# Maple integration test file: "4 Trig functions\4.3 Tangent\4.3.0 (a trg)^m (b tan)^n.txt"

lst:=[

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

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

# Integrands of the form Tan[c+d x]^n
[tan(c+d*x),x,1,-log(cos(c+d*x))/d],
[tan(c+d*x)^2,x,2,-x+tan(c+d*x)/d],
[tan(c+d*x)^3,x,2,log(cos(c+d*x))/d+1/2*tan(c+d*x)^2/d],
[tan(c+d*x)^4,x,3,x-tan(c+d*x)/d+1/3*tan(c+d*x)^3/d],
[tan(c+d*x)^5,x,3,-log(cos(c+d*x))/d-1/2*tan(c+d*x)^2/d+1/4*tan(c+d*x)^4/d],
[tan(c+d*x)^6,x,4,-x+tan(c+d*x)/d-1/3*tan(c+d*x)^3/d+1/5*tan(c+d*x)^5/d],
[tan(c+d*x)^7,x,4,log(cos(c+d*x))/d+1/2*tan(c+d*x)^2/d-1/4*tan(c+d*x)^4/d+1/6*tan(c+d*x)^6/d],
[tan(c+d*x)^8,x,5,x-tan(c+d*x)/d+1/3*tan(c+d*x)^3/d-1/5*tan(c+d*x)^5/d+1/7*tan(c+d*x)^7/d],

# Integrands of the form (b Tan[c+d x])^(n/2)
[(b*tan(c+d*x))^(7/2),x,13,-b^(7/2)*arctan(1-sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))/(d*sqrt(2))+b^(7/2)*arctan(1+sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))/(d*sqrt(2))-1/2*b^(7/2)*log(sqrt(b)-sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))/(d*sqrt(2))+1/2*b^(7/2)*log(sqrt(b)+sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))/(d*sqrt(2))-2*b^3*sqrt(b*tan(c+d*x))/d+2/5*b*(b*tan(c+d*x))^(5/2)/d],
[(b*tan(c+d*x))^(5/2),x,12,b^(5/2)*arctan(1-sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))/(d*sqrt(2))-b^(5/2)*arctan(1+sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))/(d*sqrt(2))-1/2*b^(5/2)*log(sqrt(b)-sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))/(d*sqrt(2))+1/2*b^(5/2)*log(sqrt(b)+sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))/(d*sqrt(2))+2/3*b*(b*tan(c+d*x))^(3/2)/d],
[(b*tan(c+d*x))^(3/2),x,12,b^(3/2)*arctan(1-sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))/(d*sqrt(2))-b^(3/2)*arctan(1+sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))/(d*sqrt(2))+1/2*b^(3/2)*log(sqrt(b)-sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))/(d*sqrt(2))-1/2*b^(3/2)*log(sqrt(b)+sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))/(d*sqrt(2))+2*b*sqrt(b*tan(c+d*x))/d],
[(b*tan(c+d*x))^(1/2),x,11,-arctan(1-sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))*sqrt(b)/(d*sqrt(2))+arctan(1+sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))*sqrt(b)/(d*sqrt(2))+1/2*log(sqrt(b)-sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))*sqrt(b)/(d*sqrt(2))-1/2*log(sqrt(b)+sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))*sqrt(b)/(d*sqrt(2))],
[1/(b*tan(c+d*x))^(1/2),x,11,-arctan(1-sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))/(d*sqrt(2)*sqrt(b))+arctan(1+sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))/(d*sqrt(2)*sqrt(b))-1/2*log(sqrt(b)-sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))/(d*sqrt(2)*sqrt(b))+1/2*log(sqrt(b)+sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))/(d*sqrt(2)*sqrt(b))],
[1/(b*tan(c+d*x))^(3/2),x,12,arctan(1-sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))/(b^(3/2)*d*sqrt(2))-arctan(1+sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))/(b^(3/2)*d*sqrt(2))-1/2*log(sqrt(b)-sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))/(b^(3/2)*d*sqrt(2))+1/2*log(sqrt(b)+sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))/(b^(3/2)*d*sqrt(2))+(-2)/(b*d*sqrt(b*tan(c+d*x)))],
[1/(b*tan(c+d*x))^(5/2),x,12,arctan(1-sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))/(b^(5/2)*d*sqrt(2))-arctan(1+sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))/(b^(5/2)*d*sqrt(2))+1/2*log(sqrt(b)-sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))/(b^(5/2)*d*sqrt(2))-1/2*log(sqrt(b)+sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))/(b^(5/2)*d*sqrt(2))+(-2/3)/(b*d*(b*tan(c+d*x))^(3/2))],
[1/(b*tan(c+d*x))^(7/2),x,13,-arctan(1-sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))/(b^(7/2)*d*sqrt(2))+arctan(1+sqrt(2)*sqrt(b*tan(c+d*x))/sqrt(b))/(b^(7/2)*d*sqrt(2))+1/2*log(sqrt(b)-sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))/(b^(7/2)*d*sqrt(2))-1/2*log(sqrt(b)+sqrt(2)*sqrt(b*tan(c+d*x))+sqrt(b)*tan(c+d*x))/(b^(7/2)*d*sqrt(2))+2/(b^3*d*sqrt(b*tan(c+d*x)))+(-2/5)/(b*d*(b*tan(c+d*x))^(5/2))],

# Integrands of the form (b Tan[c+d x])^(n/3)
[(b*tan(c+d*x))^(4/3),x,13,-b^(4/3)*arctan((b*tan(c+d*x))^(1/3)/b^(1/3))/d+1/2*b^(4/3)*arctan(sqrt(3)-2*(b*tan(c+d*x))^(1/3)/b^(1/3))/d-1/2*b^(4/3)*arctan(sqrt(3)+2*(b*tan(c+d*x))^(1/3)/b^(1/3))/d+1/4*b^(4/3)*log(b^(2/3)-b^(1/3)*sqrt(3)*(b*tan(c+d*x))^(1/3)+(b*tan(c+d*x))^(2/3))*sqrt(3)/d-1/4*b^(4/3)*log(b^(2/3)+b^(1/3)*sqrt(3)*(b*tan(c+d*x))^(1/3)+(b*tan(c+d*x))^(2/3))*sqrt(3)/d+3*b*(b*tan(c+d*x))^(1/3)/d],
[(b*tan(c+d*x))^(2/3),x,12,b^(2/3)*arctan((b*tan(c+d*x))^(1/3)/b^(1/3))/d-1/2*b^(2/3)*arctan(sqrt(3)-2*(b*tan(c+d*x))^(1/3)/b^(1/3))/d+1/2*b^(2/3)*arctan(sqrt(3)+2*(b*tan(c+d*x))^(1/3)/b^(1/3))/d+1/4*b^(2/3)*log(b^(2/3)-b^(1/3)*sqrt(3)*(b*tan(c+d*x))^(1/3)+(b*tan(c+d*x))^(2/3))*sqrt(3)/d-1/4*b^(2/3)*log(b^(2/3)+b^(1/3)*sqrt(3)*(b*tan(c+d*x))^(1/3)+(b*tan(c+d*x))^(2/3))*sqrt(3)/d],
[(b*tan(c+d*x))^(1/3),x,9,-1/2*b^(1/3)*log(b^(2/3)+(b*tan(c+d*x))^(2/3))/d+1/4*b^(1/3)*log(b^(4/3)-b^(2/3)*(b*tan(c+d*x))^(2/3)+(b*tan(c+d*x))^(4/3))/d-1/2*b^(1/3)*arctan((b^(2/3)-2*(b*tan(c+d*x))^(2/3))/(b^(2/3)*sqrt(3)))*sqrt(3)/d],
[1/(b*tan(c+d*x))^(1/3),x,9,1/2*log(b^(2/3)+(b*tan(c+d*x))^(2/3))/(b^(1/3)*d)-1/4*log(b^(4/3)-b^(2/3)*(b*tan(c+d*x))^(2/3)+(b*tan(c+d*x))^(4/3))/(b^(1/3)*d)-1/2*arctan((b^(2/3)-2*(b*tan(c+d*x))^(2/3))/(b^(2/3)*sqrt(3)))*sqrt(3)/(b^(1/3)*d)],
[1/(b*tan(c+d*x))^(2/3),x,12,arctan((b*tan(c+d*x))^(1/3)/b^(1/3))/(b^(2/3)*d)-1/2*arctan(sqrt(3)-2*(b*tan(c+d*x))^(1/3)/b^(1/3))/(b^(2/3)*d)+1/2*arctan(sqrt(3)+2*(b*tan(c+d*x))^(1/3)/b^(1/3))/(b^(2/3)*d)-1/4*log(b^(2/3)-b^(1/3)*sqrt(3)*(b*tan(c+d*x))^(1/3)+(b*tan(c+d*x))^(2/3))*sqrt(3)/(b^(2/3)*d)+1/4*log(b^(2/3)+b^(1/3)*sqrt(3)*(b*tan(c+d*x))^(1/3)+(b*tan(c+d*x))^(2/3))*sqrt(3)/(b^(2/3)*d)],
[1/(b*tan(c+d*x))^(4/3),x,13,-arctan((b*tan(c+d*x))^(1/3)/b^(1/3))/(b^(4/3)*d)+1/2*arctan(sqrt(3)-2*(b*tan(c+d*x))^(1/3)/b^(1/3))/(b^(4/3)*d)-1/2*arctan(sqrt(3)+2*(b*tan(c+d*x))^(1/3)/b^(1/3))/(b^(4/3)*d)-1/4*log(b^(2/3)-b^(1/3)*sqrt(3)*(b*tan(c+d*x))^(1/3)+(b*tan(c+d*x))^(2/3))*sqrt(3)/(b^(4/3)*d)+1/4*log(b^(2/3)+b^(1/3)*sqrt(3)*(b*tan(c+d*x))^(1/3)+(b*tan(c+d*x))^(2/3))*sqrt(3)/(b^(4/3)*d)+(-3)/(b*d*(b*tan(c+d*x))^(1/3))],

# Integrands of the form (b Tan[c+d x])^n with n symbolic
[(b*tan(c+d*x))^n,x,2,hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(c+d*x)^2)*(b*tan(c+d*x))^(1+n)/(b*d*(1+n))],

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

# Integrands of the form (b Tan[c+d x]^p)^(n/2) with p positive integer
[(b*tan(c+d*x)^2)^(5/2),x,4,-b^2*cot(c+d*x)*log(cos(c+d*x))*sqrt(b*tan(c+d*x)^2)/d-1/2*b^2*sqrt(b*tan(c+d*x)^2)*tan(c+d*x)/d+1/4*b^2*sqrt(b*tan(c+d*x)^2)*tan(c+d*x)^3/d],
[(b*tan(c+d*x)^2)^(3/2),x,3,b*cot(c+d*x)*log(cos(c+d*x))*sqrt(b*tan(c+d*x)^2)/d+1/2*b*sqrt(b*tan(c+d*x)^2)*tan(c+d*x)/d],
[(b*tan(c+d*x)^2)^(1/2),x,2,-cot(c+d*x)*log(cos(c+d*x))*sqrt(b*tan(c+d*x)^2)/d],
[1/(b*tan(c+d*x)^2)^(1/2),x,2,log(sin(c+d*x))*tan(c+d*x)/(d*sqrt(b*tan(c+d*x)^2))],
[1/(b*tan(c+d*x)^2)^(3/2),x,3,-1/2*cot(c+d*x)/(b*d*sqrt(b*tan(c+d*x)^2))-log(sin(c+d*x))*tan(c+d*x)/(b*d*sqrt(b*tan(c+d*x)^2))],
[1/(b*tan(c+d*x)^2)^(5/2),x,4,1/2*cot(c+d*x)/(b^2*d*sqrt(b*tan(c+d*x)^2))-1/4*cot(c+d*x)^3/(b^2*d*sqrt(b*tan(c+d*x)^2))+log(sin(c+d*x))*tan(c+d*x)/(b^2*d*sqrt(b*tan(c+d*x)^2))],
[(b*tan(c+d*x)^3)^(5/2),x,16,-2*b^2*cot(c+d*x)*sqrt(b*tan(c+d*x)^3)/d-b^2*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))*sqrt(b*tan(c+d*x)^3)/(d*sqrt(2)*tan(c+d*x)^(3/2))+b^2*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))*sqrt(b*tan(c+d*x)^3)/(d*sqrt(2)*tan(c+d*x)^(3/2))-1/2*b^2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*sqrt(b*tan(c+d*x)^3)/(d*sqrt(2)*tan(c+d*x)^(3/2))+1/2*b^2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*sqrt(b*tan(c+d*x)^3)/(d*sqrt(2)*tan(c+d*x)^(3/2))+2/5*b^2*sqrt(b*tan(c+d*x)^3)*tan(c+d*x)/d-2/9*b^2*sqrt(b*tan(c+d*x)^3)*tan(c+d*x)^3/d+2/13*b^2*sqrt(b*tan(c+d*x)^3)*tan(c+d*x)^5/d],
[(b*tan(c+d*x)^3)^(3/2),x,14,-2/3*b*sqrt(b*tan(c+d*x)^3)/d-b*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))*sqrt(b*tan(c+d*x)^3)/(d*sqrt(2)*tan(c+d*x)^(3/2))+b*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))*sqrt(b*tan(c+d*x)^3)/(d*sqrt(2)*tan(c+d*x)^(3/2))+1/2*b*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*sqrt(b*tan(c+d*x)^3)/(d*sqrt(2)*tan(c+d*x)^(3/2))-1/2*b*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*sqrt(b*tan(c+d*x)^3)/(d*sqrt(2)*tan(c+d*x)^(3/2))+2/7*b*sqrt(b*tan(c+d*x)^3)*tan(c+d*x)^2/d],
[(b*tan(c+d*x)^3)^(1/2),x,13,2*cot(c+d*x)*sqrt(b*tan(c+d*x)^3)/d+arctan(1-sqrt(2)*sqrt(tan(c+d*x)))*sqrt(b*tan(c+d*x)^3)/(d*sqrt(2)*tan(c+d*x)^(3/2))-arctan(1+sqrt(2)*sqrt(tan(c+d*x)))*sqrt(b*tan(c+d*x)^3)/(d*sqrt(2)*tan(c+d*x)^(3/2))+1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*sqrt(b*tan(c+d*x)^3)/(d*sqrt(2)*tan(c+d*x)^(3/2))-1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*sqrt(b*tan(c+d*x)^3)/(d*sqrt(2)*tan(c+d*x)^(3/2))],
[1/(b*tan(c+d*x)^3)^(1/2),x,13,-2*tan(c+d*x)/(d*sqrt(b*tan(c+d*x)^3))+arctan(1-sqrt(2)*sqrt(tan(c+d*x)))*tan(c+d*x)^(3/2)/(d*sqrt(2)*sqrt(b*tan(c+d*x)^3))-arctan(1+sqrt(2)*sqrt(tan(c+d*x)))*tan(c+d*x)^(3/2)/(d*sqrt(2)*sqrt(b*tan(c+d*x)^3))-1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(3/2)/(d*sqrt(2)*sqrt(b*tan(c+d*x)^3))+1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(3/2)/(d*sqrt(2)*sqrt(b*tan(c+d*x)^3))],
[1/(b*tan(c+d*x)^3)^(3/2),x,14,2/3/(b*d*sqrt(b*tan(c+d*x)^3))-2/7*cot(c+d*x)^2/(b*d*sqrt(b*tan(c+d*x)^3))-arctan(1-sqrt(2)*sqrt(tan(c+d*x)))*tan(c+d*x)^(3/2)/(b*d*sqrt(2)*sqrt(b*tan(c+d*x)^3))+arctan(1+sqrt(2)*sqrt(tan(c+d*x)))*tan(c+d*x)^(3/2)/(b*d*sqrt(2)*sqrt(b*tan(c+d*x)^3))-1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(3/2)/(b*d*sqrt(2)*sqrt(b*tan(c+d*x)^3))+1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(3/2)/(b*d*sqrt(2)*sqrt(b*tan(c+d*x)^3))],
[1/(b*tan(c+d*x)^3)^(5/2),x,16,-2/5*cot(c+d*x)/(b^2*d*sqrt(b*tan(c+d*x)^3))+2/9*cot(c+d*x)^3/(b^2*d*sqrt(b*tan(c+d*x)^3))-2/13*cot(c+d*x)^5/(b^2*d*sqrt(b*tan(c+d*x)^3))+2*tan(c+d*x)/(b^2*d*sqrt(b*tan(c+d*x)^3))-arctan(1-sqrt(2)*sqrt(tan(c+d*x)))*tan(c+d*x)^(3/2)/(b^2*d*sqrt(2)*sqrt(b*tan(c+d*x)^3))+arctan(1+sqrt(2)*sqrt(tan(c+d*x)))*tan(c+d*x)^(3/2)/(b^2*d*sqrt(2)*sqrt(b*tan(c+d*x)^3))+1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(3/2)/(b^2*d*sqrt(2)*sqrt(b*tan(c+d*x)^3))-1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(3/2)/(b^2*d*sqrt(2)*sqrt(b*tan(c+d*x)^3))],
[(b*tan(c+d*x)^4)^(5/2),x,7,b^2*cot(c+d*x)*sqrt(b*tan(c+d*x)^4)/d-b^2*x*cot(c+d*x)^2*sqrt(b*tan(c+d*x)^4)-1/3*b^2*sqrt(b*tan(c+d*x)^4)*tan(c+d*x)/d+1/5*b^2*sqrt(b*tan(c+d*x)^4)*tan(c+d*x)^3/d-1/7*b^2*sqrt(b*tan(c+d*x)^4)*tan(c+d*x)^5/d+1/9*b^2*sqrt(b*tan(c+d*x)^4)*tan(c+d*x)^7/d],
[(b*tan(c+d*x)^4)^(3/2),x,5,b*cot(c+d*x)*sqrt(b*tan(c+d*x)^4)/d-b*x*cot(c+d*x)^2*sqrt(b*tan(c+d*x)^4)-1/3*b*sqrt(b*tan(c+d*x)^4)*tan(c+d*x)/d+1/5*b*sqrt(b*tan(c+d*x)^4)*tan(c+d*x)^3/d],
[(b*tan(c+d*x)^4)^(1/2),x,3,cot(c+d*x)*sqrt(b*tan(c+d*x)^4)/d-x*cot(c+d*x)^2*sqrt(b*tan(c+d*x)^4)],
[1/(b*tan(c+d*x)^4)^(1/2),x,3,-tan(c+d*x)/(d*sqrt(b*tan(c+d*x)^4))-x*tan(c+d*x)^2/sqrt(b*tan(c+d*x)^4)],
[1/(b*tan(c+d*x)^4)^(3/2),x,5,1/3*cot(c+d*x)/(b*d*sqrt(b*tan(c+d*x)^4))-1/5*cot(c+d*x)^3/(b*d*sqrt(b*tan(c+d*x)^4))-tan(c+d*x)/(b*d*sqrt(b*tan(c+d*x)^4))-x*tan(c+d*x)^2/(b*sqrt(b*tan(c+d*x)^4))],
[1/(b*tan(c+d*x)^4)^(5/2),x,7,1/3*cot(c+d*x)/(b^2*d*sqrt(b*tan(c+d*x)^4))-1/5*cot(c+d*x)^3/(b^2*d*sqrt(b*tan(c+d*x)^4))+1/7*cot(c+d*x)^5/(b^2*d*sqrt(b*tan(c+d*x)^4))-1/9*cot(c+d*x)^7/(b^2*d*sqrt(b*tan(c+d*x)^4))-tan(c+d*x)/(b^2*d*sqrt(b*tan(c+d*x)^4))-x*tan(c+d*x)^2/(b^2*sqrt(b*tan(c+d*x)^4))],

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

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

# Integrands of the form (a (b Tan[c+d x])^p)^n with n symbolic
[(a*(b*tan(c+d*x))^p)^n,x,3,hypergeom([1,1/2*(1+n*p)],[1/2*(3+n*p)],-tan(c+d*x)^2)*tan(c+d*x)*(a*(b*tan(c+d*x))^p)^n/(d*(1+n*p))],

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

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

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

# n>0
[sin(a+b*x)^4*(d*tan(a+b*x))^(1/2),x,13,-21/32*arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))*sqrt(d)/(b*sqrt(2))+21/32*arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))*sqrt(d)/(b*sqrt(2))+21/64*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))*sqrt(d)/(b*sqrt(2))-21/64*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))*sqrt(d)/(b*sqrt(2))-7/16*cos(a+b*x)^2*(d*tan(a+b*x))^(3/2)/(b*d)-1/4*cos(a+b*x)^4*(d*tan(a+b*x))^(7/2)/(b*d^3)],
[sin(a+b*x)^2*(d*tan(a+b*x))^(1/2),x,12,-3/4*arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))*sqrt(d)/(b*sqrt(2))+3/4*arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))*sqrt(d)/(b*sqrt(2))+3/8*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))*sqrt(d)/(b*sqrt(2))-3/8*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))*sqrt(d)/(b*sqrt(2))-1/2*cos(a+b*x)^2*(d*tan(a+b*x))^(3/2)/(b*d)],
[csc(a+b*x)^2*(d*tan(a+b*x))^(1/2),x,2,-2*d/(b*sqrt(d*tan(a+b*x)))],
[csc(a+b*x)^4*(d*tan(a+b*x))^(1/2),x,3,-2*d/(b*sqrt(d*tan(a+b*x)))-2/5*d^3/(b*(d*tan(a+b*x))^(5/2))],
[csc(a+b*x)^6*(d*tan(a+b*x))^(1/2),x,3,-2*d/(b*sqrt(d*tan(a+b*x)))-2/9*d^5/(b*(d*tan(a+b*x))^(9/2))-4/5*d^3/(b*(d*tan(a+b*x))^(5/2))],
[sin(a+b*x)^3*(d*tan(a+b*x))^(1/2),x,5,-5/6*d*sin(a+b*x)/(b*sqrt(d*tan(a+b*x)))-1/3*d*sin(a+b*x)^3/(b*sqrt(d*tan(a+b*x)))+5/12*csc(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x))/b],
[sin(a+b*x)*(d*tan(a+b*x))^(1/2),x,4,-d*sin(a+b*x)/(b*sqrt(d*tan(a+b*x)))+1/2*csc(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x))/b],
[csc(a+b*x)*(d*tan(a+b*x))^(1/2),x,3,csc(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x))/b],
[csc(a+b*x)^3*(d*tan(a+b*x))^(1/2),x,4,-2/3*d*csc(a+b*x)/(b*sqrt(d*tan(a+b*x)))+2/3*csc(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x))/b],
[csc(a+b*x)^5*(d*tan(a+b*x))^(1/2),x,5,-4/7*d*csc(a+b*x)/(b*sqrt(d*tan(a+b*x)))-2/7*d*csc(a+b*x)^3/(b*sqrt(d*tan(a+b*x)))+4/7*csc(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x))/b],
[sin(a+b*x)^4*(d*tan(a+b*x))^(3/2),x,14,45/32*d^(3/2)*arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2))-45/32*d^(3/2)*arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2))+45/64*d^(3/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2))-45/64*d^(3/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2))+45/16*d*sqrt(d*tan(a+b*x))/b-9/16*cos(a+b*x)^2*(d*tan(a+b*x))^(5/2)/(b*d)-1/4*cos(a+b*x)^4*(d*tan(a+b*x))^(9/2)/(b*d^3)],
[sin(a+b*x)^2*(d*tan(a+b*x))^(3/2),x,13,5/4*d^(3/2)*arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2))-5/4*d^(3/2)*arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2))+5/8*d^(3/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2))-5/8*d^(3/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2))+5/2*d*sqrt(d*tan(a+b*x))/b-1/2*cos(a+b*x)^2*(d*tan(a+b*x))^(5/2)/(b*d)],
[csc(a+b*x)^2*(d*tan(a+b*x))^(3/2),x,2,2*d*sqrt(d*tan(a+b*x))/b],
[csc(a+b*x)^4*(d*tan(a+b*x))^(3/2),x,3,2*d*sqrt(d*tan(a+b*x))/b-2/3*d^3/(b*(d*tan(a+b*x))^(3/2))],
[csc(a+b*x)^6*(d*tan(a+b*x))^(3/2),x,3,2*d*sqrt(d*tan(a+b*x))/b-2/7*d^5/(b*(d*tan(a+b*x))^(7/2))-4/3*d^3/(b*(d*tan(a+b*x))^(3/2))],
[sin(a+b*x)^3*(d*tan(a+b*x))^(3/2),x,5,-7/2*d^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sin(a+b*x)/(b*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x)))+2*d*sin(a+b*x)^3*sqrt(d*tan(a+b*x))/b+7/3*d^3*sin(a+b*x)^3/(b*(d*tan(a+b*x))^(3/2))],
[sin(a+b*x)*(d*tan(a+b*x))^(3/2),x,4,-3*d^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sin(a+b*x)/(b*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x)))+2*d*sin(a+b*x)*sqrt(d*tan(a+b*x))/b],
[csc(a+b*x)*(d*tan(a+b*x))^(3/2),x,4,-2*d^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sin(a+b*x)/(b*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x)))+2*d*sin(a+b*x)*sqrt(d*tan(a+b*x))/b],
[csc(a+b*x)^3*(d*tan(a+b*x))^(3/2),x,5,-4*d^2*cos(a+b*x)/(b*sqrt(d*tan(a+b*x)))-4*d^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sin(a+b*x)/(b*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x)))+2*d*csc(a+b*x)*sqrt(d*tan(a+b*x))/b],
[sin(a+b*x)^4*(d*tan(a+b*x))^(5/2),x,14,77/32*d^(5/2)*arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2))-77/32*d^(5/2)*arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2))-77/64*d^(5/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2))+77/64*d^(5/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2))+77/48*d*(d*tan(a+b*x))^(3/2)/b-11/16*cos(a+b*x)^2*(d*tan(a+b*x))^(7/2)/(b*d)-1/4*cos(a+b*x)^4*(d*tan(a+b*x))^(11/2)/(b*d^3)],
[sin(a+b*x)^2*(d*tan(a+b*x))^(5/2),x,13,7/4*d^(5/2)*arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2))-7/4*d^(5/2)*arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2))-7/8*d^(5/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2))+7/8*d^(5/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2))+7/6*d*(d*tan(a+b*x))^(3/2)/b-1/2*cos(a+b*x)^2*(d*tan(a+b*x))^(7/2)/(b*d)],
[csc(a+b*x)^2*(d*tan(a+b*x))^(5/2),x,2,2/3*d*(d*tan(a+b*x))^(3/2)/b],
[csc(a+b*x)^4*(d*tan(a+b*x))^(5/2),x,3,-2*d^3/(b*sqrt(d*tan(a+b*x)))+2/3*d*(d*tan(a+b*x))^(3/2)/b],
[csc(a+b*x)^6*(d*tan(a+b*x))^(5/2),x,3,-4*d^3/(b*sqrt(d*tan(a+b*x)))-2/5*d^5/(b*(d*tan(a+b*x))^(5/2))+2/3*d*(d*tan(a+b*x))^(3/2)/b],
[sin(a+b*x)^3*(d*tan(a+b*x))^(5/2),x,6,5/2*d^3*sin(a+b*x)/(b*sqrt(d*tan(a+b*x)))+d^3*sin(a+b*x)^3/(b*sqrt(d*tan(a+b*x)))-5/4*d^2*csc(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x))/b+2/3*d*sin(a+b*x)^3*(d*tan(a+b*x))^(3/2)/b],
[sin(a+b*x)*(d*tan(a+b*x))^(5/2),x,5,5/3*d^3*sin(a+b*x)/(b*sqrt(d*tan(a+b*x)))-5/6*d^2*csc(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x))/b+2/3*d*sin(a+b*x)*(d*tan(a+b*x))^(3/2)/b],
[csc(a+b*x)*(d*tan(a+b*x))^(5/2),x,4,-1/3*d^2*csc(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x))/b+2/3*d*csc(a+b*x)*(d*tan(a+b*x))^(3/2)/b],
[csc(a+b*x)^3*(d*tan(a+b*x))^(5/2),x,4,2/3*d^2*csc(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x))/b+2/3*d*csc(a+b*x)*(d*tan(a+b*x))^(3/2)/b],
[csc(a+b*x)^5*(d*tan(a+b*x))^(5/2),x,5,-4/3*d^3*csc(a+b*x)/(b*sqrt(d*tan(a+b*x)))+4/3*d^2*csc(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x))/b+2/3*d*csc(a+b*x)^3*(d*tan(a+b*x))^(3/2)/b],
[csc(a+b*x)^7*(d*tan(a+b*x))^(5/2),x,6,-40/21*d^3*csc(a+b*x)/(b*sqrt(d*tan(a+b*x)))-20/21*d^3*csc(a+b*x)^3/(b*sqrt(d*tan(a+b*x)))+40/21*d^2*csc(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x))/b+2/3*d*csc(a+b*x)^5*(d*tan(a+b*x))^(3/2)/b],

# n<0
[sin(a+b*x)^4/(d*tan(a+b*x))^(1/2),x,13,-5/32*arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2)*sqrt(d))+5/32*arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2)*sqrt(d))-5/64*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2)*sqrt(d))+5/64*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2)*sqrt(d))-5/16*cos(a+b*x)^2*sqrt(d*tan(a+b*x))/(b*d)-1/4*cos(a+b*x)^4*(d*tan(a+b*x))^(5/2)/(b*d^3)],
[sin(a+b*x)^2/(d*tan(a+b*x))^(1/2),x,12,-1/4*arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2)*sqrt(d))+1/4*arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2)*sqrt(d))-1/8*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2)*sqrt(d))+1/8*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2)*sqrt(d))-1/2*cos(a+b*x)^2*sqrt(d*tan(a+b*x))/(b*d)],
[csc(a+b*x)^2/(d*tan(a+b*x))^(1/2),x,2,-2/3*d/(b*(d*tan(a+b*x))^(3/2))],
[csc(a+b*x)^4/(d*tan(a+b*x))^(1/2),x,3,-2/7*d^3/(b*(d*tan(a+b*x))^(7/2))-2/3*d/(b*(d*tan(a+b*x))^(3/2))],
[csc(a+b*x)^6/(d*tan(a+b*x))^(1/2),x,3,-2/11*d^5/(b*(d*tan(a+b*x))^(11/2))-4/7*d^3/(b*(d*tan(a+b*x))^(7/2))-2/3*d/(b*(d*tan(a+b*x))^(3/2))],
[sin(a+b*x)^5/(d*tan(a+b*x))^(1/2),x,5,7/20*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sin(a+b*x)/(b*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x)))-7/30*d*sin(a+b*x)^3/(b*(d*tan(a+b*x))^(3/2))-1/5*d*sin(a+b*x)^5/(b*(d*tan(a+b*x))^(3/2))],
[sin(a+b*x)^3/(d*tan(a+b*x))^(1/2),x,4,1/2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sin(a+b*x)/(b*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x)))-1/3*d*sin(a+b*x)^3/(b*(d*tan(a+b*x))^(3/2))],
[sin(a+b*x)/(d*tan(a+b*x))^(1/2),x,3,sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sin(a+b*x)/(b*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x)))],
[csc(a+b*x)/(d*tan(a+b*x))^(1/2),x,4,-2*cos(a+b*x)/(b*sqrt(d*tan(a+b*x)))-2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sin(a+b*x)/(b*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x)))],
[csc(a+b*x)^3/(d*tan(a+b*x))^(1/2),x,5,-4/5*cos(a+b*x)/(b*sqrt(d*tan(a+b*x)))-4/5*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sin(a+b*x)/(b*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x)))-2/5*d*csc(a+b*x)/(b*(d*tan(a+b*x))^(3/2))],
[sin(a+b*x)^4/(d*tan(a+b*x))^(3/2),x,13,-3/32*arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*d^(3/2)*sqrt(2))+3/32*arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*d^(3/2)*sqrt(2))+3/64*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*d^(3/2)*sqrt(2))-3/64*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*d^(3/2)*sqrt(2))+3/16*cos(a+b*x)^2*(d*tan(a+b*x))^(3/2)/(b*d^3)-1/4*cos(a+b*x)^4*(d*tan(a+b*x))^(3/2)/(b*d^3)],
[sin(a+b*x)^2/(d*tan(a+b*x))^(3/2),x,12,-1/4*arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*d^(3/2)*sqrt(2))+1/4*arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*d^(3/2)*sqrt(2))+1/8*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*d^(3/2)*sqrt(2))-1/8*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*d^(3/2)*sqrt(2))+1/2*cos(a+b*x)^2*(d*tan(a+b*x))^(3/2)/(b*d^3)],
[csc(a+b*x)^2/(d*tan(a+b*x))^(3/2),x,2,-2/5*d/(b*(d*tan(a+b*x))^(5/2))],
[csc(a+b*x)^4/(d*tan(a+b*x))^(3/2),x,3,-2/9*d^3/(b*(d*tan(a+b*x))^(9/2))-2/5*d/(b*(d*tan(a+b*x))^(5/2))],
[csc(a+b*x)^6/(d*tan(a+b*x))^(3/2),x,3,-2/13*d^5/(b*(d*tan(a+b*x))^(13/2))-4/9*d^3/(b*(d*tan(a+b*x))^(9/2))-2/5*d/(b*(d*tan(a+b*x))^(5/2))],
[sin(a+b*x)^3/(d*tan(a+b*x))^(3/2),x,5,-1/6*sin(a+b*x)/(b*d*sqrt(d*tan(a+b*x)))+1/3*sin(a+b*x)^3/(b*d*sqrt(d*tan(a+b*x)))+1/12*csc(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x))/(b*d^2)],
[sin(a+b*x)/(d*tan(a+b*x))^(3/2),x,4,sin(a+b*x)/(b*d*sqrt(d*tan(a+b*x)))+1/2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sec(a+b*x)*sqrt(sin(2*a+2*b*x))/(b*d*sqrt(d*tan(a+b*x)))],
[csc(a+b*x)/(d*tan(a+b*x))^(3/2),x,4,-2/3*csc(a+b*x)/(b*d*sqrt(d*tan(a+b*x)))-1/3*csc(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x))/(b*d^2)],
[csc(a+b*x)^3/(d*tan(a+b*x))^(3/2),x,5,2/21*csc(a+b*x)/(b*d*sqrt(d*tan(a+b*x)))-2/7*csc(a+b*x)^3/(b*d*sqrt(d*tan(a+b*x)))-2/21*csc(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x))/(b*d^2)],
[sin(a+b*x)^4/(d*tan(a+b*x))^(5/2),x,13,-3/32*arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*d^(5/2)*sqrt(2))+3/32*arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*d^(5/2)*sqrt(2))-3/64*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*d^(5/2)*sqrt(2))+3/64*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*d^(5/2)*sqrt(2))+1/16*cos(a+b*x)^2*sqrt(d*tan(a+b*x))/(b*d^3)-1/4*cos(a+b*x)^4*sqrt(d*tan(a+b*x))/(b*d^3)],
[sin(a+b*x)^2/(d*tan(a+b*x))^(5/2),x,12,-3/4*arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*d^(5/2)*sqrt(2))+3/4*arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*d^(5/2)*sqrt(2))-3/8*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*d^(5/2)*sqrt(2))+3/8*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*d^(5/2)*sqrt(2))+1/2*cos(a+b*x)^2*sqrt(d*tan(a+b*x))/(b*d^3)],
[csc(a+b*x)^2/(d*tan(a+b*x))^(5/2),x,2,-2/7*d/(b*(d*tan(a+b*x))^(7/2))],
[csc(a+b*x)^4/(d*tan(a+b*x))^(5/2),x,3,-2/11*d^3/(b*(d*tan(a+b*x))^(11/2))-2/7*d/(b*(d*tan(a+b*x))^(7/2))],
[csc(a+b*x)^6/(d*tan(a+b*x))^(5/2),x,3,-2/15*d^5/(b*(d*tan(a+b*x))^(15/2))-4/11*d^3/(b*(d*tan(a+b*x))^(11/2))-2/7*d/(b*(d*tan(a+b*x))^(7/2))],
[sin(a+b*x)^7/(d*tan(a+b*x))^(5/2),x,6,3/40*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sin(a+b*x)/(b*d^2*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x)))-1/20*sin(a+b*x)^3/(b*d*(d*tan(a+b*x))^(3/2))-3/70*sin(a+b*x)^5/(b*d*(d*tan(a+b*x))^(3/2))+1/7*sin(a+b*x)^7/(b*d*(d*tan(a+b*x))^(3/2))],
[sin(a+b*x)^5/(d*tan(a+b*x))^(5/2),x,5,3/20*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sin(a+b*x)/(b*d^2*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x)))-1/10*sin(a+b*x)^3/(b*d*(d*tan(a+b*x))^(3/2))+1/5*sin(a+b*x)^5/(b*d*(d*tan(a+b*x))^(3/2))],
[sin(a+b*x)^3/(d*tan(a+b*x))^(5/2),x,4,1/2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sin(a+b*x)/(b*d^2*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x)))+1/3*sin(a+b*x)^3/(b*d*(d*tan(a+b*x))^(3/2))],
[sin(a+b*x)/(d*tan(a+b*x))^(5/2),x,4,-3*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sin(a+b*x)/(b*d^2*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x)))-2*sin(a+b*x)/(b*d*(d*tan(a+b*x))^(3/2))],
[csc(a+b*x)/(d*tan(a+b*x))^(5/2),x,5,6/5*cos(a+b*x)/(b*d^2*sqrt(d*tan(a+b*x)))+6/5*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sin(a+b*x)/(b*d^2*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x)))-2/5*csc(a+b*x)/(b*d*(d*tan(a+b*x))^(3/2))],
[csc(a+b*x)^3/(d*tan(a+b*x))^(5/2),x,6,4/15*cos(a+b*x)/(b*d^2*sqrt(d*tan(a+b*x)))+4/15*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sin(a+b*x)/(b*d^2*sqrt(sin(2*a+2*b*x))*sqrt(d*tan(a+b*x)))+2/15*csc(a+b*x)/(b*d*(d*tan(a+b*x))^(3/2))-2/9*csc(a+b*x)^3/(b*d*(d*tan(a+b*x))^(3/2))],

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

# n>0
[(a*sin(e+f*x))^(5/2)*sqrt(b*tan(e+f*x)),x,2,-2/5*b*(a*sin(e+f*x))^(5/2)/(f*sqrt(b*tan(e+f*x)))-8/5*a^2*b*sqrt(a*sin(e+f*x))/(f*sqrt(b*tan(e+f*x)))],
[(a*sin(e+f*x))^(3/2)*sqrt(b*tan(e+f*x)),x,3,-2/3*b*(a*sin(e+f*x))^(3/2)/(f*sqrt(b*tan(e+f*x)))+4/3*a^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x))/(f*sqrt(a*sin(e+f*x)))],
[sqrt(a*sin(e+f*x))*sqrt(b*tan(e+f*x)),x,1,-2*b*sqrt(a*sin(e+f*x))/(f*sqrt(b*tan(e+f*x)))],
[sqrt(b*tan(e+f*x))/sqrt(a*sin(e+f*x)),x,2,2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x))/(f*sqrt(a*sin(e+f*x)))],
[sqrt(b*tan(e+f*x))/(a*sin(e+f*x))^(3/2),x,7,-arctan(sqrt(cos(e+f*x)))*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x))/(a*f*sqrt(a*sin(e+f*x)))-arctanh(sqrt(cos(e+f*x)))*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x))/(a*f*sqrt(a*sin(e+f*x)))],
[sqrt(b*tan(e+f*x))/(a*sin(e+f*x))^(5/2),x,3,-b/(a^2*f*sqrt(a*sin(e+f*x))*sqrt(b*tan(e+f*x)))+sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x))/(a^2*f*sqrt(a*sin(e+f*x)))],
[(a*sin(e+f*x))^(5/2)*(b*tan(e+f*x))^(3/2),x,4,-24/5*a^2*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(a*sin(e+f*x))/(f*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x)))-2/5*b*(a*sin(e+f*x))^(5/2)*sqrt(b*tan(e+f*x))/f+12/5*a^2*b*sqrt(a*sin(e+f*x))*sqrt(b*tan(e+f*x))/f],
[(a*sin(e+f*x))^(3/2)*(b*tan(e+f*x))^(3/2),x,2,-2/3*b*(a*sin(e+f*x))^(3/2)*sqrt(b*tan(e+f*x))/f+8/3*a^2*b*sqrt(b*tan(e+f*x))/(f*sqrt(a*sin(e+f*x)))],
[sqrt(a*sin(e+f*x))*(b*tan(e+f*x))^(3/2),x,3,-4*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(a*sin(e+f*x))/(f*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x)))+2*b*sqrt(a*sin(e+f*x))*sqrt(b*tan(e+f*x))/f],
[(b*tan(e+f*x))^(3/2)/sqrt(a*sin(e+f*x)),x,1,2*b*sqrt(b*tan(e+f*x))/(f*sqrt(a*sin(e+f*x)))],
[(b*tan(e+f*x))^(3/2)/(a*sin(e+f*x))^(3/2),x,3,-2*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(a*sin(e+f*x))/(a^2*f*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x)))+2*b*sqrt(a*sin(e+f*x))*sqrt(b*tan(e+f*x))/(a^2*f)],
[(b*tan(e+f*x))^(3/2)/(a*sin(e+f*x))^(5/2),x,8,b^2*arctan(sqrt(cos(e+f*x)))*sqrt(a*sin(e+f*x))/(a^3*f*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x)))-b^2*arctanh(sqrt(cos(e+f*x)))*sqrt(a*sin(e+f*x))/(a^3*f*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x)))+2*b*sqrt(b*tan(e+f*x))/(a^2*f*sqrt(a*sin(e+f*x)))],

# n<0
[(a*sin(e+f*x))^(9/2)/sqrt(b*tan(e+f*x)),x,4,8/15*a^4*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(a*sin(e+f*x))/(f*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x)))-4/15*a^2*b*(a*sin(e+f*x))^(5/2)/(f*(b*tan(e+f*x))^(3/2))-2/9*b*(a*sin(e+f*x))^(9/2)/(f*(b*tan(e+f*x))^(3/2))],
[(a*sin(e+f*x))^(7/2)/sqrt(b*tan(e+f*x)),x,2,-8/21*a^2*b*(a*sin(e+f*x))^(3/2)/(f*(b*tan(e+f*x))^(3/2))-2/7*b*(a*sin(e+f*x))^(7/2)/(f*(b*tan(e+f*x))^(3/2))],
[(a*sin(e+f*x))^(5/2)/sqrt(b*tan(e+f*x)),x,3,4/5*a^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(a*sin(e+f*x))/(f*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x)))-2/5*b*(a*sin(e+f*x))^(5/2)/(f*(b*tan(e+f*x))^(3/2))],
[(a*sin(e+f*x))^(3/2)/sqrt(b*tan(e+f*x)),x,1,-2/3*b*(a*sin(e+f*x))^(3/2)/(f*(b*tan(e+f*x))^(3/2))],
[sqrt(a*sin(e+f*x))/sqrt(b*tan(e+f*x)),x,2,2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(a*sin(e+f*x))/(f*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x)))],
[1/(sqrt(a*sin(e+f*x))*sqrt(b*tan(e+f*x))),x,7,arctan(sqrt(cos(e+f*x)))*sqrt(a*sin(e+f*x))/(a*f*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x)))-arctanh(sqrt(cos(e+f*x)))*sqrt(a*sin(e+f*x))/(a*f*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x)))],
[1/((a*sin(e+f*x))^(3/2)*sqrt(b*tan(e+f*x))),x,3,-sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(a*sin(e+f*x))/(a^2*f*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x)))-b*sqrt(a*sin(e+f*x))/(a^2*f*(b*tan(e+f*x))^(3/2))],
[1/((a*sin(e+f*x))^(5/2)*sqrt(b*tan(e+f*x))),x,8,1/4*arctan(sqrt(cos(e+f*x)))*sqrt(a*sin(e+f*x))/(a^3*f*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x)))-1/4*arctanh(sqrt(cos(e+f*x)))*sqrt(a*sin(e+f*x))/(a^3*f*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x)))-1/2*b/(a^2*f*sqrt(a*sin(e+f*x))*(b*tan(e+f*x))^(3/2))],
[(a*sin(e+f*x))^(13/2)/(b*tan(e+f*x))^(3/2),x,4,-16/585*a^4*(a*sin(e+f*x))^(5/2)/(b*f*sqrt(b*tan(e+f*x)))-2/117*a^2*(a*sin(e+f*x))^(9/2)/(b*f*sqrt(b*tan(e+f*x)))+2/13*(a*sin(e+f*x))^(13/2)/(b*f*sqrt(b*tan(e+f*x)))-64/585*a^6*sqrt(a*sin(e+f*x))/(b*f*sqrt(b*tan(e+f*x)))],
[(a*sin(e+f*x))^(9/2)/(b*tan(e+f*x))^(3/2),x,3,-2/45*a^2*(a*sin(e+f*x))^(5/2)/(b*f*sqrt(b*tan(e+f*x)))+2/9*(a*sin(e+f*x))^(9/2)/(b*f*sqrt(b*tan(e+f*x)))-8/45*a^4*sqrt(a*sin(e+f*x))/(b*f*sqrt(b*tan(e+f*x)))],
[(a*sin(e+f*x))^(5/2)/(b*tan(e+f*x))^(3/2),x,1,-2/5*b*(a*sin(e+f*x))^(5/2)/(f*(b*tan(e+f*x))^(5/2))],
[sqrt(a*sin(e+f*x))/(b*tan(e+f*x))^(3/2),x,8,2*sqrt(a*sin(e+f*x))/(b*f*sqrt(b*tan(e+f*x)))-a*arctan(sqrt(cos(e+f*x)))*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x))/(b^2*f*sqrt(a*sin(e+f*x)))-a*arctanh(sqrt(cos(e+f*x)))*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x))/(b^2*f*sqrt(a*sin(e+f*x)))],
[1/((a*sin(e+f*x))^(3/2)*(b*tan(e+f*x))^(3/2)),x,8,(-1/2)/(b*f*(a*sin(e+f*x))^(3/2)*sqrt(b*tan(e+f*x)))+1/4*arctan(sqrt(cos(e+f*x)))*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x))/(a*b^2*f*sqrt(a*sin(e+f*x)))+1/4*arctanh(sqrt(cos(e+f*x)))*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x))/(a*b^2*f*sqrt(a*sin(e+f*x)))],
[(a*sin(e+f*x))^(11/2)/(b*tan(e+f*x))^(3/2),x,5,-4/77*a^4*(a*sin(e+f*x))^(3/2)/(b*f*sqrt(b*tan(e+f*x)))-2/77*a^2*(a*sin(e+f*x))^(7/2)/(b*f*sqrt(b*tan(e+f*x)))+2/11*(a*sin(e+f*x))^(11/2)/(b*f*sqrt(b*tan(e+f*x)))+8/77*a^6*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x))/(b^2*f*sqrt(a*sin(e+f*x)))],
[(a*sin(e+f*x))^(7/2)/(b*tan(e+f*x))^(3/2),x,4,-2/21*a^2*(a*sin(e+f*x))^(3/2)/(b*f*sqrt(b*tan(e+f*x)))+2/7*(a*sin(e+f*x))^(7/2)/(b*f*sqrt(b*tan(e+f*x)))+4/21*a^4*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x))/(b^2*f*sqrt(a*sin(e+f*x)))],
[(a*sin(e+f*x))^(3/2)/(b*tan(e+f*x))^(3/2),x,3,2/3*(a*sin(e+f*x))^(3/2)/(b*f*sqrt(b*tan(e+f*x)))+2/3*a^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x))/(b^2*f*sqrt(a*sin(e+f*x)))],
[1/(sqrt(a*sin(e+f*x))*(b*tan(e+f*x))^(3/2)),x,3,(-1)/(b*f*sqrt(a*sin(e+f*x))*sqrt(b*tan(e+f*x)))-sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x))/(b^2*f*sqrt(a*sin(e+f*x)))],
[1/((a*sin(e+f*x))^(5/2)*(b*tan(e+f*x))^(3/2)),x,4,(-1/3)/(b*f*(a*sin(e+f*x))^(5/2)*sqrt(b*tan(e+f*x)))+1/6/(a^2*b*f*sqrt(a*sin(e+f*x))*sqrt(b*tan(e+f*x)))-1/6*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x))/(a^2*b^2*f*sqrt(a*sin(e+f*x)))],
[1/((a*sin(e+f*x))^(9/2)*(b*tan(e+f*x))^(3/2)),x,5,(-1/5)/(b*f*(a*sin(e+f*x))^(9/2)*sqrt(b*tan(e+f*x)))+1/30/(a^2*b*f*(a*sin(e+f*x))^(5/2)*sqrt(b*tan(e+f*x)))+1/12/(a^4*b*f*sqrt(a*sin(e+f*x))*sqrt(b*tan(e+f*x)))-1/12*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*tan(e+f*x))/(a^4*b^2*f*sqrt(a*sin(e+f*x)))],

# Integrands of the form (a Sin[e+f x])^(m/3) (b Tan[e+f x])^(n/2)
[(b*sin(e+f*x))^(4/3)*sqrt(d*tan(e+f*x)),x,2,6/17*(cos(e+f*x)^2)^(3/4)*hypergeom([3/4,17/12],[29/12],sin(e+f*x)^2)*(b*sin(e+f*x))^(4/3)*(d*tan(e+f*x))^(3/2)/(d*f)],
[(b*sin(e+f*x))^(1/3)*sqrt(d*tan(e+f*x)),x,2,6/11*(cos(e+f*x)^2)^(3/4)*hypergeom([3/4,11/12],[23/12],sin(e+f*x)^2)*(b*sin(e+f*x))^(1/3)*(d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(d*tan(e+f*x))/(b*sin(e+f*x))^(1/3),x,2,6/7*(cos(e+f*x)^2)^(3/4)*hypergeom([7/12,3/4],[19/12],sin(e+f*x)^2)*(d*tan(e+f*x))^(3/2)/(d*f*(b*sin(e+f*x))^(1/3))],
[sqrt(d*tan(e+f*x))/(b*sin(e+f*x))^(4/3),x,2,6*(cos(e+f*x)^2)^(3/4)*hypergeom([1/12,3/4],[13/12],sin(e+f*x)^2)*(d*tan(e+f*x))^(3/2)/(d*f*(b*sin(e+f*x))^(4/3))],
[(b*sin(e+f*x))^(4/3)*(d*tan(e+f*x))^(3/2),x,2,6/23*(cos(e+f*x)^2)^(5/4)*hypergeom([5/4,23/12],[35/12],sin(e+f*x)^2)*(b*sin(e+f*x))^(4/3)*(d*tan(e+f*x))^(5/2)/(d*f)],
[(b*sin(e+f*x))^(1/3)*(d*tan(e+f*x))^(3/2),x,2,6/17*(cos(e+f*x)^2)^(5/4)*hypergeom([5/4,17/12],[29/12],sin(e+f*x)^2)*(b*sin(e+f*x))^(1/3)*(d*tan(e+f*x))^(5/2)/(d*f)],
[(d*tan(e+f*x))^(3/2)/(b*sin(e+f*x))^(1/3),x,2,6/13*(cos(e+f*x)^2)^(5/4)*hypergeom([13/12,5/4],[25/12],sin(e+f*x)^2)*(d*tan(e+f*x))^(5/2)/(d*f*(b*sin(e+f*x))^(1/3))],
[(d*tan(e+f*x))^(3/2)/(b*sin(e+f*x))^(4/3),x,2,6/7*(cos(e+f*x)^2)^(5/4)*hypergeom([7/12,5/4],[19/12],sin(e+f*x)^2)*(d*tan(e+f*x))^(5/2)/(d*f*(b*sin(e+f*x))^(4/3))],

# Integrands of the form (a Sin[e+f x])^(m/2) (b Tan[e+f x])^(n/3)
[sqrt(b*sin(e+f*x))*(d*tan(e+f*x))^(4/3),x,2,6/17*(cos(e+f*x)^2)^(7/6)*hypergeom([7/6,17/12],[29/12],sin(e+f*x)^2)*sqrt(b*sin(e+f*x))*(d*tan(e+f*x))^(7/3)/(d*f)],
[sqrt(b*sin(e+f*x))*(d*tan(e+f*x))^(1/3),x,2,6/11*(cos(e+f*x)^2)^(2/3)*hypergeom([2/3,11/12],[23/12],sin(e+f*x)^2)*sqrt(b*sin(e+f*x))*(d*tan(e+f*x))^(4/3)/(d*f)],
[sqrt(b*sin(e+f*x))/(d*tan(e+f*x))^(1/3),x,2,6/7*(cos(e+f*x)^2)^(1/3)*hypergeom([1/3,7/12],[19/12],sin(e+f*x)^2)*sqrt(b*sin(e+f*x))*(d*tan(e+f*x))^(2/3)/(d*f)],
[sqrt(b*sin(e+f*x))/(d*tan(e+f*x))^(4/3),x,2,6*hypergeom([-1/6,1/12],[13/12],sin(e+f*x)^2)*sqrt(b*sin(e+f*x))/(d*f*(cos(e+f*x)^2)^(1/6)*(d*tan(e+f*x))^(1/3))],
[(b*sin(e+f*x))^(3/2)*(d*tan(e+f*x))^(4/3),x,2,6/23*(cos(e+f*x)^2)^(7/6)*hypergeom([7/6,23/12],[35/12],sin(e+f*x)^2)*(b*sin(e+f*x))^(3/2)*(d*tan(e+f*x))^(7/3)/(d*f)],
[(b*sin(e+f*x))^(3/2)*(d*tan(e+f*x))^(1/3),x,2,6/17*(cos(e+f*x)^2)^(2/3)*hypergeom([2/3,17/12],[29/12],sin(e+f*x)^2)*(b*sin(e+f*x))^(3/2)*(d*tan(e+f*x))^(4/3)/(d*f)],
[(b*sin(e+f*x))^(3/2)/(d*tan(e+f*x))^(1/3),x,2,6/13*(cos(e+f*x)^2)^(1/3)*hypergeom([1/3,13/12],[25/12],sin(e+f*x)^2)*(b*sin(e+f*x))^(3/2)*(d*tan(e+f*x))^(2/3)/(d*f)],
[(b*sin(e+f*x))^(3/2)/(d*tan(e+f*x))^(4/3),x,2,6/7*hypergeom([-1/6,7/12],[19/12],sin(e+f*x)^2)*(b*sin(e+f*x))^(3/2)/(d*f*(cos(e+f*x)^2)^(1/6)*(d*tan(e+f*x))^(1/3))],

# Integrands of the form (a Sin[e+f x])^m (b Tan[e+f x])^n with m symbolic
[(a*sin(e+f*x))^m*tan(e+f*x)^3,x,2,hypergeom([2,1/2*(4+m)],[1/2*(6+m)],sin(e+f*x)^2)*(a*sin(e+f*x))^(4+m)/(a^4*f*(4+m))],
[(a*sin(e+f*x))^m*tan(e+f*x),x,2,hypergeom([1,1/2*(2+m)],[1/2*(4+m)],sin(e+f*x)^2)*(a*sin(e+f*x))^(2+m)/(a^2*f*(2+m))],
[cot(e+f*x)*(a*sin(e+f*x))^m,x,2,(a*sin(e+f*x))^m/(f*m)],
[cot(e+f*x)^3*(a*sin(e+f*x))^m,x,3,-a^2*(a*sin(e+f*x))^(-2+m)/(f*(2-m))-(a*sin(e+f*x))^m/(f*m)],
[cot(e+f*x)^5*(a*sin(e+f*x))^m,x,3,-a^4*(a*sin(e+f*x))^(-4+m)/(f*(4-m))+2*a^2*(a*sin(e+f*x))^(-2+m)/(f*(2-m))+(a*sin(e+f*x))^m/(f*m)],
[(a*sin(e+f*x))^m*tan(e+f*x)^4,x,2,hypergeom([5/2,1/2*(5+m)],[1/2*(7+m)],sin(e+f*x)^2)*sec(e+f*x)*(a*sin(e+f*x))^(5+m)*sqrt(cos(e+f*x)^2)/(a^5*f*(5+m))],
[(a*sin(e+f*x))^m*tan(e+f*x)^2,x,2,hypergeom([3/2,1/2*(3+m)],[1/2*(5+m)],sin(e+f*x)^2)*sec(e+f*x)*(a*sin(e+f*x))^(3+m)*sqrt(cos(e+f*x)^2)/(a^3*f*(3+m))],
[cot(e+f*x)^2*(a*sin(e+f*x))^m,x,2,-a*cos(e+f*x)*hypergeom([-1/2,1/2*(-1+m)],[1/2*(1+m)],sin(e+f*x)^2)*(a*sin(e+f*x))^(-1+m)/(f*(1-m)*sqrt(cos(e+f*x)^2))],
[cot(e+f*x)^4*(a*sin(e+f*x))^m,x,2,-a^3*cos(e+f*x)*hypergeom([-3/2,1/2*(-3+m)],[1/2*(-1+m)],sin(e+f*x)^2)*(a*sin(e+f*x))^(-3+m)/(f*(3-m)*sqrt(cos(e+f*x)^2))],
[(a*sin(e+f*x))^m*(b*tan(e+f*x))^(3/2),x,2,2*(cos(e+f*x)^2)^(5/4)*hypergeom([5/4,1/4*(5+2*m)],[1/4*(9+2*m)],sin(e+f*x)^2)*(a*sin(e+f*x))^m*(b*tan(e+f*x))^(5/2)/(b*f*(5+2*m))],
[(a*sin(e+f*x))^m*(b*tan(e+f*x))^(1/2),x,2,2*(cos(e+f*x)^2)^(3/4)*hypergeom([3/4,1/4*(3+2*m)],[1/4*(7+2*m)],sin(e+f*x)^2)*(a*sin(e+f*x))^m*(b*tan(e+f*x))^(3/2)/(b*f*(3+2*m))],
[(a*sin(e+f*x))^m/(b*tan(e+f*x))^(1/2),x,2,2*(cos(e+f*x)^2)^(1/4)*hypergeom([1/4,1/4*(1+2*m)],[1/4*(5+2*m)],sin(e+f*x)^2)*(a*sin(e+f*x))^m*sqrt(b*tan(e+f*x))/(b*f*(1+2*m))],
[(a*sin(e+f*x))^m/(b*tan(e+f*x))^(3/2),x,2,-2*hypergeom([-1/4,1/4*(-1+2*m)],[1/4*(3+2*m)],sin(e+f*x)^2)*(a*sin(e+f*x))^m/(b*f*(1-2*m)*(cos(e+f*x)^2)^(1/4)*sqrt(b*tan(e+f*x)))],

# Integrands of the form (a Sin[e+f x])^m (b Tan[e+f x])^n with n symbolic
[(a*sin(e+f*x))^m*(b*tan(e+f*x))^n,x,2,(cos(e+f*x)^2)^(1/2*(1+n))*hypergeom([1/2*(1+n),1/2*(1+m+n)],[1/2*(3+m+n)],sin(e+f*x)^2)*(a*sin(e+f*x))^m*(b*tan(e+f*x))^(1+n)/(b*f*(1+m+n))],
[sin(e+f*x)^4*(b*tan(e+f*x))^n,x,2,hypergeom([3,1/2*(5+n)],[1/2*(7+n)],-tan(e+f*x)^2)*(b*tan(e+f*x))^(5+n)/(b^5*f*(5+n))],
[sin(e+f*x)^2*(b*tan(e+f*x))^n,x,2,hypergeom([2,1/2*(3+n)],[1/2*(5+n)],-tan(e+f*x)^2)*(b*tan(e+f*x))^(3+n)/(b^3*f*(3+n))],
[csc(e+f*x)^2*(b*tan(e+f*x))^n,x,2,-b*(b*tan(e+f*x))^(-1+n)/(f*(1-n))],
[csc(e+f*x)^4*(b*tan(e+f*x))^n,x,3,-b^3*(b*tan(e+f*x))^(-3+n)/(f*(3-n))-b*(b*tan(e+f*x))^(-1+n)/(f*(1-n))],
[csc(e+f*x)^6*(b*tan(e+f*x))^n,x,3,-b^5*(b*tan(e+f*x))^(-5+n)/(f*(5-n))-2*b^3*(b*tan(e+f*x))^(-3+n)/(f*(3-n))-b*(b*tan(e+f*x))^(-1+n)/(f*(1-n))],
[sin(e+f*x)^3*(b*tan(e+f*x))^n,x,2,(cos(e+f*x)^2)^(1/2*(1+n))*hypergeom([1/2*(1+n),1/2*(4+n)],[1/2*(6+n)],sin(e+f*x)^2)*sin(e+f*x)^3*(b*tan(e+f*x))^(1+n)/(b*f*(4+n))],
[sin(e+f*x)*(b*tan(e+f*x))^n,x,2,(cos(e+f*x)^2)^(1/2*(1+n))*hypergeom([1/2*(1+n),1/2*(2+n)],[1/2*(4+n)],sin(e+f*x)^2)*sin(e+f*x)*(b*tan(e+f*x))^(1+n)/(b*f*(2+n))],
[csc(e+f*x)*(b*tan(e+f*x))^n,x,2,-cos(e+f*x)*hypergeom([1/2*(1-n),1/2*(2-n)],[1/2*(3-n)],cos(e+f*x)^2)*(b*tan(e+f*x))^n/(f*(1-n)*(sin(e+f*x)^2)^(1/2*n))],
[csc(e+f*x)^3*(b*tan(e+f*x))^n,x,2,-cos(e+f*x)*hypergeom([1/2*(1-n),1/2*(4-n)],[1/2*(3-n)],cos(e+f*x)^2)*(b*tan(e+f*x))^n/(f*(1-n)*(sin(e+f*x)^2)^(1/2*n))],
[csc(e+f*x)^5*(b*tan(e+f*x))^n,x,2,-cos(e+f*x)*hypergeom([1/2*(1-n),1/2*(6-n)],[1/2*(3-n)],cos(e+f*x)^2)*(b*tan(e+f*x))^n/(f*(1-n)*(sin(e+f*x)^2)^(1/2*n))],
[(a*sin(e+f*x))^(3/2)*(b*tan(e+f*x))^n,x,2,2*(cos(e+f*x)^2)^(1/2*(1+n))*hypergeom([1/2*(1+n),1/4*(5+2*n)],[1/4*(9+2*n)],sin(e+f*x)^2)*(a*sin(e+f*x))^(3/2)*(b*tan(e+f*x))^(1+n)/(b*f*(5+2*n))],
[(a*sin(e+f*x))^(1/2)*(b*tan(e+f*x))^n,x,2,2*(cos(e+f*x)^2)^(1/2*(1+n))*hypergeom([1/2*(1+n),1/4*(3+2*n)],[1/4*(7+2*n)],sin(e+f*x)^2)*sqrt(a*sin(e+f*x))*(b*tan(e+f*x))^(1+n)/(b*f*(3+2*n))],
[(b*tan(e+f*x))^n/(a*sin(e+f*x))^(1/2),x,2,2*(cos(e+f*x)^2)^(1/2*(1+n))*hypergeom([1/2*(1+n),1/4*(1+2*n)],[1/4*(5+2*n)],sin(e+f*x)^2)*(b*tan(e+f*x))^(1+n)/(b*f*(1+2*n)*sqrt(a*sin(e+f*x)))],
[(b*tan(e+f*x))^n/(a*sin(e+f*x))^(3/2),x,2,-2*(cos(e+f*x)^2)^(1/2*(1+n))*hypergeom([1/2*(1+n),1/4*(-1+2*n)],[1/4*(3+2*n)],sin(e+f*x)^2)*(b*tan(e+f*x))^(1+n)/(b*f*(1-2*n)*(a*sin(e+f*x))^(3/2))],

# Integrands of the form (a Cos[e+f x])^m (b Tan[e+f x])^n
[(a*cos(e+f*x))^m*(b*tan(e+f*x))^n,x,2,(a*cos(e+f*x))^m*(cos(e+f*x)^2)^(1/2*(1-m+n))*hypergeom([1/2*(1+n),1/2*(1-m+n)],[1/2*(3+n)],sin(e+f*x)^2)*(b*tan(e+f*x))^(1+n)/(b*f*(1+n))],

# Integrands of the form (a Tan[e+f x])^m (b Tan[e+f x])^n
[(a*tan(e+f*x))^m*(b*tan(e+f*x))^n,x,3,hypergeom([1,1/2*(1+m+n)],[1/2*(3+m+n)],-tan(e+f*x)^2)*(a*tan(e+f*x))^(1+m)*(b*tan(e+f*x))^n/(a*f*(1+m+n))],

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

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

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

# m>0
[sqrt(d*cot(e+f*x))*tan(e+f*x)^4,x,14,2/5*d^3/(f*(d*cot(e+f*x))^(5/2))+arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))-arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))-2*d/(f*sqrt(d*cot(e+f*x)))],
[sqrt(d*cot(e+f*x))*tan(e+f*x)^3,x,13,2/3*d^2/(f*(d*cot(e+f*x))^(3/2))-arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))+arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))],
[sqrt(d*cot(e+f*x))*tan(e+f*x)^2,x,13,-arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))+arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))+2*d/(f*sqrt(d*cot(e+f*x)))],
[sqrt(d*cot(e+f*x))*tan(e+f*x),x,12,arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))-arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))],
[sqrt(d*cot(e+f*x)),x,11,arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))-arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))],
[cot(e+f*x)*sqrt(d*cot(e+f*x)),x,13,-arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))+arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))-2*sqrt(d*cot(e+f*x))/f],
[cot(e+f*x)^2*sqrt(d*cot(e+f*x)),x,13,-2/3*(d*cot(e+f*x))^(3/2)/(d*f)-arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))+arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))],
[cot(e+f*x)^3*sqrt(d*cot(e+f*x)),x,14,-2/5*(d*cot(e+f*x))^(5/2)/(d^2*f)+arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))-arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))*sqrt(d)/(f*sqrt(2))+2*sqrt(d*cot(e+f*x))/f],
[(d*cot(e+f*x))^(3/2)*tan(e+f*x)^5,x,14,2/5*d^4/(f*(d*cot(e+f*x))^(5/2))+d^(3/2)*arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))-d^(3/2)*arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))-1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))+1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))-2*d^2/(f*sqrt(d*cot(e+f*x)))],
[(d*cot(e+f*x))^(3/2)*tan(e+f*x)^4,x,13,2/3*d^3/(f*(d*cot(e+f*x))^(3/2))-d^(3/2)*arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))+d^(3/2)*arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))-1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))+1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))],
[(d*cot(e+f*x))^(3/2)*tan(e+f*x)^3,x,13,-d^(3/2)*arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))+d^(3/2)*arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))+1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))-1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))+2*d^2/(f*sqrt(d*cot(e+f*x)))],
[(d*cot(e+f*x))^(3/2)*tan(e+f*x)^2,x,12,d^(3/2)*arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))-d^(3/2)*arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))+1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))-1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))],
[(d*cot(e+f*x))^(3/2)*tan(e+f*x),x,12,d^(3/2)*arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))-d^(3/2)*arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))-1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))+1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))],
[(d*cot(e+f*x))^(3/2),x,12,-d^(3/2)*arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))+d^(3/2)*arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))-1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))+1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))-2*d*sqrt(d*cot(e+f*x))/f],
[cot(e+f*x)*(d*cot(e+f*x))^(3/2),x,13,-2/3*(d*cot(e+f*x))^(3/2)/f-d^(3/2)*arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))+d^(3/2)*arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))+1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))-1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))],
[cot(e+f*x)^2*(d*cot(e+f*x))^(3/2),x,14,-2/5*(d*cot(e+f*x))^(5/2)/(d*f)+d^(3/2)*arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))-d^(3/2)*arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2))+1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))-1/2*d^(3/2)*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2))+2*d*sqrt(d*cot(e+f*x))/f],

# m<0
[tan(e+f*x)^3/sqrt(d*cot(e+f*x)),x,14,2/5*d^2/(f*(d*cot(e+f*x))^(5/2))+arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2)*sqrt(d))-arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2)*sqrt(d))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2)*sqrt(d))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2)*sqrt(d))+(-2)/(f*sqrt(d*cot(e+f*x)))],
[tan(e+f*x)^2/sqrt(d*cot(e+f*x)),x,13,2/3*d/(f*(d*cot(e+f*x))^(3/2))-arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2)*sqrt(d))+arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2)*sqrt(d))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2)*sqrt(d))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2)*sqrt(d))],
[tan(e+f*x)/sqrt(d*cot(e+f*x)),x,13,-arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2)*sqrt(d))+arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2)*sqrt(d))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2)*sqrt(d))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2)*sqrt(d))+2/(f*sqrt(d*cot(e+f*x)))],
[1/sqrt(d*cot(e+f*x)),x,11,arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2)*sqrt(d))-arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2)*sqrt(d))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2)*sqrt(d))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2)*sqrt(d))],
[cot(e+f*x)/sqrt(d*cot(e+f*x)),x,12,arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2)*sqrt(d))-arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2)*sqrt(d))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2)*sqrt(d))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2)*sqrt(d))],
[cot(e+f*x)^2/sqrt(d*cot(e+f*x)),x,13,-arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2)*sqrt(d))+arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2)*sqrt(d))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2)*sqrt(d))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2)*sqrt(d))-2*sqrt(d*cot(e+f*x))/(d*f)],
[cot(e+f*x)^3/sqrt(d*cot(e+f*x)),x,13,-2/3*(d*cot(e+f*x))^(3/2)/(d^2*f)-arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2)*sqrt(d))+arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(f*sqrt(2)*sqrt(d))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2)*sqrt(d))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(f*sqrt(2)*sqrt(d))],
[tan(e+f*x)^2/(d*cot(e+f*x))^(3/2),x,14,2/5*d/(f*(d*cot(e+f*x))^(5/2))+arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))-arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))+(-2)/(d*f*sqrt(d*cot(e+f*x)))],
[tan(e+f*x)/(d*cot(e+f*x))^(3/2),x,13,2/3/(f*(d*cot(e+f*x))^(3/2))-arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))+arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))],
[1/(d*cot(e+f*x))^(3/2),x,12,-arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))+arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))+2/(d*f*sqrt(d*cot(e+f*x)))],
[cot(e+f*x)/(d*cot(e+f*x))^(3/2),x,12,arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))-arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))],
[cot(e+f*x)^2/(d*cot(e+f*x))^(3/2),x,12,arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))-arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))],
[cot(e+f*x)^3/(d*cot(e+f*x))^(3/2),x,13,-arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))+arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))-2*sqrt(d*cot(e+f*x))/(d^2*f)],
[cot(e+f*x)^4/(d*cot(e+f*x))^(3/2),x,13,-2/3*(d*cot(e+f*x))^(3/2)/(d^3*f)-arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))+arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))],
[cot(e+f*x)^5/(d*cot(e+f*x))^(3/2),x,14,-2/5*(d*cot(e+f*x))^(5/2)/(d^4*f)+arctan(1-sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))-arctan(1+sqrt(2)*sqrt(d*cot(e+f*x))/sqrt(d))/(d^(3/2)*f*sqrt(2))+1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)-sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))-1/2*log(sqrt(d)+cot(e+f*x)*sqrt(d)+sqrt(2)*sqrt(d*cot(e+f*x)))/(d^(3/2)*f*sqrt(2))+2*sqrt(d*cot(e+f*x))/(d^2*f)],

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

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

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

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

# n>0
[sec(e+f*x)^6*sqrt(d*tan(e+f*x)),x,3,2/3*(d*tan(e+f*x))^(3/2)/(d*f)+4/7*(d*tan(e+f*x))^(7/2)/(d^3*f)+2/11*(d*tan(e+f*x))^(11/2)/(d^5*f)],
[sec(e+f*x)^4*sqrt(d*tan(e+f*x)),x,3,2/3*(d*tan(e+f*x))^(3/2)/(d*f)+2/7*(d*tan(e+f*x))^(7/2)/(d^3*f)],
[sec(e+f*x)^2*sqrt(d*tan(e+f*x)),x,2,2/3*(d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(d*tan(e+f*x)),x,11,-arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))+arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))+1/2*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))*sqrt(d)/(f*sqrt(2))-1/2*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))*sqrt(d)/(f*sqrt(2))],
[cos(e+f*x)^2*sqrt(d*tan(e+f*x)),x,12,-1/4*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))+1/4*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))*sqrt(d)/(f*sqrt(2))+1/8*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))*sqrt(d)/(f*sqrt(2))-1/8*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))*sqrt(d)/(f*sqrt(2))+1/2*cos(e+f*x)^2*(d*tan(e+f*x))^(3/2)/(d*f)],
[sec(e+f*x)^3*sqrt(d*tan(e+f*x)),x,5,-4/5*cos(e+f*x)*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(d*tan(e+f*x))/(f*sqrt(sin(2*e+2*f*x)))+4/5*cos(e+f*x)*(d*tan(e+f*x))^(3/2)/(d*f)+2/5*sec(e+f*x)*(d*tan(e+f*x))^(3/2)/(d*f)],
[sec(e+f*x)*sqrt(d*tan(e+f*x)),x,4,-2*cos(e+f*x)*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(d*tan(e+f*x))/(f*sqrt(sin(2*e+2*f*x)))+2*cos(e+f*x)*(d*tan(e+f*x))^(3/2)/(d*f)],
[cos(e+f*x)*sqrt(d*tan(e+f*x)),x,3,cos(e+f*x)*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(d*tan(e+f*x))/(f*sqrt(sin(2*e+2*f*x)))],
[cos(e+f*x)^3*sqrt(d*tan(e+f*x)),x,4,1/2*cos(e+f*x)*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(d*tan(e+f*x))/(f*sqrt(sin(2*e+2*f*x)))+1/3*cos(e+f*x)^3*(d*tan(e+f*x))^(3/2)/(d*f)],
[cos(e+f*x)^5*sqrt(d*tan(e+f*x)),x,5,7/20*cos(e+f*x)*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(d*tan(e+f*x))/(f*sqrt(sin(2*e+2*f*x)))+7/30*cos(e+f*x)^3*(d*tan(e+f*x))^(3/2)/(d*f)+1/5*cos(e+f*x)^5*(d*tan(e+f*x))^(3/2)/(d*f)],
[sec(a+b*x)^6*(d*tan(a+b*x))^(3/2),x,3,2/5*(d*tan(a+b*x))^(5/2)/(b*d)+4/9*(d*tan(a+b*x))^(9/2)/(b*d^3)+2/13*(d*tan(a+b*x))^(13/2)/(b*d^5)],
[sec(a+b*x)^4*(d*tan(a+b*x))^(3/2),x,3,2/5*(d*tan(a+b*x))^(5/2)/(b*d)+2/9*(d*tan(a+b*x))^(9/2)/(b*d^3)],
[sec(a+b*x)^2*(d*tan(a+b*x))^(3/2),x,2,2/5*(d*tan(a+b*x))^(5/2)/(b*d)],
[(d*tan(a+b*x))^(3/2),x,12,d^(3/2)*arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2))-d^(3/2)*arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2))+1/2*d^(3/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2))-1/2*d^(3/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2))+2*d*sqrt(d*tan(a+b*x))/b],
[cos(a+b*x)^2*(d*tan(a+b*x))^(3/2),x,12,-1/4*d^(3/2)*arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2))+1/4*d^(3/2)*arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*sqrt(2))-1/8*d^(3/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2))+1/8*d^(3/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*sqrt(2))-1/2*d*cos(a+b*x)^2*sqrt(d*tan(a+b*x))/b],
[sec(a+b*x)^5*(d*tan(a+b*x))^(3/2),x,6,-4/77*d^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sec(a+b*x)*sqrt(sin(2*a+2*b*x))/(b*sqrt(d*tan(a+b*x)))-4/77*d*sec(a+b*x)*sqrt(d*tan(a+b*x))/b-2/77*d*sec(a+b*x)^3*sqrt(d*tan(a+b*x))/b+2/11*d*sec(a+b*x)^5*sqrt(d*tan(a+b*x))/b],
[sec(a+b*x)^3*(d*tan(a+b*x))^(3/2),x,5,-2/21*d^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sec(a+b*x)*sqrt(sin(2*a+2*b*x))/(b*sqrt(d*tan(a+b*x)))-2/21*d*sec(a+b*x)*sqrt(d*tan(a+b*x))/b+2/7*d*sec(a+b*x)^3*sqrt(d*tan(a+b*x))/b],
[sec(a+b*x)*(d*tan(a+b*x))^(3/2),x,4,-1/3*d^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sec(a+b*x)*sqrt(sin(2*a+2*b*x))/(b*sqrt(d*tan(a+b*x)))+2/3*d*sec(a+b*x)*sqrt(d*tan(a+b*x))/b],
[cos(a+b*x)*(d*tan(a+b*x))^(3/2),x,4,1/2*d^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sec(a+b*x)*sqrt(sin(2*a+2*b*x))/(b*sqrt(d*tan(a+b*x)))-d*cos(a+b*x)*sqrt(d*tan(a+b*x))/b],
[cos(a+b*x)^3*(d*tan(a+b*x))^(3/2),x,5,1/12*d^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sec(a+b*x)*sqrt(sin(2*a+2*b*x))/(b*sqrt(d*tan(a+b*x)))+1/6*d*cos(a+b*x)*sqrt(d*tan(a+b*x))/b-1/3*d*cos(a+b*x)^3*sqrt(d*tan(a+b*x))/b],
[cos(a+b*x)^5*(d*tan(a+b*x))^(3/2),x,6,1/24*d^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sec(a+b*x)*sqrt(sin(2*a+2*b*x))/(b*sqrt(d*tan(a+b*x)))+1/12*d*cos(a+b*x)*sqrt(d*tan(a+b*x))/b+1/30*d*cos(a+b*x)^3*sqrt(d*tan(a+b*x))/b-1/5*d*cos(a+b*x)^5*sqrt(d*tan(a+b*x))/b],
[sec(e+f*x)^6*(d*tan(e+f*x))^(5/2),x,3,2/7*(d*tan(e+f*x))^(7/2)/(d*f)+4/11*(d*tan(e+f*x))^(11/2)/(d^3*f)+2/15*(d*tan(e+f*x))^(15/2)/(d^5*f)],
[sec(e+f*x)^4*(d*tan(e+f*x))^(5/2),x,3,2/7*(d*tan(e+f*x))^(7/2)/(d*f)+2/11*(d*tan(e+f*x))^(11/2)/(d^3*f)],
[sec(e+f*x)^2*(d*tan(e+f*x))^(5/2),x,2,2/7*(d*tan(e+f*x))^(7/2)/(d*f)],
[(d*tan(e+f*x))^(5/2),x,12,d^(5/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(f*sqrt(2))-d^(5/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(f*sqrt(2))-1/2*d^(5/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(f*sqrt(2))+1/2*d^(5/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(f*sqrt(2))+2/3*d*(d*tan(e+f*x))^(3/2)/f],
[cos(e+f*x)^2*(d*tan(e+f*x))^(5/2),x,12,-3/4*d^(5/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(f*sqrt(2))+3/4*d^(5/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(f*sqrt(2))+3/8*d^(5/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(f*sqrt(2))-3/8*d^(5/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(f*sqrt(2))-1/2*d*cos(e+f*x)^2*(d*tan(e+f*x))^(3/2)/f],
[cos(e+f*x)^4*(d*tan(e+f*x))^(5/2),x,13,-3/32*d^(5/2)*arctan(1-sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(f*sqrt(2))+3/32*d^(5/2)*arctan(1+sqrt(2)*sqrt(d*tan(e+f*x))/sqrt(d))/(f*sqrt(2))+3/64*d^(5/2)*log(sqrt(d)-sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(f*sqrt(2))-3/64*d^(5/2)*log(sqrt(d)+sqrt(2)*sqrt(d*tan(e+f*x))+sqrt(d)*tan(e+f*x))/(f*sqrt(2))+3/16*d*cos(e+f*x)^2*(d*tan(e+f*x))^(3/2)/f-1/4*d*cos(e+f*x)^4*(d*tan(e+f*x))^(3/2)/f],

# n<0
[sec(e+f*x)^5/sqrt(d*tan(e+f*x)),x,5,4/7*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sec(e+f*x)*sqrt(sin(2*e+2*f*x))/(f*sqrt(d*tan(e+f*x)))+4/7*sec(e+f*x)*sqrt(d*tan(e+f*x))/(d*f)+2/7*sec(e+f*x)^3*sqrt(d*tan(e+f*x))/(d*f)],
[sec(e+f*x)^3/sqrt(d*tan(e+f*x)),x,4,2/3*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sec(e+f*x)*sqrt(sin(2*e+2*f*x))/(f*sqrt(d*tan(e+f*x)))+2/3*sec(e+f*x)*sqrt(d*tan(e+f*x))/(d*f)],
[sec(e+f*x)/sqrt(d*tan(e+f*x)),x,3,sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sec(e+f*x)*sqrt(sin(2*e+2*f*x))/(f*sqrt(d*tan(e+f*x)))],
[cos(e+f*x)/sqrt(d*tan(e+f*x)),x,4,1/2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sec(e+f*x)*sqrt(sin(2*e+2*f*x))/(f*sqrt(d*tan(e+f*x)))+cos(e+f*x)*sqrt(d*tan(e+f*x))/(d*f)],
[cos(e+f*x)^3/sqrt(d*tan(e+f*x)),x,5,5/12*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sec(e+f*x)*sqrt(sin(2*e+2*f*x))/(f*sqrt(d*tan(e+f*x)))+5/6*cos(e+f*x)*sqrt(d*tan(e+f*x))/(d*f)+1/3*cos(e+f*x)^3*sqrt(d*tan(e+f*x))/(d*f)],
[sec(a+b*x)^6/(d*tan(a+b*x))^(3/2),x,3,(-2)/(b*d*sqrt(d*tan(a+b*x)))+4/3*(d*tan(a+b*x))^(3/2)/(b*d^3)+2/7*(d*tan(a+b*x))^(7/2)/(b*d^5)],
[sec(a+b*x)^4/(d*tan(a+b*x))^(3/2),x,3,(-2)/(b*d*sqrt(d*tan(a+b*x)))+2/3*(d*tan(a+b*x))^(3/2)/(b*d^3)],
[sec(a+b*x)^2/(d*tan(a+b*x))^(3/2),x,2,(-2)/(b*d*sqrt(d*tan(a+b*x)))],
[1/(d*tan(a+b*x))^(3/2),x,12,arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*d^(3/2)*sqrt(2))-arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*d^(3/2)*sqrt(2))-1/2*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*d^(3/2)*sqrt(2))+1/2*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*d^(3/2)*sqrt(2))+(-2)/(b*d*sqrt(d*tan(a+b*x)))],
[cos(a+b*x)^2/(d*tan(a+b*x))^(3/2),x,13,5/4*arctan(1-sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*d^(3/2)*sqrt(2))-5/4*arctan(1+sqrt(2)*sqrt(d*tan(a+b*x))/sqrt(d))/(b*d^(3/2)*sqrt(2))-5/8*log(sqrt(d)-sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*d^(3/2)*sqrt(2))+5/8*log(sqrt(d)+sqrt(2)*sqrt(d*tan(a+b*x))+sqrt(d)*tan(a+b*x))/(b*d^(3/2)*sqrt(2))+(-5/2)/(b*d*sqrt(d*tan(a+b*x)))+1/2*cos(a+b*x)^2/(b*d*sqrt(d*tan(a+b*x)))],
[sec(a+b*x)^5/(d*tan(a+b*x))^(3/2),x,6,-2*sec(a+b*x)^3/(b*d*sqrt(d*tan(a+b*x)))-24/5*cos(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*tan(a+b*x))/(b*d^2*sqrt(sin(2*a+2*b*x)))+24/5*cos(a+b*x)*(d*tan(a+b*x))^(3/2)/(b*d^3)+12/5*sec(a+b*x)*(d*tan(a+b*x))^(3/2)/(b*d^3)],
[sec(a+b*x)^3/(d*tan(a+b*x))^(3/2),x,5,-2*sec(a+b*x)/(b*d*sqrt(d*tan(a+b*x)))-4*cos(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*tan(a+b*x))/(b*d^2*sqrt(sin(2*a+2*b*x)))+4*cos(a+b*x)*(d*tan(a+b*x))^(3/2)/(b*d^3)],
[sec(a+b*x)/(d*tan(a+b*x))^(3/2),x,4,-2*cos(a+b*x)/(b*d*sqrt(d*tan(a+b*x)))-2*cos(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*tan(a+b*x))/(b*d^2*sqrt(sin(2*a+2*b*x)))],
[cos(a+b*x)/(d*tan(a+b*x))^(3/2),x,4,-2*cos(a+b*x)/(b*d*sqrt(d*tan(a+b*x)))-3*cos(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*tan(a+b*x))/(b*d^2*sqrt(sin(2*a+2*b*x)))],
[cos(a+b*x)^3/(d*tan(a+b*x))^(3/2),x,5,-2*cos(a+b*x)^3/(b*d*sqrt(d*tan(a+b*x)))-7/2*cos(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*tan(a+b*x))/(b*d^2*sqrt(sin(2*a+2*b*x)))-7/3*cos(a+b*x)^3*(d*tan(a+b*x))^(3/2)/(b*d^3)],
[cos(a+b*x)^5/(d*tan(a+b*x))^(3/2),x,6,-2*cos(a+b*x)^5/(b*d*sqrt(d*tan(a+b*x)))-77/20*cos(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*tan(a+b*x))/(b*d^2*sqrt(sin(2*a+2*b*x)))-77/30*cos(a+b*x)^3*(d*tan(a+b*x))^(3/2)/(b*d^3)-11/5*cos(a+b*x)^5*(d*tan(a+b*x))^(3/2)/(b*d^3)],
[sec(a+b*x)/(d*tan(a+b*x))^(5/2),x,4,-1/3*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sec(a+b*x)*sqrt(sin(2*a+2*b*x))/(b*d^2*sqrt(d*tan(a+b*x)))-2/3*sec(a+b*x)/(b*d*(d*tan(a+b*x))^(3/2))],
[sec(a+b*x)^3/(d*tan(a+b*x))^(7/2),x,5,-4/5*cos(a+b*x)/(b*d^3*sqrt(d*tan(a+b*x)))-4/5*cos(a+b*x)*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*tan(a+b*x))/(b*d^4*sqrt(sin(2*a+2*b*x)))-2/5*sec(a+b*x)/(b*d*(d*tan(a+b*x))^(5/2))],

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

# n>0
[sec(e+f*x)^(4/3)*tan(e+f*x)^2,x,2,3/7*hypergeom([-7/6,-1/2],[-1/6],cos(e+f*x)^2)*sec(e+f*x)^(7/3)*sin(e+f*x)/(f*sqrt(sin(e+f*x)^2))],
[sec(e+f*x)^(2/3)*tan(e+f*x)^2,x,2,3/5*hypergeom([-5/6,-1/2],[1/6],cos(e+f*x)^2)*sec(e+f*x)^(5/3)*sin(e+f*x)/(f*sqrt(sin(e+f*x)^2))],
[sec(e+f*x)^(1/3)*tan(e+f*x)^2,x,2,3/4*hypergeom([-2/3,-1/2],[1/3],cos(e+f*x)^2)*sec(e+f*x)^(4/3)*sin(e+f*x)/(f*sqrt(sin(e+f*x)^2))],
[tan(e+f*x)^2/sec(e+f*x)^(1/3),x,2,3/2*hypergeom([-1/2,-1/3],[2/3],cos(e+f*x)^2)*sec(e+f*x)^(2/3)*sin(e+f*x)/(f*sqrt(sin(e+f*x)^2))],
[tan(e+f*x)^2/sec(e+f*x)^(2/3),x,2,3*hypergeom([-1/2,-1/6],[5/6],cos(e+f*x)^2)*sec(e+f*x)^(1/3)*sin(e+f*x)/(f*sqrt(sin(e+f*x)^2))],
[sec(e+f*x)^(4/3)*tan(e+f*x)^4,x,2,3/13*hypergeom([-13/6,-3/2],[-7/6],cos(e+f*x)^2)*sec(e+f*x)^(13/3)*sin(e+f*x)/(f*sqrt(sin(e+f*x)^2))],
[sec(e+f*x)^(2/3)*tan(e+f*x)^4,x,2,3/11*hypergeom([-11/6,-3/2],[-5/6],cos(e+f*x)^2)*sec(e+f*x)^(11/3)*sin(e+f*x)/(f*sqrt(sin(e+f*x)^2))],
[sec(e+f*x)^(1/3)*tan(e+f*x)^4,x,2,3/10*hypergeom([-5/3,-3/2],[-2/3],cos(e+f*x)^2)*sec(e+f*x)^(10/3)*sin(e+f*x)/(f*sqrt(sin(e+f*x)^2))],
[tan(e+f*x)^4/sec(e+f*x)^(1/3),x,2,3/8*hypergeom([-3/2,-4/3],[-1/3],cos(e+f*x)^2)*sec(e+f*x)^(8/3)*sin(e+f*x)/(f*sqrt(sin(e+f*x)^2))],
[tan(e+f*x)^4/sec(e+f*x)^(2/3),x,2,3/7*hypergeom([-3/2,-7/6],[-1/6],cos(e+f*x)^2)*sec(e+f*x)^(7/3)*sin(e+f*x)/(f*sqrt(sin(e+f*x)^2))],
[(d*sec(e+f*x))^(4/3)*tan(e+f*x)^2,x,1,1/3*(cos(e+f*x)^2)^(13/6)*hypergeom([3/2,13/6],[5/2],sin(e+f*x)^2)*(d*sec(e+f*x))^(4/3)*tan(e+f*x)^3/f],
[(d*sec(e+f*x))^(2/3)*tan(e+f*x)^2,x,1,1/3*(cos(e+f*x)^2)^(11/6)*hypergeom([3/2,11/6],[5/2],sin(e+f*x)^2)*(d*sec(e+f*x))^(2/3)*tan(e+f*x)^3/f],
[(d*sec(e+f*x))^(1/3)*tan(e+f*x)^2,x,1,1/3*(cos(e+f*x)^2)^(5/3)*hypergeom([3/2,5/3],[5/2],sin(e+f*x)^2)*(d*sec(e+f*x))^(1/3)*tan(e+f*x)^3/f],
[tan(e+f*x)^2/(d*sec(e+f*x))^(1/3),x,1,1/3*(cos(e+f*x)^2)^(4/3)*hypergeom([4/3,3/2],[5/2],sin(e+f*x)^2)*tan(e+f*x)^3/(f*(d*sec(e+f*x))^(1/3))],
[tan(e+f*x)^2/(d*sec(e+f*x))^(2/3),x,1,1/3*(cos(e+f*x)^2)^(7/6)*hypergeom([7/6,3/2],[5/2],sin(e+f*x)^2)*tan(e+f*x)^3/(f*(d*sec(e+f*x))^(2/3))],
[(d*sec(e+f*x))^(4/3)*tan(e+f*x)^4,x,1,1/5*(cos(e+f*x)^2)^(19/6)*hypergeom([5/2,19/6],[7/2],sin(e+f*x)^2)*(d*sec(e+f*x))^(4/3)*tan(e+f*x)^5/f],
[(d*sec(e+f*x))^(2/3)*tan(e+f*x)^4,x,1,1/5*(cos(e+f*x)^2)^(17/6)*hypergeom([5/2,17/6],[7/2],sin(e+f*x)^2)*(d*sec(e+f*x))^(2/3)*tan(e+f*x)^5/f],
[(d*sec(e+f*x))^(1/3)*tan(e+f*x)^4,x,1,1/5*(cos(e+f*x)^2)^(8/3)*hypergeom([5/2,8/3],[7/2],sin(e+f*x)^2)*(d*sec(e+f*x))^(1/3)*tan(e+f*x)^5/f],
[tan(e+f*x)^4/(d*sec(e+f*x))^(1/3),x,1,1/5*(cos(e+f*x)^2)^(7/3)*hypergeom([7/3,5/2],[7/2],sin(e+f*x)^2)*tan(e+f*x)^5/(f*(d*sec(e+f*x))^(1/3))],
[tan(e+f*x)^4/(d*sec(e+f*x))^(2/3),x,1,1/5*(cos(e+f*x)^2)^(13/6)*hypergeom([13/6,5/2],[7/2],sin(e+f*x)^2)*tan(e+f*x)^5/(f*(d*sec(e+f*x))^(2/3))],

# n<0

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

# n>0
[(d*sec(e+f*x))^(5/2)*sqrt(b*tan(e+f*x)),x,7,-1/4*d^3*arctan(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(b)*sqrt(b*tan(e+f*x))/(f*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x)))+1/4*d^3*arctanh(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(b)*sqrt(b*tan(e+f*x))/(f*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x)))+1/2*d^2*sqrt(d*sec(e+f*x))*(b*tan(e+f*x))^(3/2)/(b*f)],
[(d*sec(e+f*x))^(3/2)*sqrt(b*tan(e+f*x)),x,4,-d^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(b*tan(e+f*x))/(f*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x)))+d^2*(b*tan(e+f*x))^(3/2)/(b*f*sqrt(d*sec(e+f*x)))],
[(d*sec(e+f*x))^(1/2)*sqrt(b*tan(e+f*x)),x,6,-d*arctan(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(b)*sqrt(b*tan(e+f*x))/(f*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x)))+d*arctanh(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(b)*sqrt(b*tan(e+f*x))/(f*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x)))],
[sqrt(b*tan(e+f*x))/(d*sec(e+f*x))^(1/2),x,3,2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(b*tan(e+f*x))/(f*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x)))],
[sqrt(b*tan(e+f*x))/(d*sec(e+f*x))^(3/2),x,1,2/3*(b*tan(e+f*x))^(3/2)/(b*f*(d*sec(e+f*x))^(3/2))],
[sqrt(b*tan(e+f*x))/(d*sec(e+f*x))^(5/2),x,4,4/5*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(b*tan(e+f*x))/(d^2*f*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x)))+2/5*(b*tan(e+f*x))^(3/2)/(b*f*(d*sec(e+f*x))^(5/2))],
[sqrt(b*tan(e+f*x))/(d*sec(e+f*x))^(7/2),x,2,2/7*(b*tan(e+f*x))^(3/2)/(b*f*(d*sec(e+f*x))^(7/2))+8/21*(b*tan(e+f*x))^(3/2)/(b*d^2*f*(d*sec(e+f*x))^(3/2))],
[sqrt(b*tan(e+f*x))/(d*sec(e+f*x))^(9/2),x,5,8/15*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(b*tan(e+f*x))/(d^4*f*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x)))+2/9*(b*tan(e+f*x))^(3/2)/(b*f*(d*sec(e+f*x))^(9/2))+4/15*(b*tan(e+f*x))^(3/2)/(b*d^2*f*(d*sec(e+f*x))^(5/2))],
[(d*sec(e+f*x))^(5/2)*(b*tan(e+f*x))^(3/2),x,5,-1/6*b^2*d^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x))/(f*sqrt(b*tan(e+f*x)))+1/3*b*(d*sec(e+f*x))^(5/2)*sqrt(b*tan(e+f*x))/f-1/6*b*d^2*sqrt(d*sec(e+f*x))*sqrt(b*tan(e+f*x))/f],
[(d*sec(e+f*x))^(3/2)*(b*tan(e+f*x))^(3/2),x,7,-1/4*b^(3/2)*d*arctan(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x))/(f*sqrt(b*tan(e+f*x)))-1/4*b^(3/2)*d*arctanh(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x))/(f*sqrt(b*tan(e+f*x)))+1/2*b*(d*sec(e+f*x))^(3/2)*sqrt(b*tan(e+f*x))/f],
[(d*sec(e+f*x))^(1/2)*(b*tan(e+f*x))^(3/2),x,4,-b^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x))/(f*sqrt(b*tan(e+f*x)))+b*sqrt(d*sec(e+f*x))*sqrt(b*tan(e+f*x))/f],
[(b*tan(e+f*x))^(3/2)/(d*sec(e+f*x))^(1/2),x,7,-2*d*csc(e+f*x)*(b*tan(e+f*x))^(3/2)/(f*(d*sec(e+f*x))^(3/2))+b^(3/2)*d*arctan(sqrt(b*sin(e+f*x))/sqrt(b))*(b*tan(e+f*x))^(3/2)/(f*(d*sec(e+f*x))^(3/2)*(b*sin(e+f*x))^(3/2))+b^(3/2)*d*arctanh(sqrt(b*sin(e+f*x))/sqrt(b))*(b*tan(e+f*x))^(3/2)/(f*(d*sec(e+f*x))^(3/2)*(b*sin(e+f*x))^(3/2))],
[(b*tan(e+f*x))^(3/2)/(d*sec(e+f*x))^(3/2),x,4,2/3*b^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x))/(d^2*f*sqrt(b*tan(e+f*x)))-2/3*b*sqrt(b*tan(e+f*x))/(f*(d*sec(e+f*x))^(3/2))],
[(b*tan(e+f*x))^(3/2)/(d*sec(e+f*x))^(5/2),x,1,2/5*(b*tan(e+f*x))^(5/2)/(b*f*(d*sec(e+f*x))^(5/2))],
[(b*tan(e+f*x))^(3/2)/(d*sec(e+f*x))^(7/2),x,5,4/21*b^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x))/(d^4*f*sqrt(b*tan(e+f*x)))-2/7*b*sqrt(b*tan(e+f*x))/(f*(d*sec(e+f*x))^(7/2))+2/21*b*sqrt(b*tan(e+f*x))/(d^2*f*(d*sec(e+f*x))^(3/2))],
[(b*tan(e+f*x))^(3/2)/(d*sec(e+f*x))^(9/2),x,3,-2/9*b*sqrt(b*tan(e+f*x))/(f*(d*sec(e+f*x))^(9/2))+2/45*b*sqrt(b*tan(e+f*x))/(d^2*f*(d*sec(e+f*x))^(5/2))+8/45*b*sqrt(b*tan(e+f*x))/(d^4*f*sqrt(d*sec(e+f*x)))],
[(d*sec(e+f*x))^(5/2)*(b*tan(e+f*x))^(5/2),x,8,3/32*b^(5/2)*d^3*arctan(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(b*tan(e+f*x))/(f*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x)))-3/32*b^(5/2)*d^3*arctanh(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(b*tan(e+f*x))/(f*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x)))+1/4*b*(d*sec(e+f*x))^(5/2)*(b*tan(e+f*x))^(3/2)/f-3/16*b*d^2*sqrt(d*sec(e+f*x))*(b*tan(e+f*x))^(3/2)/f],
[(d*sec(e+f*x))^(3/2)*(b*tan(e+f*x))^(5/2),x,5,1/2*b^2*d^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(b*tan(e+f*x))/(f*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x)))+1/3*b*(d*sec(e+f*x))^(3/2)*(b*tan(e+f*x))^(3/2)/f-1/2*b*d^2*(b*tan(e+f*x))^(3/2)/(f*sqrt(d*sec(e+f*x)))],
[(d*sec(e+f*x))^(1/2)*(b*tan(e+f*x))^(5/2),x,7,3/4*b^(5/2)*d*arctan(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(b*tan(e+f*x))/(f*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x)))-3/4*b^(5/2)*d*arctanh(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(b*tan(e+f*x))/(f*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x)))+1/2*b*sqrt(d*sec(e+f*x))*(b*tan(e+f*x))^(3/2)/f],
[(b*tan(e+f*x))^(5/2)/(d*sec(e+f*x))^(1/2),x,4,-3*b^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(b*tan(e+f*x))/(f*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x)))+b*(b*tan(e+f*x))^(3/2)/(f*sqrt(d*sec(e+f*x)))],
[(b*tan(e+f*x))^(5/2)/(d*sec(e+f*x))^(3/2),x,7,-b^(5/2)*arctan(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(b*tan(e+f*x))/(d*f*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x)))+b^(5/2)*arctanh(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(b*tan(e+f*x))/(d*f*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x)))-2/3*b*(b*tan(e+f*x))^(3/2)/(f*(d*sec(e+f*x))^(3/2))],
[(b*tan(e+f*x))^(5/2)/(d*sec(e+f*x))^(5/2),x,4,6/5*b^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(b*tan(e+f*x))/(d^2*f*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x)))-2/5*b*(b*tan(e+f*x))^(3/2)/(f*(d*sec(e+f*x))^(5/2))],
[(b*tan(e+f*x))^(5/2)/(d*sec(e+f*x))^(7/2),x,1,2/7*(b*tan(e+f*x))^(7/2)/(b*f*(d*sec(e+f*x))^(7/2))],
[(b*tan(e+f*x))^(5/2)/(d*sec(e+f*x))^(9/2),x,5,4/15*b^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(b*tan(e+f*x))/(d^4*f*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x)))-2/9*b*(b*tan(e+f*x))^(3/2)/(f*(d*sec(e+f*x))^(9/2))+2/15*b*(b*tan(e+f*x))^(3/2)/(d^2*f*(d*sec(e+f*x))^(5/2))],

# n<0
[(d*sec(e+f*x))^(7/2)/sqrt(b*tan(e+f*x)),x,7,3/4*d^3*arctan(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x))/(f*sqrt(b)*sqrt(b*tan(e+f*x)))+3/4*d^3*arctanh(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x))/(f*sqrt(b)*sqrt(b*tan(e+f*x)))+1/2*d^2*(d*sec(e+f*x))^(3/2)*sqrt(b*tan(e+f*x))/(b*f)],
[(d*sec(e+f*x))^(5/2)/sqrt(b*tan(e+f*x)),x,4,d^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x))/(f*sqrt(b*tan(e+f*x)))+d^2*sqrt(d*sec(e+f*x))*sqrt(b*tan(e+f*x))/(b*f)],
[(d*sec(e+f*x))^(3/2)/sqrt(b*tan(e+f*x)),x,6,d*arctan(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x))/(f*sqrt(b)*sqrt(b*tan(e+f*x)))+d*arctanh(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x))/(f*sqrt(b)*sqrt(b*tan(e+f*x)))],
[(d*sec(e+f*x))^(1/2)/sqrt(b*tan(e+f*x)),x,3,2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x))/(f*sqrt(b*tan(e+f*x)))],
[1/((d*sec(e+f*x))^(1/2)*sqrt(b*tan(e+f*x))),x,1,2*sqrt(b*tan(e+f*x))/(b*f*sqrt(d*sec(e+f*x)))],
[1/((d*sec(e+f*x))^(3/2)*sqrt(b*tan(e+f*x))),x,4,4/3*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x))/(d^2*f*sqrt(b*tan(e+f*x)))+2/3*sqrt(b*tan(e+f*x))/(b*f*(d*sec(e+f*x))^(3/2))],
[1/((d*sec(e+f*x))^(5/2)*sqrt(b*tan(e+f*x))),x,2,2/5*sqrt(b*tan(e+f*x))/(b*f*(d*sec(e+f*x))^(5/2))+8/5*sqrt(b*tan(e+f*x))/(b*d^2*f*sqrt(d*sec(e+f*x)))],
[(d*sec(e+f*x))^(5/2)/(b*tan(e+f*x))^(3/2),x,7,-2*d^2*sqrt(d*sec(e+f*x))/(b*f*sqrt(b*tan(e+f*x)))-d^3*arctan(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(b*tan(e+f*x))/(b^(3/2)*f*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x)))+d^3*arctanh(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(b*tan(e+f*x))/(b^(3/2)*f*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x)))],
[(d*sec(e+f*x))^(3/2)/(b*tan(e+f*x))^(3/2),x,4,-2*d^2/(b*f*sqrt(d*sec(e+f*x))*sqrt(b*tan(e+f*x)))-2*d^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(b*tan(e+f*x))/(b^2*f*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x)))],
[(d*sec(e+f*x))^(1/2)/(b*tan(e+f*x))^(3/2),x,1,-2*sqrt(d*sec(e+f*x))/(b*f*sqrt(b*tan(e+f*x)))],
[1/((d*sec(e+f*x))^(1/2)*(b*tan(e+f*x))^(3/2)),x,4,(-2)/(b*f*sqrt(d*sec(e+f*x))*sqrt(b*tan(e+f*x)))-4*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(b*tan(e+f*x))/(b^2*f*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x)))],
[1/((d*sec(e+f*x))^(3/2)*(b*tan(e+f*x))^(3/2)),x,2,2/3/(b*f*(d*sec(e+f*x))^(3/2)*sqrt(b*tan(e+f*x)))-8/3*sqrt(d*sec(e+f*x))/(b*d^2*f*sqrt(b*tan(e+f*x))),(-2)/(b*f*(d*sec(e+f*x))^(3/2)*sqrt(b*tan(e+f*x)))-8/3*(b*tan(e+f*x))^(3/2)/(b^3*f*(d*sec(e+f*x))^(3/2))],
[1/((d*sec(e+f*x))^(5/2)*(b*tan(e+f*x))^(3/2)),x,5,(-2)/(b*f*(d*sec(e+f*x))^(5/2)*sqrt(b*tan(e+f*x)))-24/5*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(b*tan(e+f*x))/(b^2*d^2*f*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x)))-12/5*(b*tan(e+f*x))^(3/2)/(b^3*f*(d*sec(e+f*x))^(5/2))],
[(d*sec(e+f*x))^(7/2)/(b*tan(e+f*x))^(5/2),x,7,d^3*arctan(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x))/(b^(5/2)*f*sqrt(b*tan(e+f*x)))+d^3*arctanh(sqrt(b*sin(e+f*x))/sqrt(b))*sqrt(d*sec(e+f*x))*sqrt(b*sin(e+f*x))/(b^(5/2)*f*sqrt(b*tan(e+f*x)))-2/3*d^2*(d*sec(e+f*x))^(3/2)/(b*f*(b*tan(e+f*x))^(3/2))],
[(d*sec(e+f*x))^(5/2)/(b*tan(e+f*x))^(5/2),x,4,2/3*d^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x))/(b^2*f*sqrt(b*tan(e+f*x)))-2/3*d^2*sqrt(d*sec(e+f*x))/(b*f*(b*tan(e+f*x))^(3/2))],
[(d*sec(e+f*x))^(3/2)/(b*tan(e+f*x))^(5/2),x,1,-2/3*(d*sec(e+f*x))^(3/2)/(b*f*(b*tan(e+f*x))^(3/2))],
[(d*sec(e+f*x))^(1/2)/(b*tan(e+f*x))^(5/2),x,4,-4/3*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x))/(b^2*f*sqrt(b*tan(e+f*x)))-2/3*sqrt(d*sec(e+f*x))/(b*f*(b*tan(e+f*x))^(3/2))],
[1/((d*sec(e+f*x))^(1/2)*(b*tan(e+f*x))^(5/2)),x,2,-8/3*sqrt(b*tan(e+f*x))/(b^3*f*sqrt(d*sec(e+f*x)))+(-2/3)/(b*f*sqrt(d*sec(e+f*x))*(b*tan(e+f*x))^(3/2))],
[1/((d*sec(e+f*x))^(3/2)*(b*tan(e+f*x))^(5/2)),x,5,-8/3*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*sec(e+f*x))*sqrt(sin(e+f*x))/(b^2*d^2*f*sqrt(b*tan(e+f*x)))-4/3*sqrt(b*tan(e+f*x))/(b^3*f*(d*sec(e+f*x))^(3/2))+(-2/3)/(b*f*(d*sec(e+f*x))^(3/2)*(b*tan(e+f*x))^(3/2))],
[1/((d*sec(e+f*x))^(5/2)*(b*tan(e+f*x))^(5/2)),x,3,-16/15*sqrt(b*tan(e+f*x))/(b^3*f*(d*sec(e+f*x))^(5/2))-64/15*sqrt(b*tan(e+f*x))/(b^3*d^2*f*sqrt(d*sec(e+f*x)))+(-2/3)/(b*f*(d*sec(e+f*x))^(5/2)*(b*tan(e+f*x))^(3/2))],

# Integrands of the form (a Sec[e+f x])^(m/3) (b Tan[e+f x])^(n/2)
[(b*sec(e+f*x))^(4/3)*sqrt(d*tan(e+f*x)),x,1,2/3*(cos(e+f*x)^2)^(17/12)*hypergeom([3/4,17/12],[7/4],sin(e+f*x)^2)*(b*sec(e+f*x))^(4/3)*(d*tan(e+f*x))^(3/2)/(d*f)],
[(b*sec(e+f*x))^(1/3)*sqrt(d*tan(e+f*x)),x,1,2/3*(cos(e+f*x)^2)^(11/12)*hypergeom([3/4,11/12],[7/4],sin(e+f*x)^2)*(b*sec(e+f*x))^(1/3)*(d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(d*tan(e+f*x))/(b*sec(e+f*x))^(1/3),x,1,2/3*(cos(e+f*x)^2)^(7/12)*hypergeom([7/12,3/4],[7/4],sin(e+f*x)^2)*(d*tan(e+f*x))^(3/2)/(d*f*(b*sec(e+f*x))^(1/3))],
[sqrt(d*tan(e+f*x))/(b*sec(e+f*x))^(4/3),x,1,2/3*(cos(e+f*x)^2)^(1/12)*hypergeom([1/12,3/4],[7/4],sin(e+f*x)^2)*(d*tan(e+f*x))^(3/2)/(d*f*(b*sec(e+f*x))^(4/3))],
[(b*sec(e+f*x))^(4/3)*(d*tan(e+f*x))^(3/2),x,1,2/5*(cos(e+f*x)^2)^(23/12)*hypergeom([5/4,23/12],[9/4],sin(e+f*x)^2)*(b*sec(e+f*x))^(4/3)*(d*tan(e+f*x))^(5/2)/(d*f)],
[(b*sec(e+f*x))^(1/3)*(d*tan(e+f*x))^(3/2),x,1,2/5*(cos(e+f*x)^2)^(17/12)*hypergeom([5/4,17/12],[9/4],sin(e+f*x)^2)*(b*sec(e+f*x))^(1/3)*(d*tan(e+f*x))^(5/2)/(d*f)],
[(d*tan(e+f*x))^(3/2)/(b*sec(e+f*x))^(1/3),x,1,2/5*(cos(e+f*x)^2)^(13/12)*hypergeom([13/12,5/4],[9/4],sin(e+f*x)^2)*(d*tan(e+f*x))^(5/2)/(d*f*(b*sec(e+f*x))^(1/3))],
[(d*tan(e+f*x))^(3/2)/(b*sec(e+f*x))^(4/3),x,1,2/5*(cos(e+f*x)^2)^(7/12)*hypergeom([7/12,5/4],[9/4],sin(e+f*x)^2)*(d*tan(e+f*x))^(5/2)/(d*f*(b*sec(e+f*x))^(4/3))],

# Integrands of the form (a Sec[e+f x])^(m/2) (b Tan[e+f x])^(n/3)
[sqrt(b*sec(e+f*x))*(d*tan(e+f*x))^(4/3),x,1,3/7*(cos(e+f*x)^2)^(17/12)*hypergeom([7/6,17/12],[13/6],sin(e+f*x)^2)*sqrt(b*sec(e+f*x))*(d*tan(e+f*x))^(7/3)/(d*f)],
[sqrt(b*sec(e+f*x))*(d*tan(e+f*x))^(1/3),x,1,3/4*(cos(e+f*x)^2)^(11/12)*hypergeom([2/3,11/12],[5/3],sin(e+f*x)^2)*sqrt(b*sec(e+f*x))*(d*tan(e+f*x))^(4/3)/(d*f)],
[sqrt(b*sec(e+f*x))/(d*tan(e+f*x))^(1/3),x,1,3/2*(cos(e+f*x)^2)^(7/12)*hypergeom([1/3,7/12],[4/3],sin(e+f*x)^2)*sqrt(b*sec(e+f*x))*(d*tan(e+f*x))^(2/3)/(d*f)],
[sqrt(b*sec(e+f*x))/(d*tan(e+f*x))^(4/3),x,1,-3*(cos(e+f*x)^2)^(1/12)*hypergeom([-1/6,1/12],[5/6],sin(e+f*x)^2)*sqrt(b*sec(e+f*x))/(d*f*(d*tan(e+f*x))^(1/3))],
[(b*sec(e+f*x))^(3/2)*(d*tan(e+f*x))^(4/3),x,1,3/7*(cos(e+f*x)^2)^(23/12)*hypergeom([7/6,23/12],[13/6],sin(e+f*x)^2)*(b*sec(e+f*x))^(3/2)*(d*tan(e+f*x))^(7/3)/(d*f)],
[(b*sec(e+f*x))^(3/2)*(d*tan(e+f*x))^(1/3),x,1,3/4*(cos(e+f*x)^2)^(17/12)*hypergeom([2/3,17/12],[5/3],sin(e+f*x)^2)*(b*sec(e+f*x))^(3/2)*(d*tan(e+f*x))^(4/3)/(d*f)],
[(b*sec(e+f*x))^(3/2)/(d*tan(e+f*x))^(1/3),x,1,3/2*(cos(e+f*x)^2)^(13/12)*hypergeom([1/3,13/12],[4/3],sin(e+f*x)^2)*(b*sec(e+f*x))^(3/2)*(d*tan(e+f*x))^(2/3)/(d*f)],
[(b*sec(e+f*x))^(3/2)/(d*tan(e+f*x))^(4/3),x,1,-3*(cos(e+f*x)^2)^(7/12)*hypergeom([-1/6,7/12],[5/6],sin(e+f*x)^2)*(b*sec(e+f*x))^(3/2)/(d*f*(d*tan(e+f*x))^(1/3))],

# Integrands of the form (a Sec[e+f x])^m (b Tan[e+f x])^n with m symbolic
[(b*sec(e+f*x))^m*tan(e+f*x)^5,x,3,(b*sec(e+f*x))^m/(f*m)-2*(b*sec(e+f*x))^(2+m)/(b^2*f*(2+m))+(b*sec(e+f*x))^(4+m)/(b^4*f*(4+m))],
[(b*sec(e+f*x))^m*tan(e+f*x)^3,x,3,-(b*sec(e+f*x))^m/(f*m)+(b*sec(e+f*x))^(2+m)/(b^2*f*(2+m))],
[(b*sec(e+f*x))^m*tan(e+f*x),x,2,(b*sec(e+f*x))^m/(f*m)],
[cot(e+f*x)*(b*sec(e+f*x))^m,x,2,-hypergeom([1,1/2*m],[1/2*(2+m)],sec(e+f*x)^2)*(b*sec(e+f*x))^m/(f*m)],
[cot(e+f*x)^3*(b*sec(e+f*x))^m,x,2,hypergeom([2,1/2*m],[1/2*(2+m)],sec(e+f*x)^2)*(b*sec(e+f*x))^m/(f*m)],
[cot(e+f*x)^5*(b*sec(e+f*x))^m,x,2,-hypergeom([3,1/2*m],[1/2*(2+m)],sec(e+f*x)^2)*(b*sec(e+f*x))^m/(f*m)],
[(b*sec(e+f*x))^m*tan(e+f*x)^4,x,1,1/5*(cos(e+f*x)^2)^(1/2*(5+m))*hypergeom([5/2,1/2*(5+m)],[7/2],sin(e+f*x)^2)*(b*sec(e+f*x))^m*tan(e+f*x)^5/f],
[(b*sec(e+f*x))^m*tan(e+f*x)^2,x,1,1/3*(cos(e+f*x)^2)^(1/2*(3+m))*hypergeom([3/2,1/2*(3+m)],[5/2],sin(e+f*x)^2)*(b*sec(e+f*x))^m*tan(e+f*x)^3/f],
[cot(e+f*x)^2*(b*sec(e+f*x))^m,x,1,-(cos(e+f*x)^2)^(1/2*(-1+m))*cot(e+f*x)*hypergeom([-1/2,1/2*(-1+m)],[1/2],sin(e+f*x)^2)*(b*sec(e+f*x))^m/f],
[cot(e+f*x)^4*(b*sec(e+f*x))^m,x,1,-1/3*(cos(e+f*x)^2)^(1/2*(-3+m))*cot(e+f*x)^3*hypergeom([-3/2,1/2*(-3+m)],[-1/2],sin(e+f*x)^2)*(b*sec(e+f*x))^m/f],
[cot(e+f*x)^6*(b*sec(e+f*x))^m,x,1,-1/5*(cos(e+f*x)^2)^(1/2*(-5+m))*cot(e+f*x)^5*hypergeom([-5/2,1/2*(-5+m)],[-3/2],sin(e+f*x)^2)*(b*sec(e+f*x))^m/f],

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

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

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

# Integrands of the form (a Csc[e+f x])^m (b Tan[e+f x])^n with m symbolic
[(b*csc(e+f*x))^m*tan(e+f*x)^3,x,2,-(b*csc(e+f*x))^m*hypergeom([2,1/2*m],[1/2*(2+m)],csc(e+f*x)^2)/(f*m)],
[(b*csc(e+f*x))^m*tan(e+f*x),x,2,(b*csc(e+f*x))^m*hypergeom([1,1/2*m],[1/2*(2+m)],csc(e+f*x)^2)/(f*m)],
[cot(e+f*x)*(b*csc(e+f*x))^m,x,2,-(b*csc(e+f*x))^m/(f*m)],
[cot(e+f*x)^3*(b*csc(e+f*x))^m,x,3,(b*csc(e+f*x))^m/(f*m)-(b*csc(e+f*x))^(2+m)/(b^2*f*(2+m))],
[cot(e+f*x)^5*(b*csc(e+f*x))^m,x,3,-(b*csc(e+f*x))^m/(f*m)+2*(b*csc(e+f*x))^(2+m)/(b^2*f*(2+m))-(b*csc(e+f*x))^(4+m)/(b^4*f*(4+m))],
[(b*csc(e+f*x))^m*tan(e+f*x)^4,x,1,1/3*(b*csc(e+f*x))^m*hypergeom([-3/2,1/2*(-3+m)],[-1/2],cos(e+f*x)^2)*(sin(e+f*x)^2)^(1/2*(-3+m))*tan(e+f*x)^3/f],
[(b*csc(e+f*x))^m*tan(e+f*x)^2,x,1,(b*csc(e+f*x))^m*hypergeom([-1/2,1/2*(-1+m)],[1/2],cos(e+f*x)^2)*(sin(e+f*x)^2)^(1/2*(-1+m))*tan(e+f*x)/f],
[cot(e+f*x)^2*(b*csc(e+f*x))^m,x,1,-1/3*cot(e+f*x)^3*(b*csc(e+f*x))^m*hypergeom([3/2,1/2*(3+m)],[5/2],cos(e+f*x)^2)*(sin(e+f*x)^2)^(1/2*(3+m))/f],
[cot(e+f*x)^4*(b*csc(e+f*x))^m,x,1,-1/5*cot(e+f*x)^5*(b*csc(e+f*x))^m*hypergeom([5/2,1/2*(5+m)],[7/2],cos(e+f*x)^2)*(sin(e+f*x)^2)^(1/2*(5+m))/f],
[(b*csc(e+f*x))^m*(d*tan(e+f*x))^(3/2),x,3,2*(cos(e+f*x)^2)^(5/4)*(b*csc(e+f*x))^m*hypergeom([5/4,1/4*(5-2*m)],[1/4*(9-2*m)],sin(e+f*x)^2)*(d*tan(e+f*x))^(5/2)/(d*f*(5-2*m))],
[(b*csc(e+f*x))^m*(d*tan(e+f*x))^(1/2),x,3,2*(cos(e+f*x)^2)^(3/4)*(b*csc(e+f*x))^m*hypergeom([3/4,1/4*(3-2*m)],[1/4*(7-2*m)],sin(e+f*x)^2)*(d*tan(e+f*x))^(3/2)/(d*f*(3-2*m))],
[(b*csc(e+f*x))^m/(d*tan(e+f*x))^(1/2),x,3,2*(cos(e+f*x)^2)^(1/4)*(b*csc(e+f*x))^m*hypergeom([1/4,1/4*(1-2*m)],[1/4*(5-2*m)],sin(e+f*x)^2)*sqrt(d*tan(e+f*x))/(d*f*(1-2*m))],
[(b*csc(e+f*x))^m/(d*tan(e+f*x))^(3/2),x,3,-2*(b*csc(e+f*x))^m*hypergeom([-1/4,1/4*(-1-2*m)],[1/4*(3-2*m)],sin(e+f*x)^2)/(d*f*(1+2*m)*(cos(e+f*x)^2)^(1/4)*sqrt(d*tan(e+f*x)))],

# Integrands of the form (a Csc[e+f x])^m (b Tan[e+f x])^n with n symbolic
[(a*csc(e+f*x))^m*(b*tan(e+f*x))^n,x,3,(cos(e+f*x)^2)^(1/2*(1+n))*(a*csc(e+f*x))^m*hypergeom([1/2*(1+n),1/2*(1-m+n)],[1/2*(3-m+n)],sin(e+f*x)^2)*(b*tan(e+f*x))^(1+n)/(b*f*(1-m+n))]]: