f*x))^m*(a+b*tan(e+f*x)^2)^p,x,3,AppellF1(1/2*(1+m),1/2*(2+m),-p,1/2*(3+m),-tan(e+f*x)^2,-b*tan(e+f*x)^2/a)*(sec(e+f*x)^2)^(1/2*m)*(d*sin(e+f*x))^m*tan(e+f*x)*(a+b*tan(e+f*x)^2)^p/(f*(1+m)*(1+b*tan(e+f*x)^2/a)^p)],
[sin(e+f*x)^5*(a+b*tan(e+f*x)^2)^p,x,5,1/15*(10*a-7*b-2*b*p)*cos(e+f*x)^3*(a-b+b*sec(e+f*x)^2)^(1+p)/((a-b)^2*f)-1/5*cos(e+f*x)^5*(a-b+b*sec(e+f*x)^2)^(1+p)/((a-b)*f)-1/15*(15*a^2-20*a*b*(1+p)+4*b^2*(2+3*p+p^2))*cos(e+f*x)*hypergeom([-1/2,-p],[1/2],-b*sec(e+f*x)^2/(a-b))*(a-b+b*sec(e+f*x)^2)^p/((a-b)^2*f*(1+b*sec(e+f*x)^2/(a-b))^p)],
[sin(e+f*x)^3*(a+b*tan(e+f*x)^2)^p,x,4,1/3*cos(e+f*x)^3*(a-b+b*sec(e+f*x)^2)^(1+p)/((a-b)*f)-1/3*(3*a-2*b*(1+p))*cos(e+f*x)*hypergeom([-1/2,-p],[1/2],-b*sec(e+f*x)^2/(a-b))*(a-b+b*sec(e+f*x)^2)^p/((a-b)*f*(1+b*sec(e+f*x)^2/(a-b))^p)],
[sin(e+f*x)*(a+b*tan(e+f*x)^2)^p,x,3,-cos(e+f*x)*hypergeom([-1/2,-p],[1/2],-b*sec(e+f*x)^2/(a-b))*(a-b+b*sec(e+f*x)^2)^p/(f*(1+b*sec(e+f*x)^2/(a-b))^p)],
[csc(e+f*x)*(a+b*tan(e+f*x)^2)^p,x,3,-AppellF1(1/2,1,-p,3/2,sec(e+f*x)^2,-b*sec(e+f*x)^2/(a-b))*sec(e+f*x)*(a-b+b*sec(e+f*x)^2)^p/(f*(1+b*sec(e+f*x)^2/(a-b))^p)],
[csc(e+f*x)^3*(a+b*tan(e+f*x)^2)^p,x,3,1/3*AppellF1(3/2,2,-p,5/2,sec(e+f*x)^2,-b*sec(e+f*x)^2/(a-b))*sec(e+f*x)^3*(a-b+b*sec(e+f*x)^2)^p/(f*(1+b*sec(e+f*x)^2/(a-b))^p)],
[sin(e+f*x)^2*(a+b*tan(e+f*x)^2)^p,x,3,1/3*AppellF1(3/2,2,-p,5/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/a)*tan(e+f*x)^3*(a+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/a)^p)],
[(a+b*tan(e+f*x)^2)^p,x,3,AppellF1(1/2,1,-p,3/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/a)*tan(e+f*x)*(a+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/a)^p)],
[csc(e+f*x)^2*(a+b*tan(e+f*x)^2)^p,x,3,-cot(e+f*x)*hypergeom([-1/2,-p],[1/2],-b*tan(e+f*x)^2/a)*(a+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/a)^p)],
[csc(e+f*x)^4*(a+b*tan(e+f*x)^2)^p,x,4,-1/3*cot(e+f*x)^3*(a+b*tan(e+f*x)^2)^(1+p)/(a*f)-1/3*(3*a-b*(1-2*p))*cot(e+f*x)*hypergeom([-1/2,-p],[1/2],-b*tan(e+f*x)^2/a)*(a+b*tan(e+f*x)^2)^p/(a*f*(1+b*tan(e+f*x)^2/a)^p)],
[csc(e+f*x)^6*(a+b*tan(e+f*x)^2)^p,x,5,-1/15*(10*a-b*(3-2*p))*cot(e+f*x)^3*(a+b*tan(e+f*x)^2)^(1+p)/(a^2*f)-1/5*cot(e+f*x)^5*(a+b*tan(e+f*x)^2)^(1+p)/(a*f)-1/15*(15*a^2-b*(10*a-b*(3-2*p))*(1-2*p))*cot(e+f*x)*hypergeom([-1/2,-p],[1/2],-b*tan(e+f*x)^2/a)*(a+b*tan(e+f*x)^2)^p/(a^2*f*(1+b*tan(e+f*x)^2/a)^p)],

# Integrands of the form (d Sin[e+f x])^m (a+b Tan[e+f x]^3)^p

# Integrands of the form (d Sin[e+f x])^m (a+b Tan[e+f x]^4)^p

# Integrands of the form (d Sin[e+f x])^m (a+b (c Tan[e+f x])^n)^p

# Integrands of the form (d Sin[e+f x])^m (b (c Tan[e+f x])^n)^p when p symbolic
[(d*sin(e+f*x))^m*(b*(c*tan(e+f*x))^n)^p,x,3,(cos(e+f*x)^2)^(1/2*(1+n*p))*hypergeom([1/2*(1+n*p),1/2*(1+m+n*p)],[1/2*(3+m+n*p)],sin(e+f*x)^2)*(d*sin(e+f*x))^m*tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1+m+n*p))],
[sin(e+f*x)^2*(b*(c*tan(e+f*x))^n)^p,x,3,hypergeom([2,1/2*(3+n*p)],[1/2*(5+n*p)],-tan(e+f*x)^2)*tan(e+f*x)^3*(b*(c*tan(e+f*x))^n)^p/(f*(3+n*p))],
[(b*(c*tan(e+f*x))^n)^p,x,3,hypergeom([1,1/2*(1+n*p)],[1/2*(3+n*p)],-tan(e+f*x)^2)*tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1+n*p))],
[csc(e+f*x)^2*(b*(c*tan(e+f*x))^n)^p,x,3,-cot(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1-n*p))],
[csc(e+f*x)^4*(b*(c*tan(e+f*x))^n)^p,x,4,-cot(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1-n*p))-cot(e+f*x)^3*(b*(c*tan(e+f*x))^n)^p/(f*(3-n*p))],
[csc(e+f*x)^6*(b*(c*tan(e+f*x))^n)^p,x,4,-cot(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1-n*p))-2*cot(e+f*x)^3*(b*(c*tan(e+f*x))^n)^p/(f*(3-n*p))-cot(e+f*x)^5*(b*(c*tan(e+f*x))^n)^p/(f*(5-n*p))],
[sin(e+f*x)^3*(b*(c*tan(e+f*x))^n)^p,x,3,(cos(e+f*x)^2)^(1/2*(1+n*p))*hypergeom([1/2*(1+n*p),1/2*(4+n*p)],[1/2*(6+n*p)],sin(e+f*x)^2)*sin(e+f*x)^3*tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(4+n*p))],
[sin(e+f*x)*(b*(c*tan(e+f*x))^n)^p,x,3,(cos(e+f*x)^2)^(1/2*(1+n*p))*hypergeom([1/2*(1+n*p),1/2*(2+n*p)],[1/2*(4+n*p)],sin(e+f*x)^2)*sin(e+f*x)*tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(2+n*p))],
[csc(e+f*x)*(b*(c*tan(e+f*x))^n)^p,x,3,(cos(e+f*x)^2)^(1/2*(1+n*p))*hypergeom([1/2*n*p,1/2*(1+n*p)],[1/2*(2+n*p)],sin(e+f*x)^2)*sec(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*n*p)],
[csc(e+f*x)^3*(b*(c*tan(e+f*x))^n)^p,x,3,-(cos(e+f*x)^2)^(1/2*(1+n*p))*csc(e+f*x)^2*hypergeom([1/2*(-2+n*p),1/2*(1+n*p)],[1/2*n*p],sin(e+f*x)^2)*sec(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(2-n*p))],

# Integrands of the form (d Sin[e+f x])^m (a+b (c Tan[e+f x])^n)^p when p symbolic
[(d*sin(e+f*x))^m*(a+b*tan(e+f*x)^n)^p,x,0,Unintegrable((d*sin(e+f*x))^m*(a+b*tan(e+f*x)^n)^p,x)],

# Integrands of the form (d Cos[e+f x])^m (a+b (c Tan[e+f x])^n)^p

# Integrands of the form (d Cos[e+f x])^m (a+b Tan[e+f x]^2)^p

# Integrands of the form (d Cos[e+f x])^m (a+b Tan[e+f x]^2)^p when p symbolic
[(d*cos(e+f*x))^m*(b*tan(e+f*x)^2)^p,x,3,(d*cos(e+f*x))^m*(cos(e+f*x)^2)^(1/2*(1-m+2*p))*hypergeom([1/2*(1+2*p),1/2*(1-m+2*p)],[1/2*(3+2*p)],sin(e+f*x)^2)*tan(e+f*x)*(b*tan(e+f*x)^2)^p/(f*(1+2*p))],
[(d*cos(e+f*x))^m*(a+b*tan(e+f*x)^2)^p,x,4,AppellF1(1/2,1/2*(2+m),-p,3/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/a)*(d*cos(e+f*x))^m*(sec(e+f*x)^2)^(1/2*m)*tan(e+f*x)*(a+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/a)^p)],

# Integrands of the form (d Cos[e+f x])^m (a+b Tan[e+f x]^3)^p

# Integrands of the form (d Cos[e+f x])^m (a+b Tan[e+f x]^4)^p

# Integrands of the form (d Cos[e+f x])^m (a+b (c Tan[e+f x])^n)^p

# Integrands of the form (d Cos[e+f x])^m (b (cTan[e+f x])^n)^p when p symbolic
[(d*cos(e+f*x))^m*(b*(c*tan(e+f*x))^n)^p,x,3,(d*cos(e+f*x))^m*(cos(e+f*x)^2)^(1/2*(1-m+n*p))*hypergeom([1/2*(1+n*p),1/2*(1-m+n*p)],[1/2*(3+n*p)],sin(e+f*x)^2)*tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1+n*p))],

# Integrands of the form (d Cos[e+f x])^m (a+b (c Tan[e+f x])^n)^p when p symbolic
[(d*cos(e+f*x))^m*(a+b*(c*tan(e+f*x))^n)^p,x,1,(d*cos(e+f*x))^m*(sec(e+f*x)/d)^m*Unintegrable((a+b*(c*tan(e+f*x))^n)^p/(sec(e+f*x)/d)^m,x)],

# Integrands of the form (d Tan[e+f x])^m (a+b (c Tan[e+f x])^n)^p

# Integrands of the form (d Tan[e+f x])^m (a+b Tan[e+f x]^2)^p

# Integrands of the form Tan[e+f x]^m (a+b Tan[e+f x]^2)^p when a-b=0
[(a+a*tan(c+d*x)^2)^4,x,4,a^4*tan(c+d*x)/d+a^4*tan(c+d*x)^3/d+3/5*a^4*tan(c+d*x)^5/d+1/7*a^4*tan(c+d*x)^7/d],
[(a+a*tan(c+d*x)^2)^3,x,4,a^3*tan(c+d*x)/d+2/3*a^3*tan(c+d*x)^3/d+1/5*a^3*tan(c+d*x)^5/d],
[(a+a*tan(c+d*x)^2)^2,x,4,a^2*tan(c+d*x)/d+1/3*a^2*tan(c+d*x)^3/d],
[1/(a+a*tan(c+d*x)^2),x,4,1/2*x/a+1/2*cos(c+d*x)*sin(c+d*x)/(a*d)],
[1/(a+a*tan(c+d*x)^2)^2,x,5,3/8*x/a^2+3/8*cos(c+d*x)*sin(c+d*x)/(a^2*d)+1/4*cos(c+d*x)^3*sin(c+d*x)/(a^2*d)],
[1/(a+a*tan(c+d*x)^2)^3,x,6,5/16*x/a^3+5/16*cos(c+d*x)*sin(c+d*x)/(a^3*d)+5/24*cos(c+d*x)^3*sin(c+d*x)/(a^3*d)+1/6*cos(c+d*x)^5*sin(c+d*x)/(a^3*d)],

# Integrands of the form Tan[e+f x]^m (a+b Tan[e+f x]^2)^p

# p>0
[tan(e+f*x)^5*(a+b*tan(e+f*x)^2),x,4,-(a-b)*log(cos(e+f*x))/f-1/2*(a-b)*tan(e+f*x)^2/f+1/4*(a-b)*tan(e+f*x)^4/f+1/6*b*tan(e+f*x)^6/f],
[tan(e+f*x)^3*(a+b*tan(e+f*x)^2),x,3,(a-b)*log(cos(e+f*x))/f+1/2*(a-b)*tan(e+f*x)^2/f+1/4*b*tan(e+f*x)^4/f],
[tan(e+f*x)*(a+b*tan(e+f*x)^2),x,2,-(a-b)*log(cos(e+f*x))/f+1/2*b*tan(e+f*x)^2/f],
[cot(e+f*x)*(a+b*tan(e+f*x)^2),x,3,-b*log(cos(e+f*x))/f+a*log(sin(e+f*x))/f],
[cot(e+f*x)^3*(a+b*tan(e+f*x)^2),x,3,-1/2*a*cot(e+f*x)^2/f-(a-b)*log(sin(e+f*x))/f],
[cot(e+f*x)^5*(a+b*tan(e+f*x)^2),x,4,1/2*(a-b)*cot(e+f*x)^2/f-1/4*a*cot(e+f*x)^4/f+(a-b)*log(sin(e+f*x))/f],
[tan(e+f*x)^6*(a+b*tan(e+f*x)^2),x,5,-(a-b)*x+(a-b)*tan(e+f*x)/f-1/3*(a-b)*tan(e+f*x)^3/f+1/5*(a-b)*tan(e+f*x)^5/f+1/7*b*tan(e+f*x)^7/f],
[tan(e+f*x)^4*(a+b*tan(e+f*x)^2),x,4,(a-b)*x-(a-b)*tan(e+f*x)/f+1/3*(a-b)*tan(e+f*x)^3/f+1/5*b*tan(e+f*x)^5/f],
[tan(e+f*x)^2*(a+b*tan(e+f*x)^2),x,3,-(a-b)*x+(a-b)*tan(e+f*x)/f+1/3*b*tan(e+f*x)^3/f],
[a+b*tan(e+f*x)^2,x,3,a*x-b*x+b*tan(e+f*x)/f],
[cot(e+f*x)^2*(a+b*tan(e+f*x)^2),x,2,-(a-b)*x-a*cot(e+f*x)/f],
[cot(e+f*x)^4*(a+b*tan(e+f*x)^2),x,4,(a-b)*x+(a-b)*cot(e+f*x)/f-1/3*a*cot(e+f*x)^3/f],
[cot(e+f*x)^6*(a+b*tan(e+f*x)^2),x,5,-(a-b)*x-(a-b)*cot(e+f*x)/f+1/3*(a-b)*cot(e+f*x)^3/f-1/5*a*cot(e+f*x)^5/f],
[tan(e+f*x)^5*(a+b*tan(e+f*x)^2)^2,x,4,-(a-b)^2*log(cos(e+f*x))/f-1/2*(a-b)^2*tan(e+f*x)^2/f+1/4*(a-b)^2*tan(e+f*x)^4/f+1/6*(2*a-b)*b*tan(e+f*x)^6/f+1/8*b^2*tan(e+f*x)^8/f],
[tan(e+f*x)^3*(a+b*tan(e+f*x)^2)^2,x,4,(a-b)^2*log(cos(e+f*x))/f+1/2*(a-b)^2*tan(e+f*x)^2/f+1/4*(2*a-b)*b*tan(e+f*x)^4/f+1/6*b^2*tan(e+f*x)^6/f],
[tan(e+f*x)*(a+b*tan(e+f*x)^2)^2,x,4,-(a-b)^2*log(cos(e+f*x))/f+1/2*(a-b)*b*tan(e+f*x)^2/f+1/4*(a+b*tan(e+f*x)^2)^2/f],
[cot(e+f*x)*(a+b*tan(e+f*x)^2)^2,x,4,(a-b)^2*log(cos(e+f*x))/f+a^2*log(tan(e+f*x))/f+1/2*b^2*tan(e+f*x)^2/f],
[cot(e+f*x)^3*(a+b*tan(e+f*x)^2)^2,x,4,-1/2*a^2*cot(e+f*x)^2/f-(a-b)^2*log(cos(e+f*x))/f-a*(a-2*b)*log(tan(e+f*x))/f],
[cot(e+f*x)^5*(a+b*tan(e+f*x)^2)^2,x,4,1/2*a*(a-2*b)*cot(e+f*x)^2/f-1/4*a^2*cot(e+f*x)^4/f+(a-b)^2*log(cos(e+f*x))/f+(a-b)^2*log(tan(e+f*x))/f],
[tan(e+f*x)^6*(a+b*tan(e+f*x)^2)^2,x,4,-(a-b)^2*x+(a-b)^2*tan(e+f*x)/f-1/3*(a-b)^2*tan(e+f*x)^3/f+1/5*(a-b)^2*tan(e+f*x)^5/f+1/7*(2*a-b)*b*tan(e+f*x)^7/f+1/9*b^2*tan(e+f*x)^9/f],
[tan(e+f*x)^4*(a+b*tan(e+f*x)^2)^2,x,4,(a-b)^2*x-(a-b)^2*tan(e+f*x)/f+1/3*(a-b)^2*tan(e+f*x)^3/f+1/5*(2*a-b)*b*tan(e+f*x)^5/f+1/7*b^2*tan(e+f*x)^7/f],
[tan(e+f*x)^2*(a+b*tan(e+f*x)^2)^2,x,4,-(a-b)^2*x+(a-b)^2*tan(e+f*x)/f+1/3*(2*a-b)*b*tan(e+f*x)^3/f+1/5*b^2*tan(e+f*x)^5/f],
[(a+b*tan(e+f*x)^2)^2,x,4,(a-b)^2*x+(2*a-b)*b*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],
[cot(e+f*x)^2*(a+b*tan(e+f*x)^2)^2,x,4,-(a-b)^2*x-a^2*cot(e+f*x)/f+b^2*tan(e+f*x)/f],
[cot(e+f*x)^4*(a+b*tan(e+f*x)^2)^2,x,4,(a-b)^2*x+a*(a-2*b)*cot(e+f*x)/f-1/3*a^2*cot(e+f*x)^3/f],
[cot(e+f*x)^6*(a+b*tan(e+f*x)^2)^2,x,4,-(a-b)^2*x-(a-b)^2*cot(e+f*x)/f+1/3*a*(a-2*b)*cot(e+f*x)^3/f-1/5*a^2*cot(e+f*x)^5/f],

# p<0
[tan(e+f*x)^5/(a+b*tan(e+f*x)^2),x,4,-log(cos(e+f*x))/((a-b)*f)-1/2*a^2*log(a+b*tan(e+f*x)^2)/((a-b)*b^2*f)+1/2*tan(e+f*x)^2/(b*f)],
[tan(e+f*x)^3/(a+b*tan(e+f*x)^2),x,4,log(cos(e+f*x))/((a-b)*f)+1/2*a*log(a+b*tan(e+f*x)^2)/((a-b)*b*f)],
[tan(e+f*x)/(a+b*tan(e+f*x)^2),x,5,-1/2*log(a*cos(e+f*x)^2+b*sin(e+f*x)^2)/((a-b)*f)],
[cot(e+f*x)/(a+b*tan(e+f*x)^2),x,4,log(cos(e+f*x))/((a-b)*f)+log(tan(e+f*x))/(a*f)+1/2*b*log(a+b*tan(e+f*x)^2)/(a*(a-b)*f)],
[cot(e+f*x)^3/(a+b*tan(e+f*x)^2),x,4,-1/2*cot(e+f*x)^2/(a*f)-log(cos(e+f*x))/((a-b)*f)-(a+b)*log(tan(e+f*x))/(a^2*f)-1/2*b^2*log(a+b*tan(e+f*x)^2)/(a^2*(a-b)*f)],
[cot(e+f*x)^5/(a+b*tan(e+f*x)^2),x,4,1/2*(a+b)*cot(e+f*x)^2/(a^2*f)-1/4*cot(e+f*x)^4/(a*f)+log(cos(e+f*x))/((a-b)*f)+(a^2+a*b+b^2)*log(tan(e+f*x))/(a^3*f)+1/2*b^3*log(a+b*tan(e+f*x)^2)/(a^3*(a-b)*f)],
[tan(e+f*x)^6/(a+b*tan(e+f*x)^2),x,6,-x/(a-b)+a^(5/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))/((a-b)*b^(5/2)*f)-(a+b)*tan(e+f*x)/(b^2*f)+1/3*tan(e+f*x)^3/(b*f)],
[tan(e+f*x)^4/(a+b*tan(e+f*x)^2),x,5,x/(a-b)-a^(3/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))/((a-b)*b^(3/2)*f)+tan(e+f*x)/(b*f)],
[tan(e+f*x)^2/(a+b*tan(e+f*x)^2),x,4,-x/(a-b)+arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(a)/((a-b)*f*sqrt(b))],
[1/(a+b*tan(e+f*x)^2),x,3,x/(a-b)-arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/((a-b)*f*sqrt(a))],
[cot(e+f*x)^2/(a+b*tan(e+f*x)^2),x,5,-x/(a-b)+b^(3/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))/(a^(3/2)*(a-b)*f)-cot(e+f*x)/(a*f)],
[cot(e+f*x)^4/(a+b*tan(e+f*x)^2),x,6,x/(a-b)-b^(5/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))/(a^(5/2)*(a-b)*f)+(a+b)*cot(e+f*x)/(a^2*f)-1/3*cot(e+f*x)^3/(a*f)],
[cot(e+f*x)^6/(a+b*tan(e+f*x)^2),x,7,-x/(a-b)+b^(7/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))/(a^(7/2)*(a-b)*f)-(a^2+a*b+b^2)*cot(e+f*x)/(a^3*f)+1/3*(a+b)*cot(e+f*x)^3/(a^2*f)-1/5*cot(e+f*x)^5/(a*f)],
[tan(e+f*x)^5/(a+b*tan(e+f*x)^2)^2,x,4,-log(cos(e+f*x))/((a-b)^2*f)+1/2*a*(a-2*b)*log(a+b*tan(e+f*x)^2)/((a-b)^2*b^2*f)+1/2*a^2/((a-b)*b^2*f*(a+b*tan(e+f*x)^2))],
[tan(e+f*x)^3/(a+b*tan(e+f*x)^2)^2,x,4,1/2*log(a*cos(e+f*x)^2+b*sin(e+f*x)^2)/((a-b)^2*f)-1/2*a/((a-b)*b*f*(a+b*tan(e+f*x)^2))],
[tan(e+f*x)/(a+b*tan(e+f*x)^2)^2,x,4,-1/2*log(a*cos(e+f*x)^2+b*sin(e+f*x)^2)/((a-b)^2*f)+1/2/((a-b)*f*(a+b*tan(e+f*x)^2))],
[cot(e+f*x)/(a+b*tan(e+f*x)^2)^2,x,4,log(cos(e+f*x))/((a-b)^2*f)+log(tan(e+f*x))/(a^2*f)+1/2*(2*a-b)*b*log(a+b*tan(e+f*x)^2)/(a^2*(a-b)^2*f)-1/2*b/(a*(a-b)*f*(a+b*tan(e+f*x)^2))],
[cot(e+f*x)^3/(a+b*tan(e+f*x)^2)^2,x,4,-1/2*cot(e+f*x)^2/(a^2*f)-log(cos(e+f*x))/((a-b)^2*f)-(a+2*b)*log(tan(e+f*x))/(a^3*f)-1/2*(3*a-2*b)*b^2*log(a+b*tan(e+f*x)^2)/(a^3*(a-b)^2*f)+1/2*b^2/(a^2*(a-b)*f*(a+b*tan(e+f*x)^2))],
[cot(e+f*x)^5/(a+b*tan(e+f*x)^2)^2,x,4,1/2*(a+2*b)*cot(e+f*x)^2/(a^3*f)-1/4*cot(e+f*x)^4/(a^2*f)+log(cos(e+f*x))/((a-b)^2*f)+(a^2+2*a*b+3*b^2)*log(tan(e+f*x))/(a^4*f)+1/2*(4*a-3*b)*b^3*log(a+b*tan(e+f*x)^2)/(a^4*(a-b)^2*f)-1/2*b^3/(a^3*(a-b)*f*(a+b*tan(e+f*x)^2))],
[tan(e+f*x)^6/(a+b*tan(e+f*x)^2)^2,x,6,-x/(a-b)^2-1/2*a^(3/2)*(3*a-5*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))/((a-b)^2*b^(5/2)*f)+1/2*(3*a-2*b)*tan(e+f*x)/((a-b)*b^2*f)-1/2*a*tan(e+f*x)^3/((a-b)*b*f*(a+b*tan(e+f*x)^2))],
[tan(e+f*x)^4/(a+b*tan(e+f*x)^2)^2,x,5,x/(a-b)^2+1/2*(a-3*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(a)/((a-b)^2*b^(3/2)*f)-1/2*a*tan(e+f*x)/((a-b)*b*f*(a+b*tan(e+f*x)^2))],
[tan(e+f*x)^2/(a+b*tan(e+f*x)^2)^2,x,5,-x/(a-b)^2+1/2*(a+b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))/((a-b)^2*f*sqrt(a)*sqrt(b))+1/2*tan(e+f*x)/((a-b)*f*(a+b*tan(e+f*x)^2))],
[1/(a+b*tan(e+f*x)^2)^2,x,5,x/(a-b)^2-1/2*(3*a-b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/(a^(3/2)*(a-b)^2*f)-1/2*b*tan(e+f*x)/(a*(a-b)*f*(a+b*tan(e+f*x)^2))],
[cot(e+f*x)^2/(a+b*tan(e+f*x)^2)^2,x,6,-x/(a-b)^2+1/2*(5*a-3*b)*b^(3/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))/(a^(5/2)*(a-b)^2*f)-1/2*(2*a-3*b)*cot(e+f*x)/(a^2*(a-b)*f)-1/2*b*cot(e+f*x)/(a*(a-b)*f*(a+b*tan(e+f*x)^2))],
[cot(e+f*x)^4/(a+b*tan(e+f*x)^2)^2,x,7,x/(a-b)^2-1/2*(7*a-5*b)*b^(5/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))/(a^(7/2)*(a-b)^2*f)+1/2*(2*a^2+2*a*b-5*b^2)*cot(e+f*x)/(a^3*(a-b)*f)-1/6*(2*a-5*b)*cot(e+f*x)^3/(a^2*(a-b)*f)-1/2*b*cot(e+f*x)^3/(a*(a-b)*f*(a+b*tan(e+f*x)^2))],
[cot(e+f*x)^6/(a+b*tan(e+f*x)^2)^2,x,8,-x/(a-b)^2+1/2*(9*a-7*b)*b^(7/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))/(a^(9/2)*(a-b)^2*f)-1/2*(2*a^3+2*a^2*b+2*a*b^2-7*b^3)*cot(e+f*x)/(a^4*(a-b)*f)+1/6*(2*a^2+2*a*b-7*b^2)*cot(e+f*x)^3/(a^3*(a-b)*f)-1/10*(2*a-7*b)*cot(e+f*x)^5/(a^2*(a-b)*f)-1/2*b*cot(e+f*x)^5/(a*(a-b)*f*(a+b*tan(e+f*x)^2))],
[tan(e+f*x)^5/(a+b*tan(e+f*x)^2)^3,x,4,-1/2*log(a*cos(e+f*x)^2+b*sin(e+f*x)^2)/((a-b)^3*f)+1/4*a^2/((a-b)*b^2*f*(a+b*tan(e+f*x)^2)^2)-1/2*a*(a-2*b)/((a-b)^2*b^2*f*(a+b*tan(e+f*x)^2))],
[tan(e+f*x)^3/(a+b*tan(e+f*x)^2)^3,x,4,1/2*log(a*cos(e+f*x)^2+b*sin(e+f*x)^2)/((a-b)^3*f)-1/4*a/((a-b)*b*f*(a+b*tan(e+f*x)^2)^2)+(-1/2)/((a-b)^2*f*(a+b*tan(e+f*x)^2))],
[tan(e+f*x)/(a+b*tan(e+f*x)^2)^3,x,4,-1/2*log(a*cos(e+f*x)^2+b*sin(e+f*x)^2)/((a-b)^3*f)+1/4/((a-b)*f*(a+b*tan(e+f*x)^2)^2)+1/2/((a-b)^2*f*(a+b*tan(e+f*x)^2))],
[cot(e+f*x)/(a+b*tan(e+f*x)^2)^3,x,4,log(cos(e+f*x))/((a-b)^3*f)+log(tan(e+f*x))/(a^3*f)+1/2*b*(3*a^2-3*a*b+b^2)*log(a+b*tan(e+f*x)^2)/(a^3*(a-b)^3*f)-1/4*b/(a*(a-b)*f*(a+b*tan(e+f*x)^2)^2)-1/2*(2*a-b)*b/(a^2*(a-b)^2*f*(a+b*tan(e+f*x)^2))],
[cot(e+f*x)^3/(a+b*tan(e+f*x)^2)^3,x,4,-1/2*cot(e+f*x)^2/(a^3*f)-log(cos(e+f*x))/((a-b)^3*f)-(a+3*b)*log(tan(e+f*x))/(a^4*f)-1/2*b^2*(6*a^2-8*a*b+3*b^2)*log(a+b*tan(e+f*x)^2)/(a^4*(a-b)^3*f)+1/4*b^2/(a^2*(a-b)*f*(a+b*tan(e+f*x)^2)^2)+1/2*(3*a-2*b)*b^2/(a^3*(a-b)^2*f*(a+b*tan(e+f*x)^2))],
[cot(e+f*x)^5/(a+b*tan(e+f*x)^2)^3,x,4,1/2*(a+3*b)*cot(e+f*x)^2/(a^4*f)-1/4*cot(e+f*x)^4/(a^3*f)+log(cos(e+f*x))/((a-b)^3*f)+(a^2+3*a*b+6*b^2)*log(tan(e+f*x))/(a^5*f)+1/2*b^3*(10*a^2-15*a*b+6*b^2)*log(a+b*tan(e+f*x)^2)/(a^5*(a-b)^3*f)-1/4*b^3/(a^3*(a-b)*f*(a+b*tan(e+f*x)^2)^2)-1/2*(4*a-3*b)*b^3/(a^4*(a-b)^2*f*(a+b*tan(e+f*x)^2))],
[tan(e+f*x)^6/(a+b*tan(e+f*x)^2)^3,x,6,-x/(a-b)^3+1/8*(3*a^2-10*a*b+15*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(a)/((a-b)^3*b^(5/2)*f)-1/4*a*tan(e+f*x)^3/((a-b)*b*f*(a+b*tan(e+f*x)^2)^2)-1/8*a*(3*a-7*b)*tan(e+f*x)/((a-b)^2*b^2*f*(a+b*tan(e+f*x)^2))],
[tan(e+f*x)^4/(a+b*tan(e+f*x)^2)^3,x,6,x/(a-b)^3+1/8*(a^2-6*a*b-3*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))/((a-b)^3*b^(3/2)*f*sqrt(a))-1/4*a*tan(e+f*x)/((a-b)*b*f*(a+b*tan(e+f*x)^2)^2)+1/8*(a-5*b)*tan(e+f*x)/((a-b)^2*b*f*(a+b*tan(e+f*x)^2))],
[tan(e+f*x)^2/(a+b*tan(e+f*x)^2)^3,x,6,-x/(a-b)^3+1/8*(3*a^2+6*a*b-b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))/(a^(3/2)*(a-b)^3*f*sqrt(b))+1/4*tan(e+f*x)/((a-b)*f*(a+b*tan(e+f*x)^2)^2)+1/8*(3*a+b)*tan(e+f*x)/(a*(a-b)^2*f*(a+b*tan(e+f*x)^2))],
[1/(a+b*tan(e+f*x)^2)^3,x,6,x/(a-b)^3-1/8*(15*a^2-10*a*b+3*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))*sqrt(b)/(a^(5/2)*(a-b)^3*f)-1/4*b*tan(e+f*x)/(a*(a-b)*f*(a+b*tan(e+f*x)^2)^2)-1/8*(7*a-3*b)*b*tan(e+f*x)/(a^2*(a-b)^2*f*(a+b*tan(e+f*x)^2))],
[cot(e+f*x)^2/(a+b*tan(e+f*x)^2)^3,x,7,-x/(a-b)^3+1/8*b^(3/2)*(35*a^2-42*a*b+15*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))/(a^(7/2)*(a-b)^3*f)-1/8*(8*a^2-27*a*b+15*b^2)*cot(e+f*x)/(a^3*(a-b)^2*f)-1/4*b*cot(e+f*x)/(a*(a-b)*f*(a+b*tan(e+f*x)^2)^2)-1/8*(9*a-5*b)*b*cot(e+f*x)/(a^2*(a-b)^2*f*(a+b*tan(e+f*x)^2))],
[cot(e+f*x)^4/(a+b*tan(e+f*x)^2)^3,x,8,x/(a-b)^3-1/8*b^(5/2)*(63*a^2-90*a*b+35*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))/(a^(9/2)*(a-b)^3*f)+1/8*(8*a^3+8*a^2*b-55*a*b^2+35*b^3)*cot(e+f*x)/(a^4*(a-b)^2*f)-1/24*(8*a^2-55*a*b+35*b^2)*cot(e+f*x)^3/(a^3*(a-b)^2*f)-1/4*b*cot(e+f*x)^3/(a*(a-b)*f*(a+b*tan(e+f*x)^2)^2)-1/8*(11*a-7*b)*b*cot(e+f*x)^3/(a^2*(a-b)^2*f*(a+b*tan(e+f*x)^2))],
[cot(e+f*x)^6/(a+b*tan(e+f*x)^2)^3,x,9,-x/(a-b)^3+1/8*b^(7/2)*(99*a^2-154*a*b+63*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a))/(a^(11/2)*(a-b)^3*f)-1/8*(8*a^4+8*a^3*b+8*a^2*b^2-91*a*b^3+63*b^4)*cot(e+f*x)/(a^5*(a-b)^2*f)+1/24*(8*a^3+8*a^2*b-91*a*b^2+63*b^3)*cot(e+f*x)^3/(a^4*(a-b)^2*f)-1/40*(8*a^2-91*a*b+63*b^2)*cot(e+f*x)^5/(a^3*(a-b)^2*f)-1/4*b*cot(e+f*x)^5/(a*(a-b)*f*(a+b*tan(e+f*x)^2)^2)-1/8*(13*a-9*b)*b*cot(e+f*x)^5/(a^2*(a-b)^2*f*(a+b*tan(e+f*x)^2))],
[(a+b*tan(c+d*x)^2)^4,x,4,(a-b)^4*x+(2*a-b)*b*(2*a^2-2*a*b+b^2)*tan(c+d*x)/d+1/3*b^2*(6*a^2-4*a*b+b^2)*tan(c+d*x)^3/d+1/5*(4*a-b)*b^3*tan(c+d*x)^5/d+1/7*b^4*tan(c+d*x)^7/d],
[(a+b*tan(c+d*x)^2)^3,x,4,(a-b)^3*x+b*(3*a^2-3*a*b+b^2)*tan(c+d*x)/d+1/3*(3*a-b)*b^2*tan(c+d*x)^3/d+1/5*b^3*tan(c+d*x)^5/d],
[(a+b*tan(c+d*x)^2)^2,x,4,(a-b)^2*x+(2*a-b)*b*tan(c+d*x)/d+1/3*b^2*tan(c+d*x)^3/d],
[a+b*tan(c+d*x)^2,x,3,a*x-b*x+b*tan(c+d*x)/d],
[1/(a+b*tan(c+d*x)^2),x,3,x/(a-b)-arctan(sqrt(b)*tan(c+d*x)/sqrt(a))*sqrt(b)/((a-b)*d*sqrt(a))],
[1/(a+b*tan(c+d*x)^2)^2,x,5,x/(a-b)^2-1/2*(3*a-b)*arctan(sqrt(b)*tan(c+d*x)/sqrt(a))*sqrt(b)/(a^(3/2)*(a-b)^2*d)-1/2*b*tan(c+d*x)/(a*(a-b)*d*(a+b*tan(c+d*x)^2))],
[1/(a+b*tan(c+d*x)^2)^3,x,6,x/(a-b)^3-1/8*(15*a^2-10*a*b+3*b^2)*arctan(sqrt(b)*tan(c+d*x)/sqrt(a))*sqrt(b)/(a^(5/2)*(a-b)^3*d)-1/4*b*tan(c+d*x)/(a*(a-b)*d*(a+b*tan(c+d*x)^2)^2)-1/8*(7*a-3*b)*b*tan(c+d*x)/(a^2*(a-b)^2*d*(a+b*tan(c+d*x)^2))],

# Integrands of the form Tan[e+f x]^m (a+b Tan[e+f x]^2)^(p/2) when a-b=0

# p>0
[sqrt(a+a*tan(x)^2)*tan(x)^4,x,5,3/8*arctanh(sin(x))*cos(x)*sqrt(a*sec(x)^2)-3/8*sqrt(a*sec(x)^2)*tan(x)+1/4*sqrt(a*sec(x)^2)*tan(x)^3],
[sqrt(a+a*tan(x)^2)*tan(x)^3,x,4,1/3*(a*sec(x)^2)^(3/2)/a-sqrt(a*sec(x)^2)],
[sqrt(a+a*tan(x)^2)*tan(x)^2,x,4,-1/2*arctanh(sin(x))*cos(x)*sqrt(a*sec(x)^2)+1/2*sqrt(a*sec(x)^2)*tan(x)],
[sqrt(a+a*tan(x)^2)*tan(x),x,3,sqrt(a*sec(x)^2)],
[cot(x)*sqrt(a+a*tan(x)^2),x,4,-arctanh(sqrt(a*sec(x)^2)/sqrt(a))*sqrt(a)],
[cot(x)^2*sqrt(a+a*tan(x)^2),x,4,-cot(x)*sqrt(a*sec(x)^2)],
[cot(x)^3*sqrt(a+a*tan(x)^2),x,5,1/2*arctanh(sqrt(a*sec(x)^2)/sqrt(a))*sqrt(a)-1/2*cot(x)^2*sqrt(a*sec(x)^2)],
[cot(x)^4*sqrt(a+a*tan(x)^2),x,4,cot(x)*sqrt(a*sec(x)^2)-1/3*cot(x)*csc(x)^2*sqrt(a*sec(x)^2)],
[(a+a*tan(c+d*x)^2)^(1/2),x,4,arctanh(sqrt(a)*tan(c+d*x)/sqrt(a*sec(c+d*x)^2))*sqrt(a)/d],
[tan(x)^3*(a+a*tan(x)^2)^(3/2),x,4,-1/3*(a*sec(x)^2)^(3/2)+1/5*(a*sec(x)^2)^(5/2)/a],
[tan(x)^2*(a+a*tan(x)^2)^(3/2),x,5,-1/8*a*arctanh(sin(x))*cos(x)*sqrt(a*sec(x)^2)-1/8*a*sqrt(a*sec(x)^2)*tan(x)+1/4*a*sec(x)^2*sqrt(a*sec(x)^2)*tan(x)],
[tan(x)*(a+a*tan(x)^2)^(3/2),x,3,1/3*(a*sec(x)^2)^(3/2)],
[cot(x)*(a+a*tan(x)^2)^(3/2),x,5,-a^(3/2)*arctanh(sqrt(a*sec(x)^2)/sqrt(a))+a*sqrt(a*sec(x)^2)],
[cot(x)^2*(a+a*tan(x)^2)^(3/2),x,5,a*arctanh(sin(x))*cos(x)*sqrt(a*sec(x)^2)-a*cot(x)*sqrt(a*sec(x)^2)],
[(a+a*tan(c+d*x)^2)^(3/2),x,5,1/2*a^(3/2)*arctanh(sqrt(a)*tan(c+d*x)/sqrt(a*sec(c+d*x)^2))/d+1/2*a*sqrt(a*sec(c+d*x)^2)*tan(c+d*x)/d],
[(a+a*tan(c+d*x)^2)^(5/2),x,6,3/8*a^(5/2)*arctanh(sqrt(a)*tan(c+d*x)/sqrt(a*sec(c+d*x)^2))/d+1/4*a*(a*sec(c+d*x)^2)^(3/2)*tan(c+d*x)/d+3/8*a^2*sqrt(a*sec(c+d*x)^2)*tan(c+d*x)/d],

# p<0
[tan(x)^3/sqrt(a+a*tan(x)^2),x,4,1/sqrt(a*sec(x)^2)+sqrt(a*sec(x)^2)/a],
[tan(x)^2/sqrt(a+a*tan(x)^2),x,5,arctanh(sin(x))*sec(x)/sqrt(a*sec(x)^2)-tan(x)/sqrt(a*sec(x)^2)],
[tan(x)/sqrt(a+a*tan(x)^2),x,3,(-1)/sqrt(a*sec(x)^2)],
[cot(x)/sqrt(a+a*tan(x)^2),x,5,-arctanh(sqrt(a*sec(x)^2)/sqrt(a))/sqrt(a)+1/sqrt(a*sec(x)^2)],
[cot(x)^2/sqrt(a+a*tan(x)^2),x,5,-csc(x)*sec(x)/sqrt(a*sec(x)^2)-tan(x)/sqrt(a*sec(x)^2)],
[tan(x)^3/(a+a*tan(x)^2)^(3/2),x,4,1/3/(a*sec(x)^2)^(3/2)+(-1)/(a*sqrt(a*sec(x)^2))],
[tan(x)^2/(a+a*tan(x)^2)^(3/2),x,4,1/3*sin(x)^2*tan(x)/(a*sqrt(a*sec(x)^2))],
[tan(x)/(a+a*tan(x)^2)^(3/2),x,3,(-1/3)/(a*sec(x)^2)^(3/2)],
[cot(x)/(a+a*tan(x)^2)^(3/2),x,6,-arctanh(sqrt(a*sec(x)^2)/sqrt(a))/a^(3/2)+1/3/(a*sec(x)^2)^(3/2)+1/(a*sqrt(a*sec(x)^2))],
[cot(x)^2/(a+a*tan(x)^2)^(3/2),x,5,-csc(x)*sec(x)/(a*sqrt(a*sec(x)^2))-2*tan(x)/(a*sqrt(a*sec(x)^2))+1/3*sin(x)^2*tan(x)/(a*sqrt(a*sec(x)^2))],
[1/(a+a*tan(c+d*x)^2)^(1/2),x,3,tan(c+d*x)/(d*sqrt(a*sec(c+d*x)^2))],
[1/(a+a*tan(c+d*x)^2)^(3/2),x,4,1/3*tan(c+d*x)/(d*(a*sec(c+d*x)^2)^(3/2))+2/3*tan(c+d*x)/(a*d*sqrt(a*sec(c+d*x)^2))],
[1/(a+a*tan(c+d*x)^2)^(5/2),x,5,1/5*tan(c+d*x)/(d*(a*sec(c+d*x)^2)^(5/2))+4/15*tan(c+d*x)/(a*d*(a*sec(c+d*x)^2)^(3/2))+8/15*tan(c+d*x)/(a^2*d*sqrt(a*sec(c+d*x)^2))],
[1/(a+a*tan(c+d*x)^2)^(7/2),x,6,1/7*tan(c+d*x)/(d*(a*sec(c+d*x)^2)^(7/2))+6/35*tan(c+d*x)/(a*d*(a*sec(c+d*x)^2)^(5/2))+8/35*tan(c+d*x)/(a^2*d*(a*sec(c+d*x)^2)^(3/2))+16/35*tan(c+d*x)/(a^3*d*sqrt(a*sec(c+d*x)^2))],
[(1+tan(x)^2)^(3/2),x,4,1/2*arcsinh(tan(x))+1/2*sqrt(sec(x)^2)*tan(x)],
[sqrt(1+tan(x)^2),x,3,arcsinh(tan(x))],
[1/sqrt(1+tan(x)^2),x,3,tan(x)/sqrt(sec(x)^2)],
[(-1-tan(x)^2)^(3/2),x,5,1/2*arctan(tan(x)/sqrt(-sec(x)^2))-1/2*sqrt(-sec(x)^2)*tan(x)],
[sqrt(-1-tan(x)^2),x,4,-arctan(tan(x)/sqrt(-sec(x)^2))],
[1/sqrt(-1-tan(x)^2),x,3,tan(x)/sqrt(-sec(x)^2)],

# Integrands of the form Tan[e+f x]^m (a+b Tan[e+f x]^2)^(p/2)

# p>0
[sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)^5,x,7,-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))*sqrt(a-b)/f+sqrt(a+b*tan(e+f*x)^2)/f-1/3*(a+b)*(a+b*tan(e+f*x)^2)^(3/2)/(b^2*f)+1/5*(a+b*tan(e+f*x)^2)^(5/2)/(b^2*f)],
[sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)^3,x,6,arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))*sqrt(a-b)/f-sqrt(a+b*tan(e+f*x)^2)/f+1/3*(a+b*tan(e+f*x)^2)^(3/2)/(b*f)],
[sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x),x,5,-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))*sqrt(a-b)/f+sqrt(a+b*tan(e+f*x)^2)/f],
[cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2),x,7,-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a))*sqrt(a)/f+arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))*sqrt(a-b)/f],
[cot(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2),x,8,1/2*(2*a-b)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a))/(f*sqrt(a))-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))*sqrt(a-b)/f-1/2*cot(e+f*x)^2*sqrt(a+b*tan(e+f*x)^2)/f],
[cot(e+f*x)^5*sqrt(a+b*tan(e+f*x)^2),x,9,-1/8*(8*a^2-4*a*b-b^2)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a))/(a^(3/2)*f)+arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))*sqrt(a-b)/f+1/8*(4*a-b)*cot(e+f*x)^2*sqrt(a+b*tan(e+f*x)^2)/(a*f)-1/4*cot(e+f*x)^4*sqrt(a+b*tan(e+f*x)^2)/f],
[sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)^6,x,9,1/16*(a^3+2*a^2*b+8*a*b^2-16*b^3)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(b^(5/2)*f)-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(a-b)/f-1/16*(a-2*b)*(a+4*b)*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/(b^2*f)+1/24*(a-6*b)*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)^3/(b*f)+1/6*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)^5/f],
[sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)^4,x,8,-1/8*(a^2+4*a*b-8*b^2)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(b^(3/2)*f)+arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(a-b)/f+1/8*(a-4*b)*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/(b*f)+1/4*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)^3/f],
[sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)^2,x,7,-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(a-b)/f+1/2*(a-2*b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(f*sqrt(b))+1/2*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/f],
[sqrt(a+b*tan(e+f*x)^2),x,6,arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(a-b)/f+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(b)/f],
[cot(e+f*x)^2*sqrt(a+b*tan(e+f*x)^2),x,5,-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(a-b)/f-cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/f],
[cot(e+f*x)^4*sqrt(a+b*tan(e+f*x)^2),x,6,arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(a-b)/f+1/3*(3*a-b)*cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/(a*f)-1/3*cot(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2)/f],
[cot(e+f*x)^6*sqrt(a+b*tan(e+f*x)^2),x,7,-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(a-b)/f-1/15*(15*a^2-5*a*b-2*b^2)*cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/(a^2*f)+1/15*(5*a-b)*cot(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2)/(a*f)-1/5*cot(e+f*x)^5*sqrt(a+b*tan(e+f*x)^2)/f],
[tan(e+f*x)^5*(a+b*tan(e+f*x)^2)^(3/2),x,8,-(a-b)^(3/2)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/f+(a-b)*sqrt(a+b*tan(e+f*x)^2)/f+1/3*(a+b*tan(e+f*x)^2)^(3/2)/f-1/5*(a+b)*(a+b*tan(e+f*x)^2)^(5/2)/(b^2*f)+1/7*(a+b*tan(e+f*x)^2)^(7/2)/(b^2*f)],
[tan(e+f*x)^3*(a+b*tan(e+f*x)^2)^(3/2),x,7,(a-b)^(3/2)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/f-(a-b)*sqrt(a+b*tan(e+f*x)^2)/f-1/3*(a+b*tan(e+f*x)^2)^(3/2)/f+1/5*(a+b*tan(e+f*x)^2)^(5/2)/(b*f)],
[tan(e+f*x)*(a+b*tan(e+f*x)^2)^(3/2),x,6,-(a-b)^(3/2)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/f+(a-b)*sqrt(a+b*tan(e+f*x)^2)/f+1/3*(a+b*tan(e+f*x)^2)^(3/2)/f],
[cot(e+f*x)*(a+b*tan(e+f*x)^2)^(3/2),x,8,-a^(3/2)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a))/f+(a-b)^(3/2)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/f+b*sqrt(a+b*tan(e+f*x)^2)/f],
[cot(e+f*x)^3*(a+b*tan(e+f*x)^2)^(3/2),x,8,-(a-b)^(3/2)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/f+1/2*(2*a-3*b)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a))*sqrt(a)/f-1/2*a*cot(e+f*x)^2*sqrt(a+b*tan(e+f*x)^2)/f],
[cot(e+f*x)^5*(a+b*tan(e+f*x)^2)^(3/2),x,9,(a-b)^(3/2)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/f-1/8*(8*a^2-12*a*b+3*b^2)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a))/(f*sqrt(a))+1/8*(4*a-5*b)*cot(e+f*x)^2*sqrt(a+b*tan(e+f*x)^2)/f-1/4*a*cot(e+f*x)^4*sqrt(a+b*tan(e+f*x)^2)/f],
[tan(e+f*x)^6*(a+b*tan(e+f*x)^2)^(3/2),x,10,-(a-b)^(3/2)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/f+1/128*(3*a^4+8*a^3*b+48*a^2*b^2-192*a*b^3+128*b^4)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(b^(5/2)*f)-1/128*(3*a^3+8*a^2*b-80*a*b^2+64*b^3)*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/(b^2*f)+1/192*(3*a^2-56*a*b+48*b^2)*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)^3/(b*f)+1/48*(9*a-8*b)*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)^5/f+1/8*b*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)^7/f],
[tan(e+f*x)^4*(a+b*tan(e+f*x)^2)^(3/2),x,9,(a-b)^(3/2)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/f-1/16*(a^3+6*a^2*b-24*a*b^2+16*b^3)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(b^(3/2)*f)+1/16*(a^2-10*a*b+8*b^2)*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/(b*f)+1/24*(7*a-6*b)*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)^3/f+1/6*b*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)^5/f],
[tan(e+f*x)^2*(a+b*tan(e+f*x)^2)^(3/2),x,8,-(a-b)^(3/2)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/f+1/8*(3*a^2-12*a*b+8*b^2)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(f*sqrt(b))+1/8*(5*a-4*b)*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/f+1/4*b*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)^3/f],
[(a+b*tan(e+f*x)^2)^(3/2),x,7,(a-b)^(3/2)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/f+1/2*(3*a-2*b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))*sqrt(b)/f+1/2*b*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/f],
[cot(e+f*x)^2*(a+b*tan(e+f*x)^2)^(3/2),x,7,-(a-b)^(3/2)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/f+b^(3/2)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/f-a*cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/f],
[cot(e+f*x)^4*(a+b*tan(e+f*x)^2)^(3/2),x,6,(a-b)^(3/2)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/f+1/3*(3*a-4*b)*cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/f-1/3*a*cot(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2)/f],
[cot(e+f*x)^6*(a+b*tan(e+f*x)^2)^(3/2),x,7,-(a-b)^(3/2)*arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/f-1/15*(15*a^2-20*a*b+3*b^2)*cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/(a*f)+1/15*(5*a-6*b)*cot(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2)/f-1/5*a*cot(e+f*x)^5*sqrt(a+b*tan(e+f*x)^2)/f],
[(a+b*tan(c+d*x)^2)^(5/2),x,8,(a-b)^(5/2)*arctan(sqrt(a-b)*tan(c+d*x)/sqrt(a+b*tan(c+d*x)^2))/d+1/8*(15*a^2-20*a*b+8*b^2)*arctanh(sqrt(b)*tan(c+d*x)/sqrt(a+b*tan(c+d*x)^2))*sqrt(b)/d+1/8*(7*a-4*b)*b*sqrt(a+b*tan(c+d*x)^2)*tan(c+d*x)/d+1/4*b*tan(c+d*x)*(a+b*tan(c+d*x)^2)^(3/2)/d],

# p<0
[tan(e+f*x)^5/sqrt(a+b*tan(e+f*x)^2),x,6,-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/(f*sqrt(a-b))-(a+b)*sqrt(a+b*tan(e+f*x)^2)/(b^2*f)+1/3*(a+b*tan(e+f*x)^2)^(3/2)/(b^2*f)],
[tan(e+f*x)^3/sqrt(a+b*tan(e+f*x)^2),x,5,arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/(f*sqrt(a-b))+sqrt(a+b*tan(e+f*x)^2)/(b*f)],
[tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2),x,4,-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/(f*sqrt(a-b))],
[cot(e+f*x)/sqrt(a+b*tan(e+f*x)^2),x,7,-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a))/(f*sqrt(a))+arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/(f*sqrt(a-b))],
[cot(e+f*x)^3/sqrt(a+b*tan(e+f*x)^2),x,8,1/2*(2*a+b)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a))/(a^(3/2)*f)-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/(f*sqrt(a-b))-1/2*cot(e+f*x)^2*sqrt(a+b*tan(e+f*x)^2)/(a*f)],
[cot(e+f*x)^5/sqrt(a+b*tan(e+f*x)^2),x,9,-1/8*(8*a^2+4*a*b+3*b^2)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a))/(a^(5/2)*f)+arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/(f*sqrt(a-b))+1/8*(4*a+3*b)*cot(e+f*x)^2*sqrt(a+b*tan(e+f*x)^2)/(a^2*f)-1/4*cot(e+f*x)^4*sqrt(a+b*tan(e+f*x)^2)/(a*f)],
[tan(e+f*x)^6/sqrt(a+b*tan(e+f*x)^2),x,8,1/8*(3*a^2+4*a*b+8*b^2)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(b^(5/2)*f)-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(f*sqrt(a-b))-1/8*(3*a+4*b)*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/(b^2*f)+1/4*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)^3/(b*f)],
[tan(e+f*x)^4/sqrt(a+b*tan(e+f*x)^2),x,7,-1/2*(a+2*b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(b^(3/2)*f)+arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(f*sqrt(a-b))+1/2*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/(b*f)],
[tan(e+f*x)^2/sqrt(a+b*tan(e+f*x)^2),x,6,-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(f*sqrt(a-b))+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(f*sqrt(b))],
[1/sqrt(a+b*tan(e+f*x)^2),x,3,arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(f*sqrt(a-b))],
[cot(e+f*x)^2/sqrt(a+b*tan(e+f*x)^2),x,5,-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(f*sqrt(a-b))-cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/(a*f)],
[cot(e+f*x)^4/sqrt(a+b*tan(e+f*x)^2),x,6,arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(f*sqrt(a-b))+1/3*(3*a+2*b)*cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/(a^2*f)-1/3*cot(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2)/(a*f)],
[cot(e+f*x)^6/sqrt(a+b*tan(e+f*x)^2),x,7,-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(f*sqrt(a-b))-1/15*(15*a^2+10*a*b+8*b^2)*cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/(a^3*f)+1/15*(5*a+4*b)*cot(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2)/(a^2*f)-1/5*cot(e+f*x)^5*sqrt(a+b*tan(e+f*x)^2)/(a*f)],
[tan(e+f*x)^5/(a+b*tan(e+f*x)^2)^(3/2),x,6,-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/((a-b)^(3/2)*f)+a^2/((a-b)*b^2*f*sqrt(a+b*tan(e+f*x)^2))+sqrt(a+b*tan(e+f*x)^2)/(b^2*f)],
[tan(e+f*x)^3/(a+b*tan(e+f*x)^2)^(3/2),x,5,arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/((a-b)^(3/2)*f)-a/((a-b)*b*f*sqrt(a+b*tan(e+f*x)^2))],
[tan(e+f*x)/(a+b*tan(e+f*x)^2)^(3/2),x,5,-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/((a-b)^(3/2)*f)+1/((a-b)*f*sqrt(a+b*tan(e+f*x)^2))],
[cot(e+f*x)/(a+b*tan(e+f*x)^2)^(3/2),x,8,-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a))/(a^(3/2)*f)+arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/((a-b)^(3/2)*f)-b/(a*(a-b)*f*sqrt(a+b*tan(e+f*x)^2))],
[cot(e+f*x)^3/(a+b*tan(e+f*x)^2)^(3/2),x,9,1/2*(2*a+3*b)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a))/(a^(5/2)*f)-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/((a-b)^(3/2)*f)-1/2*(a-3*b)*b/(a^2*(a-b)*f*sqrt(a+b*tan(e+f*x)^2))-1/2*cot(e+f*x)^2/(a*f*sqrt(a+b*tan(e+f*x)^2))],
[cot(e+f*x)^5/(a+b*tan(e+f*x)^2)^(3/2),x,10,-1/8*(8*a^2+12*a*b+15*b^2)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a))/(a^(7/2)*f)+arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/((a-b)^(3/2)*f)+1/8*b*(4*a^2+3*a*b-15*b^2)/(a^3*(a-b)*f*sqrt(a+b*tan(e+f*x)^2))+1/8*(4*a+5*b)*cot(e+f*x)^2/(a^2*f*sqrt(a+b*tan(e+f*x)^2))-1/4*cot(e+f*x)^4/(a*f*sqrt(a+b*tan(e+f*x)^2))],
[tan(e+f*x)^6/(a+b*tan(e+f*x)^2)^(3/2),x,8,-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(3/2)*f)-1/2*(3*a+2*b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(b^(5/2)*f)+1/2*(3*a-b)*sqrt(a+b*tan(e+f*x)^2)*tan(e+f*x)/((a-b)*b^2*f)-a*tan(e+f*x)^3/((a-b)*b*f*sqrt(a+b*tan(e+f*x)^2))],
[tan(e+f*x)^4/(a+b*tan(e+f*x)^2)^(3/2),x,7,arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(3/2)*f)+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(b^(3/2)*f)-a*tan(e+f*x)/((a-b)*b*f*sqrt(a+b*tan(e+f*x)^2))],
[tan(e+f*x)^2/(a+b*tan(e+f*x)^2)^(3/2),x,4,-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(3/2)*f)+tan(e+f*x)/((a-b)*f*sqrt(a+b*tan(e+f*x)^2))],
[1/(a+b*tan(e+f*x)^2)^(3/2),x,4,arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(3/2)*f)-b*tan(e+f*x)/(a*(a-b)*f*sqrt(a+b*tan(e+f*x)^2))],
[cot(e+f*x)^2/(a+b*tan(e+f*x)^2)^(3/2),x,6,-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(3/2)*f)-b*cot(e+f*x)/(a*(a-b)*f*sqrt(a+b*tan(e+f*x)^2))-(a-2*b)*cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/(a^2*(a-b)*f)],
[cot(e+f*x)^4/(a+b*tan(e+f*x)^2)^(3/2),x,7,arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(3/2)*f)-b*cot(e+f*x)^3/(a*(a-b)*f*sqrt(a+b*tan(e+f*x)^2))+1/3*(3*a-4*b)*(a+2*b)*cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/(a^3*(a-b)*f)-1/3*(a-4*b)*cot(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2)/(a^2*(a-b)*f)],
[cot(e+f*x)^6/(a+b*tan(e+f*x)^2)^(3/2),x,8,-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(3/2)*f)-b*cot(e+f*x)^5/(a*(a-b)*f*sqrt(a+b*tan(e+f*x)^2))-1/15*(15*a^3+10*a^2*b+8*a*b^2-48*b^3)*cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/(a^4*(a-b)*f)+1/15*(5*a^2+4*a*b-24*b^2)*cot(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2)/(a^3*(a-b)*f)-1/5*(a-6*b)*cot(e+f*x)^5*sqrt(a+b*tan(e+f*x)^2)/(a^2*(a-b)*f)],
[tan(e+f*x)^5/(a+b*tan(e+f*x)^2)^(5/2),x,6,-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/((a-b)^(5/2)*f)-a*(a-2*b)/((a-b)^2*b^2*f*sqrt(a+b*tan(e+f*x)^2))+1/3*a^2/((a-b)*b^2*f*(a+b*tan(e+f*x)^2)^(3/2))],
[tan(e+f*x)^3/(a+b*tan(e+f*x)^2)^(5/2),x,6,arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/((a-b)^(5/2)*f)+(-1)/((a-b)^2*f*sqrt(a+b*tan(e+f*x)^2))-1/3*a/((a-b)*b*f*(a+b*tan(e+f*x)^2)^(3/2))],
[tan(e+f*x)/(a+b*tan(e+f*x)^2)^(5/2),x,6,-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/((a-b)^(5/2)*f)+1/((a-b)^2*f*sqrt(a+b*tan(e+f*x)^2))+1/3/((a-b)*f*(a+b*tan(e+f*x)^2)^(3/2))],
[cot(e+f*x)/(a+b*tan(e+f*x)^2)^(5/2),x,9,-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a))/(a^(5/2)*f)+arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/((a-b)^(5/2)*f)-(2*a-b)*b/(a^2*(a-b)^2*f*sqrt(a+b*tan(e+f*x)^2))-1/3*b/(a*(a-b)*f*(a+b*tan(e+f*x)^2)^(3/2))],
[cot(e+f*x)^3/(a+b*tan(e+f*x)^2)^(5/2),x,10,1/2*(2*a+5*b)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a))/(a^(7/2)*f)-arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/((a-b)^(5/2)*f)-1/2*b*(a^2-8*a*b+5*b^2)/(a^3*(a-b)^2*f*sqrt(a+b*tan(e+f*x)^2))-1/6*(3*a-5*b)*b/(a^2*(a-b)*f*(a+b*tan(e+f*x)^2)^(3/2))-1/2*cot(e+f*x)^2/(a*f*(a+b*tan(e+f*x)^2)^(3/2))],
[cot(e+f*x)^5/(a+b*tan(e+f*x)^2)^(5/2),x,11,-1/8*(8*a^2+20*a*b+35*b^2)*arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a))/(a^(9/2)*f)+arctanh(sqrt(a+b*tan(e+f*x)^2)/sqrt(a-b))/((a-b)^(5/2)*f)+1/8*b*(4*a^3+3*a^2*b-50*a*b^2+35*b^3)/(a^4*(a-b)^2*f*sqrt(a+b*tan(e+f*x)^2))+1/24*b*(12*a^2+15*a*b-35*b^2)/(a^3*(a-b)*f*(a+b*tan(e+f*x)^2)^(3/2))+1/8*(4*a+7*b)*cot(e+f*x)^2/(a^2*f*(a+b*tan(e+f*x)^2)^(3/2))-1/4*cot(e+f*x)^4/(a*f*(a+b*tan(e+f*x)^2)^(3/2))],
[tan(e+f*x)^6/(a+b*tan(e+f*x)^2)^(5/2),x,8,-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(5/2)*f)+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/(b^(5/2)*f)-a*(a-2*b)*tan(e+f*x)/((a-b)^2*b^2*f*sqrt(a+b*tan(e+f*x)^2))-1/3*a*tan(e+f*x)^3/((a-b)*b*f*(a+b*tan(e+f*x)^2)^(3/2))],
[tan(e+f*x)^4/(a+b*tan(e+f*x)^2)^(5/2),x,6,arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(5/2)*f)+1/3*(a-4*b)*tan(e+f*x)/((a-b)^2*b*f*sqrt(a+b*tan(e+f*x)^2))-1/3*a*tan(e+f*x)/((a-b)*b*f*(a+b*tan(e+f*x)^2)^(3/2))],
[tan(e+f*x)^2/(a+b*tan(e+f*x)^2)^(5/2),x,6,-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(5/2)*f)+1/3*(2*a+b)*tan(e+f*x)/(a*(a-b)^2*f*sqrt(a+b*tan(e+f*x)^2))+1/3*tan(e+f*x)/((a-b)*f*(a+b*tan(e+f*x)^2)^(3/2))],
[1/(a+b*tan(e+f*x)^2)^(5/2),x,6,arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(5/2)*f)-1/3*(5*a-2*b)*b*tan(e+f*x)/(a^2*(a-b)^2*f*sqrt(a+b*tan(e+f*x)^2))-1/3*b*tan(e+f*x)/(a*(a-b)*f*(a+b*tan(e+f*x)^2)^(3/2))],
[cot(e+f*x)^2/(a+b*tan(e+f*x)^2)^(5/2),x,7,-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(5/2)*f)-1/3*(7*a-4*b)*b*cot(e+f*x)/(a^2*(a-b)^2*f*sqrt(a+b*tan(e+f*x)^2))-1/3*(a-4*b)*(3*a-2*b)*cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/(a^3*(a-b)^2*f)-1/3*b*cot(e+f*x)/(a*(a-b)*f*(a+b*tan(e+f*x)^2)^(3/2))],
[cot(e+f*x)^4/(a+b*tan(e+f*x)^2)^(5/2),x,8,arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(5/2)*f)-(3*a-2*b)*b*cot(e+f*x)^3/(a^2*(a-b)^2*f*sqrt(a+b*tan(e+f*x)^2))+1/3*(a-2*b)*(3*a^2+8*a*b-8*b^2)*cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/(a^4*(a-b)^2*f)-1/3*(a^2-12*a*b+8*b^2)*cot(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2)/(a^3*(a-b)^2*f)-1/3*b*cot(e+f*x)^3/(a*(a-b)*f*(a+b*tan(e+f*x)^2)^(3/2))],
[cot(e+f*x)^6/(a+b*tan(e+f*x)^2)^(5/2),x,9,-arctan(sqrt(a-b)*tan(e+f*x)/sqrt(a+b*tan(e+f*x)^2))/((a-b)^(5/2)*f)-1/3*(11*a-8*b)*b*cot(e+f*x)^5/(a^2*(a-b)^2*f*sqrt(a+b*tan(e+f*x)^2))-1/15*(15*a^4+10*a^3*b+8*a^2*b^2-176*a*b^3+128*b^4)*cot(e+f*x)*sqrt(a+b*tan(e+f*x)^2)/(a^5*(a-b)^2*f)+1/15*(5*a^3+4*a^2*b-88*a*b^2+64*b^3)*cot(e+f*x)^3*sqrt(a+b*tan(e+f*x)^2)/(a^4*(a-b)^2*f)-1/5*(a^2-22*a*b+16*b^2)*cot(e+f*x)^5*sqrt(a+b*tan(e+f*x)^2)/(a^3*(a-b)^2*f)-1/3*b*cot(e+f*x)^5/(a*(a-b)*f*(a+b*tan(e+f*x)^2)^(3/2))],

# Integrands of the form (d Tan[e+f x])^m (a+b Tan[e+f x]^2)^p when p symbolic
[(d*tan(e+f*x))^m*(b*tan(e+f*x)^2)^p,x,4,hypergeom([1,1/2*(1+m+2*p)],[1/2*(3+m+2*p)],-tan(e+f*x)^2)*tan(e+f*x)*(d*tan(e+f*x))^m*(b*tan(e+f*x)^2)^p/(f*(1+m+2*p))],
[(d*tan(e+f*x))^m*(a+b*tan(e+f*x)^2)^p,x,3,AppellF1(1/2*(1+m),1,-p,1/2*(3+m),-tan(e+f*x)^2,-b*tan(e+f*x)^2/a)*(d*tan(e+f*x))^(1+m)*(a+b*tan(e+f*x)^2)^p/(d*f*(1+m)*(1+b*tan(e+f*x)^2/a)^p)],
[tan(e+f*x)^5*(a+b*tan(e+f*x)^2)^p,x,5,-1/2*(a+b)*(a+b*tan(e+f*x)^2)^(1+p)/(b^2*f*(1+p))-1/2*hypergeom([1,1+p],[2+p],(a+b*tan(e+f*x)^2)/(a-b))*(a+b*tan(e+f*x)^2)^(1+p)/((a-b)*f*(1+p))+1/2*(a+b*tan(e+f*x)^2)^(2+p)/(b^2*f*(2+p))],
[tan(e+f*x)^3*(a+b*tan(e+f*x)^2)^p,x,4,1/2*(a+b*tan(e+f*x)^2)^(1+p)/(b*f*(1+p))+1/2*hypergeom([1,1+p],[2+p],(a+b*tan(e+f*x)^2)/(a-b))*(a+b*tan(e+f*x)^2)^(1+p)/((a-b)*f*(1+p))],
[tan(e+f*x)*(a+b*tan(e+f*x)^2)^p,x,3,-1/2*hypergeom([1,1+p],[2+p],(a+b*tan(e+f*x)^2)/(a-b))*(a+b*tan(e+f*x)^2)^(1+p)/((a-b)*f*(1+p))],
[cot(e+f*x)*(a+b*tan(e+f*x)^2)^p,x,5,1/2*hypergeom([1,1+p],[2+p],(a+b*tan(e+f*x)^2)/(a-b))*(a+b*tan(e+f*x)^2)^(1+p)/((a-b)*f*(1+p))-1/2*hypergeom([1,1+p],[2+p],1+b*tan(e+f*x)^2/a)*(a+b*tan(e+f*x)^2)^(1+p)/(a*f*(1+p))],
[cot(e+f*x)^3*(a+b*tan(e+f*x)^2)^p,x,6,-1/2*cot(e+f*x)^2*(a+b*tan(e+f*x)^2)^(1+p)/(a*f)-1/2*hypergeom([1,1+p],[2+p],(a+b*tan(e+f*x)^2)/(a-b))*(a+b*tan(e+f*x)^2)^(1+p)/((a-b)*f*(1+p))+1/2*(a-b*p)*hypergeom([1,1+p],[2+p],1+b*tan(e+f*x)^2/a)*(a+b*tan(e+f*x)^2)^(1+p)/(a^2*f*(1+p))],
[cot(e+f*x)^5*(a+b*tan(e+f*x)^2)^p,x,7,1/4*(2*a+b-b*p)*cot(e+f*x)^2*(a+b*tan(e+f*x)^2)^(1+p)/(a^2*f)-1/4*cot(e+f*x)^4*(a+b*tan(e+f*x)^2)^(1+p)/(a*f)+1/2*hypergeom([1,1+p],[2+p],(a+b*tan(e+f*x)^2)/(a-b))*(a+b*tan(e+f*x)^2)^(1+p)/((a-b)*f*(1+p))-1/4*(2*a^2-2*a*b*p-b^2*(1-p)*p)*hypergeom([1,1+p],[2+p],1+b*tan(e+f*x)^2/a)*(a+b*tan(e+f*x)^2)^(1+p)/(a^3*f*(1+p))],
[tan(e+f*x)^6*(a+b*tan(e+f*x)^2)^p,x,3,1/7*AppellF1(7/2,1,-p,9/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/a)*tan(e+f*x)^7*(a+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/a)^p)],
[tan(e+f*x)^4*(a+b*tan(e+f*x)^2)^p,x,3,1/5*AppellF1(5/2,1,-p,7/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/a)*tan(e+f*x)^5*(a+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/a)^p)],
[tan(e+f*x)^2*(a+b*tan(e+f*x)^2)^p,x,3,1/3*AppellF1(3/2,1,-p,5/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/a)*tan(e+f*x)^3*(a+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/a)^p)],
[(a+b*tan(e+f*x)^2)^p,x,3,AppellF1(1/2,1,-p,3/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/a)*tan(e+f*x)*(a+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/a)^p)],
[cot(e+f*x)^2*(a+b*tan(e+f*x)^2)^p,x,3,-AppellF1(-1/2,1,-p,1/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/a)*cot(e+f*x)*(a+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/a)^p)],
[cot(e+f*x)^4*(a+b*tan(e+f*x)^2)^p,x,3,-1/3*AppellF1(-3/2,1,-p,-1/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/a)*cot(e+f*x)^3*(a+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/a)^p)],
[cot(e+f*x)^6*(a+b*tan(e+f*x)^2)^p,x,3,-1/5*AppellF1(-5/2,1,-p,-3/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/a)*cot(e+f*x)^5*(a+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/a)^p)],

# Integrands of the form (d Tan[e+f x])^m (a+b Tan[e+f x]^3)^p

# Integrands of the form (a+b Tan[e+f x]^3)^p
[(a+b*tan(c+d*x)^3)^4,x,6,(a^4-6*a^2*b^2+b^4)*x+4*a*b*(a^2-b^2)*log(cos(c+d*x))/d+b^2*(6*a^2-b^2)*tan(c+d*x)/d+2*a*b*(a^2-b^2)*tan(c+d*x)^2/d-1/3*b^2*(6*a^2-b^2)*tan(c+d*x)^3/d+a*b^3*tan(c+d*x)^4/d+1/5*b^2*(6*a^2-b^2)*tan(c+d*x)^5/d-2/3*a*b^3*tan(c+d*x)^6/d+1/7*b^4*tan(c+d*x)^7/d+1/2*a*b^3*tan(c+d*x)^8/d-1/9*b^4*tan(c+d*x)^9/d+1/11*b^4*tan(c+d*x)^11/d],
[(a+b*tan(c+d*x)^3)^3,x,6,a*(a^2-3*b^2)*x+b*(3*a^2-b^2)*log(cos(c+d*x))/d+3*a*b^2*tan(c+d*x)/d+1/2*b*(3*a^2-b^2)*tan(c+d*x)^2/d-a*b^2*tan(c+d*x)^3/d+1/4*b^3*tan(c+d*x)^4/d+3/5*a*b^2*tan(c+d*x)^5/d-1/6*b^3*tan(c+d*x)^6/d+1/8*b^3*tan(c+d*x)^8/d],
[(a+b*tan(c+d*x)^3)^2,x,6,(a^2-b^2)*x+2*a*b*log(cos(c+d*x))/d+b^2*tan(c+d*x)/d+a*b*tan(c+d*x)^2/d-1/3*b^2*tan(c+d*x)^3/d+1/5*b^2*tan(c+d*x)^5/d],
[a+b*tan(c+d*x)^3,x,3,a*x+b*log(cos(c+d*x))/d+1/2*b*tan(c+d*x)^2/d],
[1/(a+b*tan(c+d*x)^3),x,14,a*x/(a^2+b^2)-1/3*b*log(a*cos(c+d*x)^3+b*sin(c+d*x)^3)/((a^2+b^2)*d)+1/3*b^(1/3)*(a^(4/3)+b^(4/3))*log(a^(1/3)+b^(1/3)*tan(c+d*x))/(a^(2/3)*(a^2+b^2)*d)-1/6*b^(1/3)*(a^(4/3)+b^(4/3))*log(a^(2/3)-a^(1/3)*b^(1/3)*tan(c+d*x)+b^(2/3)*tan(c+d*x)^2)/(a^(2/3)*(a^2+b^2)*d)+b^(1/3)*(a^(4/3)-b^(4/3))*arctan((a^(1/3)-2*b^(1/3)*tan(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(2/3)*(a^2+b^2)*d*sqrt(3))],
[1/(a+b*tan(c+d*x)^3)^2,x,21,(a^2-b^2)*x/(a^2+b^2)^2-2/3*a*b*log(a*cos(c+d*x)^3+b*sin(c+d*x)^3)/((a^2+b^2)^2*d)+1/3*b^(1/3)*(a^2+2*a^(2/3)*b^(4/3)-b^2)*log(a^(1/3)+b^(1/3)*tan(c+d*x))/(a^(1/3)*(a^2+b^2)^2*d)+1/9*b^(1/3)*(a^(4/3)+2*b^(4/3))*log(a^(1/3)+b^(1/3)*tan(c+d*x))/(a^(5/3)*(a^2+b^2)*d)-1/6*b^(1/3)*(a^2+2*a^(2/3)*b^(4/3)-b^2)*log(a^(2/3)-a^(1/3)*b^(1/3)*tan(c+d*x)+b^(2/3)*tan(c+d*x)^2)/(a^(1/3)*(a^2+b^2)^2*d)-1/18*b^(1/3)*(a^(4/3)+2*b^(4/3))*log(a^(2/3)-a^(1/3)*b^(1/3)*tan(c+d*x)+b^(2/3)*tan(c+d*x)^2)/(a^(5/3)*(a^2+b^2)*d)+b^(1/3)*(a^2-2*a^(2/3)*b^(4/3)-b^2)*arctan((a^(1/3)-2*b^(1/3)*tan(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(1/3)*(a^2+b^2)^2*d*sqrt(3))+1/3*b^(1/3)*(a^(4/3)-2*b^(4/3))*arctan((a^(1/3)-2*b^(1/3)*tan(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(5/3)*(a^2+b^2)*d*sqrt(3))+1/3*b*(a+tan(c+d*x)*(b-a*tan(c+d*x)))/(a*(a^2+b^2)*d*(a+b*tan(c+d*x)^3))],
[1/(1+tan(x)^3),x,7,1/2*x-1/2*log(cos(x))+1/6*log(1+tan(x))-1/3*log(1-tan(x)+tan(x)^2)],

# Integrands of the form (d Tan[e+f x])^m (a+b Tan[e+f x]^4)^p

# Integrands of the form (a+b Tan[e+f x]^4)^p
[(a+b*tan(c+d*x)^4)^4,x,4,(a+b)^4*x-b*(2*a+b)*(2*a^2+2*a*b+b^2)*tan(c+d*x)/d+1/3*b*(2*a+b)*(2*a^2+2*a*b+b^2)*tan(c+d*x)^3/d-1/5*b^2*(6*a^2+4*a*b+b^2)*tan(c+d*x)^5/d+1/7*b^2*(6*a^2+4*a*b+b^2)*tan(c+d*x)^7/d-1/9*b^3*(4*a+b)*tan(c+d*x)^9/d+1/11*b^3*(4*a+b)*tan(c+d*x)^11/d-1/13*b^4*tan(c+d*x)^13/d+1/15*b^4*tan(c+d*x)^15/d],
[(a+b*tan(c+d*x)^4)^3,x,4,(a+b)^3*x-b*(3*a^2+3*a*b+b^2)*tan(c+d*x)/d+1/3*b*(3*a^2+3*a*b+b^2)*tan(c+d*x)^3/d-1/5*b^2*(3*a+b)*tan(c+d*x)^5/d+1/7*b^2*(3*a+b)*tan(c+d*x)^7/d-1/9*b^3*tan(c+d*x)^9/d+1/11*b^3*tan(c+d*x)^11/d],
[(a+b*tan(c+d*x)^4)^2,x,4,(a+b)^2*x-b*(2*a+b)*tan(c+d*x)/d+1/3*b*(2*a+b)*tan(c+d*x)^3/d-1/5*b^2*tan(c+d*x)^5/d+1/7*b^2*tan(c+d*x)^7/d],
[a+b*tan(c+d*x)^4,x,4,a*x+b*x-b*tan(c+d*x)/d+1/3*b*tan(c+d*x)^3/d],
[1/(a+b*tan(c+d*x)^4),x,13,x/(a+b)+1/2*b^(1/4)*arctan(1-b^(1/4)*sqrt(2)*tan(c+d*x)/a^(1/4))*(sqrt(a)-sqrt(b))/(a^(3/4)*(a+b)*d*sqrt(2))-1/2*b^(1/4)*arctan(1+b^(1/4)*sqrt(2)*tan(c+d*x)/a^(1/4))*(sqrt(a)-sqrt(b))/(a^(3/4)*(a+b)*d*sqrt(2))-1/4*b^(1/4)*log(sqrt(a)-a^(1/4)*b^(1/4)*sqrt(2)*tan(c+d*x)+sqrt(b)*tan(c+d*x)^2)*(sqrt(a)+sqrt(b))/(a^(3/4)*(a+b)*d*sqrt(2))+1/4*b^(1/4)*log(sqrt(a)+a^(1/4)*b^(1/4)*sqrt(2)*tan(c+d*x)+sqrt(b)*tan(c+d*x)^2)*(sqrt(a)+sqrt(b))/(a^(3/4)*(a+b)*d*sqrt(2))],
[1/(a+b*tan(c+d*x)^4)^2,x,23,x/(a+b)^2+1/8*b^(1/4)*arctan(1-b^(1/4)*sqrt(2)*tan(c+d*x)/a^(1/4))*(sqrt(a)-3*sqrt(b))/(a^(7/4)*(a+b)*d*sqrt(2))-1/8*b^(1/4)*arctan(1+b^(1/4)*sqrt(2)*tan(c+d*x)/a^(1/4))*(sqrt(a)-3*sqrt(b))/(a^(7/4)*(a+b)*d*sqrt(2))+1/2*b^(1/4)*arctan(1-b^(1/4)*sqrt(2)*tan(c+d*x)/a^(1/4))*(sqrt(a)-sqrt(b))/(a^(3/4)*(a+b)^2*d*sqrt(2))-1/2*b^(1/4)*arctan(1+b^(1/4)*sqrt(2)*tan(c+d*x)/a^(1/4))*(sqrt(a)-sqrt(b))/(a^(3/4)*(a+b)^2*d*sqrt(2))-1/4*b^(1/4)*log(sqrt(a)-a^(1/4)*b^(1/4)*sqrt(2)*tan(c+d*x)+sqrt(b)*tan(c+d*x)^2)*(sqrt(a)+sqrt(b))/(a^(3/4)*(a+b)^2*d*sqrt(2))+1/4*b^(1/4)*log(sqrt(a)+a^(1/4)*b^(1/4)*sqrt(2)*tan(c+d*x)+sqrt(b)*tan(c+d*x)^2)*(sqrt(a)+sqrt(b))/(a^(3/4)*(a+b)^2*d*sqrt(2))-1/16*b^(1/4)*log(sqrt(a)-a^(1/4)*b^(1/4)*sqrt(2)*tan(c+d*x)+sqrt(b)*tan(c+d*x)^2)*(sqrt(a)+3*sqrt(b))/(a^(7/4)*(a+b)*d*sqrt(2))+1/16*b^(1/4)*log(sqrt(a)+a^(1/4)*b^(1/4)*sqrt(2)*tan(c+d*x)+sqrt(b)*tan(c+d*x)^2)*(sqrt(a)+3*sqrt(b))/(a^(7/4)*(a+b)*d*sqrt(2))+1/4*b*tan(c+d*x)*(1-tan(c+d*x)^2)/(a*(a+b)*d*(a+b*tan(c+d*x)^4))],

# Integrands of the form (a+b Tan[e+f x]^4)^(p/2)
[(a+b*tan(c+d*x)^4)^(1/2),x,8,1/2*arctan(sqrt(a+b)*tan(c+d*x)/sqrt(a+b*tan(c+d*x)^4))*sqrt(a+b)/d+sqrt(b)*sqrt(a+b*tan(c+d*x)^4)*tan(c+d*x)/(d*(sqrt(a)+sqrt(b)*tan(c+d*x)^2))-a^(1/4)*b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4))),sqrt(1/2))*sqrt((a+b*tan(c+d*x)^4)/(sqrt(a)+sqrt(b)*tan(c+d*x)^2)^2)*(sqrt(a)+sqrt(b)*tan(c+d*x)^2)/(d*sqrt(a+b*tan(c+d*x)^4))-1/2*b^(1/4)*(a+b)*sqrt(cos(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4))),sqrt(1/2))*sqrt((a+b*tan(c+d*x)^4)/(sqrt(a)+sqrt(b)*tan(c+d*x)^2)^2)*(sqrt(a)+sqrt(b)*tan(c+d*x)^2)/(a^(1/4)*d*(sqrt(a)-sqrt(b))*sqrt(a+b*tan(c+d*x)^4))+1/2*b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4))),sqrt(1/2))*(sqrt(a)-sqrt(b))*sqrt((a+b*tan(c+d*x)^4)/(sqrt(a)+sqrt(b)*tan(c+d*x)^2)^2)*(sqrt(a)+sqrt(b)*tan(c+d*x)^2)/(a^(1/4)*d*sqrt(a+b*tan(c+d*x)^4))+1/4*(a+b)*sqrt(cos(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4)))*EllipticPi(sin(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4))),-1/4*(sqrt(a)-sqrt(b))^2/(sqrt(a)*sqrt(b)),sqrt(1/2))*(sqrt(a)+sqrt(b))*sqrt((a+b*tan(c+d*x)^4)/(sqrt(a)+sqrt(b)*tan(c+d*x)^2)^2)*(sqrt(a)+sqrt(b)*tan(c+d*x)^2)/(a^(1/4)*b^(1/4)*d*(sqrt(a)-sqrt(b))*sqrt(a+b*tan(c+d*x)^4))],
[1/(a+b*tan(c+d*x)^4)^(1/2),x,4,1/2*arctan(sqrt(a+b)*tan(c+d*x)/sqrt(a+b*tan(c+d*x)^4))/(d*sqrt(a+b))-1/2*b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4))),sqrt(1/2))*sqrt((a+b*tan(c+d*x)^4)/(sqrt(a)+sqrt(b)*tan(c+d*x)^2)^2)*(sqrt(a)+sqrt(b)*tan(c+d*x)^2)/(a^(1/4)*d*(sqrt(a)-sqrt(b))*sqrt(a+b*tan(c+d*x)^4))+1/4*sqrt(cos(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4)))*EllipticPi(sin(2*arctan(b^(1/4)*tan(c+d*x)/a^(1/4))),-1/4*(sqrt(a)-sqrt(b))^2/(sqrt(a)*sqrt(b)),sqrt(1/2))*(sqrt(a)+sqrt(b))*sqrt((a+b*tan(c+d*x)^4)/(sqrt(a)+sqrt(b)*tan(c+d*x)^2)^2)*(sqrt(a)+sqrt(b)*tan(c+d*x)^2)/(a^(1/4)*b^(1/4)*d*(sqrt(a)-sqrt(b))*sqrt(a+b*tan(c+d*x)^4))],

# Integrands of the form Tan[e+f x]^m (a+b Tan[e+f x]^4)^(p/2)

# p>0
[sqrt(a+b*tan(x)^4)*tan(x)^3,x,8,1/4*(a+2*b)*arctanh(sqrt(b)*tan(x)^2/sqrt(a+b*tan(x)^4))/sqrt(b)+1/2*arctanh((a-b*tan(x)^2)/(sqrt(a+b)*sqrt(a+b*tan(x)^4)))*sqrt(a+b)-1/4*sqrt(a+b*tan(x)^4)*(2-tan(x)^2)],
[sqrt(a+b*tan(x)^4)*tan(x),x,8,-1/2*arctanh(sqrt(b)*tan(x)^2/sqrt(a+b*tan(x)^4))*sqrt(b)-1/2*arctanh((a-b*tan(x)^2)/(sqrt(a+b)*sqrt(a+b*tan(x)^4)))*sqrt(a+b)+1/2*sqrt(a+b*tan(x)^4)],
[cot(x)*sqrt(a+b*tan(x)^4),x,11,-1/2*arctanh(sqrt(a+b*tan(x)^4)/sqrt(a))*sqrt(a)+1/2*arctanh(sqrt(b)*tan(x)^2/sqrt(a+b*tan(x)^4))*sqrt(b)+1/2*arctanh((a-b*tan(x)^2)/(sqrt(a+b)*sqrt(a+b*tan(x)^4)))*sqrt(a+b)],
[sqrt(a+b*tan(x)^4)*tan(x)^2,x,12,-1/2*arctan(sqrt(a+b)*tan(x)/sqrt(a+b*tan(x)^4))*sqrt(a+b)+1/3*sqrt(a+b*tan(x)^4)*tan(x)-sqrt(b)*sqrt(a+b*tan(x)^4)*tan(x)/(sqrt(a)+sqrt(b)*tan(x)^2)+a^(1/4)*b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*tan(x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*tan(x)/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*tan(x)/a^(1/4))),sqrt(1/2))*sqrt((a+b*tan(x)^4)/(sqrt(a)+sqrt(b)*tan(x)^2)^2)*(sqrt(a)+sqrt(b)*tan(x)^2)/sqrt(a+b*tan(x)^4)+1/3*a^(3/4)*sqrt(cos(2*arctan(b^(1/4)*tan(x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*tan(x)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*tan(x)/a^(1/4))),sqrt(1/2))*sqrt((a+b*tan(x)^4)/(sqrt(a)+sqrt(b)*tan(x)^2)^2)*(sqrt(a)+sqrt(b)*tan(x)^2)/(b^(1/4)*sqrt(a+b*tan(x)^4))+1/2*b^(1/4)*(a+b)*sqrt(cos(2*arctan(b^(1/4)*tan(x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*tan(x)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*tan(x)/a^(1/4))),sqrt(1/2))*sqrt((a+b*tan(x)^4)/(sqrt(a)+sqrt(b)*tan(x)^2)^2)*(sqrt(a)+sqrt(b)*tan(x)^2)/(a^(1/4)*(sqrt(a)-sqrt(b))*sqrt(a+b*tan(x)^4))-1/2*b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*tan(x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*tan(x)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*tan(x)/a^(1/4))),sqrt(1/2))*(sqrt(a)-sqrt(b))*sqrt((a+b*tan(x)^4)/(sqrt(a)+sqrt(b)*tan(x)^2)^2)*(sqrt(a)+sqrt(b)*tan(x)^2)/(a^(1/4)*sqrt(a+b*tan(x)^4))-1/4*(a+b)*sqrt(cos(2*arctan(b^(1/4)*tan(x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*tan(x)/a^(1/4)))*EllipticPi(sin(2*arctan(b^(1/4)*tan(x)/a^(1/4))),-1/4*(sqrt(a)-sqrt(b))^2/(sqrt(a)*sqrt(b)),sqrt(1/2))*(sqrt(a)+sqrt(b))*sqrt((a+b*tan(x)^4)/(sqrt(a)+sqrt(b)*tan(x)^2)^2)*(sqrt(a)+sqrt(b)*tan(x)^2)/(a^(1/4)*b^(1/4)*(sqrt(a)-sqrt(b))*sqrt(a+b*tan(x)^4))],
[tan(x)^3*(a+b*tan(x)^4)^(3/2),x,9,1/2*(a+b)^(3/2)*arctanh((a-b*tan(x)^2)/(sqrt(a+b)*sqrt(a+b*tan(x)^4)))+1/16*(3*a^2+12*a*b+8*b^2)*arctanh(sqrt(b)*tan(x)^2/sqrt(a+b*tan(x)^4))/sqrt(b)-1/16*sqrt(a+b*tan(x)^4)*(8*(a+b)-(3*a+4*b)*tan(x)^2)-1/24*(4-3*tan(x)^2)*(a+b*tan(x)^4)^(3/2)],
[tan(x)*(a+b*tan(x)^4)^(3/2),x,9,-1/2*(a+b)^(3/2)*arctanh((a-b*tan(x)^2)/(sqrt(a+b)*sqrt(a+b*tan(x)^4)))-1/4*(3*a+2*b)*arctanh(sqrt(b)*tan(x)^2/sqrt(a+b*tan(x)^4))*sqrt(b)+1/4*sqrt(a+b*tan(x)^4)*(2*(a+b)-b*tan(x)^2)+1/6*(a+b*tan(x)^4)^(3/2)],
[cot(x)*(a+b*tan(x)^4)^(3/2),x,13,-1/2*a^(3/2)*arctanh(sqrt(a+b*tan(x)^4)/sqrt(a))+1/2*(a+b)^(3/2)*arctanh((a-b*tan(x)^2)/(sqrt(a+b)*sqrt(a+b*tan(x)^4)))+1/4*(3*a+2*b)*arctanh(sqrt(b)*tan(x)^2/sqrt(a+b*tan(x)^4))*sqrt(b)+1/2*a*sqrt(a+b*tan(x)^4)-1/4*sqrt(a+b*tan(x)^4)*(2*(a+b)-b*tan(x)^2)],

# p<0
[tan(x)^3/sqrt(a+b*tan(x)^4),x,7,1/2*arctanh(sqrt(b)*tan(x)^2/sqrt(a+b*tan(x)^4))/sqrt(b)+1/2*arctanh((a-b*tan(x)^2)/(sqrt(a+b)*sqrt(a+b*tan(x)^4)))/sqrt(a+b)],
[tan(x)/sqrt(a+b*tan(x)^4),x,4,-1/2*arctanh((a-b*tan(x)^2)/(sqrt(a+b)*sqrt(a+b*tan(x)^4)))/sqrt(a+b)],
[cot(x)/sqrt(a+b*tan(x)^4),x,9,-1/2*arctanh(sqrt(a+b*tan(x)^4)/sqrt(a))/sqrt(a)+1/2*arctanh((a-b*tan(x)^2)/(sqrt(a+b)*sqrt(a+b*tan(x)^4)))/sqrt(a+b)],
[tan(x)^2/sqrt(a+b*tan(x)^4),x,4,-1/2*arctan(sqrt(a+b)*tan(x)/sqrt(a+b*tan(x)^4))/sqrt(a+b)+1/2*a^(1/4)*sqrt(cos(2*arctan(b^(1/4)*tan(x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*tan(x)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*tan(x)/a^(1/4))),sqrt(1/2))*sqrt((a+b*tan(x)^4)/(sqrt(a)+sqrt(b)*tan(x)^2)^2)*(sqrt(a)+sqrt(b)*tan(x)^2)/(b^(1/4)*(sqrt(a)-sqrt(b))*sqrt(a+b*tan(x)^4))-1/4*sqrt(cos(2*arctan(b^(1/4)*tan(x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*tan(x)/a^(1/4)))*EllipticPi(sin(2*arctan(b^(1/4)*tan(x)/a^(1/4))),-1/4*(sqrt(a)-sqrt(b))^2/(sqrt(a)*sqrt(b)),sqrt(1/2))*(sqrt(a)+sqrt(b))*sqrt((a+b*tan(x)^4)/(sqrt(a)+sqrt(b)*tan(x)^2)^2)*(sqrt(a)+sqrt(b)*tan(x)^2)/(a^(1/4)*b^(1/4)*(sqrt(a)-sqrt(b))*sqrt(a+b*tan(x)^4))],
[tan(x)^3/(a+b*tan(x)^4)^(3/2),x,6,1/2*arctanh((a-b*tan(x)^2)/(sqrt(a+b)*sqrt(a+b*tan(x)^4)))/(a+b)^(3/2)+1/2*(-1+tan(x)^2)/((a+b)*sqrt(a+b*tan(x)^4))],
[tan(x)/(a+b*tan(x)^4)^(3/2),x,6,-1/2*arctanh((a-b*tan(x)^2)/(sqrt(a+b)*sqrt(a+b*tan(x)^4)))/(a+b)^(3/2)+1/2*(a+b*tan(x)^2)/(a*(a+b)*sqrt(a+b*tan(x)^4))],
[cot(x)/(a+b*tan(x)^4)^(3/2),x,12,-1/2*arctanh(sqrt(a+b*tan(x)^4)/sqrt(a))/a^(3/2)+1/2*arctanh((a-b*tan(x)^2)/(sqrt(a+b)*sqrt(a+b*tan(x)^4)))/(a+b)^(3/2)+1/2/(a*sqrt(a+b*tan(x)^4))+1/2*(-a-b*tan(x)^2)/(a*(a+b)*sqrt(a+b*tan(x)^4))],
[tan(x)^3/(a+b*tan(x)^4)^(5/2),x,7,1/2*arctanh((a-b*tan(x)^2)/(sqrt(a+b)*sqrt(a+b*tan(x)^4)))/(a+b)^(5/2)+1/6*(-3*a-(-2*a+b)*tan(x)^2)/(a*(a+b)^2*sqrt(a+b*tan(x)^4))+1/6*(-1+tan(x)^2)/((a+b)*(a+b*tan(x)^4)^(3/2))],
[tan(x)/(a+b*tan(x)^4)^(5/2),x,7,-1/2*arctanh((a-b*tan(x)^2)/(sqrt(a+b)*sqrt(a+b*tan(x)^4)))/(a+b)^(5/2)+1/6*(3*a^2+b*(5*a+2*b)*tan(x)^2)/(a^2*(a+b)^2*sqrt(a+b*tan(x)^4))+1/6*(a+b*tan(x)^2)/(a*(a+b)*(a+b*tan(x)^4)^(3/2))],
[cot(x)/(a+b*tan(x)^4)^(5/2),x,14,-1/2*arctanh(sqrt(a+b*tan(x)^4)/sqrt(a))/a^(5/2)+1/2*arctanh((a-b*tan(x)^2)/(sqrt(a+b)*sqrt(a+b*tan(x)^4)))/(a+b)^(5/2)+1/2/(a^2*sqrt(a+b*tan(x)^4))+1/6*(-3*a^2-b*(5*a+2*b)*tan(x)^2)/(a^2*(a+b)^2*sqrt(a+b*tan(x)^4))+1/6/(a*(a+b*tan(x)^4)^(3/2))+1/6*(-a-b*tan(x)^2)/(a*(a+b)*(a+b*tan(x)^4)^(3/2))],

# Integrands of the form (d Tan[e+f x])^m (a+b (c Tan[e+f x])^n)^p

# Integrands of the form Tan[e+f x]^m (a+b Tan[e+f x]^(n/2))^p
[(a+b*sqrt(c*tan(e+f*x)))^2*(d*tan(e+f*x))^m,x,9,1/2*hypergeom([1,1+m],[2+m],-c*tan(e+f*x)/sqrt(-c^2))*(a^2-b^2*sqrt(-c^2))*tan(e+f*x)*(d*tan(e+f*x))^m/(f*(1+m))+1/2*hypergeom([1,1+m],[2+m],c*tan(e+f*x)/sqrt(-c^2))*(a^2+b^2*sqrt(-c^2))*tan(e+f*x)*(d*tan(e+f*x))^m/(f*(1+m))+4*a*b*hypergeom([1,1/4*(3+2*m)],[1/4*(7+2*m)],-tan(e+f*x)^2)*(c*tan(e+f*x))^(3/2)*(d*tan(e+f*x))^m/(c*f*(3+2*m))],
[(a+b*sqrt(c*tan(e+f*x)))*(d*tan(e+f*x))^m,x,7,a*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(e+f*x)^2)*tan(e+f*x)*(d*tan(e+f*x))^m/(f*(1+m))+2*b*hypergeom([1,1/4*(3+2*m)],[1/4*(7+2*m)],-tan(e+f*x)^2)*(c*tan(e+f*x))^(3/2)*(d*tan(e+f*x))^m/(c*f*(3+2*m))],
[(d*tan(e+f*x))^m/(a+b*sqrt(c*tan(e+f*x))),x,14,b^4*c^2*hypergeom([1,2*(1+m)],[3+2*m],-b*sqrt(c*tan(e+f*x))/a)*tan(e+f*x)*(d*tan(e+f*x))^m/(a*(a^4+b^4*c^2)*f*(1+m))+1/2*a*hypergeom([1,1+m],[2+m],-c*tan(e+f*x)/sqrt(-c^2))*(a^2-b^2*sqrt(-c^2))*tan(e+f*x)*(d*tan(e+f*x))^m/((a^4+b^4*c^2)*f*(1+m))+1/2*a*hypergeom([1,1+m],[2+m],c*tan(e+f*x)/sqrt(-c^2))*(a^2+b^2*sqrt(-c^2))*tan(e+f*x)*(d*tan(e+f*x))^m/((a^4+b^4*c^2)*f*(1+m))-b*hypergeom([1,1/2*(3+2*m)],[1/2*(5+2*m)],-c*tan(e+f*x)/sqrt(-c^2))*(a^2-b^2*sqrt(-c^2))*(c*tan(e+f*x))^(3/2)*(d*tan(e+f*x))^m/(c*(a^4+b^4*c^2)*f*(3+2*m))-b*hypergeom([1,1/2*(3+2*m)],[1/2*(5+2*m)],c*tan(e+f*x)/sqrt(-c^2))*(a^2+b^2*sqrt(-c^2))*(c*tan(e+f*x))^(3/2)*(d*tan(e+f*x))^m/(c*(a^4+b^4*c^2)*f*(3+2*m))],
[(d*tan(e+f*x))^m/(a+b*sqrt(c*tan(e+f*x)))^2,x,15,4*a^2*b^4*c^2*hypergeom([1,2*(1+m)],[3+2*m],-b*sqrt(c*tan(e+f*x))/a)*tan(e+f*x)*(d*tan(e+f*x))^m/((a^4+b^4*c^2)^2*f*(1+m))+b^4*c^2*hypergeom([2,2*(1+m)],[3+2*m],-b*sqrt(c*tan(e+f*x))/a)*tan(e+f*x)*(d*tan(e+f*x))^m/(a^2*(a^4+b^4*c^2)*f*(1+m))+1/2*hypergeom([1,1+m],[2+m],-c*tan(e+f*x)/sqrt(-c^2))*(a^6-3*a^2*b^4*c^2-b^6*(-c^2)^(3/2)-3*a^4*b^2*sqrt(-c^2))*tan(e+f*x)*(d*tan(e+f*x))^m/((a^4+b^4*c^2)^2*f*(1+m))+1/2*hypergeom([1,1+m],[2+m],c*tan(e+f*x)/sqrt(-c^2))*(a^6-3*a^2*b^4*c^2+b^6*(-c^2)^(3/2)+3*a^4*b^2*sqrt(-c^2))*tan(e+f*x)*(d*tan(e+f*x))^m/((a^4+b^4*c^2)^2*f*(1+m))-2*a*b*hypergeom([1,1/2*(3+2*m)],[1/2*(5+2*m)],-c*tan(e+f*x)/sqrt(-c^2))*(a^4-b^4*c^2-2*a^2*b^2*sqrt(-c^2))*(c*tan(e+f*x))^(3/2)*(d*tan(e+f*x))^m/(c*(a^4+b^4*c^2)^2*f*(3+2*m))-2*a*b*hypergeom([1,1/2*(3+2*m)],[1/2*(5+2*m)],c*tan(e+f*x)/sqrt(-c^2))*(a^4-b^4*c^2+2*a^2*b^2*sqrt(-c^2))*(c*tan(e+f*x))^(3/2)*(d*tan(e+f*x))^m/(c*(a^4+b^4*c^2)^2*f*(3+2*m))],

# Integrands of the form (d Tan[e+f x])^m (b (c Tan[e+f x])^n)^p when p symbolic
[(d*tan(e+f*x))^m*(b*(c*tan(e+f*x))^n)^p,x,4,hypergeom([1,1/2*(1+m+n*p)],[1/2*(3+m+n*p)],-tan(e+f*x)^2)*tan(e+f*x)*(d*tan(e+f*x))^m*(b*(c*tan(e+f*x))^n)^p/(f*(1+m+n*p))],
[tan(e+f*x)^2*(b*(c*tan(e+f*x))^n)^p,x,4,hypergeom([1,1/2*(3+n*p)],[1/2*(5+n*p)],-tan(e+f*x)^2)*tan(e+f*x)^3*(b*(c*tan(e+f*x))^n)^p/(f*(3+n*p))],
[(b*(c*tan(e+f*x))^n)^p,x,3,hypergeom([1,1/2*(1+n*p)],[1/2*(3+n*p)],-tan(e+f*x)^2)*tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1+n*p))],
[cot(e+f*x)^2*(b*(c*tan(e+f*x))^n)^p,x,4,-cot(e+f*x)*hypergeom([1,1/2*(-1+n*p)],[1/2*(1+n*p)],-tan(e+f*x)^2)*(b*(c*tan(e+f*x))^n)^p/(f*(1-n*p))],
[cot(e+f*x)^4*(b*(c*tan(e+f*x))^n)^p,x,4,-cot(e+f*x)^3*hypergeom([1,1/2*(-3+n*p)],[1/2*(-1+n*p)],-tan(e+f*x)^2)*(b*(c*tan(e+f*x))^n)^p/(f*(3-n*p))],
[cot(e+f*x)^6*(b*(c*tan(e+f*x))^n)^p,x,4,-cot(e+f*x)^5*hypergeom([1,1/2*(-5+n*p)],[1/2*(-3+n*p)],-tan(e+f*x)^2)*(b*(c*tan(e+f*x))^n)^p/(f*(5-n*p))],
[tan(e+f*x)^3*(b*(c*tan(e+f*x))^n)^p,x,4,hypergeom([1,1/2*(4+n*p)],[1/2*(6+n*p)],-tan(e+f*x)^2)*tan(e+f*x)^4*(b*(c*tan(e+f*x))^n)^p/(f*(4+n*p))],
[tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p,x,4,hypergeom([1,1/2*(2+n*p)],[1/2*(4+n*p)],-tan(e+f*x)^2)*tan(e+f*x)^2*(b*(c*tan(e+f*x))^n)^p/(f*(2+n*p))],
[cot(e+f*x)*(b*(c*tan(e+f*x))^n)^p,x,4,hypergeom([1,1/2*n*p],[1+1/2*n*p],-tan(e+f*x)^2)*(b*(c*tan(e+f*x))^n)^p/(f*n*p)],
[cot(e+f*x)^3*(b*(c*tan(e+f*x))^n)^p,x,4,-cot(e+f*x)^2*hypergeom([1,1/2*(-2+n*p)],[1/2*n*p],-tan(e+f*x)^2)*(b*(c*tan(e+f*x))^n)^p/(f*(2-n*p))],

# Integrands of the form (d Tan[e+f x])^m (a+b (c Tan[e+f x])^n)^p when p symbolic
[(d*tan(e+f*x))^m*(a+b*(c*tan(e+f*x))^n)^p,x,0,Unintegrable((d*tan(e+f*x))^m*(a+b*(c*tan(e+f*x))^n)^p,x)],

# Integrands of the form (d Cot[e+f x])^m (a+b (c Tan[e+f x])^n)^p

# Integrands of the form (d Cot[e+f x])^m (a+b Tan[e+f x]^2)^p

# Integrands of the form (d Cot[e+f x])^m (a+b Tan[e+f x]^2)^p when p symbolic
[(d*cot(e+f*x))^m*(b*tan(e+f*x)^2)^p,x,4,(d*cot(e+f*x))^m*hypergeom([1,1/2*(1-m+2*p)],[1/2*(3-m+2*p)],-tan(e+f*x)^2)*tan(e+f*x)*(b*tan(e+f*x)^2)^p/(f*(1-m+2*p))],
[(d*cot(e+f*x))^m*(a+b*tan(e+f*x)^2)^p,x,4,AppellF1(1/2*(1-m),1,-p,1/2*(3-m),-tan(e+f*x)^2,-b*tan(e+f*x)^2/a)*(d*cot(e+f*x))^m*tan(e+f*x)*(a+b*tan(e+f*x)^2)^p/(f*(1-m)*(1+b*tan(e+f*x)^2/a)^p)],

# Integrands of the form (d Cot[e+f x])^m (a+b Tan[e+f x]^3)^p

# Integrands of the form (d Cot[e+f x])^m (a+b Tan[e+f x]^4)^p

# Integrands of the form (d Cot[e+f x])^m (a+b (c Tan[e+f x])^n)^p

# Integrands of the form (d Cot[e+f x])^m (b (cTan[e+f x])^n)^p when p symbolic
[(d*cot(e+f*x))^m*(b*(c*tan(e+f*x))^n)^p,x,4,(d*cot(e+f*x))^m*hypergeom([1,1/2*(1-m+n*p)],[1/2*(3-m+n*p)],-tan(e+f*x)^2)*tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1-m+n*p))],

# Integrands of the form (d Cot[e+f x])^m (a+b (c Tan[e+f x])^n)^p when p symbolic
[(d*cot(e+f*x))^m*(a+b*(c*tan(e+f*x))^n)^p,x,1,(d*cot(e+f*x))^m*(tan(e+f*x)/d)^m*Unintegrable((a+b*(c*tan(e+f*x))^n)^p/(tan(e+f*x)/d)^m,x)],

# Integrands of the form (d Sec[e+f x])^m (a+b (c Tan[e+f x])^n)^p

# Integrands of the form (d Sec[e+f x])^m (a+b Tan[e+f x]^2)^p

# Integrands of the form Sec[e+f x]^m (a+b Tan[e+f x]^2)^p

# p>0
[sec(c+d*x)^3*(a+b*tan(c+d*x)^2),x,4,1/8*(4*a-b)*arctanh(sin(c+d*x))/d+1/8*(4*a-b)*sec(c+d*x)*tan(c+d*x)/d+1/4*b*sec(c+d*x)^3*tan(c+d*x)/d],
[sec(c+d*x)*(a+b*tan(c+d*x)^2),x,3,1/2*(2*a-b)*arctanh(sin(c+d*x))/d+1/2*b*sec(c+d*x)*tan(c+d*x)/d],
[cos(c+d*x)*(a+b*tan(c+d*x)^2),x,3,b*arctanh(sin(c+d*x))/d+(a-b)*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+b*tan(c+d*x)^2),x,2,a*sin(c+d*x)/d-1/3*(a-b)*sin(c+d*x)^3/d],
[cos(c+d*x)^5*(a+b*tan(c+d*x)^2),x,3,a*sin(c+d*x)/d-1/3*(2*a-b)*sin(c+d*x)^3/d+1/5*(a-b)*sin(c+d*x)^5/d],
[cos(c+d*x)^7*(a+b*tan(c+d*x)^2),x,3,a*sin(c+d*x)/d-1/3*(3*a-b)*sin(c+d*x)^3/d+1/5*(3*a-2*b)*sin(c+d*x)^5/d-1/7*(a-b)*sin(c+d*x)^7/d],
[sec(c+d*x)^6*(a+b*tan(c+d*x)^2),x,3,a*tan(c+d*x)/d+1/3*(2*a+b)*tan(c+d*x)^3/d+1/5*(a+2*b)*tan(c+d*x)^5/d+1/7*b*tan(c+d*x)^7/d],
[sec(c+d*x)^4*(a+b*tan(c+d*x)^2),x,3,a*tan(c+d*x)/d+1/3*(a+b)*tan(c+d*x)^3/d+1/5*b*tan(c+d*x)^5/d],
[sec(c+d*x)^2*(a+b*tan(c+d*x)^2),x,2,a*tan(c+d*x)/d+1/3*b*tan(c+d*x)^3/d],
[cos(c+d*x)^2*(a+b*tan(c+d*x)^2),x,3,1/2*(a+b)*x+1/2*(a-b)*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^4*(a+b*tan(c+d*x)^2),x,4,1/8*(3*a+b)*x+1/8*(3*a+b)*cos(c+d*x)*sin(c+d*x)/d+1/4*(a-b)*cos(c+d*x)^3*sin(c+d*x)/d],
[cos(c+d*x)^6*(a+b*tan(c+d*x)^2),x,5,1/16*(5*a+b)*x+1/16*(5*a+b)*cos(c+d*x)*sin(c+d*x)/d+1/24*(5*a+b)*cos(c+d*x)^3*sin(c+d*x)/d+1/6*(a-b)*cos(c+d*x)^5*sin(c+d*x)/d],
[sec(c+d*x)^3*(a+b*tan(c+d*x)^2)^2,x,5,1/16*(8*a^2-4*a*b+b^2)*arctanh(sin(c+d*x))/d+1/16*(8*a^2-4*a*b+b^2)*sec(c+d*x)*tan(c+d*x)/d+1/24*(8*a-3*b)*b*sec(c+d*x)^3*tan(c+d*x)/d+1/6*b*sec(c+d*x)^5*(a-(a-b)*sin(c+d*x)^2)*tan(c+d*x)/d],
[sec(c+d*x)*(a+b*tan(c+d*x)^2)^2,x,4,1/8*(8*a^2-8*a*b+3*b^2)*arctanh(sin(c+d*x))/d+3/8*(2*a-b)*b*sec(c+d*x)*tan(c+d*x)/d+1/4*b*sec(c+d*x)^3*(a-(a-b)*sin(c+d*x)^2)*tan(c+d*x)/d],
[cos(c+d*x)*(a+b*tan(c+d*x)^2)^2,x,5,1/2*(4*a-3*b)*b*arctanh(sin(c+d*x))/d+(a-b)^2*sin(c+d*x)/d+1/2*b^2*sec(c+d*x)*tan(c+d*x)/d],
[cos(c+d*x)^3*(a+b*tan(c+d*x)^2)^2,x,4,b^2*arctanh(sin(c+d*x))/d+(a^2-b^2)*sin(c+d*x)/d-1/3*(a-b)^2*sin(c+d*x)^3/d],
[cos(c+d*x)^5*(a+b*tan(c+d*x)^2)^2,x,3,a^2*sin(c+d*x)/d-2/3*a*(a-b)*sin(c+d*x)^3/d+1/5*(a-b)^2*sin(c+d*x)^5/d],
[cos(c+d*x)^7*(a+b*tan(c+d*x)^2)^2,x,3,a^2*sin(c+d*x)/d-1/3*a*(3*a-2*b)*sin(c+d*x)^3/d+1/5*(a-b)*(3*a-b)*sin(c+d*x)^5/d-1/7*(a-b)^2*sin(c+d*x)^7/d],
[cos(c+d*x)^9*(a+b*tan(c+d*x)^2)^2,x,3,a^2*sin(c+d*x)/d-2/3*a*(2*a-b)*sin(c+d*x)^3/d+1/5*(6*a^2-6*a*b+b^2)*sin(c+d*x)^5/d-2/7*(a-b)*(2*a-b)*sin(c+d*x)^7/d+1/9*(a-b)^2*sin(c+d*x)^9/d],
[sec(c+d*x)^6*(a+b*tan(c+d*x)^2)^2,x,3,a^2*tan(c+d*x)/d+2/3*a*(a+b)*tan(c+d*x)^3/d+1/5*(a^2+4*a*b+b^2)*tan(c+d*x)^5/d+2/7*b*(a+b)*tan(c+d*x)^7/d+1/9*b^2*tan(c+d*x)^9/d],
[sec(c+d*x)^4*(a+b*tan(c+d*x)^2)^2,x,3,a^2*tan(c+d*x)/d+1/3*a*(a+2*b)*tan(c+d*x)^3/d+1/5*b*(2*a+b)*tan(c+d*x)^5/d+1/7*b^2*tan(c+d*x)^7/d],
[sec(c+d*x)^2*(a+b*tan(c+d*x)^2)^2,x,3,a^2*tan(c+d*x)/d+2/3*a*b*tan(c+d*x)^3/d+1/5*b^2*tan(c+d*x)^5/d],
[cos(c+d*x)^2*(a+b*tan(c+d*x)^2)^2,x,5,1/2*(a-b)*(a+3*b)*x+1/2*(a-b)^2*cos(c+d*x)*sin(c+d*x)/d+b^2*tan(c+d*x)/d],
[cos(c+d*x)^4*(a+b*tan(c+d*x)^2)^2,x,4,1/8*(3*a^2+2*a*b+3*b^2)*x+3/8*(a^2-b^2)*cos(c+d*x)*sin(c+d*x)/d+1/4*(a-b)*cos(c+d*x)^3*sin(c+d*x)*(a+b*tan(c+d*x)^2)/d],
[cos(c+d*x)^6*(a+b*tan(c+d*x)^2)^2,x,5,1/16*(5*a^2+2*a*b+b^2)*x+1/16*(5*a^2+2*a*b+b^2)*cos(c+d*x)*sin(c+d*x)/d+1/24*(a-b)*(5*a+3*b)*cos(c+d*x)^3*sin(c+d*x)/d+1/6*(a-b)*cos(c+d*x)^5*sin(c+d*x)*(a+b*tan(c+d*x)^2)/d],

# p<0
[sec(c+d*x)^5/(a+b*tan(c+d*x)^2),x,5,-1/2*(2*a-3*b)*arctanh(sin(c+d*x))/(b^2*d)+(a-b)^(3/2)*arctanh(sin(c+d*x)*sqrt(a-b)/sqrt(a))/(b^2*d*sqrt(a))+1/2*sec(c+d*x)*tan(c+d*x)/(b*d)],
[sec(c+d*x)^3/(a+b*tan(c+d*x)^2),x,4,arctanh(sin(c+d*x))/(b*d)-arctanh(sin(c+d*x)*sqrt(a-b)/sqrt(a))*sqrt(a-b)/(b*d*sqrt(a))],
[sec(c+d*x)/(a+b*tan(c+d*x)^2),x,2,arctanh(sin(c+d*x)*sqrt(a-b)/sqrt(a))/(d*sqrt(a)*sqrt(a-b))],
[cos(c+d*x)/(a+b*tan(c+d*x)^2),x,3,sin(c+d*x)/((a-b)*d)-b*arctanh(sin(c+d*x)*sqrt(a-b)/sqrt(a))/((a-b)^(3/2)*d*sqrt(a))],
[cos(c+d*x)^3/(a+b*tan(c+d*x)^2),x,4,(a-2*b)*sin(c+d*x)/((a-b)^2*d)-1/3*sin(c+d*x)^3/((a-b)*d)+b^2*arctanh(sin(c+d*x)*sqrt(a-b)/sqrt(a))/((a-b)^(5/2)*d*sqrt(a))],
[cos(c+d*x)^5/(a+b*tan(c+d*x)^2),x,4,(a^2-3*a*b+3*b^2)*sin(c+d*x)/((a-b)^3*d)-1/3*(2*a-3*b)*sin(c+d*x)^3/((a-b)^2*d)+1/5*sin(c+d*x)^5/((a-b)*d)-b^3*arctanh(sin(c+d*x)*sqrt(a-b)/sqrt(a))/((a-b)^(7/2)*d*sqrt(a))],
[sec(c+d*x)^8/(a+b*tan(c+d*x)^2),x,4,-(a-b)^3*arctan(sqrt(b)*tan(c+d*x)/sqrt(a))/(b^(7/2)*d*sqrt(a))+(a^2-3*a*b+3*b^2)*tan(c+d*x)/(b^3*d)-1/3*(a-3*b)*tan(c+d*x)^3/(b^2*d)+1/5*tan(c+d*x)^5/(b*d)],
[sec(c+d*x)^6/(a+b*tan(c+d*x)^2),x,4,(a-b)^2*arctan(sqrt(b)*tan(c+d*x)/sqrt(a))/(b^(5/2)*d*sqrt(a))-(a-2*b)*tan(c+d*x)/(b^2*d)+1/3*tan(c+d*x)^3/(b*d)],
[sec(c+d*x)^4/(a+b*tan(c+d*x)^2),x,3,-(a-b)*arctan(sqrt(b)*tan(c+d*x)/sqrt(a))/(b^(3/2)*d*sqrt(a))+tan(c+d*x)/(b*d)],
[sec(c+d*x)^2/(a+b*tan(c+d*x)^2),x,2,arctan(sqrt(b)*tan(c+d*x)/sqrt(a))/(d*sqrt(a)*sqrt(b))],
[cos(c+d*x)^2/(a+b*tan(c+d*x)^2),x,5,1/2*(a-3*b)*x/(a-b)^2+1/2*cos(c+d*x)*sin(c+d*x)/((a-b)*d)+b^(3/2)*arctan(sqrt(b)*tan(c+d*x)/sqrt(a))/((a-b)^2*d*sqrt(a))],
[cos(c+d*x)^4/(a+b*tan(c+d*x)^2),x,6,1/8*(3*a^2-10*a*b+15*b^2)*x/(a-b)^3+1/8*(3*a-7*b)*cos(c+d*x)*sin(c+d*x)/((a-b)^2*d)+1/4*cos(c+d*x)^3*sin(c+d*x)/((a-b)*d)-b^(5/2)*arctan(sqrt(b)*tan(c+d*x)/sqrt(a))/((a-b)^3*d*sqrt(a))],
[sec(c+d*x)^7/(a+b*tan(c+d*x)^2)^2,x,6,-1/2*(4*a-5*b)*arctanh(sin(c+d*x))/(b^3*d)+1/2*(a-b)^(3/2)*(4*a+b)*arctanh(sin(c+d*x)*sqrt(a-b)/sqrt(a))/(a^(3/2)*b^3*d)+1/2*(a-b)*(2*a-b)*sin(c+d*x)/(a*b^2*d*(a-(a-b)*sin(c+d*x)^2))+1/2*sec(c+d*x)*tan(c+d*x)/(b*d*(a-(a-b)*sin(c+d*x)^2))],
[sec(c+d*x)^5/(a+b*tan(c+d*x)^2)^2,x,5,arctanh(sin(c+d*x))/(b^2*d)-1/2*(a-b)*sin(c+d*x)/(a*b*d*(a-(a-b)*sin(c+d*x)^2))-1/2*(2*a+b)*arctanh(sin(c+d*x)*sqrt(a-b)/sqrt(a))*sqrt(a-b)/(a^(3/2)*b^2*d)],
[sec(c+d*x)^3/(a+b*tan(c+d*x)^2)^2,x,3,1/2*sin(c+d*x)/(a*d*(a-(a-b)*sin(c+d*x)^2))+1/2*arctanh(sin(c+d*x)*sqrt(a-b)/sqrt(a))/(a^(3/2)*d*sqrt(a-b))],
[sec(c+d*x)/(a+b*tan(c+d*x)^2)^2,x,3,1/2*(2*a-b)*arctanh(sin(c+d*x)*sqrt(a-b)/sqrt(a))/(a^(3/2)*(a-b)^(3/2)*d)-1/2*b*sin(c+d*x)/(a*(a-b)*d*(a-(a-b)*sin(c+d*x)^2))],
[cos(c+d*x)/(a+b*tan(c+d*x)^2)^2,x,5,-1/2*(4*a-b)*b*arctanh(sin(c+d*x)*sqrt(a-b)/sqrt(a))/(a^(3/2)*(a-b)^(5/2)*d)+sin(c+d*x)/((a-b)^2*d)+1/2*b^2*sin(c+d*x)/(a*(a-b)^2*d*(a-(a-b)*sin(c+d*x)^2))],
[cos(c+d*x)^3/(a+b*tan(c+d*x)^2)^2,x,5,1/2*(6*a-b)*b^2*arctanh(sin(c+d*x)*sqrt(a-b)/sqrt(a))/(a^(3/2)*(a-b)^(7/2)*d)+(a-3*b)*sin(c+d*x)/((a-b)^3*d)-1/3*sin(c+d*x)^3/((a-b)^2*d)-1/2*b^3*sin(c+d*x)/(a*(a-b)^3*d*(a-(a-b)*sin(c+d*x)^2))],
[sec(c+d*x)^8/(a+b*tan(c+d*x)^2)^2,x,5,1/2*(a-b)^2*(5*a+b)*arctan(sqrt(b)*tan(c+d*x)/sqrt(a))/(a^(3/2)*b^(7/2)*d)-(2*a-3*b)*tan(c+d*x)/(b^3*d)+1/3*tan(c+d*x)^3/(b^2*d)-1/2*(a-b)^3*tan(c+d*x)/(a*b^3*d*(a+b*tan(c+d*x)^2))],
[sec(c+d*x)^6/(a+b*tan(c+d*x)^2)^2,x,5,-1/2*(3*a^2-2*a*b-b^2)*arctan(sqrt(b)*tan(c+d*x)/sqrt(a))/(a^(3/2)*b^(5/2)*d)+tan(c+d*x)/(b^2*d)+1/2*(a-b)^2*tan(c+d*x)/(a*b^2*d*(a+b*tan(c+d*x)^2))],
[sec(c+d*x)^4/(a+b*tan(c+d*x)^2)^2,x,3,1/2*(a+b)*arctan(sqrt(b)*tan(c+d*x)/sqrt(a))/(a^(3/2)*b^(3/2)*d)-1/2*(a-b)*tan(c+d*x)/(a*b*d*(a+b*tan(c+d*x)^2))],
[sec(c+d*x)^2/(a+b*tan(c+d*x)^2)^2,x,3,1/2*arctan(sqrt(b)*tan(c+d*x)/sqrt(a))/(a^(3/2)*d*sqrt(b))+1/2*tan(c+d*x)/(a*d*(a+b*tan(c+d*x)^2))],
[cos(c+d*x)^2/(a+b*tan(c+d*x)^2)^2,x,6,1/2*(a-5*b)*x/(a-b)^3+1/2*(5*a-b)*b^(3/2)*arctan(sqrt(b)*tan(c+d*x)/sqrt(a))/(a^(3/2)*(a-b)^3*d)+1/2*cos(c+d*x)*sin(c+d*x)/((a-b)*d*(a+b*tan(c+d*x)^2))+1/2*b*(a+b)*tan(c+d*x)/(a*(a-b)^2*d*(a+b*tan(c+d*x)^2))],
[cos(c+d*x)^4/(a+b*tan(c+d*x)^2)^2,x,7,1/8*(3*a^2-14*a*b+35*b^2)*x/(a-b)^4-1/2*(7*a-b)*b^(5/2)*arctan(sqrt(b)*tan(c+d*x)/sqrt(a))/(a^(3/2)*(a-b)^4*d)+3/8*(a-3*b)*cos(c+d*x)*sin(c+d*x)/((a-b)^2*d*(a+b*tan(c+d*x)^2))+1/4*cos(c+d*x)^3*sin(c+d*x)/((a-b)*d*(a+b*tan(c+d*x)^2))+1/8*(a-4*b)*b*(3*a+b)*tan(c+d*x)/(a*(a-b)^3*d*(a+b*tan(c+d*x)^2))],

# Integrands of the form Sec[e+f x]^m (a+b Tan[e+f x]^2)^(p/2)

# Integrands of the form (d Sec[e+f x])^m (a+b Tan[e+f x]^2)^p when p symbolic
[(d*sec(e+f*x))^m*(b*tan(e+f*x)^2)^p,x,2,(cos(e+f*x)^2)^(1/2*(1+m+2*p))*hypergeom([1/2*(1+2*p),1/2*(1+m+2*p)],[1/2*(3+2*p)],sin(e+f*x)^2)*(d*sec(e+f*x))^m*tan(e+f*x)*(b*tan(e+f*x)^2)^p/(f*(1+2*p))],
[(d*sec(e+f*x))^m*(a+b*tan(e+f*x)^2)^p,x,3,AppellF1(1/2,1-1/2*m,-p,3/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/a)*(d*sec(e+f*x))^m*tan(e+f*x)*(a+b*tan(e+f*x)^2)^p/(f*(sec(e+f*x)^2)^(1/2*m)*(1+b*tan(e+f*x)^2/a)^p)],

# Integrands of the form (d Sec[e+f x])^m (a+b Tan[e+f x]^3)^p

# Integrands of the form (d Sec[e+f x])^m (a+b Tan[e+f x]^4)^p

# Integrands of the form (d Sec[e+f x])^m (a+b (c Tan[e+f x])^n)^p

# Integrands of the form (d Sec[e+f x])^m (b (cTan[e+f x])^n)^p when p symbolic
[(d*sec(e+f*x))^m*(b*(c*tan(e+f*x))^n)^p,x,2,(cos(e+f*x)^2)^(1/2*(1+m+n*p))*hypergeom([1/2*(1+n*p),1/2*(1+m+n*p)],[1/2*(3+n*p)],sin(e+f*x)^2)*(d*sec(e+f*x))^m*tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1+n*p))],
[sec(e+f*x)^6*(b*(c*tan(e+f*x))^n)^p,x,4,tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1+n*p))+2*tan(e+f*x)^3*(b*(c*tan(e+f*x))^n)^p/(f*(3+n*p))+tan(e+f*x)^5*(b*(c*tan(e+f*x))^n)^p/(f*(5+n*p))],
[sec(e+f*x)^4*(b*(c*tan(e+f*x))^n)^p,x,4,tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1+n*p))+tan(e+f*x)^3*(b*(c*tan(e+f*x))^n)^p/(f*(3+n*p))],
[sec(e+f*x)^2*(b*(c*tan(e+f*x))^n)^p,x,3,tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1+n*p))],
[(b*(c*tan(e+f*x))^n)^p,x,3,hypergeom([1,1/2*(1+n*p)],[1/2*(3+n*p)],-tan(e+f*x)^2)*tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1+n*p))],
[cos(e+f*x)^2*(b*(c*tan(e+f*x))^n)^p,x,3,hypergeom([2,1/2*(1+n*p)],[1/2*(3+n*p)],-tan(e+f*x)^2)*tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1+n*p))],
[sec(e+f*x)^3*(b*(c*tan(e+f*x))^n)^p,x,2,(cos(e+f*x)^2)^(1/2*(4+n*p))*hypergeom([1/2*(1+n*p),1/2*(4+n*p)],[1/2*(3+n*p)],sin(e+f*x)^2)*sec(e+f*x)^3*tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1+n*p))],
[sec(e+f*x)*(b*(c*tan(e+f*x))^n)^p,x,2,(cos(e+f*x)^2)^(1/2*(2+n*p))*hypergeom([1/2*(1+n*p),1/2*(2+n*p)],[1/2*(3+n*p)],sin(e+f*x)^2)*sec(e+f*x)*tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1+n*p))],
[cos(e+f*x)*(b*(c*tan(e+f*x))^n)^p,x,2,(cos(e+f*x)^2)^(1/2*n*p)*hypergeom([1/2*n*p,1/2*(1+n*p)],[1/2*(3+n*p)],sin(e+f*x)^2)*sin(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1+n*p))],
[cos(e+f*x)^3*(b*(c*tan(e+f*x))^n)^p,x,2,(cos(e+f*x)^2)^(1/2*n*p)*hypergeom([1/2*(-2+n*p),1/2*(1+n*p)],[1/2*(3+n*p)],sin(e+f*x)^2)*sin(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1+n*p))],

# Integrands of the form (d Sec[e+f x])^m (a+b (c Tan[e+f x])^n)^p when p symbolic
[(d*sec(e+f*x))^m*(a+b*(c*tan(e+f*x))^n)^p,x,0,Unintegrable((d*sec(e+f*x))^m*(a+b*(c*tan(e+f*x))^n)^p,x)],
[sec(e+f*x)^3*(a+b*(c*tan(e+f*x))^n)^p,x,0,Unintegrable(sec(e+f*x)^3*(a+b*(c*tan(e+f*x))^n)^p,x)],
[sec(e+f*x)*(a+b*(c*tan(e+f*x))^n)^p,x,0,Unintegrable(sec(e+f*x)*(a+b*(c*tan(e+f*x))^n)^p,x)],
[cos(e+f*x)*(a+b*(c*tan(e+f*x))^n)^p,x,0,Unintegrable(cos(e+f*x)*(a+b*(c*tan(e+f*x))^n)^p,x)],
[cos(e+f*x)^3*(a+b*(c*tan(e+f*x))^n)^p,x,0,Unintegrable(cos(e+f*x)^3*(a+b*(c*tan(e+f*x))^n)^p,x)],
[sec(e+f*x)^6*(a+b*(c*tan(e+f*x))^n)^p,x,9,hypergeom([1/n,-p],[1+1/n],-b*(c*tan(e+f*x))^n/a)*tan(e+f*x)*(a+b*(c*tan(e+f*x))^n)^p/(f*(1+b*(c*tan(e+f*x))^n/a)^p)+2/3*hypergeom([3/n,-p],[(3+n)/n],-b*(c*tan(e+f*x))^n/a)*tan(e+f*x)^3*(a+b*(c*tan(e+f*x))^n)^p/(f*(1+b*(c*tan(e+f*x))^n/a)^p)+1/5*hypergeom([5/n,-p],[(5+n)/n],-b*(c*tan(e+f*x))^n/a)*tan(e+f*x)^5*(a+b*(c*tan(e+f*x))^n)^p/(f*(1+b*(c*tan(e+f*x))^n/a)^p)],
[sec(e+f*x)^4*(a+b*(c*tan(e+f*x))^n)^p,x,7,hypergeom([1/n,-p],[1+1/n],-b*(c*tan(e+f*x))^n/a)*tan(e+f*x)*(a+b*(c*tan(e+f*x))^n)^p/(f*(1+b*(c*tan(e+f*x))^n/a)^p)+1/3*hypergeom([3/n,-p],[(3+n)/n],-b*(c*tan(e+f*x))^n/a)*tan(e+f*x)^3*(a+b*(c*tan(e+f*x))^n)^p/(f*(1+b*(c*tan(e+f*x))^n/a)^p)],
[sec(e+f*x)^2*(a+b*(c*tan(e+f*x))^n)^p,x,3,hypergeom([1/n,-p],[1+1/n],-b*(c*tan(e+f*x))^n/a)*tan(e+f*x)*(a+b*(c*tan(e+f*x))^n)^p/(f*(1+b*(c*tan(e+f*x))^n/a)^p)],
[(a+b*(c*tan(e+f*x))^n)^p,x,0,Unintegrable((a+b*(c*tan(e+f*x))^n)^p,x)],
[cos(e+f*x)^2*(a+b*(c*tan(e+f*x))^n)^p,x,0,Unintegrable(cos(e+f*x)^2*(a+b*(c*tan(e+f*x))^n)^p,x)],

# Integrands of the form (d Csc[e+f x])^m (a+b (c Tan[e+f x])^n)^p

# Integrands of the form (d Csc[e+f x])^m (a+b Tan[e+f x]^2)^p

# Integrands of the form (d Csc[e+f x])^m (a+b Tan[e+f x]^2)^p when p symbolic
[(d*csc(e+f*x))^m*(b*tan(e+f*x)^2)^p,x,4,(cos(e+f*x)^2)^(1/2+p)*(d*csc(e+f*x))^m*hypergeom([1/2*(1+2*p),1/2*(1-m+2*p)],[1/2*(3-m+2*p)],sin(e+f*x)^2)*tan(e+f*x)*(b*tan(e+f*x)^2)^p/(f*(1-m+2*p))],
[(d*csc(e+f*x))^m*(a+b*tan(e+f*x)^2)^p,x,4,AppellF1(1/2*(1-m),1-1/2*m,-p,1/2*(3-m),-tan(e+f*x)^2,-b*tan(e+f*x)^2/a)*(d*csc(e+f*x))^m*tan(e+f*x)*(a+b*tan(e+f*x)^2)^p/(f*(1-m)*(sec(e+f*x)^2)^(1/2*m)*(1+b*tan(e+f*x)^2/a)^p)],

# Integrands of the form (d Csc[e+f x])^m (a+b Tan[e+f x]^3)^p

# Integrands of the form (d Csc[e+f x])^m (a+b Tan[e+f x]^4)^p

# Integrands of the form (d Csc[e+f x])^m (a+b (c Tan[e+f x])^n)^p

# Integrands of the form (d Csc[e+f x])^m (b (cTan[e+f x])^n)^p when p symbolic
[(d*csc(e+f*x))^m*(b*(c*tan(e+f*x))^n)^p,x,4,(cos(e+f*x)^2)^(1/2*(1+n*p))*(d*csc(e+f*x))^m*hypergeom([1/2*(1+n*p),1/2*(1-m+n*p)],[1/2*(3-m+n*p)],sin(e+f*x)^2)*tan(e+f*x)*(b*(c*tan(e+f*x))^n)^p/(f*(1-m+n*p))],

# Integrands of the form (d Csc[e+f x])^m (a+b (c Tan[e+f x])^n)^p when p symbolic
[(d*csc(e+f*x))^m*(a+b*(c*tan(e+f*x))^n)^p,x,1,(d*csc(e+f*x))^m*(sin(e+f*x)/d)^m*Unintegrable((a+b*(c*tan(e+f*x))^n)^p/(sin(e+f*x)/d)^m,x)]]:
