# Maple integration test file: "4 Trig functions\4.3 Tangent\4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).txt"

lst:=[

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

# Integrands of the form (d Tan[e+f x])^m (a+a I Tan[e+f x])^n (A+B Tan[e+f x])

# Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+i a Tan[c+d x])^n

# n>0
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,4,-a*(A-I*B)*x+a*(I*A+B)*log(cos(c+d*x))/d+a*(A-I*B)*tan(c+d*x)/d+1/2*a*(I*A+B)*tan(c+d*x)^2/d+1/3*I*a*B*tan(c+d*x)^3/d],
[tan(c+d*x)*(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,3,-a*(I*A+B)*x-a*(A-I*B)*log(cos(c+d*x))/d+a*(I*A+B)*tan(c+d*x)/d+1/2*I*a*B*tan(c+d*x)^2/d],
[(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,2,a*(A-I*B)*x-a*(I*A+B)*log(cos(c+d*x))/d+I*a*B*tan(c+d*x)/d],
[cot(c+d*x)*(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,4,a*(I*A+B)*x-I*a*B*log(cos(c+d*x))/d+a*A*log(sin(c+d*x))/d],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,3,-a*(A-I*B)*x-a*A*cot(c+d*x)/d+a*(I*A+B)*log(sin(c+d*x))/d],
[cot(c+d*x)^3*(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,4,-a*(I*A+B)*x-a*(I*A+B)*cot(c+d*x)/d-1/2*a*A*cot(c+d*x)^2/d-a*(A-I*B)*log(sin(c+d*x))/d],
[cot(c+d*x)^4*(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,5,a*(A-I*B)*x+a*(A-I*B)*cot(c+d*x)/d-1/2*a*(I*A+B)*cot(c+d*x)^2/d-1/3*a*A*cot(c+d*x)^3/d-a*(I*A+B)*log(sin(c+d*x))/d],
[cot(c+d*x)^5*(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,6,a*(I*A+B)*x+a*(I*A+B)*cot(c+d*x)/d+1/2*a*(A-I*B)*cot(c+d*x)^2/d-1/3*a*(I*A+B)*cot(c+d*x)^3/d-1/4*a*A*cot(c+d*x)^4/d+a*(A-I*B)*log(sin(c+d*x))/d],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,5,-2*a^2*(A-I*B)*x+2*a^2*(I*A+B)*log(cos(c+d*x))/d+2*a^2*(A-I*B)*tan(c+d*x)/d+a^2*(I*A+B)*tan(c+d*x)^2/d-1/12*a^2*(4*A-5*I*B)*tan(c+d*x)^3/d+1/4*I*B*tan(c+d*x)^3*(a^2+I*a^2*tan(c+d*x))/d],
[tan(c+d*x)*(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,4,-2*a^2*(I*A+B)*x-2*a^2*(A-I*B)*log(cos(c+d*x))/d+a^2*(I*A+B)*tan(c+d*x)/d+1/2*A*(a+I*a*tan(c+d*x))^2/d-1/3*I*B*(a+I*a*tan(c+d*x))^3/(a*d)],
[(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,3,2*a^2*(A-I*B)*x-2*a^2*(I*A+B)*log(cos(c+d*x))/d-a^2*(A-I*B)*tan(c+d*x)/d+1/2*B*(a+I*a*tan(c+d*x))^2/d],
[cot(c+d*x)*(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,5,2*a^2*(I*A+B)*x+a^2*(A-2*I*B)*log(cos(c+d*x))/d+a^2*A*log(sin(c+d*x))/d+I*B*(a^2+I*a^2*tan(c+d*x))/d],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,5,-2*a^2*(A-I*B)*x+a^2*B*log(cos(c+d*x))/d+a^2*(2*I*A+B)*log(sin(c+d*x))/d-A*cot(c+d*x)*(a^2+I*a^2*tan(c+d*x))/d],
[cot(c+d*x)^3*(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,4,-2*a^2*(I*A+B)*x-1/2*a^2*(3*I*A+2*B)*cot(c+d*x)/d-2*a^2*(A-I*B)*log(sin(c+d*x))/d-1/2*A*cot(c+d*x)^2*(a^2+I*a^2*tan(c+d*x))/d],
[cot(c+d*x)^4*(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,5,2*a^2*(A-I*B)*x+2*a^2*(A-I*B)*cot(c+d*x)/d-1/6*a^2*(4*I*A+3*B)*cot(c+d*x)^2/d-2*a^2*(I*A+B)*log(sin(c+d*x))/d-1/3*A*cot(c+d*x)^3*(a^2+I*a^2*tan(c+d*x))/d],
[cot(c+d*x)^5*(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,6,2*a^2*(I*A+B)*x+2*a^2*(I*A+B)*cot(c+d*x)/d+a^2*(A-I*B)*cot(c+d*x)^2/d-1/12*a^2*(5*I*A+4*B)*cot(c+d*x)^3/d+2*a^2*(A-I*B)*log(sin(c+d*x))/d-1/4*A*cot(c+d*x)^4*(a^2+I*a^2*tan(c+d*x))/d],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,6,-4*a^3*(A-I*B)*x+4*a^3*(I*A+B)*log(cos(c+d*x))/d+4*a^3*(A-I*B)*tan(c+d*x)/d+2*a^3*(I*A+B)*tan(c+d*x)^2/d-1/60*a^3*(45*A-47*I*B)*tan(c+d*x)^3/d+1/5*I*a*B*tan(c+d*x)^3*(a+I*a*tan(c+d*x))^2/d-1/20*(5*A-7*I*B)*tan(c+d*x)^3*(a^3+I*a^3*tan(c+d*x))/d],
[tan(c+d*x)*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,5,-4*a^3*(I*A+B)*x-4*a^3*(A-I*B)*log(cos(c+d*x))/d+2*a^3*(I*A+B)*tan(c+d*x)/d+1/2*a*(A-I*B)*(a+I*a*tan(c+d*x))^2/d+1/3*A*(a+I*a*tan(c+d*x))^3/d-1/4*I*B*(a+I*a*tan(c+d*x))^4/(a*d)],
[(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,4,4*a^3*(A-I*B)*x-4*a^3*(I*A+B)*log(cos(c+d*x))/d-2*a^3*(A-I*B)*tan(c+d*x)/d+1/2*a*(I*A+B)*(a+I*a*tan(c+d*x))^2/d+1/3*B*(a+I*a*tan(c+d*x))^3/d],
[cot(c+d*x)*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,6,4*a^3*(I*A+B)*x+a^3*(3*A-4*I*B)*log(cos(c+d*x))/d+a^3*A*log(sin(c+d*x))/d+1/2*I*a*B*(a+I*a*tan(c+d*x))^2/d-(A-2*I*B)*(a^3+I*a^3*tan(c+d*x))/d],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,6,-4*a^3*(A-I*B)*x+a^3*(I*A+3*B)*log(cos(c+d*x))/d+a^3*(3*I*A+B)*log(sin(c+d*x))/d-a*A*cot(c+d*x)*(a+I*a*tan(c+d*x))^2/d+(I*A-B)*(a^3+I*a^3*tan(c+d*x))/d],
[cot(c+d*x)^3*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,6,-4*a^3*(I*A+B)*x+I*a^3*B*log(cos(c+d*x))/d-a^3*(4*A-3*I*B)*log(sin(c+d*x))/d-1/2*a*A*cot(c+d*x)^2*(a+I*a*tan(c+d*x))^2/d-(2*I*A+B)*cot(c+d*x)*(a^3+I*a^3*tan(c+d*x))/d],
[cot(c+d*x)^4*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,5,4*a^3*(A-I*B)*x+1/6*a^3*(17*A-15*I*B)*cot(c+d*x)/d-4*a^3*(I*A+B)*log(sin(c+d*x))/d-1/3*a*A*cot(c+d*x)^3*(a+I*a*tan(c+d*x))^2/d-1/6*(5*I*A+3*B)*cot(c+d*x)^2*(a^3+I*a^3*tan(c+d*x))/d],
[cot(c+d*x)^5*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,6,4*a^3*(I*A+B)*x+4*a^3*(I*A+B)*cot(c+d*x)/d+1/12*a^3*(15*A-14*I*B)*cot(c+d*x)^2/d+4*a^3*(A-I*B)*log(sin(c+d*x))/d-1/4*a*A*cot(c+d*x)^4*(a+I*a*tan(c+d*x))^2/d-1/6*(3*I*A+2*B)*cot(c+d*x)^3*(a^3+I*a^3*tan(c+d*x))/d],
[cot(c+d*x)^6*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,7,-4*a^3*(A-I*B)*x-4*a^3*(A-I*B)*cot(c+d*x)/d+2*a^3*(I*A+B)*cot(c+d*x)^2/d+1/60*a^3*(47*A-45*I*B)*cot(c+d*x)^3/d+4*a^3*(I*A+B)*log(sin(c+d*x))/d-1/5*a*A*cot(c+d*x)^5*(a+I*a*tan(c+d*x))^2/d-1/20*(7*I*A+5*B)*cot(c+d*x)^4*(a^3+I*a^3*tan(c+d*x))/d],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,7,-8*a^4*(A-I*B)*x+8*a^4*(I*A+B)*log(cos(c+d*x))/d+8*a^4*(A-I*B)*tan(c+d*x)/d+4*a^4*(I*A+B)*tan(c+d*x)^2/d-1/60*a^4*(92*A-93*I*B)*tan(c+d*x)^3/d+1/6*I*a*B*tan(c+d*x)^3*(a+I*a*tan(c+d*x))^3/d-1/10*(2*A-3*I*B)*tan(c+d*x)^3*(a^2+I*a^2*tan(c+d*x))^2/d-1/20*(12*A-13*I*B)*tan(c+d*x)^3*(a^4+I*a^4*tan(c+d*x))/d],
[tan(c+d*x)*(a+I*a*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,6,-8*a^4*(I*A+B)*x-8*a^4*(A-I*B)*log(cos(c+d*x))/d+4*a^4*(I*A+B)*tan(c+d*x)/d+1/3*a*(A-I*B)*(a+I*a*tan(c+d*x))^3/d+1/4*A*(a+I*a*tan(c+d*x))^4/d-1/5*I*B*(a+I*a*tan(c+d*x))^5/(a*d)+(A-I*B)*(a^2+I*a^2*tan(c+d*x))^2/d],
[(a+I*a*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,5,8*a^4*(A-I*B)*x-8*a^4*(I*A+B)*log(cos(c+d*x))/d-4*a^4*(A-I*B)*tan(c+d*x)/d+1/3*a*(I*A+B)*(a+I*a*tan(c+d*x))^3/d+1/4*B*(a+I*a*tan(c+d*x))^4/d+(I*A+B)*(a^2+I*a^2*tan(c+d*x))^2/d],
[cot(c+d*x)*(a+I*a*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,7,8*a^4*(I*A+B)*x+a^4*(7*A-8*I*B)*log(cos(c+d*x))/d+a^4*A*log(sin(c+d*x))/d+1/3*I*a*B*(a+I*a*tan(c+d*x))^3/d-1/2*(A-2*I*B)*(a^2+I*a^2*tan(c+d*x))^2/d-(3*A-4*I*B)*(a^4+I*a^4*tan(c+d*x))/d],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,7,-8*a^4*(A-I*B)*x+a^4*(4*I*A+7*B)*log(cos(c+d*x))/d+a^4*(4*I*A+B)*log(sin(c+d*x))/d-a*A*cot(c+d*x)*(a+I*a*tan(c+d*x))^3/d+1/2*(2*I*A-B)*(a^2+I*a^2*tan(c+d*x))^2/d-3*B*(a^4+I*a^4*tan(c+d*x))/d],
[cot(c+d*x)^3*(a+I*a*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,7,-8*a^4*(I*A+B)*x-a^4*(A-4*I*B)*log(cos(c+d*x))/d-a^4*(7*A-4*I*B)*log(sin(c+d*x))/d-1/2*a*A*cot(c+d*x)^2*(a+I*a*tan(c+d*x))^3/d-1/2*(5*I*A+2*B)*cot(c+d*x)*(a^2+I*a^2*tan(c+d*x))^2/d-3*A*(a^4+I*a^4*tan(c+d*x))/d],
[cot(c+d*x)^4*(a+I*a*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,7,8*a^4*(A-I*B)*x-a^4*B*log(cos(c+d*x))/d-a^4*(8*I*A+7*B)*log(sin(c+d*x))/d-1/3*a*A*cot(c+d*x)^3*(a+I*a*tan(c+d*x))^3/d-1/2*(2*I*A+B)*cot(c+d*x)^2*(a^2+I*a^2*tan(c+d*x))^2/d+(4*A-3*I*B)*cot(c+d*x)*(a^4+I*a^4*tan(c+d*x))/d],
[cot(c+d*x)^5*(a+I*a*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,6,8*a^4*(I*A+B)*x+1/12*a^4*(67*I*A+64*B)*cot(c+d*x)/d+8*a^4*(A-I*B)*log(sin(c+d*x))/d-1/4*a*A*cot(c+d*x)^4*(a+I*a*tan(c+d*x))^3/d-1/12*(7*I*A+4*B)*cot(c+d*x)^3*(a^2+I*a^2*tan(c+d*x))^2/d+1/12*(19*A-16*I*B)*cot(c+d*x)^2*(a^4+I*a^4*tan(c+d*x))/d],
[cot(c+d*x)^6*(a+I*a*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,7,-8*a^4*(A-I*B)*x-8*a^4*(A-I*B)*cot(c+d*x)/d+1/60*a^4*(148*I*A+145*B)*cot(c+d*x)^2/d+8*a^4*(I*A+B)*log(sin(c+d*x))/d-1/5*a*A*cot(c+d*x)^5*(a+I*a*tan(c+d*x))^3/d-1/20*(8*I*A+5*B)*cot(c+d*x)^4*(a^2+I*a^2*tan(c+d*x))^2/d+1/30*(28*A-25*I*B)*cot(c+d*x)^3*(a^4+I*a^4*tan(c+d*x))/d],
[cot(c+d*x)^7*(a+I*a*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,8,-8*a^4*(I*A+B)*x-8*a^4*(I*A+B)*cot(c+d*x)/d-4*a^4*(A-I*B)*cot(c+d*x)^2/d+1/60*a^4*(93*I*A+92*B)*cot(c+d*x)^3/d-8*a^4*(A-I*B)*log(sin(c+d*x))/d-1/6*a*A*cot(c+d*x)^6*(a+I*a*tan(c+d*x))^3/d-1/10*(3*I*A+2*B)*cot(c+d*x)^5*(a^2+I*a^2*tan(c+d*x))^2/d+1/20*(13*A-12*I*B)*cot(c+d*x)^4*(a^4+I*a^4*tan(c+d*x))/d],

# n<0
[tan(c+d*x)^3*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x)),x,4,3/2*(I*A-B)*x/a-(A+2*I*B)*log(cos(c+d*x))/(a*d)-3/2*(I*A-B)*tan(c+d*x)/(a*d)-1/2*(A+2*I*B)*tan(c+d*x)^2/(a*d)+1/2*(I*A-B)*tan(c+d*x)^3/(d*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^2*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x)),x,3,1/2*(A+3*I*B)*x/a+(I*A-B)*log(cos(c+d*x))/(a*d)-1/2*(A+3*I*B)*tan(c+d*x)/(a*d)+1/2*(I*A-B)*tan(c+d*x)^2/(d*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x)),x,5,-1/2*(I*A-B)*x/a+I*B*log(cos(c+d*x))/(a*d)+1/2*(-A-I*B)/(a*d*(1+I*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x)),x,2,1/2*(A-I*B)*x/a+1/2*(I*A-B)/(d*(a+I*a*tan(c+d*x)))],
[cot(c+d*x)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x)),x,3,-1/2*(I*A-B)*x/a+A*log(sin(c+d*x))/(a*d)+1/2*(A+I*B)/(d*(a+I*a*tan(c+d*x)))],
[cot(c+d*x)^2*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x)),x,4,-1/2*(3*A+I*B)*x/a-1/2*(3*A+I*B)*cot(c+d*x)/(a*d)-(I*A-B)*log(sin(c+d*x))/(a*d)+1/2*(A+I*B)*cot(c+d*x)/(d*(a+I*a*tan(c+d*x)))],
[cot(c+d*x)^3*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x)),x,5,3/2*(I*A-B)*x/a+3/2*(I*A-B)*cot(c+d*x)/(a*d)-1/2*(2*A+I*B)*cot(c+d*x)^2/(a*d)-(2*A+I*B)*log(sin(c+d*x))/(a*d)+1/2*(A+I*B)*cot(c+d*x)^2/(d*(a+I*a*tan(c+d*x)))],
[cot(c+d*x)^4*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x)),x,6,1/2*(5*A+3*I*B)*x/a+1/2*(5*A+3*I*B)*cot(c+d*x)/(a*d)+(I*A-B)*cot(c+d*x)^2/(a*d)-1/6*(5*A+3*I*B)*cot(c+d*x)^3/(a*d)+2*(I*A-B)*log(sin(c+d*x))/(a*d)+1/2*(A+I*B)*cot(c+d*x)^3/(d*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^3*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^2,x,4,-3/4*(I*A-3*B)*x/a^2+(A+2*I*B)*log(cos(c+d*x))/(a^2*d)+3/4*(I*A-3*B)*tan(c+d*x)/(a^2*d)+1/2*(A+2*I*B)*tan(c+d*x)^2/(a^2*d*(1+I*tan(c+d*x)))+1/4*(I*A-B)*tan(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)^2*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^2,x,6,-1/4*(A+3*I*B)*x/a^2+B*log(cos(c+d*x))/(a^2*d)+1/4*(I*A-3*B)/(a^2*d*(1+I*tan(c+d*x)))+1/4*(I*A-B)*tan(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^2,x,3,-1/4*(I*A+B)*x/a^2+1/4*(A+3*I*B)/(a^2*d*(1+I*tan(c+d*x)))+1/4*(-A-I*B)/(d*(a+I*a*tan(c+d*x))^2)],
[(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^2,x,3,1/4*(A-I*B)*x/a^2+1/4*(I*A-B)/(d*(a+I*a*tan(c+d*x))^2)+1/4*(I*A+B)/(d*(a^2+I*a^2*tan(c+d*x)))],
[cot(c+d*x)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^2,x,4,-1/4*(3*I*A-B)*x/a^2+A*log(sin(c+d*x))/(a^2*d)+1/4*(3*A+I*B)/(a^2*d*(1+I*tan(c+d*x)))+1/4*(A+I*B)/(d*(a+I*a*tan(c+d*x))^2)],
[cot(c+d*x)^2*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^2,x,5,-3/4*(3*A+I*B)*x/a^2-3/4*(3*A+I*B)*cot(c+d*x)/(a^2*d)-(2*I*A-B)*log(sin(c+d*x))/(a^2*d)+1/2*(2*A+I*B)*cot(c+d*x)/(a^2*d*(1+I*tan(c+d*x)))+1/4*(A+I*B)*cot(c+d*x)/(d*(a+I*a*tan(c+d*x))^2)],
[cot(c+d*x)^3*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^2,x,6,3/4*(5*I*A-3*B)*x/a^2+3/4*(5*I*A-3*B)*cot(c+d*x)/(a^2*d)-(2*A+I*B)*cot(c+d*x)^2/(a^2*d)-2*(2*A+I*B)*log(sin(c+d*x))/(a^2*d)+1/4*(5*A+3*I*B)*cot(c+d*x)^2/(a^2*d*(1+I*tan(c+d*x)))+1/4*(A+I*B)*cot(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)^4*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,5,-1/8*(7*A+25*I*B)*x/a^3-(I*A-3*B)*log(cos(c+d*x))/(a^3*d)+1/8*(7*A+25*I*B)*tan(c+d*x)/(a^3*d)+1/6*(I*A-B)*tan(c+d*x)^4/(d*(a+I*a*tan(c+d*x))^3)+1/24*(5*A+11*I*B)*tan(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^2)-1/2*(I*A-3*B)*tan(c+d*x)^2/(d*(a^3+I*a^3*tan(c+d*x)))],
[tan(c+d*x)^3*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,7,1/8*(I*A-7*B)*x/a^3-I*B*log(cos(c+d*x))/(a^3*d)+1/6*(I*A-B)*tan(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^3)+1/8*(A+3*I*B)*tan(c+d*x)^2/(a*d*(a+I*a*tan(c+d*x))^2)+1/8*(A+7*I*B)/(d*(a^3+I*a^3*tan(c+d*x)))],
[tan(c+d*x)^2*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,4,-1/8*(A-I*B)*x/a^3+1/6*(I*A-B)*tan(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^3)+1/24*(I*A-7*B)/(a*d*(a+I*a*tan(c+d*x))^2)+1/24*(I*A+17*B)/(d*(a^3+I*a^3*tan(c+d*x)))],
[tan(c+d*x)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,4,-1/8*(I*A+B)*x/a^3+1/6*(-A-I*B)/(d*(a+I*a*tan(c+d*x))^3)+1/8*(A+3*I*B)/(a*d*(a+I*a*tan(c+d*x))^2)+1/8*(A-I*B)/(d*(a^3+I*a^3*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,4,1/8*(A-I*B)*x/a^3+1/6*(I*A-B)/(d*(a+I*a*tan(c+d*x))^3)+1/8*(I*A+B)/(a*d*(a+I*a*tan(c+d*x))^2)+1/8*(I*A+B)/(d*(a^3+I*a^3*tan(c+d*x)))],
[cot(c+d*x)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,5,-1/8*(7*I*A-B)*x/a^3+A*log(sin(c+d*x))/(a^3*d)+1/6*(A+I*B)/(d*(a+I*a*tan(c+d*x))^3)+1/8*(3*A+I*B)/(a*d*(a+I*a*tan(c+d*x))^2)+1/8*(7*A+I*B)/(d*(a^3+I*a^3*tan(c+d*x)))],
[cot(c+d*x)^2*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,6,-1/8*(25*A+7*I*B)*x/a^3-1/8*(25*A+7*I*B)*cot(c+d*x)/(a^3*d)-(3*I*A-B)*log(sin(c+d*x))/(a^3*d)+1/6*(A+I*B)*cot(c+d*x)/(d*(a+I*a*tan(c+d*x))^3)+1/24*(11*A+5*I*B)*cot(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^2)+1/2*(3*A+I*B)*cot(c+d*x)/(d*(a^3+I*a^3*tan(c+d*x)))],
[cot(c+d*x)^3*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,7,5/8*(11*I*A-5*B)*x/a^3+5/8*(11*I*A-5*B)*cot(c+d*x)/(a^3*d)-1/2*(7*A+3*I*B)*cot(c+d*x)^2/(a^3*d)-(7*A+3*I*B)*log(sin(c+d*x))/(a^3*d)+1/6*(A+I*B)*cot(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^3)+1/24*(13*A+7*I*B)*cot(c+d*x)^2/(a*d*(a+I*a*tan(c+d*x))^2)+5/24*(11*A+5*I*B)*cot(c+d*x)^2/(d*(a^3+I*a^3*tan(c+d*x)))],
[tan(c+d*x)^4*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^4,x,8,1/16*(A+15*I*B)*x/a^4-B*log(cos(c+d*x))/(a^4*d)+1/16*(-I*A+15*B)/(a^4*d*(1+I*tan(c+d*x)))-1/16*(I*A-7*B)*tan(c+d*x)^2/(a^4*d*(1+I*tan(c+d*x))^2)+1/8*(I*A-B)*tan(c+d*x)^4/(d*(a+I*a*tan(c+d*x))^4)+1/12*(A+3*I*B)*tan(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^3)],
[tan(c+d*x)^3*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^4,x,5,1/16*(I*A+B)*x/a^4+1/48*(-A+13*I*B)/(a^4*d*(1+I*tan(c+d*x))^2)+1/48*(5*A-29*I*B)/(a^4*d*(1+I*tan(c+d*x)))+1/8*(I*A-B)*tan(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^4)+1/24*(A+5*I*B)*tan(c+d*x)^2/(a*d*(a+I*a*tan(c+d*x))^3)],
[tan(c+d*x)^2*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^4,x,5,-1/16*(A-I*B)*x/a^4+1/16*(I*A+5*B)/(a^4*d*(1+I*tan(c+d*x))^2)+1/16*(-I*A-B)/(a^4*d*(1+I*tan(c+d*x)))+1/8*(I*A-B)*tan(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^4)-1/6*B/(a*d*(a+I*a*tan(c+d*x))^3)],
[tan(c+d*x)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^4,x,5,-1/16*(I*A+B)*x/a^4+1/8*(-A-I*B)/(d*(a+I*a*tan(c+d*x))^4)+1/12*(A+3*I*B)/(a*d*(a+I*a*tan(c+d*x))^3)+1/16*(A-I*B)/(d*(a^2+I*a^2*tan(c+d*x))^2)+1/16*(A-I*B)/(d*(a^4+I*a^4*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^4,x,5,1/16*(A-I*B)*x/a^4+1/8*(I*A-B)/(d*(a+I*a*tan(c+d*x))^4)+1/12*(I*A+B)/(a*d*(a+I*a*tan(c+d*x))^3)+1/16*(I*A+B)/(d*(a^2+I*a^2*tan(c+d*x))^2)+1/16*(I*A+B)/(d*(a^4+I*a^4*tan(c+d*x)))],
[cot(c+d*x)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^4,x,6,-1/16*(15*I*A-B)*x/a^4+A*log(sin(c+d*x))/(a^4*d)+1/16*(7*A+I*B)/(a^4*d*(1+I*tan(c+d*x))^2)+1/16*(15*A+I*B)/(a^4*d*(1+I*tan(c+d*x)))+1/8*(A+I*B)/(d*(a+I*a*tan(c+d*x))^4)+1/12*(3*A+I*B)/(a*d*(a+I*a*tan(c+d*x))^3)],
[cot(c+d*x)^2*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^4,x,7,-5/16*(13*A+3*I*B)*x/a^4-5/16*(13*A+3*I*B)*cot(c+d*x)/(a^4*d)-(4*I*A-B)*log(sin(c+d*x))/(a^4*d)+1/48*(31*A+9*I*B)*cot(c+d*x)/(a^4*d*(1+I*tan(c+d*x))^2)+1/2*(4*A+I*B)*cot(c+d*x)/(a^4*d*(1+I*tan(c+d*x)))+1/8*(A+I*B)*cot(c+d*x)/(d*(a+I*a*tan(c+d*x))^4)+1/24*(7*A+3*I*B)*cot(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^3)],
[cot(c+d*x)^3*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^4,x,8,5/16*(35*I*A-13*B)*x/a^4+5/16*(35*I*A-13*B)*cot(c+d*x)/(a^4*d)-1/2*(11*A+4*I*B)*cot(c+d*x)^2/(a^4*d)-(11*A+4*I*B)*log(sin(c+d*x))/(a^4*d)+1/48*(43*A+17*I*B)*cot(c+d*x)^2/(a^4*d*(1+I*tan(c+d*x))^2)+5/48*(35*A+13*I*B)*cot(c+d*x)^2/(a^4*d*(1+I*tan(c+d*x)))+1/8*(A+I*B)*cot(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^4)+1/6*(2*A+I*B)*cot(c+d*x)^2/(a*d*(a+I*a*tan(c+d*x))^3)],

# Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+i a Tan[c+d x])^(n/2)

# n>0
[sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^3*(A+B*tan(c+d*x)),x,6,(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d-8/35*(7*A-I*B)*sqrt(a+I*a*tan(c+d*x))/d+2/35*(7*A-I*B)*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^2/d+2/7*B*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^3/d-2/105*(7*A-31*I*B)*(a+I*a*tan(c+d*x))^(3/2)/(a*d)],
[sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^2*(A+B*tan(c+d*x)),x,5,(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d-8/5*B*sqrt(a+I*a*tan(c+d*x))/d+2/5*B*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^2/d-2/15*(5*I*A+B)*(a+I*a*tan(c+d*x))^(3/2)/(a*d)],
[sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)*(A+B*tan(c+d*x)),x,4,-(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d+2*A*sqrt(a+I*a*tan(c+d*x))/d-2/3*I*B*(a+I*a*tan(c+d*x))^(3/2)/(a*d)],
[sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,3,-(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d+2*B*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,6,-2*A*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))*sqrt(a)/d+(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d],
[cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,7,-(I*A+2*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))*sqrt(a)/d+(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d-A*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,8,1/4*(7*A-4*I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))*sqrt(a)/d-(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d-1/4*(I*A+4*B)*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-1/2*A*cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^4*sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,9,1/8*(9*I*A+14*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))*sqrt(a)/d-(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d+1/8*(7*A-2*I*B)*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-1/12*(I*A+6*B)*cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/d-1/3*A*cot(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/d],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,6,2*a^(3/2)*(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-8/35*a*(7*I*A+8*B)*sqrt(a+I*a*tan(c+d*x))/d+2/35*a*(7*I*A+8*B)*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^2/d+2/7*I*a*B*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^3/d-4/105*(21*I*A+19*B)*(a+I*a*tan(c+d*x))^(3/2)/d],
[tan(c+d*x)*(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,5,-2*a^(3/2)*(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+2*a*(A-I*B)*sqrt(a+I*a*tan(c+d*x))/d+2/3*A*(a+I*a*tan(c+d*x))^(3/2)/d-2/5*I*B*(a+I*a*tan(c+d*x))^(5/2)/(a*d)],
[(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,4,-2*a^(3/2)*(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+2*a*(I*A+B)*sqrt(a+I*a*tan(c+d*x))/d+2/3*B*(a+I*a*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)*(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,7,-2*a^(3/2)*A*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d+2*a^(3/2)*(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+2*I*a*B*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,7,-a^(3/2)*(3*I*A+2*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d+2*a^(3/2)*(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-a*A*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^3*(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,8,1/4*a^(3/2)*(11*A-12*I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d-2*a^(3/2)*(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-1/4*a*(5*I*A+4*B)*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-1/2*a*A*cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^4*(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,9,1/8*a^(3/2)*(23*I*A+22*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d-2*a^(3/2)*(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+1/8*a*(9*A-10*I*B)*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-1/12*a*(7*I*A+6*B)*cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/d-1/3*a*A*cot(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/d],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,7,4*a^(5/2)*(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-8/105*a^2*(45*I*A+46*B)*sqrt(a+I*a*tan(c+d*x))/d+2/105*a^2*(45*I*A+46*B)*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^2/d-2/21*a^2*(3*A-4*I*B)*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^3/d-8/315*a*(60*I*A+59*B)*(a+I*a*tan(c+d*x))^(3/2)/d+2/9*I*a*B*tan(c+d*x)^3*(a+I*a*tan(c+d*x))^(3/2)/d],
[tan(c+d*x)*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,6,-4*a^(5/2)*(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+4*a^2*(A-I*B)*sqrt(a+I*a*tan(c+d*x))/d+2/3*a*(A-I*B)*(a+I*a*tan(c+d*x))^(3/2)/d+2/5*A*(a+I*a*tan(c+d*x))^(5/2)/d-2/7*I*B*(a+I*a*tan(c+d*x))^(7/2)/(a*d)],
[(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,5,-4*a^(5/2)*(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+4*a^2*(I*A+B)*sqrt(a+I*a*tan(c+d*x))/d+2/3*a*(I*A+B)*(a+I*a*tan(c+d*x))^(3/2)/d+2/5*B*(a+I*a*tan(c+d*x))^(5/2)/d],
[cot(c+d*x)*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,8,-2*a^(5/2)*A*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d+4*a^(5/2)*(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-2*a^2*(A-2*I*B)*sqrt(a+I*a*tan(c+d*x))/d+2/3*I*a*B*(a+I*a*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,8,-a^(5/2)*(5*I*A+2*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d+4*a^(5/2)*(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+a^2*(I*A-2*B)*sqrt(a+I*a*tan(c+d*x))/d-a*A*cot(c+d*x)*(a+I*a*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^3*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,8,1/4*a^(5/2)*(23*A-20*I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d-4*a^(5/2)*(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-1/4*a^2*(7*I*A+4*B)*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-1/2*a*A*cot(c+d*x)^2*(a+I*a*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^4*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,9,1/8*a^(5/2)*(45*I*A+46*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d-4*a^(5/2)*(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+1/8*a^2*(19*A-18*I*B)*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d-1/4*a^2*(3*I*A+2*B)*cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/d-1/3*a*A*cot(c+d*x)^3*(a+I*a*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^5*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,10,-3/64*a^(5/2)*(121*A-120*I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/d+4*a^(5/2)*(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+1/64*a^2*(149*I*A+152*B)*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/d+1/96*a^2*(107*A-104*I*B)*cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/d-1/24*a^2*(11*I*A+8*B)*cot(c+d*x)^3*sqrt(a+I*a*tan(c+d*x))/d-1/4*a*A*cot(c+d*x)^4*(a+I*a*tan(c+d*x))^(3/2)/d],

# n<0
[tan(c+d*x)^3*(A+B*tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)),x,6,(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+4/5*(5*A+7*I*B)*sqrt(a+I*a*tan(c+d*x))/(a*d)-1/5*(5*A+7*I*B)*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^2/(a*d)+(I*A-B)*tan(c+d*x)^3/(d*sqrt(a+I*a*tan(c+d*x)))-1/15*(25*A+23*I*B)*(a+I*a*tan(c+d*x))^(3/2)/(a^2*d)],
[tan(c+d*x)^2*(A+B*tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)),x,5,(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))-4*(I*A-B)*sqrt(a+I*a*tan(c+d*x))/(a*d)+(I*A-B)*tan(c+d*x)^2/(d*sqrt(a+I*a*tan(c+d*x)))+1/3*(3*I*A-5*B)*(a+I*a*tan(c+d*x))^(3/2)/(a^2*d)],
[tan(c+d*x)*(A+B*tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)),x,4,-(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+(-A-I*B)/(d*sqrt(a+I*a*tan(c+d*x)))-2*I*B*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[(A+B*tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)),x,3,-(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+(I*A-B)/(d*sqrt(a+I*a*tan(c+d*x)))],
[cot(c+d*x)*(A+B*tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)),x,7,-2*A*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(d*sqrt(a))+(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+(A+I*B)/(d*sqrt(a+I*a*tan(c+d*x)))],
[cot(c+d*x)^2*(A+B*tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)),x,8,(I*A-2*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(d*sqrt(a))+(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+(A+I*B)*cot(c+d*x)/(d*sqrt(a+I*a*tan(c+d*x)))-(2*A+I*B)*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[cot(c+d*x)^3*(A+B*tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)),x,9,1/4*(11*A+4*I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(d*sqrt(a))-(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+(A+I*B)*cot(c+d*x)^2/(d*sqrt(a+I*a*tan(c+d*x)))+1/4*(7*I*A-8*B)*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a*d)-1/2*(3*A+2*I*B)*cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[tan(c+d*x)^3*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(3/2),x,6,1/2*(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))-2*(3*A+5*I*B)*sqrt(a+I*a*tan(c+d*x))/(a^2*d)+1/2*(3*A+5*I*B)*tan(c+d*x)^2/(a*d*sqrt(a+I*a*tan(c+d*x)))+1/3*(I*A-B)*tan(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^(3/2))+1/6*(11*A+21*I*B)*(a+I*a*tan(c+d*x))^(3/2)/(a^3*d)],
[tan(c+d*x)^2*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(3/2),x,5,1/2*(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+1/6*(5*I*A-11*B)/(a*d*sqrt(a+I*a*tan(c+d*x)))+1/3*(I*A-7*B)*sqrt(a+I*a*tan(c+d*x))/(a^2*d)+1/3*(I*A-B)*tan(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(3/2),x,4,-1/2*(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+1/2*(A+3*I*B)/(a*d*sqrt(a+I*a*tan(c+d*x)))+1/3*(-A-I*B)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(3/2),x,4,-1/2*(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+1/2*(I*A+B)/(a*d*sqrt(a+I*a*tan(c+d*x)))+1/3*(I*A-B)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[cot(c+d*x)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(3/2),x,8,-2*A*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(a^(3/2)*d)+1/2*(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+1/2*(3*A+I*B)/(a*d*sqrt(a+I*a*tan(c+d*x)))+1/3*(A+I*B)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[cot(c+d*x)^2*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(3/2),x,9,(3*I*A-2*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(a^(3/2)*d)+1/2*(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+1/6*(13*A+7*I*B)*cot(c+d*x)/(a*d*sqrt(a+I*a*tan(c+d*x)))-1/2*(7*A+3*I*B)*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a^2*d)+1/3*(A+I*B)*cot(c+d*x)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[cot(c+d*x)^3*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(3/2),x,10,1/4*(23*A+12*I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(a^(3/2)*d)-1/2*(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+1/6*(17*A+11*I*B)*cot(c+d*x)^2/(a*d*sqrt(a+I*a*tan(c+d*x)))+7/4*(3*I*A-2*B)*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a^2*d)-1/6*(22*A+13*I*B)*cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/(a^2*d)+1/3*(A+I*B)*cot(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^4*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,7,-1/4*(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+1/5*(39*I*A-89*B)*sqrt(a+I*a*tan(c+d*x))/(a^3*d)-1/20*(39*I*A-89*B)*tan(c+d*x)^2/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/5*(I*A-B)*tan(c+d*x)^4/(d*(a+I*a*tan(c+d*x))^(5/2))+1/30*(11*A+21*I*B)*tan(c+d*x)^3/(a*d*(a+I*a*tan(c+d*x))^(3/2))-1/60*(151*I*A-361*B)*(a+I*a*tan(c+d*x))^(3/2)/(a^4*d)],
[tan(c+d*x)^3*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,6,1/4*(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+1/60*(41*A+151*I*B)/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/30*(13*A+83*I*B)*sqrt(a+I*a*tan(c+d*x))/(a^3*d)+1/5*(I*A-B)*tan(c+d*x)^3/(d*(a+I*a*tan(c+d*x))^(5/2))+1/30*(7*A+17*I*B)*tan(c+d*x)^2/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^2*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,5,1/4*(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+1/20*(-I*A+31*B)/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/5*(I*A-B)*tan(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^(5/2))+1/30*(3*I*A-13*B)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,5,-1/4*(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+1/4*(A-I*B)/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/5*(-A-I*B)/(d*(a+I*a*tan(c+d*x))^(5/2))+1/6*(A+3*I*B)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,5,-1/4*(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+1/4*(I*A+B)/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/5*(I*A-B)/(d*(a+I*a*tan(c+d*x))^(5/2))+1/6*(I*A+B)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[cot(c+d*x)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,9,-2*A*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(a^(5/2)*d)+1/4*(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+1/4*(7*A+I*B)/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/5*(A+I*B)/(d*(a+I*a*tan(c+d*x))^(5/2))+1/6*(3*A+I*B)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[cot(c+d*x)^2*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,10,(5*I*A-2*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(a^(5/2)*d)+1/4*(I*A+B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+1/12*(41*A+15*I*B)*cot(c+d*x)/(a^2*d*sqrt(a+I*a*tan(c+d*x)))-7/4*(3*A+I*B)*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a^3*d)+1/5*(A+I*B)*cot(c+d*x)/(d*(a+I*a*tan(c+d*x))^(5/2))+1/30*(19*A+9*I*B)*cot(c+d*x)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[cot(c+d*x)^3*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,11,1/4*(43*A+20*I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/sqrt(a))/(a^(5/2)*d)-1/4*(A-I*B)*arctanh(sqrt(a+I*a*tan(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+1/60*(337*A+167*I*B)*cot(c+d*x)^2/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+21/4*(2*I*A-B)*cot(c+d*x)*sqrt(a+I*a*tan(c+d*x))/(a^3*d)-1/12*(85*A+41*I*B)*cot(c+d*x)^2*sqrt(a+I*a*tan(c+d*x))/(a^3*d)+1/5*(A+I*B)*cot(c+d*x)^2/(d*(a+I*a*tan(c+d*x))^(5/2))+1/30*(23*A+13*I*B)*cot(c+d*x)^2/(a*d*(a+I*a*tan(c+d*x))^(3/2))],

# Integrands of the form Tan[c+d x]^(m/2) (A+B Tan[c+d x]) (a+i a Tan[c+d x])^n

# n>0
[tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,6,-2*(-1)^(1/4)*a*(I*A+B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d-2*a*(I*A+B)*sqrt(tan(c+d*x))/d+2/3*a*(A-I*B)*tan(c+d*x)^(3/2)/d+2/5*a*(I*A+B)*tan(c+d*x)^(5/2)/d+2/7*I*a*B*tan(c+d*x)^(7/2)/d],
[tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,5,2*(-1)^(1/4)*a*(A-I*B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d+2*a*(A-I*B)*sqrt(tan(c+d*x))/d+2/3*a*(I*A+B)*tan(c+d*x)^(3/2)/d+2/5*I*a*B*tan(c+d*x)^(5/2)/d],
[sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,4,2*(-1)^(1/4)*a*(I*A+B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d+2*a*(I*A+B)*sqrt(tan(c+d*x))/d+2/3*I*a*B*tan(c+d*x)^(3/2)/d],
[(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x))/sqrt(tan(c+d*x)),x,3,-2*(-1)^(1/4)*a*(A-I*B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d+2*I*a*B*sqrt(tan(c+d*x))/d],
[(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x))/tan(c+d*x)^(3/2),x,3,-2*(-1)^(1/4)*a*(I*A+B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d-2*a*A/(d*sqrt(tan(c+d*x)))],
[(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x))/tan(c+d*x)^(5/2),x,4,2*(-1)^(1/4)*a*(A-I*B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d-2*a*(I*A+B)/(d*sqrt(tan(c+d*x)))-2/3*a*A/(d*tan(c+d*x)^(3/2))],
[(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x))/tan(c+d*x)^(7/2),x,5,2*(-1)^(1/4)*a*(I*A+B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d+2*a*(A-I*B)/(d*sqrt(tan(c+d*x)))-2/5*a*A/(d*tan(c+d*x)^(5/2))-2/3*a*(I*A+B)/(d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,7,-4*(-1)^(1/4)*a^2*(I*A+B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d-4*a^2*(I*A+B)*sqrt(tan(c+d*x))/d+4/3*a^2*(A-I*B)*tan(c+d*x)^(3/2)/d+4/5*a^2*(I*A+B)*tan(c+d*x)^(5/2)/d-2/63*a^2*(9*A-11*I*B)*tan(c+d*x)^(7/2)/d+2/9*I*B*tan(c+d*x)^(7/2)*(a^2+I*a^2*tan(c+d*x))/d],
[tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,6,4*(-1)^(1/4)*a^2*(A-I*B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d+4*a^2*(A-I*B)*sqrt(tan(c+d*x))/d+4/3*a^2*(I*A+B)*tan(c+d*x)^(3/2)/d-2/35*a^2*(7*A-9*I*B)*tan(c+d*x)^(5/2)/d+2/7*I*B*tan(c+d*x)^(5/2)*(a^2+I*a^2*tan(c+d*x))/d],
[sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,5,4*(-1)^(1/4)*a^2*(I*A+B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d+4*a^2*(I*A+B)*sqrt(tan(c+d*x))/d-2/15*a^2*(5*A-7*I*B)*tan(c+d*x)^(3/2)/d+2/5*I*B*tan(c+d*x)^(3/2)*(a^2+I*a^2*tan(c+d*x))/d],
[(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x))/sqrt(tan(c+d*x)),x,4,-4*(-1)^(1/4)*a^2*(A-I*B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d-2/3*a^2*(3*A-5*I*B)*sqrt(tan(c+d*x))/d+2/3*I*B*sqrt(tan(c+d*x))*(a^2+I*a^2*tan(c+d*x))/d],
[(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x))/tan(c+d*x)^(3/2),x,4,-4*(-1)^(1/4)*a^2*(I*A+B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d+2*a^2*(I*A-B)*sqrt(tan(c+d*x))/d-2*A*(a^2+I*a^2*tan(c+d*x))/(d*sqrt(tan(c+d*x)))],
[(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x))/tan(c+d*x)^(5/2),x,4,4*(-1)^(1/4)*a^2*(A-I*B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d-2/3*a^2*(5*I*A+3*B)/(d*sqrt(tan(c+d*x)))-2/3*A*(a^2+I*a^2*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x))/tan(c+d*x)^(7/2),x,5,4*(-1)^(1/4)*a^2*(I*A+B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d+4*a^2*(A-I*B)/(d*sqrt(tan(c+d*x)))-2/15*a^2*(7*I*A+5*B)/(d*tan(c+d*x)^(3/2))-2/5*A*(a^2+I*a^2*tan(c+d*x))/(d*tan(c+d*x)^(5/2))],
[(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x))/tan(c+d*x)^(9/2),x,6,-4*(-1)^(1/4)*a^2*(A-I*B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d+4*a^2*(I*A+B)/(d*sqrt(tan(c+d*x)))-2/35*a^2*(9*I*A+7*B)/(d*tan(c+d*x)^(5/2))+4/3*a^2*(A-I*B)/(d*tan(c+d*x)^(3/2))-2/7*A*(a^2+I*a^2*tan(c+d*x))/(d*tan(c+d*x)^(7/2))],
[tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,7,8*(-1)^(1/4)*a^3*(A-I*B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d+8*a^3*(A-I*B)*sqrt(tan(c+d*x))/d+8/3*a^3*(I*A+B)*tan(c+d*x)^(3/2)/d-16/315*a^3*(18*A-19*I*B)*tan(c+d*x)^(5/2)/d+2/9*I*a*B*tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^2/d-2/63*(9*A-13*I*B)*tan(c+d*x)^(5/2)*(a^3+I*a^3*tan(c+d*x))/d],
[sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,6,8*(-1)^(1/4)*a^3*(I*A+B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d+8*a^3*(I*A+B)*sqrt(tan(c+d*x))/d-8/105*a^3*(21*A-23*I*B)*tan(c+d*x)^(3/2)/d+2/7*I*a*B*tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^2/d-2/35*(7*A-11*I*B)*tan(c+d*x)^(3/2)*(a^3+I*a^3*tan(c+d*x))/d],
[(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x))/sqrt(tan(c+d*x)),x,5,-8*(-1)^(1/4)*a^3*(A-I*B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d-16/15*a^3*(5*A-6*I*B)*sqrt(tan(c+d*x))/d+2/5*I*a*B*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^2/d-2/15*(5*A-9*I*B)*sqrt(tan(c+d*x))*(a^3+I*a^3*tan(c+d*x))/d],
[(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x))/tan(c+d*x)^(3/2),x,5,-8*(-1)^(1/4)*a^3*(I*A+B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d-16/3*a^3*B*sqrt(tan(c+d*x))/d-2*a*A*(a+I*a*tan(c+d*x))^2/(d*sqrt(tan(c+d*x)))+2/3*(3*I*A-B)*sqrt(tan(c+d*x))*(a^3+I*a^3*tan(c+d*x))/d],
[(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x))/tan(c+d*x)^(5/2),x,5,8*(-1)^(1/4)*a^3*(A-I*B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d-16/3*a^3*A*sqrt(tan(c+d*x))/d-2/3*a*A*(a+I*a*tan(c+d*x))^2/(d*tan(c+d*x)^(3/2))-2/3*(7*I*A+3*B)*(a^3+I*a^3*tan(c+d*x))/(d*sqrt(tan(c+d*x)))],
[(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x))/tan(c+d*x)^(7/2),x,5,8*(-1)^(1/4)*a^3*(I*A+B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d+16/15*a^3*(6*A-5*I*B)/(d*sqrt(tan(c+d*x)))-2/5*a*A*(a+I*a*tan(c+d*x))^2/(d*tan(c+d*x)^(5/2))-2/15*(9*I*A+5*B)*(a^3+I*a^3*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x))/tan(c+d*x)^(9/2),x,6,-8*(-1)^(1/4)*a^3*(A-I*B)*arctan((-1)^(3/4)*sqrt(tan(c+d*x)))/d+8*a^3*(I*A+B)/(d*sqrt(tan(c+d*x)))+8/105*a^3*(23*A-21*I*B)/(d*tan(c+d*x)^(3/2))-2/7*a*A*(a+I*a*tan(c+d*x))^2/(d*tan(c+d*x)^(7/2))-2/35*(11*I*A+7*B)*(a^3+I*a^3*tan(c+d*x))/(d*tan(c+d*x)^(5/2))],

# n<0
[tan(c+d*x)^(5/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x)),x,13,(-1/4-1/4*I)*((4+I)*A+(1+6*I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a*d*sqrt(2))+(1/4+1/4*I)*((4+I)*A+(1+6*I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a*d*sqrt(2))+(-1/8-1/8*I)*((1+4*I)*A+(-6-I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*sqrt(2))-1/8*((3-5*I)*A+(5+7*I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*sqrt(2))-5/2*(I*A-B)*sqrt(tan(c+d*x))/(a*d)-1/6*(3*A+7*I*B)*tan(c+d*x)^(3/2)/(a*d)+1/2*(I*A-B)*tan(c+d*x)^(5/2)/(d*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x)),x,12,-1/4*((1-3*I)*A+(3+5*I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a*d*sqrt(2))+(-1/4-1/4*I)*((1+2*I)*A+(-4-I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a*d*sqrt(2))+(-1/8-1/8*I)*((2+I)*A+(1+4*I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*sqrt(2))+(1/8+1/8*I)*((2+I)*A+(1+4*I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*sqrt(2))-1/2*(A+5*I*B)*sqrt(tan(c+d*x))/(a*d)+1/2*(I*A-B)*tan(c+d*x)^(3/2)/(d*(a+I*a*tan(c+d*x)))],
[sqrt(tan(c+d*x))*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x)),x,11,(-1/4+1/4*I)*(A+(2-I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a*d*sqrt(2))+(1/4-1/4*I)*(A+(2-I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a*d*sqrt(2))+(1/8+1/8*I)*(A+(-2-I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*sqrt(2))+(-1/8-1/8*I)*(A+(-2-I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*sqrt(2))+1/2*(I*A-B)*sqrt(tan(c+d*x))/(d*(a+I*a*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))),x,11,(-1/4+1/4*I)*((2+I)*A+B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a*d*sqrt(2))+(1/4-1/4*I)*((2+I)*A+B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a*d*sqrt(2))-1/8*((3+I)*A+(-1-I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*sqrt(2))+1/8*((3+I)*A+(-1-I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*sqrt(2))+1/2*(A+I*B)*sqrt(tan(c+d*x))/(d*(a+I*a*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))),x,12,1/4*((5+3*I)*A+(-3+I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a*d*sqrt(2))+1/4*((-5-3*I)*A+(3-I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a*d*sqrt(2))+(-1/8+1/8*I)*((4+I)*A+(1+2*I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*sqrt(2))+1/8*((5-3*I)*A+(3+I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*sqrt(2))+1/2*(-5*A-I*B)/(a*d*sqrt(tan(c+d*x)))+1/2*(A+I*B)/(d*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))),x,13,1/4*((7-5*I)*A+(5+3*I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a*d*sqrt(2))+(-1/4+1/4*I)*((6+I)*A+(1+4*I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a*d*sqrt(2))+1/8*((7+5*I)*A+(-5+3*I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*sqrt(2))+1/8*((-7-5*I)*A+(5-3*I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*sqrt(2))+5/2*(I*A-B)/(a*d*sqrt(tan(c+d*x)))+1/6*(-7*A-3*I*B)/(a*d*tan(c+d*x)^(3/2))+1/2*(A+I*B)/(d*tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^(5/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^2,x,13,1/16*((9+5*I)*A+(-25+21*I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*sqrt(2))-1/16*((9+5*I)*A+(-25+21*I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*sqrt(2))+(-1/32+1/32*I)*((7+2*I)*A+(2+23*I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*sqrt(2))+(1/32-1/32*I)*((7+2*I)*A+(2+23*I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*sqrt(2))+5/8*(I*A-5*B)*sqrt(tan(c+d*x))/(a^2*d)+1/8*(3*A+7*I*B)*tan(c+d*x)^(3/2)/(a^2*d*(1+I*tan(c+d*x)))+1/4*(I*A-B)*tan(c+d*x)^(5/2)/(d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^2,x,12,1/16*((1+3*I)*A+(9+5*I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*sqrt(2))-1/16*((1+3*I)*A+(9+5*I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*sqrt(2))+1/32*((1-3*I)*A+(-9+5*I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*sqrt(2))-1/32*((1-3*I)*A+(-9+5*I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*sqrt(2))+1/8*(A+5*I*B)*sqrt(tan(c+d*x))/(a^2*d*(1+I*tan(c+d*x)))+1/4*(I*A-B)*tan(c+d*x)^(3/2)/(d*(a+I*a*tan(c+d*x))^2)],
[sqrt(tan(c+d*x))*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^2,x,12,1/16*((-1+3*I)*A+(1+3*I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*sqrt(2))-1/16*((-1+3*I)*A+(1+3*I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*sqrt(2))+1/32*((1+3*I)*A+(1-3*I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*sqrt(2))-1/32*((1+3*I)*A+(1-3*I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*sqrt(2))+1/8*(I*A+3*B)*sqrt(tan(c+d*x))/(a^2*d*(1+I*tan(c+d*x)))+1/4*(I*A-B)*sqrt(tan(c+d*x))/(d*(a+I*a*tan(c+d*x))^2)],
[(A+B*tan(c+d*x))/(sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^2),x,12,(1/16+1/16*I)*((-2+7*I)*A+(1+2*I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*sqrt(2))+1/16*((9-5*I)*A+(1-3*I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*sqrt(2))+(1/32+1/32*I)*((-7+2*I)*A+(2+I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*sqrt(2))+1/32*((9+5*I)*A+(-1-3*I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*sqrt(2))+1/8*(5*A+I*B)*sqrt(tan(c+d*x))/(a^2*d*(1+I*tan(c+d*x)))+1/4*(A+I*B)*sqrt(tan(c+d*x))/(d*(a+I*a*tan(c+d*x))^2)],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^2),x,13,1/16*((25+21*I)*A+(-9+5*I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*sqrt(2))+(-1/16+1/16*I)*((2+23*I)*A+(-7-2*I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*sqrt(2))+(-1/32+1/32*I)*((23+2*I)*A+(2+7*I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*sqrt(2))+(1/32-1/32*I)*((23+2*I)*A+(2+7*I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*sqrt(2))-5/8*(5*A+I*B)/(a^2*d*sqrt(tan(c+d*x)))+1/8*(7*A+3*I*B)/(a^2*d*sqrt(tan(c+d*x))*(1+I*tan(c+d*x)))+1/4*(A+I*B)/(d*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^2)],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^2),x,14,(1/16-1/16*I)*((47+2*I)*A+(2+23*I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*sqrt(2))+(-1/16+1/16*I)*((47+2*I)*A+(2+23*I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*sqrt(2))+1/32*((49+45*I)*A+(-25+21*I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*sqrt(2))+(-1/32+1/32*I)*((2+47*I)*A+(-23-2*I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*sqrt(2))+5/8*(9*I*A-5*B)/(a^2*d*sqrt(tan(c+d*x)))-7/24*(7*A+3*I*B)/(a^2*d*tan(c+d*x)^(3/2))+1/8*(9*A+5*I*B)/(a^2*d*(1+I*tan(c+d*x))*tan(c+d*x)^(3/2))+1/4*(A+I*B)/(d*tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)^(9/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,15,(1/16+1/16*I)*((29+I)*A+(1+76*I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))+(-1/16-1/16*I)*((29+I)*A+(1+76*I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))-1/32*((28-30*I)*A+(75+77*I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))+(-1/32-1/32*I)*((1+29*I)*A+(-76-I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))+15/8*(2*I*A-5*B)*sqrt(tan(c+d*x))/(a^3*d)+7/24*(4*A+11*I*B)*tan(c+d*x)^(3/2)/(a^3*d)+1/6*(I*A-B)*tan(c+d*x)^(9/2)/(d*(a+I*a*tan(c+d*x))^3)+1/4*(A+2*I*B)*tan(c+d*x)^(7/2)/(a*d*(a+I*a*tan(c+d*x))^2)-3/8*(2*I*A-5*B)*tan(c+d*x)^(5/2)/(d*(a^3+I*a^3*tan(c+d*x)))],
[tan(c+d*x)^(7/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,14,(-1/16-1/16*I)*((1+6*I)*A+(-29-I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))-1/16*((5-7*I)*A+(28+30*I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))+(1/32+1/32*I)*((6+I)*A+(1+29*I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))+(-1/32-1/32*I)*((6+I)*A+(1+29*I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))+5/8*(A+6*I*B)*sqrt(tan(c+d*x))/(a^3*d)+1/6*(I*A-B)*tan(c+d*x)^(7/2)/(d*(a+I*a*tan(c+d*x))^3)+1/12*(2*A+5*I*B)*tan(c+d*x)^(5/2)/(a*d*(a+I*a*tan(c+d*x))^2)-7/24*(I*A-4*B)*tan(c+d*x)^(3/2)/(d*(a^3+I*a^3*tan(c+d*x)))],
[tan(c+d*x)^(5/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,13,1/16*(2*A+(5-7*I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))-1/16*(2*A+(5-7*I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))-1/32*(2*A+(-5-7*I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))+1/32*(2*A+(-5-7*I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))+1/6*(I*A-B)*tan(c+d*x)^(5/2)/(d*(a+I*a*tan(c+d*x))^3)+1/12*(A+4*I*B)*tan(c+d*x)^(3/2)/(a*d*(a+I*a*tan(c+d*x))^2)+5/8*B*sqrt(tan(c+d*x))/(d*(a^3+I*a^3*tan(c+d*x)))],
[tan(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,13,1/16*((1+I)*A+2*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))-1/16*((1+I)*A+2*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))-1/32*((-1+I)*A+2*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))+1/32*((-1+I)*A+2*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))+1/6*(I*A-B)*tan(c+d*x)^(3/2)/(d*(a+I*a*tan(c+d*x))^3)+1/4*I*B*sqrt(tan(c+d*x))/(a*d*(a+I*a*tan(c+d*x))^2)+1/8*(A-2*I*B)*sqrt(tan(c+d*x))/(d*(a^3+I*a^3*tan(c+d*x)))],
[sqrt(tan(c+d*x))*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,13,(1/16+1/16*I)*((1+I)*A+B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))+(-1/16-1/16*I)*((1+I)*A+B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))+1/32*(2*I*A+(1-I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))-1/32*(2*I*A+(1-I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))+1/6*(I*A-B)*sqrt(tan(c+d*x))/(d*(a+I*a*tan(c+d*x))^3)+1/12*(I*A+2*B)*sqrt(tan(c+d*x))/(a*d*(a+I*a*tan(c+d*x))^2)+1/8*B*sqrt(tan(c+d*x))/(d*(a^3+I*a^3*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^3),x,13,-1/16*((7-5*I)*A-2*I*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))+1/16*((7-5*I)*A-2*I*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))-1/32*((7+5*I)*A-2*I*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))+1/32*((7+5*I)*A-2*I*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))+1/6*(A+I*B)*sqrt(tan(c+d*x))/(d*(a+I*a*tan(c+d*x))^3)+1/12*(4*A+I*B)*sqrt(tan(c+d*x))/(a*d*(a+I*a*tan(c+d*x))^2)+5/8*A*sqrt(tan(c+d*x))/(d*(a^3+I*a^3*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^3),x,14,1/16*((30+28*I)*A+(-7+5*I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))+(-1/16+1/16*I)*((1+29*I)*A+(-6-I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))+(-1/32+1/32*I)*((29+I)*A+(1+6*I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))+(1/32-1/32*I)*((29+I)*A+(1+6*I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))-5/8*(6*A+I*B)/(a^3*d*sqrt(tan(c+d*x)))+1/6*(A+I*B)/(d*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^3)+1/12*(5*A+2*I*B)/(a*d*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^2)+7/24*(4*A+I*B)/(d*sqrt(tan(c+d*x))*(a^3+I*a^3*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^3),x,15,(1/16-1/16*I)*((76+I)*A+(1+29*I)*B)*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))+(-1/16+1/16*I)*((76+I)*A+(1+29*I)*B)*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^3*d*sqrt(2))+1/32*((77+75*I)*A+(-30+28*I)*B)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))+(-1/32+1/32*I)*((1+76*I)*A+(-29-I)*B)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^3*d*sqrt(2))+15/8*(5*I*A-2*B)/(a^3*d*sqrt(tan(c+d*x)))-7/24*(11*A+4*I*B)/(a^3*d*tan(c+d*x)^(3/2))+1/6*(A+I*B)/(d*tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^3)+1/4*(2*A+I*B)/(a*d*tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^2)+3/8*(5*A+2*I*B)/(d*tan(c+d*x)^(3/2)*(a^3+I*a^3*tan(c+d*x)))],

# Integrands of the form Tan[c+d x]^(m/2) (A+B Tan[c+d x]) (a+i a Tan[c+d x])^(n/2)

# n>0
[sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2)*(A+B*tan(c+d*x)),x,9,1/4*(-1)^(3/4)*(4*I*A+7*B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d+(1+I)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d+1/4*(4*A-I*B)*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d+1/2*B*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2)/d],
[sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,8,-(-1)^(3/4)*(2*A-I*B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d+(-1-I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d+B*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x))/sqrt(tan(c+d*x)),x,7,-2*(-1)^(3/4)*B*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d+(-1-I)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d],
[sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x))/tan(c+d*x)^(3/2),x,4,(1+I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d-2*A*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))],
[sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x))/tan(c+d*x)^(5/2),x,5,(1+I)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d-2/3*(I*A+3*B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/3*A*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x))/tan(c+d*x)^(7/2),x,6,(-1-I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d+2/15*(13*A-5*I*B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/5*A*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(5/2))-2/15*(I*A+5*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x))/tan(c+d*x)^(9/2),x,7,(1-I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)/d+2/105*(43*I*A+91*B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/7*A*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(7/2))-2/35*(I*A+7*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(5/2))+2/105*(31*A-7*I*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,10,1/8*(-1)^(3/4)*a^(3/2)*(22*I*A+23*B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(2+2*I)*a^(3/2)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+1/8*a*(10*A-9*I*B)*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d+1/12*a*(6*I*A+7*B)*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2)/d+1/3*I*a*B*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(5/2)/d],
[sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,9,-1/4*(-1)^(3/4)*a^(3/2)*(12*A-11*I*B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(-2-2*I)*a^(3/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+1/4*a*(4*I*A+5*B)*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d+1/2*I*a*B*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2)/d],
[(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x))/sqrt(tan(c+d*x)),x,8,-(-1)^(3/4)*a^(3/2)*(2*I*A+3*B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(-2-2*I)*a^(3/2)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+I*a*B*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(3/2),x,8,2*(-1)^(1/4)*a^(3/2)*B*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(2+2*I)*a^(3/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-2*a*A*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))],
[(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(5/2),x,5,(2+2*I)*a^(3/2)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-2/3*a*(4*I*A+3*B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/3*a*A*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(7/2),x,6,(-2-2*I)*a^(3/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+4/15*a*(9*A-10*I*B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/5*a*A*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(5/2))-2/15*a*(6*I*A+5*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(9/2),x,7,(2-2*I)*a^(3/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+4/105*a*(67*I*A+63*B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/7*a*A*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(7/2))-2/35*a*(8*I*A+7*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(5/2))+4/105*a*(19*A-21*I*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(11/2),x,8,(2+2*I)*a^(3/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-4/315*a*(193*A-201*I*B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/9*a*A*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(9/2))-2/63*a*(10*I*A+9*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(7/2))+4/105*a*(11*A-12*I*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(5/2))+4/315*a*(61*I*A+57*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,11,3/64*(-1)^(3/4)*a^(5/2)*(120*I*A+121*B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(4+4*I)*a^(5/2)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+1/64*a^2*(152*A-149*I*B)*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d+1/96*a^2*(104*I*A+107*B)*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2)/d-1/24*a^2*(8*A-11*I*B)*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(5/2)/d+1/4*I*a*B*tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(3/2)/d],
[sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,10,-1/8*(-1)^(3/4)*a^(5/2)*(46*A-45*I*B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(-4-4*I)*a^(5/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+1/8*a^2*(18*I*A+19*B)*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d-1/4*a^2*(2*A-3*I*B)*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2)/d+1/3*I*a*B*tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2)/d],
[(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/sqrt(tan(c+d*x)),x,9,-1/4*(-1)^(3/4)*a^(5/2)*(20*I*A+23*B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(4-4*I)*a^(5/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-1/4*a^2*(4*A-7*I*B)*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d+1/2*I*a*B*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^(3/2)/d],
[(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(3/2),x,9,(-1)^(3/4)*a^(5/2)*(2*A-5*I*B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(4+4*I)*a^(5/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+a^2*(2*I*A-B)*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d-2*a*A*(a+I*a*tan(c+d*x))^(3/2)/(d*sqrt(tan(c+d*x)))],
[(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(5/2),x,9,2*(-1)^(3/4)*a^(5/2)*B*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(4+4*I)*a^(5/2)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-2*a^2*(2*I*A+B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/3*a*A*(a+I*a*tan(c+d*x))^(3/2)/(d*tan(c+d*x)^(3/2))],
[(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(7/2),x,6,(-4-4*I)*a^(5/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+2/15*a^2*(38*A-35*I*B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/15*a^2*(8*I*A+5*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))-2/5*a*A*(a+I*a*tan(c+d*x))^(3/2)/(d*tan(c+d*x)^(5/2))],
[(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(9/2),x,7,(4-4*I)*a^(5/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+4/105*a^2*(130*I*A+133*B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/35*a^2*(10*I*A+7*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(5/2))+2/105*a^2*(80*A-77*I*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))-2/7*a*A*(a+I*a*tan(c+d*x))^(3/2)/(d*tan(c+d*x)^(7/2))],
[(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(11/2),x,8,(4+4*I)*a^(5/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-8/315*a^2*(197*A-195*I*B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/21*a^2*(4*I*A+3*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(7/2))+2/105*a^2*(46*A-45*I*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(5/2))+8/315*a^2*(59*I*A+60*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))-2/9*a*A*(a+I*a*tan(c+d*x))^(3/2)/(d*tan(c+d*x)^(9/2))],
[(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(13/2),x,9,(4+4*I)*a^(5/2)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-8/3465*a^2*(2155*I*A+2167*B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/99*a^2*(14*I*A+11*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(9/2))+2/693*a^2*(212*A-209*I*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(7/2))+4/1155*a^2*(250*I*A+253*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(5/2))-8/3465*a^2*(655*A-649*I*B)*sqrt(a+I*a*tan(c+d*x))/(d*tan(c+d*x)^(3/2))-2/11*a*A*(a+I*a*tan(c+d*x))^(3/2)/(d*tan(c+d*x)^(11/2))],
[(a+I*a*tan(c+d*x))^(5/2)*(3/2*b*B/a+B*tan(c+d*x))/tan(c+d*x)^(5/2),x,9,2*(-1)^(3/4)*a^(5/2)*B*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d+(2+2*I)*a^(3/2)*(2*a+3*I*b)*B*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/d-2*a*(a+3*I*b)*B*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-b*B*(a+I*a*tan(c+d*x))^(3/2)/(d*tan(c+d*x)^(3/2))],

# n<0
[tan(c+d*x)^(3/2)*(A+B*tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)),x,9,(-1)^(3/4)*(2*I*A-B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+(-1/2+1/2*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))-(A+2*I*B)*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a*d)+(I*A-B)*tan(c+d*x)^(3/2)/(d*sqrt(a+I*a*tan(c+d*x)))],
[sqrt(tan(c+d*x))*(A+B*tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)),x,8,-2*(-1)^(1/4)*B*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+(-1/2-1/2*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+(I*A-B)*sqrt(tan(c+d*x))/(d*sqrt(a+I*a*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))),x,4,(1/2-1/2*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+(A+I*B)*sqrt(tan(c+d*x))/(d*sqrt(a+I*a*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2)),x,5,(1/2+1/2*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+(A+I*B)/(d*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-(3*A+I*B)*sqrt(a+I*a*tan(c+d*x))/(a*d*sqrt(tan(c+d*x)))],
[(A+B*tan(c+d*x))/(sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(5/2)),x,6,(1/2+1/2*I)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+1/3*(7*I*A-9*B)*sqrt(a+I*a*tan(c+d*x))/(a*d*sqrt(tan(c+d*x)))+(A+I*B)/(d*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2))-1/3*(5*A+3*I*B)*sqrt(a+I*a*tan(c+d*x))/(a*d*tan(c+d*x)^(3/2))],
[(A+B*tan(c+d*x))/(sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(7/2)),x,7,(-1/2-1/2*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(d*sqrt(a))+1/15*(61*A+35*I*B)*sqrt(a+I*a*tan(c+d*x))/(a*d*sqrt(tan(c+d*x)))+(A+I*B)/(d*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(5/2))-1/5*(7*A+5*I*B)*sqrt(a+I*a*tan(c+d*x))/(a*d*tan(c+d*x)^(5/2))+1/15*(23*I*A-25*B)*sqrt(a+I*a*tan(c+d*x))/(a*d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(3/2),x,9,2*(-1)^(3/4)*B*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(3/2)*d)+(-1/4+1/4*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(3/2)*d)+1/2*(A+3*I*B)*sqrt(tan(c+d*x))/(a*d*sqrt(a+I*a*tan(c+d*x)))+1/3*(I*A-B)*tan(c+d*x)^(3/2)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sqrt(tan(c+d*x))*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(3/2),x,5,(-1/4-1/4*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(3/2)*d)+1/6*(I*A+5*B)*sqrt(tan(c+d*x))/(a*d*sqrt(a+I*a*tan(c+d*x)))+1/3*(I*A-B)*sqrt(tan(c+d*x))/(d*(a+I*a*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^(3/2)),x,5,(1/4-1/4*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(3/2)*d)+1/6*(7*A+I*B)*sqrt(tan(c+d*x))/(a*d*sqrt(a+I*a*tan(c+d*x)))+1/3*(A+I*B)*sqrt(tan(c+d*x))/(d*(a+I*a*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2)),x,6,(1/4+1/4*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(3/2)*d)+1/6*(11*A+5*I*B)/(a*d*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-1/6*(25*A+7*I*B)*sqrt(a+I*a*tan(c+d*x))/(a^2*d*sqrt(tan(c+d*x)))+1/3*(A+I*B)/(d*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(3/2)),x,7,(1/4+1/4*I)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(3/2)*d)+1/6*(39*I*A-25*B)*sqrt(a+I*a*tan(c+d*x))/(a^2*d*sqrt(tan(c+d*x)))+1/2*(5*A+3*I*B)/(a*d*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2))-1/6*(21*A+11*I*B)*sqrt(a+I*a*tan(c+d*x))/(a^2*d*tan(c+d*x)^(3/2))+1/3*(A+I*B)/(d*tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(5/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,10,2*(-1)^(1/4)*B*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)+(1/8+1/8*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)-1/4*(I*A-7*B)*sqrt(tan(c+d*x))/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/5*(I*A-B)*tan(c+d*x)^(5/2)/(d*(a+I*a*tan(c+d*x))^(5/2))+1/6*(A+3*I*B)*tan(c+d*x)^(3/2)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,6,(-1/8+1/8*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)+1/60*(13*A-37*I*B)*sqrt(tan(c+d*x))/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/5*(I*A-B)*tan(c+d*x)^(3/2)/(d*(a+I*a*tan(c+d*x))^(5/2))+1/30*(A+11*I*B)*sqrt(tan(c+d*x))/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[sqrt(tan(c+d*x))*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,6,(-1/8-1/8*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)-1/60*(3*I*A-13*B)*sqrt(tan(c+d*x))/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/5*(I*A-B)*sqrt(tan(c+d*x))/(d*(a+I*a*tan(c+d*x))^(5/2))+1/30*(3*I*A+7*B)*sqrt(tan(c+d*x))/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^(5/2)),x,6,(1/8-1/8*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)+1/60*(67*A-3*I*B)*sqrt(tan(c+d*x))/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/5*(A+I*B)*sqrt(tan(c+d*x))/(d*(a+I*a*tan(c+d*x))^(5/2))+1/30*(13*A+3*I*B)*sqrt(tan(c+d*x))/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(5/2)),x,7,(1/8+1/8*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)+1/60*(151*A+41*I*B)/(a^2*d*sqrt(tan(c+d*x))*sqrt(a+I*a*tan(c+d*x)))-1/60*(317*A+67*I*B)*sqrt(a+I*a*tan(c+d*x))/(a^3*d*sqrt(tan(c+d*x)))+1/5*(A+I*B)/(d*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^(5/2))+1/30*(17*A+7*I*B)/(a*d*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(5/2)),x,8,(1/8+1/8*I)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))/(a^(5/2)*d)+1/60*(707*I*A-317*B)*sqrt(a+I*a*tan(c+d*x))/(a^3*d*sqrt(tan(c+d*x)))+1/20*(89*A+39*I*B)/(a^2*d*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(3/2))-1/60*(361*A+151*I*B)*sqrt(a+I*a*tan(c+d*x))/(a^3*d*tan(c+d*x)^(3/2))+1/5*(A+I*B)/(d*tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(5/2))+1/30*(21*A+11*I*B)/(a*d*tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2))],

# Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+i a Tan[c+d x])^(n/3)

# n>0
[(a+I*a*tan(c+d*x))^(1/3)*(A+B*tan(c+d*x)),x,6,-1/2*a^(1/3)*(A-I*B)*x/2^(2/3)+1/2*a^(1/3)*(I*A+B)*log(cos(c+d*x))/(2^(2/3)*d)+3/2*a^(1/3)*(I*A+B)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*d)-a^(1/3)*(I*A+B)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(2/3)*d)+3*B*(a+I*a*tan(c+d*x))^(1/3)/d],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))^(2/3)*(A+B*tan(c+d*x)),x,8,1/2*a^(2/3)*(A-I*B)*x/2^(1/3)-1/2*a^(2/3)*(I*A+B)*log(cos(c+d*x))/(2^(1/3)*d)-3/2*a^(2/3)*(I*A+B)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*d)-a^(2/3)*(I*A+B)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*d)-9/8*B*(a+I*a*tan(c+d*x))^(2/3)/d+3/8*B*tan(c+d*x)^2*(a+I*a*tan(c+d*x))^(2/3)/d-3/20*(4*I*A+B)*(a+I*a*tan(c+d*x))^(5/3)/(a*d)],
[tan(c+d*x)*(a+I*a*tan(c+d*x))^(2/3)*(A+B*tan(c+d*x)),x,7,1/2*a^(2/3)*(I*A+B)*x/2^(1/3)+1/2*a^(2/3)*(A-I*B)*log(cos(c+d*x))/(2^(1/3)*d)+3/2*a^(2/3)*(A-I*B)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*d)+a^(2/3)*(A-I*B)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*d)+3/2*A*(a+I*a*tan(c+d*x))^(2/3)/d-3/5*I*B*(a+I*a*tan(c+d*x))^(5/3)/(a*d)],
[(a+I*a*tan(c+d*x))^(2/3)*(A+B*tan(c+d*x)),x,6,-1/2*a^(2/3)*(A-I*B)*x/2^(1/3)+1/2*a^(2/3)*(I*A+B)*log(cos(c+d*x))/(2^(1/3)*d)+3/2*a^(2/3)*(I*A+B)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*d)+a^(2/3)*(I*A+B)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*d)+3/2*B*(a+I*a*tan(c+d*x))^(2/3)/d],
[cot(c+d*x)*(a+I*a*tan(c+d*x))^(2/3)*(A+B*tan(c+d*x)),x,11,-1/2*a^(2/3)*(I*A+B)*x/2^(1/3)-1/2*a^(2/3)*(A-I*B)*log(cos(c+d*x))/(2^(1/3)*d)-1/2*a^(2/3)*A*log(tan(c+d*x))/d+3/2*a^(2/3)*A*log(a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/d-3/2*a^(2/3)*(A-I*B)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*d)+a^(2/3)*A*arctan((a^(1/3)+2*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/d-a^(2/3)*(A-I*B)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*d)],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))^(2/3)*(A+B*tan(c+d*x)),x,12,1/2*a^(2/3)*(A-I*B)*x/2^(1/3)-1/2*a^(2/3)*(I*A+B)*log(cos(c+d*x))/(2^(1/3)*d)-1/6*a^(2/3)*(2*I*A+3*B)*log(tan(c+d*x))/d+1/2*a^(2/3)*(2*I*A+3*B)*log(a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/d-3/2*a^(2/3)*(I*A+B)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*d)+a^(2/3)*(2*I*A+3*B)*arctan((a^(1/3)+2*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))/(d*sqrt(3))-a^(2/3)*(I*A+B)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*d)-A*cot(c+d*x)*(a+I*a*tan(c+d*x))^(2/3)/d],

# n<0
[(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(1/3),x,6,-1/4*(A-I*B)*x/(2^(1/3)*a^(1/3))+1/4*(I*A+B)*log(cos(c+d*x))/(2^(1/3)*a^(1/3)*d)+3/4*(I*A+B)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(1/3)*a^(1/3)*d)+1/2*(I*A+B)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(1/3)*a^(1/3)*d)+3/2*(I*A-B)/(d*(a+I*a*tan(c+d*x))^(1/3))],
[(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(2/3),x,6,-1/4*(A-I*B)*x/(2^(2/3)*a^(2/3))+1/4*(I*A+B)*log(cos(c+d*x))/(2^(2/3)*a^(2/3)*d)+3/4*(I*A+B)*log(2^(1/3)*a^(1/3)-(a+I*a*tan(c+d*x))^(1/3))/(2^(2/3)*a^(2/3)*d)-1/2*(I*A+B)*arctan((a^(1/3)+2^(2/3)*(a+I*a*tan(c+d*x))^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/(2^(2/3)*a^(2/3)*d)+3/4*(I*A-B)/(d*(a+I*a*tan(c+d*x))^(2/3))],

# Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+i a Tan[c+d x])^n with m symbolic
[tan(c+d*x)^m*(a+I*a*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,7,-2*a^4*(A*(64+60*m+19*m^2+2*m^3)-I*B*(67+60*m+19*m^2+2*m^3))*tan(c+d*x)^(1+m)/(d*(3+m)*(4+m)*(2+3*m+m^2))+8*a^4*(A-I*B)*hypergeom([1,1+m],[2+m],I*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m))+I*a*B*tan(c+d*x)^(1+m)*(a+I*a*tan(c+d*x))^3/(d*(4+m))-(A*(4+m)-I*B*(7+m))*tan(c+d*x)^(1+m)*(a^2+I*a^2*tan(c+d*x))^2/(d*(3+m)*(4+m))-2*(A*(4+m)^2-I*B*(19+8*m+m^2))*tan(c+d*x)^(1+m)*(a^4+I*a^4*tan(c+d*x))/(d*(4+m)*(6+5*m+m^2))],
[tan(c+d*x)^m*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,6,-a^3*(A*(15+11*m+2*m^2)-I*B*(17+11*m+2*m^2))*tan(c+d*x)^(1+m)/(d*(3+m)*(2+3*m+m^2))+4*a^3*(A-I*B)*hypergeom([1,1+m],[2+m],I*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m))+I*a*B*tan(c+d*x)^(1+m)*(a+I*a*tan(c+d*x))^2/(d*(3+m))-(A*(3+m)-I*B*(5+m))*tan(c+d*x)^(1+m)*(a^3+I*a^3*tan(c+d*x))/(d*(2+m)*(3+m))],
[tan(c+d*x)^m*(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,5,I*a^2*(B+(I*A+B)*(2+m))*tan(c+d*x)^(1+m)/(d*(1+m)*(2+m))+2*a^2*(A-I*B)*hypergeom([1,1+m],[2+m],I*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m))+I*B*tan(c+d*x)^(1+m)*(a^2+I*a^2*tan(c+d*x))/(d*(2+m))],
[tan(c+d*x)^m*(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,3,I*a*B*tan(c+d*x)^(1+m)/(d*(1+m))+a*(A-I*B)*hypergeom([1,1+m],[2+m],I*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m))],
[tan(c+d*x)^m*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x)),x,6,1/2*(A*(1-m)-I*B*(1+m))*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(c+d*x)^2)*tan(c+d*x)^(1+m)/(a*d*(1+m))+1/2*(I*A-B)*m*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-tan(c+d*x)^2)*tan(c+d*x)^(2+m)/(a*d*(2+m))+1/2*(A+I*B)*tan(c+d*x)^(1+m)/(d*(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^m*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^2,x,7,1/4*(1-m)*(A*(1-m)-I*B*(1+m))*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(c+d*x)^2)*tan(c+d*x)^(1+m)/(a^2*d*(1+m))+1/4*(A*(2-m)-I*B*m)*tan(c+d*x)^(1+m)/(a^2*d*(1+I*tan(c+d*x)))+1/4*m*(I*A*(2-m)+B*m)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-tan(c+d*x)^2)*tan(c+d*x)^(2+m)/(a^2*d*(2+m))+1/4*(A+I*B)*tan(c+d*x)^(1+m)/(d*(a+I*a*tan(c+d*x))^2)],
[tan(c+d*x)^m*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,8,-1/24*(1-m)*(I*B*(3+m-2*m^2)-A*(3-7*m+2*m^2))*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(c+d*x)^2)*tan(c+d*x)^(1+m)/(a^3*d*(1+m))+1/24*(2-m)*m*(B+I*A*(5-2*m)+2*B*m)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-tan(c+d*x)^2)*tan(c+d*x)^(2+m)/(a^3*d*(2+m))+1/6*(A+I*B)*tan(c+d*x)^(1+m)/(d*(a+I*a*tan(c+d*x))^3)+1/24*(I*B*(1-2*m)+A*(7-2*m))*tan(c+d*x)^(1+m)/(a*d*(a+I*a*tan(c+d*x))^2)+1/24*(2-m)*(A*(5-2*m)-I*(B+2*B*m))*tan(c+d*x)^(1+m)/(d*(a^3+I*a^3*tan(c+d*x)))],
[tan(c+d*x)^m*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^4,x,9,-1/48*(3-4*m+m^2)*(I*B*(1-m^2)-A*(1-4*m+m^2))*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(c+d*x)^2)*tan(c+d*x)^(1+m)/(a^4*d*(1+m))-1/48*(I*B*(1+3*m-m^2)-A*(13-7*m+m^2))*tan(c+d*x)^(1+m)/(a^4*d*(1+I*tan(c+d*x))^2)-1/48*(2-m)*(I*B*(2+2*m-m^2)-A*(8-6*m+m^2))*tan(c+d*x)^(1+m)/(a^4*d*(1+I*tan(c+d*x)))+1/48*(2-m)*m*(B*(2+2*m-m^2)+I*A*(8-6*m+m^2))*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-tan(c+d*x)^2)*tan(c+d*x)^(2+m)/(a^4*d*(2+m))+1/8*(A+I*B)*tan(c+d*x)^(1+m)/(d*(a+I*a*tan(c+d*x))^4)+1/24*(I*B*(1-m)+A*(5-m))*tan(c+d*x)^(1+m)/(a*d*(a+I*a*tan(c+d*x))^3)],
[tan(c+d*x)^m*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,9,2*a^2*(2*B*(19+17*m+4*m^2)+I*A*(35+34*m+8*m^2))*hypergeom([1/2,-m],[3/2],1+I*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^m/(d*(3+2*m)*(5+2*m)*(-I*tan(c+d*x))^m)+4*a^3*(A-I*B)*AppellF1(1+m,1/2,1,2+m,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(a+I*a*tan(c+d*x)))+2*a^2*(2*I*B*(4+m)-A*(5+2*m))*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(3+2*m)*(5+2*m))+2*I*a*B*tan(c+d*x)^(1+m)*(a+I*a*tan(c+d*x))^(3/2)/(d*(5+2*m))],
[tan(c+d*x)^m*(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,8,2*a*(B+(I*A+B)*(3+2*m))*hypergeom([1/2,-m],[3/2],1+I*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^m/(d*(3+2*m)*(-I*tan(c+d*x))^m)+2*a^2*(A-I*B)*AppellF1(1+m,1/2,1,2+m,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(a+I*a*tan(c+d*x)))+2*I*a*B*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(3+2*m))],
[tan(c+d*x)^m*(a+I*a*tan(c+d*x))^(1/2)*(A+B*tan(c+d*x)),x,7,2*B*hypergeom([1/2,-m],[3/2],1+I*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^m/(d*(-I*tan(c+d*x))^m)+a*(A-I*B)*AppellF1(1+m,1/2,1,2+m,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^m*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(1/2),x,8,(I*A-B)*(1+2*m)*hypergeom([1/2,-m],[3/2],1+I*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^m/(a*d*(-I*tan(c+d*x))^m)+(A+I*B)*tan(c+d*x)^(1+m)/(d*sqrt(a+I*a*tan(c+d*x)))+1/2*(A-I*B)*AppellF1(1+m,1/2,1,2+m,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(a+I*a*tan(c+d*x)))],
[tan(c+d*x)^m*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(3/2),x,9,1/6*(1+2*m)*(B+I*A*(5-4*m)+4*B*m)*hypergeom([1/2,-m],[3/2],1+I*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^m/(a^2*d*(-I*tan(c+d*x))^m)+1/6*(A*(5-4*m)-I*(B+4*B*m))*tan(c+d*x)^(1+m)/(a*d*sqrt(a+I*a*tan(c+d*x)))+1/4*(A-I*B)*AppellF1(1+m,1/2,1,2+m,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(1+m)/(a*d*(1+m)*sqrt(a+I*a*tan(c+d*x)))+1/3*(A+I*B)*tan(c+d*x)^(1+m)/(d*(a+I*a*tan(c+d*x))^(3/2))],
[tan(c+d*x)^m*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,10,1/60*(1+2*m)*(B*(13+12*m-16*m^2)+I*A*(37-52*m+16*m^2))*hypergeom([1/2,-m],[3/2],1+I*tan(c+d*x))*sqrt(a+I*a*tan(c+d*x))*tan(c+d*x)^m/(a^3*d*(-I*tan(c+d*x))^m)-1/60*(I*B*(13+12*m-16*m^2)-A*(37-52*m+16*m^2))*tan(c+d*x)^(1+m)/(a^2*d*sqrt(a+I*a*tan(c+d*x)))+1/8*(A-I*B)*AppellF1(1+m,1/2,1,2+m,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(1+I*tan(c+d*x))*tan(c+d*x)^(1+m)/(a^2*d*(1+m)*sqrt(a+I*a*tan(c+d*x)))+1/5*(A+I*B)*tan(c+d*x)^(1+m)/(d*(a+I*a*tan(c+d*x))^(5/2))+1/30*(I*B*(1-4*m)+A*(11-4*m))*tan(c+d*x)^(1+m)/(a*d*(a+I*a*tan(c+d*x))^(3/2))],

# Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+i a Tan[c+d x])^n with n symbolic
[tan(c+d*x)^m*(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,7,(A-I*B)*AppellF1(1+m,1-n,1,2+m,-I*tan(c+d*x),I*tan(c+d*x))*tan(c+d*x)^(1+m)*(a+I*a*tan(c+d*x))^n/(d*(1+m)*(1+I*tan(c+d*x))^n)+I*B*hypergeom([1+m,1-n],[2+m],-I*tan(c+d*x))*tan(c+d*x)^(1+m)*(a+I*a*tan(c+d*x))^n/(d*(1+m)*(1+I*tan(c+d*x))^n)],
[tan(c+d*x)^3*(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,6,2*(I*B*n-A*(3+n))*(a+I*a*tan(c+d*x))^n/(d*n*(2+n)*(3+n))+1/2*(A-I*B)*hypergeom([1,n],[1+n],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^n/(d*n)-(I*B*n-A*(3+n))*tan(c+d*x)^2*(a+I*a*tan(c+d*x))^n/(d*(2+n)*(3+n))+B*tan(c+d*x)^3*(a+I*a*tan(c+d*x))^n/(d*(3+n))-(A*n*(3+n)-I*B*(6+3*n+n^2))*(a+I*a*tan(c+d*x))^(1+n)/(a*d*(3+n)*(2+3*n+n^2))],
[tan(c+d*x)^2*(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,5,-2*B*(a+I*a*tan(c+d*x))^n/(d*n*(2+n))+1/2*(I*A+B)*hypergeom([1,n],[1+n],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^n/(d*n)+B*tan(c+d*x)^2*(a+I*a*tan(c+d*x))^n/(d*(2+n))-(B*n+I*A*(2+n))*(a+I*a*tan(c+d*x))^(1+n)/(a*d*(1+n)*(2+n))],
[tan(c+d*x)*(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,4,A*(a+I*a*tan(c+d*x))^n/(d*n)-1/2*(A-I*B)*hypergeom([1,n],[1+n],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^n/(d*n)-I*B*(a+I*a*tan(c+d*x))^(1+n)/(a*d*(1+n))],
[(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,3,B*(a+I*a*tan(c+d*x))^n/(d*n)-1/2*(I*A+B)*hypergeom([1,n],[1+n],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^n/(d*n)],
[cot(c+d*x)*(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,5,1/2*(A-I*B)*hypergeom([1,n],[1+n],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^n/(d*n)-A*hypergeom([1,n],[1+n],1+I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*n)],
[cot(c+d*x)^2*(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,6,-A*cot(c+d*x)*(a+I*a*tan(c+d*x))^n/d+1/2*(I*A+B)*hypergeom([1,n],[1+n],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^n/(d*n)-(B+I*A*n)*hypergeom([1,n],[1+n],1+I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*n)],
[cot(c+d*x)^3*(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,7,-1/2*(2*B+I*A*n)*cot(c+d*x)*(a+I*a*tan(c+d*x))^n/d-1/2*A*cot(c+d*x)^2*(a+I*a*tan(c+d*x))^n/d-1/2*(A-I*B)*hypergeom([1,n],[1+n],1/2*(1+I*tan(c+d*x)))*(a+I*a*tan(c+d*x))^n/(d*n)-1/2*(2*I*B*n-A*(2-n+n^2))*hypergeom([1,n],[1+n],1+I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*n)],
[tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,11,-2*(2*I*A*n*(5+2*n)+B*(15+10*n+4*n^2))*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(5+2*n)*(3+8*n+4*n^2))+2*(I*A+B)*AppellF1(1/2,1-n,1,3/2,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(1+I*tan(c+d*x))^n)-2*(4*B*n*(9+8*n+2*n^2)+I*A*(15+36*n+32*n^2+8*n^3))*hypergeom([1/2,1-n],[3/2],-I*tan(c+d*x))*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(5+2*n)*(3+8*n+4*n^2)*(1+I*tan(c+d*x))^n)-2*(2*I*B*n-A*(5+2*n))*tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^n/(d*(3+2*n)*(5+2*n))+2*B*tan(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^n/(d*(5+2*n))],
[tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,10,-2*(2*I*B*n-A*(3+2*n))*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(3+8*n+4*n^2))-2*(A-I*B)*AppellF1(1/2,1-n,1,3/2,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(1+I*tan(c+d*x))^n)+2*(2*A*n*(3+2*n)-I*B*(3+6*n+4*n^2))*hypergeom([1/2,1-n],[3/2],-I*tan(c+d*x))*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(3+8*n+4*n^2)*(1+I*tan(c+d*x))^n)+2*B*tan(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^n/(d*(3+2*n))],
[tan(c+d*x)^(1/2)*(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,9,2*B*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(1+2*n))-2*(I*A+B)*AppellF1(1/2,1-n,1,3/2,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(1+I*tan(c+d*x))^n)+2*(2*B*n+I*A*(1+2*n))*hypergeom([1/2,1-n],[3/2],-I*tan(c+d*x))*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(1+2*n)*(1+I*tan(c+d*x))^n)],
[(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x))/tan(c+d*x)^(1/2),x,8,2*(A-I*B)*AppellF1(1/2,1-n,1,3/2,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(1+I*tan(c+d*x))^n)+2*I*B*hypergeom([1/2,1-n],[3/2],-I*tan(c+d*x))*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(1+I*tan(c+d*x))^n)],
[(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x))/tan(c+d*x)^(3/2),x,9,-2*A*(a+I*a*tan(c+d*x))^n/(d*sqrt(tan(c+d*x)))+2*(I*A+B)*AppellF1(1/2,1-n,1,3/2,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(1+I*tan(c+d*x))^n)-2*I*A*(1-2*n)*hypergeom([1/2,1-n],[3/2],-I*tan(c+d*x))*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(1+I*tan(c+d*x))^n)],
[(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x))/tan(c+d*x)^(5/2),x,10,-2/3*(3*B+2*I*A*n)*(a+I*a*tan(c+d*x))^n/(d*sqrt(tan(c+d*x)))-2*(A-I*B)*AppellF1(1/2,1-n,1,3/2,-I*tan(c+d*x),I*tan(c+d*x))*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(1+I*tan(c+d*x))^n)-2/3*(1-2*n)*(3*I*B-2*A*n)*hypergeom([1/2,1-n],[3/2],-I*tan(c+d*x))*sqrt(tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(1+I*tan(c+d*x))^n)-2/3*A*(a+I*a*tan(c+d*x))^n/(d*tan(c+d*x)^(3/2))],

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

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

# n>0
[tan(c+d*x)^2*(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,4,-(a*A-b*B)*x+(A*b+a*B)*log(cos(c+d*x))/d+(a*A-b*B)*tan(c+d*x)/d+1/2*(A*b+a*B)*tan(c+d*x)^2/d+1/3*b*B*tan(c+d*x)^3/d],
[tan(c+d*x)*(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,3,-(A*b+a*B)*x-(a*A-b*B)*log(cos(c+d*x))/d+(A*b+a*B)*tan(c+d*x)/d+1/2*b*B*tan(c+d*x)^2/d],
[(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,2,(a*A-b*B)*x-(A*b+a*B)*log(cos(c+d*x))/d+b*B*tan(c+d*x)/d],
[cot(c+d*x)*(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,4,(A*b+a*B)*x-b*B*log(cos(c+d*x))/d+a*A*log(sin(c+d*x))/d],
[cot(c+d*x)^2*(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,3,-(a*A-b*B)*x-a*A*cot(c+d*x)/d+(A*b+a*B)*log(sin(c+d*x))/d],
[cot(c+d*x)^3*(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,4,-(A*b+a*B)*x-(A*b+a*B)*cot(c+d*x)/d-1/2*a*A*cot(c+d*x)^2/d-(a*A-b*B)*log(sin(c+d*x))/d],
[cot(c+d*x)^4*(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,5,(a*A-b*B)*x+(a*A-b*B)*cot(c+d*x)/d-1/2*(A*b+a*B)*cot(c+d*x)^2/d-1/3*a*A*cot(c+d*x)^3/d-(A*b+a*B)*log(sin(c+d*x))/d],
[cot(c+d*x)^5*(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,6,(A*b+a*B)*x+(A*b+a*B)*cot(c+d*x)/d+1/2*(a*A-b*B)*cot(c+d*x)^2/d-1/3*(A*b+a*B)*cot(c+d*x)^3/d-1/4*a*A*cot(c+d*x)^4/d+(a*A-b*B)*log(sin(c+d*x))/d],
[tan(c+d*x)^2*(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,5,-(a^2*A-A*b^2-2*a*b*B)*x+(2*a*A*b+a^2*B-b^2*B)*log(cos(c+d*x))/d-b*(A*b+a*B)*tan(c+d*x)/d-1/2*B*(a+b*tan(c+d*x))^2/d+1/12*(4*A*b-a*B)*(a+b*tan(c+d*x))^3/(b^2*d)+1/4*B*tan(c+d*x)*(a+b*tan(c+d*x))^3/(b*d)],
[tan(c+d*x)*(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,4,-(2*a*A*b+a^2*B-b^2*B)*x-(a^2*A-A*b^2-2*a*b*B)*log(cos(c+d*x))/d+b*(a*A-b*B)*tan(c+d*x)/d+1/2*A*(a+b*tan(c+d*x))^2/d+1/3*B*(a+b*tan(c+d*x))^3/(b*d)],
[(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,3,(a^2*A-A*b^2-2*a*b*B)*x-(2*a*A*b+a^2*B-b^2*B)*log(cos(c+d*x))/d+b*(A*b+a*B)*tan(c+d*x)/d+1/2*B*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)*(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,4,(2*a*A*b+a^2*B-b^2*B)*x-b*(A*b+2*a*B)*log(cos(c+d*x))/d+a^2*A*log(sin(c+d*x))/d+b^2*B*tan(c+d*x)/d],
[cot(c+d*x)^2*(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,4,-(a^2*A-A*b^2-2*a*b*B)*x-a^2*A*cot(c+d*x)/d-b^2*B*log(cos(c+d*x))/d+a*(2*A*b+a*B)*log(sin(c+d*x))/d],
[cot(c+d*x)^3*(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,4,(b^2*B-a*(2*A*b+a*B))*x-a*(2*A*b+a*B)*cot(c+d*x)/d-1/2*a^2*A*cot(c+d*x)^2/d-(a^2*A-A*b^2-2*a*b*B)*log(sin(c+d*x))/d],
[cot(c+d*x)^4*(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,5,(a^2*A-A*b^2-2*a*b*B)*x+(a^2*A-A*b^2-2*a*b*B)*cot(c+d*x)/d-1/2*a*(2*A*b+a*B)*cot(c+d*x)^2/d-1/3*a^2*A*cot(c+d*x)^3/d+(b^2*B-a*(2*A*b+a*B))*log(sin(c+d*x))/d],
[cot(c+d*x)^5*(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,6,(2*a*A*b+a^2*B-b^2*B)*x-(b^2*B-a*(2*A*b+a*B))*cot(c+d*x)/d+1/2*(a^2*A-A*b^2-2*a*b*B)*cot(c+d*x)^2/d-1/3*a*(2*A*b+a*B)*cot(c+d*x)^3/d-1/4*a^2*A*cot(c+d*x)^4/d+(a^2*A-A*b^2-2*a*b*B)*log(sin(c+d*x))/d],
[tan(c+d*x)^2*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,6,-(a^3*A-3*a*A*b^2-3*a^2*b*B+b^3*B)*x+(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*log(cos(c+d*x))/d-b*(2*a*A*b+a^2*B-b^2*B)*tan(c+d*x)/d-1/2*(A*b+a*B)*(a+b*tan(c+d*x))^2/d-1/3*B*(a+b*tan(c+d*x))^3/d+1/20*(5*A*b-a*B)*(a+b*tan(c+d*x))^4/(b^2*d)+1/5*B*tan(c+d*x)*(a+b*tan(c+d*x))^4/(b*d)],
[tan(c+d*x)*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,5,-(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*x-(a^3*A-3*a*A*b^2-3*a^2*b*B+b^3*B)*log(cos(c+d*x))/d+b*(a^2*A-A*b^2-2*a*b*B)*tan(c+d*x)/d+1/2*(a*A-b*B)*(a+b*tan(c+d*x))^2/d+1/3*A*(a+b*tan(c+d*x))^3/d+1/4*B*(a+b*tan(c+d*x))^4/(b*d)],
[(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,4,(a^3*A-3*a*A*b^2-3*a^2*b*B+b^3*B)*x-(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*log(cos(c+d*x))/d+b*(2*a*A*b+a^2*B-b^2*B)*tan(c+d*x)/d+1/2*(A*b+a*B)*(a+b*tan(c+d*x))^2/d+1/3*B*(a+b*tan(c+d*x))^3/d],
[cot(c+d*x)*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,5,(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*x-b*(3*a*A*b+3*a^2*B-b^2*B)*log(cos(c+d*x))/d+a^3*A*log(sin(c+d*x))/d+b^2*(A*b+2*a*B)*tan(c+d*x)/d+1/2*b*B*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^2*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,5,-(a^3*A-3*a*A*b^2-3*a^2*b*B+b^3*B)*x-b^2*(A*b+3*a*B)*log(cos(c+d*x))/d+a^2*(3*A*b+a*B)*log(sin(c+d*x))/d+b^2*(a*A+b*B)*tan(c+d*x)/d-a*A*cot(c+d*x)*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^3*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,5,-(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*x-a^2*(2*A*b+a*B)*cot(c+d*x)/d-b^3*B*log(cos(c+d*x))/d-a*(a^2*A-3*A*b^2-3*a*b*B)*log(sin(c+d*x))/d-1/2*a*A*cot(c+d*x)^2*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^4*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,5,(a^3*A-3*a*A*b^2-3*a^2*b*B+b^3*B)*x+1/3*a*(3*a^2*A-8*A*b^2-9*a*b*B)*cot(c+d*x)/d-1/6*a^2*(5*A*b+3*a*B)*cot(c+d*x)^2/d-(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*log(sin(c+d*x))/d-1/3*a*A*cot(c+d*x)^3*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^5*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,6,(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*x+(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*cot(c+d*x)/d+1/4*a*(2*a^2*A-5*A*b^2-6*a*b*B)*cot(c+d*x)^2/d-1/6*a^2*(3*A*b+2*a*B)*cot(c+d*x)^3/d+(a^3*A-3*a*A*b^2-3*a^2*b*B+b^3*B)*log(sin(c+d*x))/d-1/4*a*A*cot(c+d*x)^4*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^6*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,7,-(a^3*A-3*a*A*b^2-3*a^2*b*B+b^3*B)*x-(a^3*A-3*a*A*b^2-3*a^2*b*B+b^3*B)*cot(c+d*x)/d+1/2*(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*cot(c+d*x)^2/d+1/15*a*(5*a^2*A-12*A*b^2-15*a*b*B)*cot(c+d*x)^3/d-1/20*a^2*(7*A*b+5*a*B)*cot(c+d*x)^4/d+(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*log(sin(c+d*x))/d-1/5*a*A*cot(c+d*x)^5*(a+b*tan(c+d*x))^2/d],
[tan(c+d*x)^2*(a+b*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,7,-(a^4*A-6*a^2*A*b^2+A*b^4-4*a^3*b*B+4*a*b^3*B)*x+(4*a^3*A*b-4*a*A*b^3+a^4*B-6*a^2*b^2*B+b^4*B)*log(cos(c+d*x))/d-b*(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*tan(c+d*x)/d-1/2*(2*a*A*b+a^2*B-b^2*B)*(a+b*tan(c+d*x))^2/d-1/3*(A*b+a*B)*(a+b*tan(c+d*x))^3/d-1/4*B*(a+b*tan(c+d*x))^4/d+1/30*(6*A*b-a*B)*(a+b*tan(c+d*x))^5/(b^2*d)+1/6*B*tan(c+d*x)*(a+b*tan(c+d*x))^5/(b*d)],
[tan(c+d*x)*(a+b*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,6,-(4*a^3*A*b-4*a*A*b^3+a^4*B-6*a^2*b^2*B+b^4*B)*x-(a^4*A-6*a^2*A*b^2+A*b^4-4*a^3*b*B+4*a*b^3*B)*log(cos(c+d*x))/d+b*(a^3*A-3*a*A*b^2-3*a^2*b*B+b^3*B)*tan(c+d*x)/d+1/2*(a^2*A-A*b^2-2*a*b*B)*(a+b*tan(c+d*x))^2/d+1/3*(a*A-b*B)*(a+b*tan(c+d*x))^3/d+1/4*A*(a+b*tan(c+d*x))^4/d+1/5*B*(a+b*tan(c+d*x))^5/(b*d)],
[(a+b*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,5,(a^4*A-6*a^2*A*b^2+A*b^4-4*a^3*b*B+4*a*b^3*B)*x-(4*a^3*A*b-4*a*A*b^3+a^4*B-6*a^2*b^2*B+b^4*B)*log(cos(c+d*x))/d+b*(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*tan(c+d*x)/d+1/2*(2*a*A*b+a^2*B-b^2*B)*(a+b*tan(c+d*x))^2/d+1/3*(A*b+a*B)*(a+b*tan(c+d*x))^3/d+1/4*B*(a+b*tan(c+d*x))^4/d],
[cot(c+d*x)*(a+b*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,6,(4*a^3*A*b-4*a*A*b^3+a^4*B-6*a^2*b^2*B+b^4*B)*x-b*(6*a^2*A*b-A*b^3+4*a^3*B-4*a*b^2*B)*log(cos(c+d*x))/d+a^4*A*log(sin(c+d*x))/d+b^2*(3*a*A*b+3*a^2*B-b^2*B)*tan(c+d*x)/d+1/2*b*(A*b+2*a*B)*(a+b*tan(c+d*x))^2/d+1/3*b*B*(a+b*tan(c+d*x))^3/d],
[cot(c+d*x)^2*(a+b*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,6,-(a^4*A-6*a^2*A*b^2+A*b^4-4*a^3*b*B+4*a*b^3*B)*x-b^2*(4*a*A*b+6*a^2*B-b^2*B)*log(cos(c+d*x))/d+a^3*(4*A*b+a*B)*log(sin(c+d*x))/d+b^2*(a^2*A+A*b^2+3*a*b*B)*tan(c+d*x)/d+1/2*b*(2*a*A+b*B)*(a+b*tan(c+d*x))^2/d-a*A*cot(c+d*x)*(a+b*tan(c+d*x))^3/d],
[cot(c+d*x)^3*(a+b*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,6,-(4*a^3*A*b-4*a*A*b^3+a^4*B-6*a^2*b^2*B+b^4*B)*x-b^3*(A*b+4*a*B)*log(cos(c+d*x))/d-a^2*(a^2*A-6*A*b^2-4*a*b*B)*log(sin(c+d*x))/d+b^2*(3*a*A*b+a^2*B+b^2*B)*tan(c+d*x)/d-1/2*a*(5*A*b+2*a*B)*cot(c+d*x)*(a+b*tan(c+d*x))^2/d-1/2*a*A*cot(c+d*x)^2*(a+b*tan(c+d*x))^3/d],
[cot(c+d*x)^4*(a+b*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,6,(a^4*A-6*a^2*A*b^2+A*b^4-4*a^3*b*B+4*a*b^3*B)*x+a^2*(a^2*A-3*A*b^2-3*a*b*B)*cot(c+d*x)/d-b^4*B*log(cos(c+d*x))/d-a*(4*a^2*A*b-4*A*b^3+a^3*B-6*a*b^2*B)*log(sin(c+d*x))/d-1/2*a*(2*A*b+a*B)*cot(c+d*x)^2*(a+b*tan(c+d*x))^2/d-1/3*a*A*cot(c+d*x)^3*(a+b*tan(c+d*x))^3/d],
[cot(c+d*x)^5*(a+b*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,6,(4*a^3*A*b-4*a*A*b^3+a^4*B-6*a^2*b^2*B+b^4*B)*x+1/6*a*(24*a^2*A*b-19*A*b^3+6*a^3*B-34*a*b^2*B)*cot(c+d*x)/d+1/12*a^2*(6*a^2*A-13*A*b^2-16*a*b*B)*cot(c+d*x)^2/d+(a^4*A-6*a^2*A*b^2+A*b^4-4*a^3*b*B+4*a*b^3*B)*log(sin(c+d*x))/d-1/12*a*(7*A*b+4*a*B)*cot(c+d*x)^3*(a+b*tan(c+d*x))^2/d-1/4*a*A*cot(c+d*x)^4*(a+b*tan(c+d*x))^3/d],
[cot(c+d*x)^6*(a+b*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,7,-(a^4*A-6*a^2*A*b^2+A*b^4-4*a^3*b*B+4*a*b^3*B)*x-(a^4*A-6*a^2*A*b^2+A*b^4-4*a^3*b*B+4*a*b^3*B)*cot(c+d*x)/d+1/20*a*(40*a^2*A*b-28*A*b^3+10*a^3*B-55*a*b^2*B)*cot(c+d*x)^2/d+1/30*a^2*(10*a^2*A-18*A*b^2-25*a*b*B)*cot(c+d*x)^3/d+(4*a^3*A*b-4*a*A*b^3+a^4*B-6*a^2*b^2*B+b^4*B)*log(sin(c+d*x))/d-1/20*a*(8*A*b+5*a*B)*cot(c+d*x)^4*(a+b*tan(c+d*x))^2/d-1/5*a*A*cot(c+d*x)^5*(a+b*tan(c+d*x))^3/d],
[cot(c+d*x)^7*(a+b*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,8,-(4*a^3*A*b-4*a*A*b^3+a^4*B-6*a^2*b^2*B+b^4*B)*x-(4*a^3*A*b-4*a*A*b^3+a^4*B-6*a^2*b^2*B+b^4*B)*cot(c+d*x)/d-1/2*(a^4*A-6*a^2*A*b^2+A*b^4-4*a^3*b*B+4*a*b^3*B)*cot(c+d*x)^2/d+1/15*a*(20*a^2*A*b-13*A*b^3+5*a^3*B-27*a*b^2*B)*cot(c+d*x)^3/d+1/20*a^2*(5*a^2*A-8*A*b^2-12*a*b*B)*cot(c+d*x)^4/d-(a^4*A-6*a^2*A*b^2+A*b^4-4*a^3*b*B+4*a*b^3*B)*log(sin(c+d*x))/d-1/10*a*(3*A*b+2*a*B)*cot(c+d*x)^5*(a+b*tan(c+d*x))^2/d-1/6*a*A*cot(c+d*x)^6*(a+b*tan(c+d*x))^3/d],

# n<0
[tan(c+d*x)^3*(A+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,6,-(A*b-a*B)*x/(a^2+b^2)+(a*A+b*B)*log(cos(c+d*x))/((a^2+b^2)*d)-a^3*(A*b-a*B)*log(a+b*tan(c+d*x))/(b^3*(a^2+b^2)*d)+(A*b-a*B)*tan(c+d*x)/(b^2*d)+1/2*B*tan(c+d*x)^2/(b*d)],
[tan(c+d*x)^2*(A+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,5,-(a*A+b*B)*x/(a^2+b^2)-(A*b-a*B)*log(cos(c+d*x))/((a^2+b^2)*d)+a^2*(A*b-a*B)*log(a+b*tan(c+d*x))/(b^2*(a^2+b^2)*d)+B*tan(c+d*x)/(b*d)],
[tan(c+d*x)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,5,(A*b-a*B)*x/(a^2+b^2)-B*log(cos(c+d*x))/(b*d)-a*(A*b-a*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/(b*(a^2+b^2)*d)],
[(A+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,2,(a*A+b*B)*x/(a^2+b^2)+(A*b-a*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)*d)],
[cot(c+d*x)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,3,-(A*b-a*B)*x/(a^2+b^2)+A*log(sin(c+d*x))/(a*d)-b*(A*b-a*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a*(a^2+b^2)*d)],
[cot(c+d*x)^2*(A+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,4,-(a*A+b*B)*x/(a^2+b^2)-A*cot(c+d*x)/(a*d)-(A*b-a*B)*log(sin(c+d*x))/(a^2*d)+b^2*(A*b-a*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^2*(a^2+b^2)*d)],
[cot(c+d*x)^3*(A+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,5,(A*b-a*B)*x/(a^2+b^2)+(A*b-a*B)*cot(c+d*x)/(a^2*d)-1/2*A*cot(c+d*x)^2/(a*d)-(a^2*A-A*b^2+a*b*B)*log(sin(c+d*x))/(a^3*d)-b^3*(A*b-a*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^3*(a^2+b^2)*d)],
[cot(c+d*x)^4*(A+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,6,(a*A+b*B)*x/(a^2+b^2)+(a^2*A-A*b^2+a*b*B)*cot(c+d*x)/(a^3*d)+1/2*(A*b-a*B)*cot(c+d*x)^2/(a^2*d)-1/3*A*cot(c+d*x)^3/(a*d)+(a^2-b^2)*(A*b-a*B)*log(sin(c+d*x))/(a^4*d)+b^4*(A*b-a*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^4*(a^2+b^2)*d)],
[tan(c+d*x)^3*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,6,-(2*a*A*b-a^2*B+b^2*B)*x/(a^2+b^2)^2+(a^2*A-A*b^2+2*a*b*B)*log(cos(c+d*x))/((a^2+b^2)^2*d)+a^2*(a^2*A*b+3*A*b^3-2*a^3*B-4*a*b^2*B)*log(a+b*tan(c+d*x))/(b^3*(a^2+b^2)^2*d)-(a*A*b-2*a^2*B-b^2*B)*tan(c+d*x)/(b^2*(a^2+b^2)*d)+a*(A*b-a*B)*tan(c+d*x)^2/(b*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^2*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,5,-(a^2*A-A*b^2+2*a*b*B)*x/(a^2+b^2)^2-(2*a*A*b-a^2*B+b^2*B)*log(cos(c+d*x))/((a^2+b^2)^2*d)-a*(2*A*b^3-a*(a^2+3*b^2)*B)*log(a+b*tan(c+d*x))/(b^2*(a^2+b^2)^2*d)-a^2*(A*b-a*B)/(b^2*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,3,(2*a*A*b-a^2*B+b^2*B)*x/(a^2+b^2)^2-(a^2*A-A*b^2+2*a*b*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^2*d)+a*(A*b-a*B)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,3,(a^2*A-A*b^2+2*a*b*B)*x/(a^2+b^2)^2+(2*a*A*b-a^2*B+b^2*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^2*d)+(-A*b+a*B)/((a^2+b^2)*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,4,-(2*a*A*b-a^2*B+b^2*B)*x/(a^2+b^2)^2+A*log(sin(c+d*x))/(a^2*d)-b*(3*a^2*A*b+A*b^3-2*a^3*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^2*(a^2+b^2)^2*d)+b*(A*b-a*B)/(a*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)^2*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,5,-(a^2*A-A*b^2+2*a*b*B)*x/(a^2+b^2)^2-(2*A*b-a*B)*log(sin(c+d*x))/(a^3*d)+b^2*(4*a^2*A*b+2*A*b^3-3*a^3*B-a*b^2*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^3*(a^2+b^2)^2*d)-b*(a^2*A+2*A*b^2-a*b*B)/(a^2*(a^2+b^2)*d*(a+b*tan(c+d*x)))-A*cot(c+d*x)/(a*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)^3*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,6,(2*a*A*b-a^2*B+b^2*B)*x/(a^2+b^2)^2-(a^2*A-3*A*b^2+2*a*b*B)*log(sin(c+d*x))/(a^4*d)-b^3*(5*a^2*A*b+3*A*b^3-4*a^3*B-2*a*b^2*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^4*(a^2+b^2)^2*d)+b*(2*a^2*A*b+3*A*b^3-a^3*B-2*a*b^2*B)/(a^3*(a^2+b^2)*d*(a+b*tan(c+d*x)))+1/2*(3*A*b-2*a*B)*cot(c+d*x)/(a^2*d*(a+b*tan(c+d*x)))-1/2*A*cot(c+d*x)^2/(a*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^4*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^3,x,7,(a^3*A-3*a*A*b^2+3*a^2*b*B-b^3*B)*x/(a^2+b^2)^3+(3*a^2*A*b-A*b^3-a^3*B+3*a*b^2*B)*log(cos(c+d*x))/((a^2+b^2)^3*d)+a^2*(a^4*A*b+3*a^2*A*b^3+6*A*b^5-3*a^5*B-9*a^3*b^2*B-10*a*b^4*B)*log(a+b*tan(c+d*x))/(b^4*(a^2+b^2)^3*d)-(a^3*A*b+3*a*A*b^3-3*a^4*B-6*a^2*b^2*B-b^4*B)*tan(c+d*x)/(b^3*(a^2+b^2)^2*d)+1/2*a*(A*b-a*B)*tan(c+d*x)^3/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+1/2*a*(a^2*A*b+5*A*b^3-3*a^3*B-7*a*b^2*B)*tan(c+d*x)^2/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^3*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^3,x,6,-(3*a^2*A*b-A*b^3-a^3*B+3*a*b^2*B)*x/(a^2+b^2)^3+(a^3*A-3*a*A*b^2+3*a^2*b*B-b^3*B)*log(cos(c+d*x))/((a^2+b^2)^3*d)+a*(a^2*A*b^3-3*A*b^5+a^5*B+3*a^3*b^2*B+6*a*b^4*B)*log(a+b*tan(c+d*x))/(b^3*(a^2+b^2)^3*d)+1/2*a*(A*b-a*B)*tan(c+d*x)^2/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)-a^2*(2*A*b^3-a*(a^2+3*b^2)*B)/(b^3*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^2*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^3,x,4,-(a^3*A-3*a*A*b^2+3*a^2*b*B-b^3*B)*x/(a^2+b^2)^3-(3*a^2*A*b-A*b^3-a^3*B+3*a*b^2*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^3*d)-1/2*a^2*(A*b-a*B)/(b^2*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+a*(2*A*b^3-a*(a^2+3*b^2)*B)/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^3,x,4,(3*a^2*A*b-A*b^3-a^3*B+3*a*b^2*B)*x/(a^2+b^2)^3-(a^3*A-3*a*A*b^2+3*a^2*b*B-b^3*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^3*d)+1/2*a*(A*b-a*B)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+(a^2*A-A*b^2+2*a*b*B)/((a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^3,x,4,(a^3*A-3*a*A*b^2+3*a^2*b*B-b^3*B)*x/(a^2+b^2)^3+(3*a^2*A*b-A*b^3-a^3*B+3*a*b^2*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^3*d)+1/2*(-A*b+a*B)/((a^2+b^2)*d*(a+b*tan(c+d*x))^2)+(-2*a*A*b+a^2*B-b^2*B)/((a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^3,x,5,-(3*a^2*A*b-A*b^3-a^3*B+3*a*b^2*B)*x/(a^2+b^2)^3+A*log(sin(c+d*x))/(a^3*d)-b*(6*a^4*A*b+3*a^2*A*b^3+A*b^5-3*a^5*B+a^3*b^2*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^3*(a^2+b^2)^3*d)+1/2*b*(A*b-a*B)/(a*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+b*(3*a^2*A*b+A*b^3-2*a^3*B)/(a^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)^2*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^3,x,6,-(a^3*A-3*a*A*b^2+3*a^2*b*B-b^3*B)*x/(a^2+b^2)^3-(3*A*b-a*B)*log(sin(c+d*x))/(a^4*d)+b^2*(10*a^4*A*b+9*a^2*A*b^3+3*A*b^5-6*a^5*B-3*a^3*b^2*B-a*b^4*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^4*(a^2+b^2)^3*d)-1/2*b*(2*a^2*A+3*A*b^2-a*b*B)/(a^2*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)-A*cot(c+d*x)/(a*d*(a+b*tan(c+d*x))^2)-b*(a^4*A+6*a^2*A*b^2+3*A*b^4-3*a^3*b*B-a*b^3*B)/(a^3*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)^3*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^3,x,7,(3*a^2*A*b-A*b^3-a^3*B+3*a*b^2*B)*x/(a^2+b^2)^3-(a^2*A-6*A*b^2+3*a*b*B)*log(sin(c+d*x))/(a^5*d)-b^3*(15*a^4*A*b+17*a^2*A*b^3+6*A*b^5-10*a^5*B-9*a^3*b^2*B-3*a*b^4*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^5*(a^2+b^2)^3*d)+1/2*b*(5*a^2*A*b+6*A*b^3-2*a^3*B-3*a*b^2*B)/(a^3*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+(2*A*b-a*B)*cot(c+d*x)/(a^2*d*(a+b*tan(c+d*x))^2)-1/2*A*cot(c+d*x)^2/(a*d*(a+b*tan(c+d*x))^2)+b*(3*a^4*A*b+11*a^2*A*b^3+6*A*b^5-a^5*B-6*a^3*b^2*B-3*a*b^4*B)/(a^4*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^4*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^4,x,7,(a^4*A-6*a^2*A*b^2+A*b^4+4*a^3*b*B-4*a*b^3*B)*x/(a^2+b^2)^4+(4*a^3*A*b-4*a*A*b^3-a^4*B+6*a^2*b^2*B-b^4*B)*log(cos(c+d*x))/((a^2+b^2)^4*d)+a*(4*a^2*A*b^5-4*A*b^7+a^7*B+4*a^5*b^2*B+5*a^3*b^4*B+10*a*b^6*B)*log(a+b*tan(c+d*x))/(b^4*(a^2+b^2)^4*d)+1/3*a*(A*b-a*B)*tan(c+d*x)^3/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^3)+1/2*a*(2*A*b^3-a*(a^2+3*b^2)*B)*tan(c+d*x)^2/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)+a^2*(a^2*A*b^3-3*A*b^5+a^5*B+3*a^3*b^2*B+6*a*b^4*B)/(b^4*(a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^3*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^4,x,5,-(4*a^3*A*b-4*a*A*b^3-a^4*B+6*a^2*b^2*B-b^4*B)*x/(a^2+b^2)^4+(a^4*A-6*a^2*A*b^2+A*b^4+4*a^3*b*B-4*a*b^3*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^4*d)+1/3*a*(A*b-a*B)*tan(c+d*x)^2/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^3)+1/6*a^2*(a^2*A*b-5*A*b^3+2*a^3*B+8*a*b^2*B)/(b^3*(a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)-1/3*a*(a^4*A*b+5*a^2*A*b^3-8*A*b^5+2*a^5*B+7*a^3*b^2*B+17*a*b^4*B)/(b^3*(a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^2*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^4,x,5,-(a^4*A-6*a^2*A*b^2+A*b^4+4*a^3*b*B-4*a*b^3*B)*x/(a^2+b^2)^4-(4*a^3*A*b-4*a*A*b^3-a^4*B+6*a^2*b^2*B-b^4*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^4*d)-1/3*a^2*(A*b-a*B)/(b^2*(a^2+b^2)*d*(a+b*tan(c+d*x))^3)+1/2*a*(2*A*b^3-a*(a^2+3*b^2)*B)/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)+(3*a^2*A*b-A*b^3-a^3*B+3*a*b^2*B)/((a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^4,x,5,(4*a^3*A*b-4*a*A*b^3-a^4*B+6*a^2*b^2*B-b^4*B)*x/(a^2+b^2)^4-(a^4*A-6*a^2*A*b^2+A*b^4+4*a^3*b*B-4*a*b^3*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^4*d)+1/3*a*(A*b-a*B)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^3)+1/2*(a^2*A-A*b^2+2*a*b*B)/((a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)+(a^3*A-3*a*A*b^2+3*a^2*b*B-b^3*B)/((a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^4,x,5,(a^4*A-6*a^2*A*b^2+A*b^4+4*a^3*b*B-4*a*b^3*B)*x/(a^2+b^2)^4+(4*a^3*A*b-4*a*A*b^3-a^4*B+6*a^2*b^2*B-b^4*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^4*d)+1/3*(-A*b+a*B)/((a^2+b^2)*d*(a+b*tan(c+d*x))^3)+1/2*(-2*a*A*b+a^2*B-b^2*B)/((a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)+(-3*a^2*A*b+A*b^3+a^3*B-3*a*b^2*B)/((a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^4,x,6,-(4*a^3*A*b-4*a*A*b^3-a^4*B+6*a^2*b^2*B-b^4*B)*x/(a^2+b^2)^4+A*log(sin(c+d*x))/(a^4*d)-b*(10*a^6*A*b+5*a^4*A*b^3+4*a^2*A*b^5+A*b^7-4*a^7*B+4*a^5*b^2*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^4*(a^2+b^2)^4*d)+1/3*b*(A*b-a*B)/(a*(a^2+b^2)*d*(a+b*tan(c+d*x))^3)+1/2*b*(3*a^2*A*b+A*b^3-2*a^3*B)/(a^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)+b*(6*a^4*A*b+3*a^2*A*b^3+A*b^5-3*a^5*B+a^3*b^2*B)/(a^3*(a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)^2*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^4,x,7,-(a^4*A-6*a^2*A*b^2+A*b^4+4*a^3*b*B-4*a*b^3*B)*x/(a^2+b^2)^4-(4*A*b-a*B)*log(sin(c+d*x))/(a^5*d)+b^2*(20*a^6*A*b+24*a^4*A*b^3+16*a^2*A*b^5+4*A*b^7-10*a^7*B-5*a^5*b^2*B-4*a^3*b^4*B-a*b^6*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^5*(a^2+b^2)^4*d)-1/3*b*(3*a^2*A+4*A*b^2-a*b*B)/(a^2*(a^2+b^2)*d*(a+b*tan(c+d*x))^3)-A*cot(c+d*x)/(a*d*(a+b*tan(c+d*x))^3)-1/2*b*(2*a^4*A+8*a^2*A*b^2+4*A*b^4-3*a^3*b*B-a*b^3*B)/(a^3*(a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)-b*(a^6*A+13*a^4*A*b^2+12*a^2*A*b^4+4*A*b^6-6*a^5*b*B-3*a^3*b^3*B-a*b^5*B)/(a^4*(a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)^3*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^4,x,8,(4*a^3*A*b-4*a*A*b^3-a^4*B+6*a^2*b^2*B-b^4*B)*x/(a^2+b^2)^4-(a^2*A-10*A*b^2+4*a*b*B)*log(sin(c+d*x))/(a^6*d)-b^3*(35*a^6*A*b+56*a^4*A*b^3+39*a^2*A*b^5+10*A*b^7-20*a^7*B-24*a^5*b^2*B-16*a^3*b^4*B-4*a*b^6*B)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^6*(a^2+b^2)^4*d)+1/3*b*(9*a^2*A*b+10*A*b^3-3*a^3*B-4*a*b^2*B)/(a^3*(a^2+b^2)*d*(a+b*tan(c+d*x))^3)+1/2*(5*A*b-2*a*B)*cot(c+d*x)/(a^2*d*(a+b*tan(c+d*x))^3)-1/2*A*cot(c+d*x)^2/(a*d*(a+b*tan(c+d*x))^3)+1/2*b*(7*a^4*A*b+19*a^2*A*b^3+10*A*b^5-2*a^5*B-8*a^3*b^2*B-4*a*b^4*B)/(a^4*(a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)+b*(4*a^6*A*b+27*a^4*A*b^3+29*a^2*A*b^5+10*A*b^7-a^7*B-13*a^5*b^2*B-12*a^3*b^4*B-4*a*b^6*B)/(a^5*(a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^3*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x)),x,3,B*log(cos(c+d*x))/d+1/2*B*tan(c+d*x)^2/d],
[tan(c+d*x)^2*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x)),x,3,-B*x+B*tan(c+d*x)/d],
[tan(c+d*x)*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x)),x,2,-B*log(cos(c+d*x))/d],
[(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x)),x,2,B*x],
[cot(c+d*x)*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x)),x,2,B*log(sin(c+d*x))/d],
[cot(c+d*x)^2*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x)),x,3,-B*x-B*cot(c+d*x)/d],
[cot(c+d*x)^3*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x)),x,3,-1/2*B*cot(c+d*x)^2/d-B*log(sin(c+d*x))/d],
[cot(c+d*x)^4*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x)),x,4,B*x+B*cot(c+d*x)/d-1/3*B*cot(c+d*x)^3/d],
[tan(c+d*x)^4*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,7,a*B*x/(a^2+b^2)+b*B*log(cos(c+d*x))/((a^2+b^2)*d)+a^4*B*log(a+b*tan(c+d*x))/(b^3*(a^2+b^2)*d)-a*B*tan(c+d*x)/(b^2*d)+1/2*B*tan(c+d*x)^2/(b*d)],
[tan(c+d*x)^3*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,6,-b*B*x/(a^2+b^2)+a*B*log(cos(c+d*x))/((a^2+b^2)*d)-a^3*B*log(a+b*tan(c+d*x))/(b^2*(a^2+b^2)*d)+B*tan(c+d*x)/(b*d)],
[tan(c+d*x)^2*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,5,-a*B*x/b^2+a^3*B*x/(b^2*(a^2+b^2))-B*log(cos(c+d*x))/(b*d)+a^2*B*log(a*cos(c+d*x)+b*sin(c+d*x))/(b*(a^2+b^2)*d)],
[tan(c+d*x)*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,3,b*B*x/(a^2+b^2)-a*B*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)*d)],
[(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,3,a*B*x/(a^2+b^2)+b*B*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)*d)],
[cot(c+d*x)*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,4,-b*B*x/(a^2+b^2)+B*log(sin(c+d*x))/(a*d)-b^2*B*log(a*cos(c+d*x)+b*sin(c+d*x))/(a*(a^2+b^2)*d)],
[cot(c+d*x)^2*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,5,-a*B*x/(a^2+b^2)-B*cot(c+d*x)/(a*d)-b*B*log(sin(c+d*x))/(a^2*d)+b^3*B*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^2*(a^2+b^2)*d)],
[cot(c+d*x)^3*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,6,b*B*x/(a^2+b^2)+b*B*cot(c+d*x)/(a^2*d)-1/2*B*cot(c+d*x)^2/(a*d)-(a^2-b^2)*B*log(sin(c+d*x))/(a^3*d)-b^4*B*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^3*(a^2+b^2)*d)],
[(3+tan(c+d*x))/(2-tan(c+d*x)),x,2,x-log(2*cos(c+d*x)-sin(c+d*x))/d],
[(b*B/a+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,2,2*b*B*x/(a^2+b^2)-(a-b^2/a)*B*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)*d)],
[(a+b*tan(c+d*x))/(b+a*tan(c+d*x))^2,x,3,-a*(a^2-3*b^2)*x/(a^2+b^2)^2+b*(3*a^2-b^2)*log(b*cos(c+d*x)+a*sin(c+d*x))/((a^2+b^2)^2*d)+(-a^2+b^2)/((a^2+b^2)*d*(b+a*tan(c+d*x)))],

# Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+b Tan[c+d x])^(n/2)

# n>0
[sqrt(a+b*tan(c+d*x))*tan(c+d*x)^3*(A+B*tan(c+d*x)),x,11,(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))*sqrt(a-I*b)/d+(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))*sqrt(a+I*b)/d-2*A*sqrt(a+b*tan(c+d*x))/d-2/105*(14*a*A*b-8*a^2*B+35*b^2*B)*(a+b*tan(c+d*x))^(3/2)/(b^3*d)+2/35*(7*A*b-4*a*B)*tan(c+d*x)*(a+b*tan(c+d*x))^(3/2)/(b^2*d)+2/7*B*tan(c+d*x)^2*(a+b*tan(c+d*x))^(3/2)/(b*d)],
[sqrt(a+b*tan(c+d*x))*tan(c+d*x)^2*(A+B*tan(c+d*x)),x,10,(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))*sqrt(a-I*b)/d-(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))*sqrt(a+I*b)/d-2*B*sqrt(a+b*tan(c+d*x))/d+2/15*(5*A*b-2*a*B)*(a+b*tan(c+d*x))^(3/2)/(b^2*d)+2/5*B*tan(c+d*x)*(a+b*tan(c+d*x))^(3/2)/(b*d)],
[sqrt(a+b*tan(c+d*x))*tan(c+d*x)*(A+B*tan(c+d*x)),x,9,-(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))*sqrt(a-I*b)/d-(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))*sqrt(a+I*b)/d+2*A*sqrt(a+b*tan(c+d*x))/d+2/3*B*(a+b*tan(c+d*x))^(3/2)/(b*d)],
[sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,8,-(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))*sqrt(a-I*b)/d+(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))*sqrt(a+I*b)/d+2*B*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)*sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,11,-2*A*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))*sqrt(a)/d+(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))*sqrt(a-I*b)/d+(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))*sqrt(a+I*b)/d],
[cot(c+d*x)^2*sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,12,-(A*b+2*a*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(d*sqrt(a))+(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))*sqrt(a-I*b)/d-(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))*sqrt(a+I*b)/d-A*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)^3*sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,13,1/4*(8*a^2*A+A*b^2-4*a*b*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(3/2)*d)-(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))*sqrt(a-I*b)/d-(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))*sqrt(a+I*b)/d-1/4*(A*b+4*a*B)*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/(a*d)-1/2*A*cot(c+d*x)^2*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)^4*sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,14,1/8*(8*a^2*A*b-A*b^3+16*a^3*B+2*a*b^2*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(5/2)*d)-(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))*sqrt(a-I*b)/d+(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))*sqrt(a+I*b)/d+1/8*(8*a^2*A+A*b^2-2*a*b*B)*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/(a^2*d)-1/12*(A*b+6*a*B)*cot(c+d*x)^2*sqrt(a+b*tan(c+d*x))/(a*d)-1/3*A*cot(c+d*x)^3*sqrt(a+b*tan(c+d*x))/d],
[tan(c+d*x)^2*(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,11,(a-I*b)^(3/2)*(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-(a+I*b)^(3/2)*(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d-2*(A*b+a*B)*sqrt(a+b*tan(c+d*x))/d-2/3*B*(a+b*tan(c+d*x))^(3/2)/d+2/35*(7*A*b-2*a*B)*(a+b*tan(c+d*x))^(5/2)/(b^2*d)+2/7*B*tan(c+d*x)*(a+b*tan(c+d*x))^(5/2)/(b*d)],
[tan(c+d*x)*(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,10,-(a-I*b)^(3/2)*(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-(a+I*b)^(3/2)*(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+2*(a*A-b*B)*sqrt(a+b*tan(c+d*x))/d+2/3*A*(a+b*tan(c+d*x))^(3/2)/d+2/5*B*(a+b*tan(c+d*x))^(5/2)/(b*d)],
[(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,9,-(a-I*b)^(3/2)*(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+(a+I*b)^(3/2)*(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+2*(A*b+a*B)*sqrt(a+b*tan(c+d*x))/d+2/3*B*(a+b*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)*(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,12,-2*a^(3/2)*A*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/d+(a-I*b)^(3/2)*(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+(a+I*b)^(3/2)*(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+2*b*B*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)^2*(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,12,(a-I*b)^(3/2)*(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-(a+I*b)^(3/2)*(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d-(3*A*b+2*a*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))*sqrt(a)/d-a*A*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)^3*(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,13,-(a-I*b)^(3/2)*(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-(a+I*b)^(3/2)*(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+1/4*(8*a^2*A-3*A*b^2-12*a*b*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(d*sqrt(a))-1/4*(5*A*b+4*a*B)*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/d-1/2*a*A*cot(c+d*x)^2*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)^4*(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,14,1/8*(24*a^2*A*b+A*b^3+16*a^3*B-6*a*b^2*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(3/2)*d)-(a-I*b)^(3/2)*(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+(a+I*b)^(3/2)*(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+1/8*(8*a^2*A-A*b^2-10*a*b*B)*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/(a*d)-1/12*(7*A*b+6*a*B)*cot(c+d*x)^2*sqrt(a+b*tan(c+d*x))/d-1/3*a*A*cot(c+d*x)^3*sqrt(a+b*tan(c+d*x))/d],
[tan(c+d*x)^2*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,12,(a-I*b)^(5/2)*(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-(a+I*b)^(5/2)*(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d-2*(2*a*A*b+a^2*B-b^2*B)*sqrt(a+b*tan(c+d*x))/d-2/3*(A*b+a*B)*(a+b*tan(c+d*x))^(3/2)/d-2/5*B*(a+b*tan(c+d*x))^(5/2)/d+2/63*(9*A*b-2*a*B)*(a+b*tan(c+d*x))^(7/2)/(b^2*d)+2/9*B*tan(c+d*x)*(a+b*tan(c+d*x))^(7/2)/(b*d)],
[tan(c+d*x)*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,11,-(a-I*b)^(5/2)*(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-(a+I*b)^(5/2)*(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+2*(a^2*A-A*b^2-2*a*b*B)*sqrt(a+b*tan(c+d*x))/d+2/3*(a*A-b*B)*(a+b*tan(c+d*x))^(3/2)/d+2/5*A*(a+b*tan(c+d*x))^(5/2)/d+2/7*B*(a+b*tan(c+d*x))^(7/2)/(b*d)],
[(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,10,-(a-I*b)^(5/2)*(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+(a+I*b)^(5/2)*(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+2*(2*a*A*b+a^2*B-b^2*B)*sqrt(a+b*tan(c+d*x))/d+2/3*(A*b+a*B)*(a+b*tan(c+d*x))^(3/2)/d+2/5*B*(a+b*tan(c+d*x))^(5/2)/d],
[cot(c+d*x)*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,13,-2*a^(5/2)*A*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/d+(a-I*b)^(5/2)*(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+(a+I*b)^(5/2)*(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+2*b*(A*b+2*a*B)*sqrt(a+b*tan(c+d*x))/d+2/3*b*B*(a+b*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^2*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,13,-a^(3/2)*(5*A*b+2*a*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/d+(a-I*b)^(5/2)*(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-(a+I*b)^(5/2)*(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+b*(a*A+2*b*B)*sqrt(a+b*tan(c+d*x))/d-a*A*cot(c+d*x)*(a+b*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^3*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,13,-(a-I*b)^(5/2)*(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-(a+I*b)^(5/2)*(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+1/4*(8*a^2*A-15*A*b^2-20*a*b*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))*sqrt(a)/d-1/4*a*(7*A*b+4*a*B)*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/d-1/2*a*A*cot(c+d*x)^2*(a+b*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^4*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,14,-(a-I*b)^(5/2)*(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+(a+I*b)^(5/2)*(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+1/8*(40*a^2*A*b-5*A*b^3+16*a^3*B-30*a*b^2*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(d*sqrt(a))+1/8*(8*a^2*A-11*A*b^2-18*a*b*B)*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/d-1/4*a*(3*A*b+2*a*B)*cot(c+d*x)^2*sqrt(a+b*tan(c+d*x))/d-1/3*a*A*cot(c+d*x)^3*(a+b*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^5*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,15,-1/64*(128*a^4*A-240*a^2*A*b^2-5*A*b^4-320*a^3*b*B+40*a*b^3*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(3/2)*d)+(a-I*b)^(5/2)*(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d+(a+I*b)^(5/2)*(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d+1/64*(144*a^2*A*b-5*A*b^3+64*a^3*B-88*a*b^2*B)*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/(a*d)+1/96*(48*a^2*A-59*A*b^2-104*a*b*B)*cot(c+d*x)^2*sqrt(a+b*tan(c+d*x))/d-1/24*a*(11*A*b+8*a*B)*cot(c+d*x)^3*sqrt(a+b*tan(c+d*x))/d-1/4*a*A*cot(c+d*x)^4*(a+b*tan(c+d*x))^(3/2)/d],
[(-a+b*tan(c+d*x))*(a+b*tan(c+d*x))^(5/2),x,10,(I*a-b)*(a-I*b)^(5/2)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/d-(a+I*b)^(5/2)*(I*a+b)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/d-2*b*(a^2+b^2)*sqrt(a+b*tan(c+d*x))/d+2/5*b*(a+b*tan(c+d*x))^(5/2)/d],
[(-a+b*tan(c+d*x))*(a+b*tan(c+d*x))^(3/2),x,13,-b*(a^2+b^2)*arctanh((sqrt(a+sqrt(a^2+b^2))-sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a-sqrt(a^2+b^2)))+b*(a^2+b^2)*arctanh((sqrt(a+sqrt(a^2+b^2))+sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a-sqrt(a^2+b^2)))-1/2*b*(a^2+b^2)*log(a+sqrt(a^2+b^2)-sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a+sqrt(a^2+b^2)))+1/2*b*(a^2+b^2)*log(a+sqrt(a^2+b^2)+sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a+sqrt(a^2+b^2)))+2/3*b*(a+b*tan(c+d*x))^(3/2)/d],
[(-a+b*tan(c+d*x))*(a+b*tan(c+d*x))^(1/2),x,13,-b*arctanh((sqrt(a+sqrt(a^2+b^2))-sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))*sqrt(a^2+b^2)/(d*sqrt(2)*sqrt(a-sqrt(a^2+b^2)))+b*arctanh((sqrt(a+sqrt(a^2+b^2))+sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))*sqrt(a^2+b^2)/(d*sqrt(2)*sqrt(a-sqrt(a^2+b^2)))+1/2*b*log(a+sqrt(a^2+b^2)-sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))*sqrt(a^2+b^2)/(d*sqrt(2)*sqrt(a+sqrt(a^2+b^2)))-1/2*b*log(a+sqrt(a^2+b^2)+sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))*sqrt(a^2+b^2)/(d*sqrt(2)*sqrt(a+sqrt(a^2+b^2)))+2*b*sqrt(a+b*tan(c+d*x))/d],

# n<0
[tan(c+d*x)^3*(A+B*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,10,(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/(d*sqrt(a-I*b))+(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/(d*sqrt(a+I*b))-2/15*(10*a*A*b-8*a^2*B+15*b^2*B)*sqrt(a+b*tan(c+d*x))/(b^3*d)+2/15*(5*A*b-4*a*B)*sqrt(a+b*tan(c+d*x))*tan(c+d*x)/(b^2*d)+2/5*B*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^2/(b*d)],
[tan(c+d*x)^2*(A+B*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,9,(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/(d*sqrt(a-I*b))-(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/(d*sqrt(a+I*b))+2/3*(3*A*b-2*a*B)*sqrt(a+b*tan(c+d*x))/(b^2*d)+2/3*B*sqrt(a+b*tan(c+d*x))*tan(c+d*x)/(b*d)],
[tan(c+d*x)*(A+B*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,8,-(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/(d*sqrt(a-I*b))-(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/(d*sqrt(a+I*b))+2*B*sqrt(a+b*tan(c+d*x))/(b*d)],
[(A+B*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,7,-(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/(d*sqrt(a-I*b))+(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/(d*sqrt(a+I*b))],
[cot(c+d*x)*(A+B*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,11,-2*A*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(d*sqrt(a))+(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/(d*sqrt(a-I*b))+(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/(d*sqrt(a+I*b))],
[cot(c+d*x)^2*(A+B*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,12,(A*b-2*a*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(3/2)*d)+(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/(d*sqrt(a-I*b))-(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/(d*sqrt(a+I*b))-A*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/(a*d)],
[cot(c+d*x)^3*(A+B*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,13,1/4*(8*a^2*A-3*A*b^2+4*a*b*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(5/2)*d)-(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/(d*sqrt(a-I*b))-(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/(d*sqrt(a+I*b))+1/4*(3*A*b-4*a*B)*cot(c+d*x)*sqrt(a+b*tan(c+d*x))/(a^2*d)-1/2*A*cot(c+d*x)^2*sqrt(a+b*tan(c+d*x))/(a*d)],
[tan(c+d*x)^3*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,10,(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)+(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)+2/3*(6*a^2*A*b+3*A*b^3-8*a^3*B-5*a*b^2*B)*sqrt(a+b*tan(c+d*x))/(b^3*(a^2+b^2)*d)-2/3*(3*a*A*b-4*a^2*B-b^2*B)*sqrt(a+b*tan(c+d*x))*tan(c+d*x)/(b^2*(a^2+b^2)*d)+2*a*(A*b-a*B)*tan(c+d*x)^2/(b*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[tan(c+d*x)^2*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,9,(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)-(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)-2*a^2*(A*b-a*B)/(b^2*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))+2*B*sqrt(a+b*tan(c+d*x))/(b^2*d)],
[tan(c+d*x)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,8,-(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)-(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)+2*a*(A*b-a*B)/(b*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,8,-(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)+(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)-2*(A*b-a*B)/((a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[cot(c+d*x)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,12,-2*A*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(3/2)*d)+(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)+(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)+2*b*(A*b-a*B)/(a*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[cot(c+d*x)^2*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,13,(3*A*b-2*a*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(5/2)*d)+(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)-(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)-b*(a^2*A+3*A*b^2-2*a*b*B)/(a^2*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))-A*cot(c+d*x)/(a*d*sqrt(a+b*tan(c+d*x)))],
[cot(c+d*x)^3*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,14,1/4*(8*a^2*A-15*A*b^2+12*a*b*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(7/2)*d)-(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)-(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)+1/4*b*(7*a^2*A*b+15*A*b^3-4*a^3*B-12*a*b^2*B)/(a^3*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))+1/4*(5*A*b-4*a*B)*cot(c+d*x)/(a^2*d*sqrt(a+b*tan(c+d*x)))-1/2*A*cot(c+d*x)^2/(a*d*sqrt(a+b*tan(c+d*x)))],
[tan(c+d*x)^4*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,11,-(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)+(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)+2/3*(8*a^4*A*b+17*a^2*A*b^3+3*A*b^5-16*a^5*B-30*a^3*b^2*B-8*a*b^4*B)*sqrt(a+b*tan(c+d*x))/(b^4*(a^2+b^2)^2*d)-2/3*(4*a^3*A*b+10*a*A*b^3-8*a^4*B-15*a^2*b^2*B-b^4*B)*sqrt(a+b*tan(c+d*x))*tan(c+d*x)/(b^3*(a^2+b^2)^2*d)+2*a*(a^2*A*b+3*A*b^3-2*a^3*B-4*a*b^2*B)*tan(c+d*x)^2/(b^2*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+2/3*a*(A*b-a*B)*tan(c+d*x)^3/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[tan(c+d*x)^3*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,10,(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)+(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)-2/3*a^2*(a^2*A*b+7*A*b^3-4*a^3*B-10*a*b^2*B)/(b^3*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))-2/3*(a*A*b-4*a^2*B-3*b^2*B)*sqrt(a+b*tan(c+d*x))/(b^3*(a^2+b^2)*d)+2/3*a*(A*b-a*B)*tan(c+d*x)^2/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[tan(c+d*x)^2*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,9,(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)-(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)+2*a*(2*A*b^3-a*(a^2+3*b^2)*B)/(b^2*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))-2/3*a^2*(A*b-a*B)/(b^2*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[tan(c+d*x)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,9,-(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)-(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)+2*(a^2*A-A*b^2+2*a*b*B)/((a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+2/3*a*(A*b-a*B)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,9,-(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)+(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)-2*(2*a*A*b-a^2*B+b^2*B)/((a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))-2/3*(A*b-a*B)/((a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[cot(c+d*x)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,13,-2*A*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(5/2)*d)+(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)+(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)+2*b*(3*a^2*A*b+A*b^3-2*a^3*B)/(a^2*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+2/3*b*(A*b-a*B)/(a*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[cot(c+d*x)^2*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,14,(5*A*b-2*a*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(7/2)*d)+(I*A+B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)-(I*A-B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)-b*(a^4*A+10*a^2*A*b^2+5*A*b^4-6*a^3*b*B-2*a*b^3*B)/(a^3*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))-1/3*b*(3*a^2*A+5*A*b^2-2*a*b*B)/(a^2*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))-A*cot(c+d*x)/(a*d*(a+b*tan(c+d*x))^(3/2))],
[cot(c+d*x)^3*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,15,1/4*(8*a^2*A-35*A*b^2+20*a*b*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(9/2)*d)-(A-I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)-(A+I*B)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)+1/4*b*(11*a^4*A*b+62*a^2*A*b^3+35*A*b^5-4*a^5*B-40*a^3*b^2*B-20*a*b^4*B)/(a^4*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+1/12*b*(27*a^2*A*b+35*A*b^3-12*a^3*B-20*a*b^2*B)/(a^3*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))+1/4*(7*A*b-4*a*B)*cot(c+d*x)/(a^2*d*(a+b*tan(c+d*x))^(3/2))-1/2*A*cot(c+d*x)^2/(a*d*(a+b*tan(c+d*x))^(3/2))],
[(a*B+b*B*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,12,b*B*arctanh((sqrt(a+sqrt(a^2+b^2))-sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a-sqrt(a^2+b^2)))-b*B*arctanh((sqrt(a+sqrt(a^2+b^2))+sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a-sqrt(a^2+b^2)))+1/2*b*B*log(a+sqrt(a^2+b^2)-sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a+sqrt(a^2+b^2)))-1/2*b*B*log(a+sqrt(a^2+b^2)+sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a+sqrt(a^2+b^2)))],
[(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,12,b*B*arctanh((sqrt(a+sqrt(a^2+b^2))-sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a-sqrt(a^2+b^2)))-b*B*arctanh((sqrt(a+sqrt(a^2+b^2))+sqrt(2)*sqrt(a+b*tan(c+d*x)))/sqrt(a-sqrt(a^2+b^2)))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a-sqrt(a^2+b^2)))-1/2*b*B*log(a+sqrt(a^2+b^2)-sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a+sqrt(a^2+b^2)))+1/2*b*B*log(a+sqrt(a^2+b^2)+sqrt(2)*sqrt(a+sqrt(a^2+b^2))*sqrt(a+b*tan(c+d*x))+b*tan(c+d*x))/(d*sqrt(2)*sqrt(a^2+b^2)*sqrt(a+sqrt(a^2+b^2)))],
[cot(c+d*x)*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,12,-2*B*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(d*sqrt(a))+B*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/(d*sqrt(a-I*b))+B*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/(d*sqrt(a+I*b))],
[(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,9,-I*B*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)+I*B*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)-2*b*B/((a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[cot(c+d*x)*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,13,-2*B*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a))/(a^(3/2)*d)+B*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)+B*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)+2*b^2*B/(a*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[(-a+b*tan(c+d*x))/(a+b*tan(c+d*x))^(1/2),x,7,(I*a-b)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/(d*sqrt(a-I*b))-(I*a+b)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/(d*sqrt(a+I*b))],
[(-a+b*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,8,(I*a-b)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(3/2)*d)-(I*a+b)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(3/2)*d)+4*a*b/((a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[(-a+b*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,9,(I*a-b)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/((a-I*b)^(5/2)*d)-(I*a+b)*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/((a+I*b)^(5/2)*d)+2*b*(3*a^2-b^2)/((a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+4/3*a*b/((a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[(1+I*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,3,-2*I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a-I*b))/(d*sqrt(a-I*b))],
[(1-I*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,3,2*I*arctanh(sqrt(a+b*tan(c+d*x))/sqrt(a+I*b))/(d*sqrt(a+I*b))],
[(3+tan(x))/sqrt(4+3*tan(x)),x,2,-arctan((1-3*tan(x))/(sqrt(2)*sqrt(4+3*tan(x))))*sqrt(2)],
[(1-3*tan(x))/sqrt(4+3*tan(x)),x,2,arctanh((3+tan(x))/(sqrt(2)*sqrt(4+3*tan(x))))*sqrt(2)],
[(4-3*tan(a+b*x))/sqrt(4+3*tan(a+b*x)),x,5,-9/5*arctan((1-3*tan(a+b*x))/(sqrt(2)*sqrt(4+3*tan(a+b*x))))/(b*sqrt(2))+13/5*arctanh((3+tan(a+b*x))/(sqrt(2)*sqrt(4+3*tan(a+b*x))))/(b*sqrt(2))],

# Integrands of the form Tan[c+d x]^(m/2) (A+B Tan[c+d x]) (a+b Tan[c+d x])^n

# n>0
[tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,14,(a*(A-B)-b*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a*(A-B)-b*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(b*(A-B)+a*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(b*(A-B)+a*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-2*(A*b+a*B)*sqrt(tan(c+d*x))/d+2/3*(a*A-b*B)*tan(c+d*x)^(3/2)/d+2/5*(A*b+a*B)*tan(c+d*x)^(5/2)/d+2/7*b*B*tan(c+d*x)^(7/2)/d],
[tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,13,(b*(A-B)+a*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(b*(A-B)+a*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a*(A-B)-b*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a*(A-B)-b*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*(a*A-b*B)*sqrt(tan(c+d*x))/d+2/3*(A*b+a*B)*tan(c+d*x)^(3/2)/d+2/5*b*B*tan(c+d*x)^(5/2)/d],
[sqrt(tan(c+d*x))*(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,12,-(a*(A-B)-b*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a*(A-B)-b*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(b*(A-B)+a*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(b*(A-B)+a*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*(A*b+a*B)*sqrt(tan(c+d*x))/d+2/3*b*B*tan(c+d*x)^(3/2)/d],
[(a+b*tan(c+d*x))*(A+B*tan(c+d*x))/sqrt(tan(c+d*x)),x,11,-(b*(A-B)+a*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(b*(A-B)+a*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(a*(A-B)-b*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(a*(A-B)-b*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*b*B*sqrt(tan(c+d*x))/d],
[(a+b*tan(c+d*x))*(A+B*tan(c+d*x))/tan(c+d*x)^(3/2),x,11,(a*(A-B)-b*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a*(A-B)-b*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(b*(A-B)+a*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(b*(A-B)+a*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-2*a*A/(d*sqrt(tan(c+d*x)))],
[(a+b*tan(c+d*x))*(A+B*tan(c+d*x))/tan(c+d*x)^(5/2),x,12,(b*(A-B)+a*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(b*(A-B)+a*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a*(A-B)-b*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a*(A-B)-b*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-2*(A*b+a*B)/(d*sqrt(tan(c+d*x)))-2/3*a*A/(d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))*(A+B*tan(c+d*x))/tan(c+d*x)^(7/2),x,13,-(a*(A-B)-b*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a*(A-B)-b*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(b*(A-B)+a*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(b*(A-B)+a*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*(a*A-b*B)/(d*sqrt(tan(c+d*x)))-2/5*a*A/(d*tan(c+d*x)^(5/2))-2/3*(A*b+a*B)/(d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,15,(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-2*(2*a*A*b+a^2*B-b^2*B)*sqrt(tan(c+d*x))/d+2/3*(a^2*A-A*b^2-2*a*b*B)*tan(c+d*x)^(3/2)/d+2/5*(2*a*A*b+a^2*B-b^2*B)*tan(c+d*x)^(5/2)/d+2/63*b*(9*A*b+11*a*B)*tan(c+d*x)^(7/2)/d+2/9*b*B*tan(c+d*x)^(7/2)*(a+b*tan(c+d*x))/d],
[tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,14,(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*(a^2*A-A*b^2-2*a*b*B)*sqrt(tan(c+d*x))/d+2/3*(2*a*A*b+a^2*B-b^2*B)*tan(c+d*x)^(3/2)/d+2/35*b*(7*A*b+9*a*B)*tan(c+d*x)^(5/2)/d+2/7*b*B*tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))/d],
[sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,13,-(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*(2*a*A*b+a^2*B-b^2*B)*sqrt(tan(c+d*x))/d+2/15*b*(5*A*b+7*a*B)*tan(c+d*x)^(3/2)/d+2/5*b*B*tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))/d],
[(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x))/sqrt(tan(c+d*x)),x,12,-(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2/3*b*(3*A*b+5*a*B)*sqrt(tan(c+d*x))/d+2/3*b*B*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))/d],
[(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x))/tan(c+d*x)^(3/2),x,12,(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-2*a^2*A/(d*sqrt(tan(c+d*x)))+2*b^2*B*sqrt(tan(c+d*x))/d],
[(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x))/tan(c+d*x)^(5/2),x,12,(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-2*a*(2*A*b+a*B)/(d*sqrt(tan(c+d*x)))-2/3*a^2*A/(d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x))/tan(c+d*x)^(7/2),x,13,-(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*(a^2*A-A*b^2-2*a*b*B)/(d*sqrt(tan(c+d*x)))-2/5*a^2*A/(d*tan(c+d*x)^(5/2))-2/3*a*(2*A*b+a*B)/(d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,15,(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*(a^3*A-3*a*A*b^2-3*a^2*b*B+b^3*B)*sqrt(tan(c+d*x))/d+2/3*(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*tan(c+d*x)^(3/2)/d+2/45*b*(27*a*A*b+22*a^2*B-9*b^2*B)*tan(c+d*x)^(5/2)/d+2/63*b^2*(9*A*b+13*a*B)*tan(c+d*x)^(7/2)/d+2/9*b*B*tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^2/d],
[sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,14,-(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*sqrt(tan(c+d*x))/d+2/21*b*(21*a*A*b+18*a^2*B-7*b^2*B)*tan(c+d*x)^(3/2)/d+2/35*b^2*(7*A*b+11*a*B)*tan(c+d*x)^(5/2)/d+2/7*b*B*tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^2/d],
[(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x))/sqrt(tan(c+d*x)),x,13,-(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2/5*b*(15*a*A*b+14*a^2*B-5*b^2*B)*sqrt(tan(c+d*x))/d+2/15*b^2*(5*A*b+9*a*B)*tan(c+d*x)^(3/2)/d+2/5*b*B*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^2/d],
[(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x))/tan(c+d*x)^(3/2),x,13,(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*b*(2*a^2*A+A*b^2+3*a*b*B)*sqrt(tan(c+d*x))/d+2/3*b^2*(3*a*A+b*B)*tan(c+d*x)^(3/2)/d-2*a*A*(a+b*tan(c+d*x))^2/(d*sqrt(tan(c+d*x)))],
[(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x))/tan(c+d*x)^(5/2),x,13,(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-2/3*a^2*(7*A*b+3*a*B)/(d*sqrt(tan(c+d*x)))+2/3*b^2*(a*A+3*b*B)*sqrt(tan(c+d*x))/d-2/3*a*A*(a+b*tan(c+d*x))^2/(d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x))/tan(c+d*x)^(7/2),x,13,-(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2/5*a*(5*a^2*A-14*A*b^2-15*a*b*B)/(d*sqrt(tan(c+d*x)))-2/15*a^2*(9*A*b+5*a*B)/(d*tan(c+d*x)^(3/2))-2/5*a*A*(a+b*tan(c+d*x))^2/(d*tan(c+d*x)^(5/2))],

# n<0
[tan(c+d*x)^(5/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,16,-2*a^(5/2)*(A*b-a*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(b^(5/2)*(a^2+b^2)*d)+(a*(A-B)+b*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-(a*(A-B)+b*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(b*(A-B)-a*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-1/2*(b*(A-B)-a*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+2*(A*b-a*B)*sqrt(tan(c+d*x))/(b^2*d)+2/3*B*tan(c+d*x)^(3/2)/(b*d)],
[tan(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,15,2*a^(3/2)*(A*b-a*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(b^(3/2)*(a^2+b^2)*d)-(b*(A-B)-a*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(b*(A-B)-a*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a*(A-B)+b*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-1/2*(a*(A-B)+b*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+2*B*sqrt(tan(c+d*x))/(b*d)],
[sqrt(tan(c+d*x))*(A+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,14,-(a*(A-B)+b*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(a*(A-B)+b*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(b*(A-B)-a*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+1/2*(b*(A-B)-a*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-2*(A*b-a*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))*sqrt(a)/((a^2+b^2)*d*sqrt(b))],
[(A+B*tan(c+d*x))/(sqrt(tan(c+d*x))*(a+b*tan(c+d*x))),x,14,(b*(A-B)-a*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-(b*(A-B)-a*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a*(A-B)+b*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+1/2*(a*(A-B)+b*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+2*(A*b-a*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))*sqrt(b)/((a^2+b^2)*d*sqrt(a))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))),x,15,-2*b^(3/2)*(A*b-a*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(3/2)*(a^2+b^2)*d)+(a*(A-B)+b*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-(a*(A-B)+b*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(b*(A-B)-a*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-1/2*(b*(A-B)-a*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-2*A/(a*d*sqrt(tan(c+d*x)))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))),x,16,2*b^(5/2)*(A*b-a*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(5/2)*(a^2+b^2)*d)-(b*(A-B)-a*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(b*(A-B)-a*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a*(A-B)+b*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-1/2*(a*(A-B)+b*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+2*(A*b-a*B)/(a^2*d*sqrt(tan(c+d*x)))-2/3*A/(a*d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(5/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,16,a^(3/2)*(a^2*A*b+5*A*b^3-3*a^3*B-7*a*b^2*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(b^(5/2)*(a^2+b^2)^2*d)+(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))-1/2*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))-(a*A*b-3*a^2*B-2*b^2*B)*sqrt(tan(c+d*x))/(b^2*(a^2+b^2)*d)+a*(A*b-a*B)*tan(c+d*x)^(3/2)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,15,-(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))+(a^2*A*b-3*A*b^3+a^3*B+5*a*b^2*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))*sqrt(a)/(b^(3/2)*(a^2+b^2)^2*d)+a*(A*b-a*B)*sqrt(tan(c+d*x))/(b*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[sqrt(tan(c+d*x))*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,15,-(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))+1/2*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))-(3*a^2*A*b-A*b^3-a^3*B+3*a*b^2*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/((a^2+b^2)^2*d*sqrt(a)*sqrt(b))-(A*b-a*B)*sqrt(tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^2),x,15,(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))+(5*a^2*A*b+A*b^3-3*a^3*B+a*b^2*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))*sqrt(b)/(a^(3/2)*(a^2+b^2)^2*d)+b*(A*b-a*B)*sqrt(tan(c+d*x))/(a*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^2),x,16,-b^(3/2)*(7*a^2*A*b+3*A*b^3-5*a^3*B-a*b^2*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(5/2)*(a^2+b^2)^2*d)+(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))-1/2*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))+(-2*a^2*A-3*A*b^2+a*b*B)/(a^2*(a^2+b^2)*d*sqrt(tan(c+d*x)))+b*(A*b-a*B)/(a*(a^2+b^2)*d*sqrt(tan(c+d*x))*(a+b*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^2),x,17,b^(5/2)*(9*a^2*A*b+5*A*b^3-7*a^3*B-3*a*b^2*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(7/2)*(a^2+b^2)^2*d)-(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^2*d*sqrt(2))+(4*a^2*A*b+5*A*b^3-2*a^3*B-3*a*b^2*B)/(a^3*(a^2+b^2)*d*sqrt(tan(c+d*x)))+1/3*(-2*a^2*A-5*A*b^2+3*a*b*B)/(a^2*(a^2+b^2)*d*tan(c+d*x)^(3/2))+b*(A*b-a*B)/(a*(a^2+b^2)*d*tan(c+d*x)^(3/2)*(a+b*tan(c+d*x)))],
[tan(c+d*x)^(7/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^3,x,17,1/4*a^(3/2)*(3*a^4*A*b+6*a^2*A*b^3+35*A*b^5-15*a^5*B-46*a^3*b^2*B-63*a*b^4*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(b^(7/2)*(a^2+b^2)^3*d)+(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))-1/4*(3*a^3*A*b+11*a*A*b^3-15*a^4*B-31*a^2*b^2*B-8*b^4*B)*sqrt(tan(c+d*x))/(b^3*(a^2+b^2)^2*d)+1/2*a*(A*b-a*B)*tan(c+d*x)^(5/2)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+1/4*a*(a^2*A*b+9*A*b^3-5*a^3*B-13*a*b^2*B)*tan(c+d*x)^(3/2)/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^(5/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^3,x,16,(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))-1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/4*(a^4*A*b+18*a^2*A*b^3-15*A*b^5+3*a^5*B+6*a^3*b^2*B+35*a*b^4*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))*sqrt(a)/(b^(5/2)*(a^2+b^2)^3*d)+1/2*a*(A*b-a*B)*tan(c+d*x)^(3/2)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)-1/4*a*(a^2*A*b-7*A*b^3+3*a^3*B+11*a*b^2*B)*sqrt(tan(c+d*x))/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^3,x,16,-(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/4*(3*a^4*A*b-26*a^2*A*b^3+3*A*b^5+a^5*B+18*a^3*b^2*B-15*a*b^4*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(b^(3/2)*(a^2+b^2)^3*d*sqrt(a))+1/2*a*(A*b-a*B)*sqrt(tan(c+d*x))/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+1/4*(3*a^2*A*b-5*A*b^3+a^3*B+9*a*b^2*B)*sqrt(tan(c+d*x))/(b*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[sqrt(tan(c+d*x))*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^3,x,16,-(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))-1/4*(15*a^4*A*b-18*a^2*A*b^3-A*b^5-3*a^5*B+26*a^3*b^2*B-3*a*b^4*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(3/2)*(a^2+b^2)^3*d*sqrt(b))-1/2*(A*b-a*B)*sqrt(tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x))^2)-1/4*(7*a^2*A*b-A*b^3-3*a^3*B+5*a*b^2*B)*sqrt(tan(c+d*x))/(a*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^3),x,16,(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/4*(35*a^4*A*b+6*a^2*A*b^3+3*A*b^5-15*a^5*B+18*a^3*b^2*B+a*b^4*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))*sqrt(b)/(a^(5/2)*(a^2+b^2)^3*d)+1/2*b*(A*b-a*B)*sqrt(tan(c+d*x))/(a*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+1/4*b*(11*a^2*A*b+3*A*b^3-7*a^3*B+a*b^2*B)*sqrt(tan(c+d*x))/(a^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^3),x,17,-1/4*b^(3/2)*(63*a^4*A*b+46*a^2*A*b^3+15*A*b^5-35*a^5*B-6*a^3*b^2*B-3*a*b^4*B)*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(7/2)*(a^2+b^2)^3*d)+(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))-1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)^3*d*sqrt(2))+1/4*(-8*a^4*A-31*a^2*A*b^2-15*A*b^4+11*a^3*b*B+3*a*b^3*B)/(a^3*(a^2+b^2)^2*d*sqrt(tan(c+d*x)))+1/2*b*(A*b-a*B)/(a*(a^2+b^2)*d*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^2)+1/4*b*(13*a^2*A*b+5*A*b^3-9*a^3*B-a*b^2*B)/(a^2*(a^2+b^2)^2*d*sqrt(tan(c+d*x))*(a+b*tan(c+d*x)))],
[tan(c+d*x)^(5/2)*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x)),x,13,B*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-B*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*B*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*B*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2/3*B*tan(c+d*x)^(3/2)/d],
[tan(c+d*x)^(3/2)*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x)),x,13,B*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-B*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*B*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*B*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+2*B*sqrt(tan(c+d*x))/d],
[sqrt(tan(c+d*x))*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x)),x,12,-B*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+B*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*B*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*B*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))],
[(a*B+b*B*tan(c+d*x))/(sqrt(tan(c+d*x))*(a+b*tan(c+d*x))),x,12,-B*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+B*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*B*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*B*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))],
[(a*B+b*B*tan(c+d*x))/(tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))),x,13,B*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-B*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-1/2*B*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))+1/2*B*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-2*B/(d*sqrt(tan(c+d*x)))],
[(a*B+b*B*tan(c+d*x))/(tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))),x,13,B*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))-B*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2))+1/2*B*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-1/2*B*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2))-2/3*B/(d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(5/2)*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,16,-2*a^(5/2)*B*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(b^(3/2)*(a^2+b^2)*d)+(a+b)*B*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-(a+b)*B*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a-b)*B*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+1/2*(a-b)*B*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+2*B*sqrt(tan(c+d*x))/(b*d)],
[tan(c+d*x)^(3/2)*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,15,(a-b)*B*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-(a-b)*B*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a+b)*B*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-1/2*(a+b)*B*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+2*a^(3/2)*B*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/((a^2+b^2)*d*sqrt(b))],
[sqrt(tan(c+d*x))*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,15,-(a+b)*B*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(a+b)*B*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a-b)*B*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-1/2*(a-b)*B*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-2*B*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))*sqrt(a)*sqrt(b)/((a^2+b^2)*d)],
[(a*B+b*B*tan(c+d*x))/(sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^2),x,15,-(a-b)*B*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(a-b)*B*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a+b)*B*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+1/2*(a+b)*B*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+2*b^(3/2)*B*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/((a^2+b^2)*d*sqrt(a))],
[(a*B+b*B*tan(c+d*x))/(tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^2),x,16,-2*b^(5/2)*B*arctan(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a))/(a^(3/2)*(a^2+b^2)*d)+(a+b)*B*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-(a+b)*B*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a-b)*B*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))+1/2*(a-b)*B*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/((a^2+b^2)*d*sqrt(2))-2*B/(a*d*sqrt(tan(c+d*x)))],

# Integrands of the form Tan[c+d x]^(m/2) (A+B Tan[c+d x]) (a+b Tan[c+d x])^(n/2)

# n>0
[sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(3/2)*(A+B*tan(c+d*x)),x,14,1/4*(4*a*A*b-a^2*B-8*b^2*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(b^(3/2)*d)+(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)/d+(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)/d+1/4*(4*A*b-a*B)*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/(b*d)+1/2*B*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2)/(b*d)],
[sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,13,(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)/d+(2*A*b+a*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(b))-(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)/d+B*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/d],
[sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x))/sqrt(tan(c+d*x)),x,12,-(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)/d+2*B*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(b)/d-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)/d],
[sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x))/tan(c+d*x)^(3/2),x,8,-(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)/d+(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)/d-2*A*sqrt(a+b*tan(c+d*x))/(d*sqrt(tan(c+d*x)))],
[sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x))/tan(c+d*x)^(5/2),x,9,(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)/d+(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)/d-2/3*(A*b+3*a*B)*sqrt(a+b*tan(c+d*x))/(a*d*sqrt(tan(c+d*x)))-2/3*A*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x))/tan(c+d*x)^(7/2),x,10,(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)/d-(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)/d+2/15*(15*a^2*A+2*A*b^2-5*a*b*B)*sqrt(a+b*tan(c+d*x))/(a^2*d*sqrt(tan(c+d*x)))-2/5*A*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(5/2))-2/15*(A*b+5*a*B)*sqrt(a+b*tan(c+d*x))/(a*d*tan(c+d*x)^(3/2))],
[sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x))/tan(c+d*x)^(9/2),x,11,-(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)/d-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)/d+2/105*(35*a^2*A*b-8*A*b^3+105*a^3*B+14*a*b^2*B)*sqrt(a+b*tan(c+d*x))/(a^3*d*sqrt(tan(c+d*x)))-2/7*A*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(7/2))-2/35*(A*b+7*a*B)*sqrt(a+b*tan(c+d*x))/(a*d*tan(c+d*x)^(5/2))+2/105*(35*a^2*A+4*A*b^2-7*a*b*B)*sqrt(a+b*tan(c+d*x))/(a^2*d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,15,(I*a-b)^(3/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+1/8*(6*a^2*A*b-16*A*b^3-a^3*B-24*a*b^2*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(b^(3/2)*d)+(I*a+b)^(3/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+1/8*(6*a*A*b-a^2*B-8*b^2*B)*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/(b*d)+1/12*(6*A*b-a*B)*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2)/(b*d)+1/3*B*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(5/2)/(b*d)],
[sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,14,(I*a+b)^(3/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+(a+I*b)^2*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))+1/4*(12*a*A*b+3*a^2*B-8*b^2*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(b))+1/4*(4*A*b+5*a*B)*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/d+1/2*b*B*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(3/2)/d],
[(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x))/sqrt(tan(c+d*x)),x,13,-(I*a-b)^(3/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-(I*a+b)^(3/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+(2*A*b+3*a*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(b)/d+b*B*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/d],
[(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(3/2),x,13,2*b^(3/2)*B*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-(I*a+b)^(3/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-(a+I*b)^2*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))-2*a*A*sqrt(a+b*tan(c+d*x))/(d*sqrt(tan(c+d*x)))],
[(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(5/2),x,9,(I*a-b)^(3/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+(I*a+b)^(3/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-2/3*(4*A*b+3*a*B)*sqrt(a+b*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/3*a*A*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(7/2),x,10,(I*a+b)^(3/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+(a+I*b)^2*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))+2/15*(15*a^2*A-3*A*b^2-20*a*b*B)*sqrt(a+b*tan(c+d*x))/(a*d*sqrt(tan(c+d*x)))-2/5*a*A*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(5/2))-2/15*(6*A*b+5*a*B)*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(9/2),x,11,-(I*a-b)^(3/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-(I*a+b)^(3/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+2/105*(140*a^2*A*b+6*A*b^3+105*a^3*B-21*a*b^2*B)*sqrt(a+b*tan(c+d*x))/(a^2*d*sqrt(tan(c+d*x)))-2/7*a*A*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(7/2))-2/35*(8*A*b+7*a*B)*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(5/2))+2/105*(35*a^2*A-3*A*b^2-42*a*b*B)*sqrt(a+b*tan(c+d*x))/(a*d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(11/2),x,12,(I*a-b)^(3/2)*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-(I*a+b)^(3/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-2/315*(315*a^4*A-63*a^2*A*b^2+8*A*b^4-420*a^3*b*B-18*a*b^3*B)*sqrt(a+b*tan(c+d*x))/(a^3*d*sqrt(tan(c+d*x)))-2/9*a*A*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(9/2))-2/63*(10*A*b+9*a*B)*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(7/2))+2/105*(21*a^2*A-A*b^2-24*a*b*B)*sqrt(a+b*tan(c+d*x))/(a*d*tan(c+d*x)^(5/2))+2/315*(126*a^2*A*b+4*A*b^3+105*a^3*B-9*a*b^2*B)*sqrt(a+b*tan(c+d*x))/(a^2*d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,16,-(I*a-b)^(5/2)*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+1/64*(40*a^3*A*b-320*a*A*b^3-5*a^4*B-240*a^2*b^2*B+128*b^4*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(b^(3/2)*d)-(I*a+b)^(5/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+1/64*(40*a^2*A*b-64*A*b^3-5*a^3*B-112*a*b^2*B)*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/(b*d)+1/96*(40*a*A*b-5*a^2*B-48*b^2*B)*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2)/(b*d)+1/24*(8*A*b-a*B)*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(5/2)/(b*d)+1/4*B*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(7/2)/(b*d)],
[sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,15,-(I*a-b)^(5/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+(I*a+b)^(5/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+1/8*(30*a^2*A*b-16*A*b^3+5*a^3*B-40*a*b^2*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(b))+1/8*(14*a*A*b+5*a^2*B-8*b^2*B)*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/d+1/4*(2*A*b+3*a*B)*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2)/d+1/3*b*B*tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(3/2)/d],
[(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/sqrt(tan(c+d*x)),x,14,(I*a-b)^(5/2)*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+(I*a+b)^(5/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+1/4*(20*a*A*b+15*a^2*B-8*b^2*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(b)/d+1/4*b*(4*A*b+7*a*B)*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/d+1/2*b*B*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2)/d],
[(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(3/2),x,14,(I*a-b)^(5/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+b^(3/2)*(2*A*b+5*a*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-(I*a+b)^(5/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+b*(2*a*A+b*B)*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/d-2*a*A*(a+b*tan(c+d*x))^(3/2)/(d*sqrt(tan(c+d*x)))],
[(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(5/2),x,14,-(I*a-b)^(5/2)*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+2*b^(5/2)*B*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-(I*a+b)^(5/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-2*a*(2*A*b+a*B)*sqrt(a+b*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/3*a*A*(a+b*tan(c+d*x))^(3/2)/(d*tan(c+d*x)^(3/2))],
[(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(7/2),x,10,-(I*a-b)^(5/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+(I*a+b)^(5/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+2/15*(15*a^2*A-23*A*b^2-35*a*b*B)*sqrt(a+b*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-2/15*a*(8*A*b+5*a*B)*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(3/2))-2/5*a*A*(a+b*tan(c+d*x))^(3/2)/(d*tan(c+d*x)^(5/2))],
[(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(9/2),x,11,(I*a-b)^(5/2)*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+(I*a+b)^(5/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d+2/105*(245*a^2*A*b-15*A*b^3+105*a^3*B-161*a*b^2*B)*sqrt(a+b*tan(c+d*x))/(a*d*sqrt(tan(c+d*x)))-2/35*a*(10*A*b+7*a*B)*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(5/2))+2/105*(35*a^2*A-45*A*b^2-77*a*b*B)*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(3/2))-2/7*a*A*(a+b*tan(c+d*x))^(3/2)/(d*tan(c+d*x)^(7/2))],
[(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(11/2),x,12,(I*a-b)^(5/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-(I*a+b)^(5/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-2/315*(315*a^4*A-483*a^2*A*b^2-10*A*b^4-735*a^3*b*B+45*a*b^3*B)*sqrt(a+b*tan(c+d*x))/(a^2*d*sqrt(tan(c+d*x)))-2/21*a*(4*A*b+3*a*B)*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(7/2))+2/105*(21*a^2*A-25*A*b^2-45*a*b*B)*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(5/2))+2/315*(231*a^2*A*b-5*A*b^3+105*a^3*B-135*a*b^2*B)*sqrt(a+b*tan(c+d*x))/(a*d*tan(c+d*x)^(3/2))-2/9*a*A*(a+b*tan(c+d*x))^(3/2)/(d*tan(c+d*x)^(9/2))],
[(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/tan(c+d*x)^(13/2),x,13,-(I*a-b)^(5/2)*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-(I*a+b)^(5/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-2/3465*(8085*a^4*A*b-495*a^2*A*b^3+40*A*b^5+3465*a^5*B-5313*a^3*b^2*B-110*a*b^4*B)*sqrt(a+b*tan(c+d*x))/(a^3*d*sqrt(tan(c+d*x)))-2/99*a*(14*A*b+11*a*B)*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(9/2))+2/693*(99*a^2*A-113*A*b^2-209*a*b*B)*sqrt(a+b*tan(c+d*x))/(d*tan(c+d*x)^(7/2))+2/1155*(495*a^2*A*b-5*A*b^3+231*a^3*B-275*a*b^2*B)*sqrt(a+b*tan(c+d*x))/(a*d*tan(c+d*x)^(5/2))-2/3465*(1155*a^4*A-1485*a^2*A*b^2-20*A*b^4-2541*a^3*b*B+55*a*b^3*B)*sqrt(a+b*tan(c+d*x))/(a^2*d*tan(c+d*x)^(3/2))-2/11*a*A*(a+b*tan(c+d*x))^(3/2)/(d*tan(c+d*x)^(11/2))],
[(a+b*tan(c+d*x))^(5/2)*(3/2*b*B/a+B*tan(c+d*x))/tan(c+d*x)^(5/2),x,14,1/2*(I*a-b)^(5/2)*(2*a-3*I*b)*B*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(a*d)+2*b^(5/2)*B*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/d-1/2*(2*a+3*I*b)*(I*a+b)^(5/2)*B*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(a*d)-2*(a^2+3*b^2)*B*sqrt(a+b*tan(c+d*x))/(d*sqrt(tan(c+d*x)))-b*B*(a+b*tan(c+d*x))^(3/2)/(d*tan(c+d*x)^(3/2))],

# n<0
[tan(c+d*x)^(3/2)*(A+B*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,13,(2*A*b-a*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(b^(3/2)*d)-(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))-(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))+B*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))/(b*d)],
[sqrt(tan(c+d*x))*(A+B*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,12,(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))+2*B*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(b))-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))],
[(A+B*tan(c+d*x))/(sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x))),x,7,(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))+(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))],
[(A+B*tan(c+d*x))/(sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(3/2)),x,8,-(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))+(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))-2*A*sqrt(a+b*tan(c+d*x))/(a*d*sqrt(tan(c+d*x)))],
[(A+B*tan(c+d*x))/(sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(5/2)),x,9,-(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))-(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))+2/3*(2*A*b-3*a*B)*sqrt(a+b*tan(c+d*x))/(a^2*d*sqrt(tan(c+d*x)))-2/3*A*sqrt(a+b*tan(c+d*x))/(a*d*tan(c+d*x)^(3/2))],
[(A+B*tan(c+d*x))/(sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(7/2)),x,10,(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))+2/15*(15*a^2*A-8*A*b^2+10*a*b*B)*sqrt(a+b*tan(c+d*x))/(a^3*d*sqrt(tan(c+d*x)))-2/5*A*sqrt(a+b*tan(c+d*x))/(a*d*tan(c+d*x)^(5/2))+2/15*(4*A*b-5*a*B)*sqrt(a+b*tan(c+d*x))/(a^2*d*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,13,-(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(3/2)*d)+2*B*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(b^(3/2)*d)-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(3/2)*d)+2*a*(A*b-a*B)*sqrt(tan(c+d*x))/(b*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[sqrt(tan(c+d*x))*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,8,-(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(3/2)*d)+(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(3/2)*d)-2*(A*b-a*B)*sqrt(tan(c+d*x))/((a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2)),x,8,(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(3/2)*d)+(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(3/2)*d)+2*b*(A*b-a*B)*sqrt(tan(c+d*x))/(a*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(3/2)),x,9,(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(3/2)*d)-(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(3/2)*d)-2*A/(a*d*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x)))-2*b*(a^2*A+2*A*b^2-a*b*B)*sqrt(tan(c+d*x))/(a^2*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(3/2)),x,10,-(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(3/2)*d)-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(3/2)*d)+2/3*(4*A*b-3*a*B)/(a^2*d*sqrt(tan(c+d*x))*sqrt(a+b*tan(c+d*x)))+2/3*b*(5*a^2*A*b+8*A*b^3-3*a^3*B-6*a*b^2*B)*sqrt(tan(c+d*x))/(a^3*(a^2+b^2)*d*sqrt(a+b*tan(c+d*x)))-2/3*A/(a*d*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(3/2))],
[tan(c+d*x)^(5/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,14,(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(5/2)*d)+2*B*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(b^(5/2)*d)-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(5/2)*d)+2*a*(2*A*b^3-a*(a^2+3*b^2)*B)*sqrt(tan(c+d*x))/(b^2*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+2/3*a*(A*b-a*B)*tan(c+d*x)^(3/2)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,9,(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(5/2)*d)+(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(5/2)*d)+2/3*(2*a^2*A*b-4*A*b^3+a^3*B+7*a*b^2*B)*sqrt(tan(c+d*x))/(b*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+2/3*a*(A*b-a*B)*sqrt(tan(c+d*x))/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[sqrt(tan(c+d*x))*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,9,-(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(5/2)*d)+(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(5/2)*d)-2/3*(5*a^2*A*b-A*b^3-2*a^3*B+4*a*b^2*B)*sqrt(tan(c+d*x))/(a*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))-2/3*(A*b-a*B)*sqrt(tan(c+d*x))/((a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(5/2)),x,9,-(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(5/2)*d)-(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(5/2)*d)+2/3*b*(8*a^2*A*b+2*A*b^3-5*a^3*B+a*b^2*B)*sqrt(tan(c+d*x))/(a^2*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+2/3*b*(A*b-a*B)*sqrt(tan(c+d*x))/(a*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(5/2)),x,10,(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(5/2)*d)-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(5/2)*d)-2/3*b*(3*a^4*A+17*a^2*A*b^2+8*A*b^4-8*a^3*b*B-2*a*b^3*B)*sqrt(tan(c+d*x))/(a^3*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))-2*A/(a*d*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2))-2/3*b*(3*a^2*A+4*A*b^2-a*b*B)*sqrt(tan(c+d*x))/(a^2*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(5/2)),x,11,(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a-b)^(5/2)*d)+(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/((I*a+b)^(5/2)*d)+2/3*b*(8*a^4*A*b+30*a^2*A*b^3+16*A*b^5-3*a^5*B-17*a^3*b^2*B-8*a*b^4*B)*sqrt(tan(c+d*x))/(a^4*(a^2+b^2)^2*d*sqrt(a+b*tan(c+d*x)))+2*(2*A*b-a*B)/(a^2*d*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2))+2/3*b*(7*a^2*A*b+8*A*b^3-3*a^3*B-4*a*b^2*B)*sqrt(tan(c+d*x))/(a^3*(a^2+b^2)*d*(a+b*tan(c+d*x))^(3/2))-2/3*A/(a*d*tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(3/2))],
[tan(c+d*x)^(3/2)*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,13,-B*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))+2*B*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(b))-B*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))],
[sqrt(tan(c+d*x))*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,8,I*B*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))-I*B*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))],
[(a*B+b*B*tan(c+d*x))/(sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^(3/2)),x,8,B*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))+B*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))],
[(a*B+b*B*tan(c+d*x))/(tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(3/2)),x,10,-I*B*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a-b))+I*B*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))/(d*sqrt(I*a+b))-2*B*sqrt(a+b*tan(c+d*x))/(a*d*sqrt(tan(c+d*x)))],

# Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+b Tan[c+d x])^(n/3)

# n>0
[(a+b*tan(c+d*x))^(2/3)*(A+B*tan(c+d*x)),x,12,-1/4*(a-I*b)^(2/3)*(A-I*B)*x-1/4*(a+I*b)^(2/3)*(A+I*B)*x-1/4*(a+I*b)^(2/3)*(I*A-B)*log(cos(c+d*x))/d+1/4*(a-I*b)^(2/3)*(I*A+B)*log(cos(c+d*x))/d+3/4*(a-I*b)^(2/3)*(I*A+B)*log((a-I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/d-3/4*(a+I*b)^(2/3)*(I*A-B)*log((a+I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/d+1/2*(a-I*b)^(2/3)*(I*A+B)*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a-I*b)^(1/3))/sqrt(3))*sqrt(3)/d-1/2*(a+I*b)^(2/3)*(I*A-B)*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a+I*b)^(1/3))/sqrt(3))*sqrt(3)/d+3/2*B*(a+b*tan(c+d*x))^(2/3)/d],
[(a+b*tan(c+d*x))^(1/3)*(A+B*tan(c+d*x)),x,12,-1/4*(a-I*b)^(1/3)*(A-I*B)*x-1/4*(a+I*b)^(1/3)*(A+I*B)*x-1/4*(a+I*b)^(1/3)*(I*A-B)*log(cos(c+d*x))/d+1/4*(a-I*b)^(1/3)*(I*A+B)*log(cos(c+d*x))/d+3/4*(a-I*b)^(1/3)*(I*A+B)*log((a-I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/d-3/4*(a+I*b)^(1/3)*(I*A-B)*log((a+I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/d-1/2*(a-I*b)^(1/3)*(I*A+B)*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a-I*b)^(1/3))/sqrt(3))*sqrt(3)/d+1/2*(a+I*b)^(1/3)*(I*A-B)*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a+I*b)^(1/3))/sqrt(3))*sqrt(3)/d+3*B*(a+b*tan(c+d*x))^(1/3)/d],

# n<0
[(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(1/3),x,11,-1/4*(A-I*B)*x/(a-I*b)^(1/3)-1/4*(A+I*B)*x/(a+I*b)^(1/3)-1/4*(I*A-B)*log(cos(c+d*x))/((a+I*b)^(1/3)*d)+1/4*(I*A+B)*log(cos(c+d*x))/((a-I*b)^(1/3)*d)+3/4*(I*A+B)*log((a-I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/((a-I*b)^(1/3)*d)-3/4*(I*A-B)*log((a+I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/((a+I*b)^(1/3)*d)+1/2*(I*A+B)*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a-I*b)^(1/3))/sqrt(3))*sqrt(3)/((a-I*b)^(1/3)*d)-1/2*(I*A-B)*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a+I*b)^(1/3))/sqrt(3))*sqrt(3)/((a+I*b)^(1/3)*d)],
[(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(2/3),x,11,-1/4*(A-I*B)*x/(a-I*b)^(2/3)-1/4*(A+I*B)*x/(a+I*b)^(2/3)-1/4*(I*A-B)*log(cos(c+d*x))/((a+I*b)^(2/3)*d)+1/4*(I*A+B)*log(cos(c+d*x))/((a-I*b)^(2/3)*d)+3/4*(I*A+B)*log((a-I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/((a-I*b)^(2/3)*d)-3/4*(I*A-B)*log((a+I*b)^(1/3)-(a+b*tan(c+d*x))^(1/3))/((a+I*b)^(2/3)*d)-1/2*(I*A+B)*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a-I*b)^(1/3))/sqrt(3))*sqrt(3)/((a-I*b)^(2/3)*d)+1/2*(I*A-B)*arctan((1+2*(a+b*tan(c+d*x))^(1/3)/(a+I*b)^(1/3))/sqrt(3))*sqrt(3)/((a+I*b)^(2/3)*d)],
[(I-tan(e+f*x))/(c+d*tan(e+f*x))^(1/3),x,5,-1/2*I*x/(c-I*d)^(1/3)-1/2*log(cos(e+f*x))/((c-I*d)^(1/3)*f)-3/2*log((c-I*d)^(1/3)-(c+d*tan(e+f*x))^(1/3))/((c-I*d)^(1/3)*f)-arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c-I*d)^(1/3))/sqrt(3))*sqrt(3)/((c-I*d)^(1/3)*f)],
[(d-c*tan(e+f*x))/(c+d*tan(e+f*x))^(2/3),x,11,-1/4*I*(c-I*d)^(1/3)*x+1/4*I*(c+I*d)^(1/3)*x-1/4*(c-I*d)^(1/3)*log(cos(e+f*x))/f-1/4*(c+I*d)^(1/3)*log(cos(e+f*x))/f-3/4*(c-I*d)^(1/3)*log((c-I*d)^(1/3)-(c+d*tan(e+f*x))^(1/3))/f-3/4*(c+I*d)^(1/3)*log((c+I*d)^(1/3)-(c+d*tan(e+f*x))^(1/3))/f+1/2*(c-I*d)^(1/3)*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c-I*d)^(1/3))/sqrt(3))*sqrt(3)/f+1/2*(c+I*d)^(1/3)*arctan((1+2*(c+d*tan(e+f*x))^(1/3)/(c+I*d)^(1/3))/sqrt(3))*sqrt(3)/f],

# Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+b Tan[c+d x])^n with m symbolic
[tan(c+d*x)^m*(a+b*tan(c+d*x))^4*(A+B*tan(c+d*x)),x,9,-b*(A*b^3*(12+7*m+m^2)+4*a*b^2*B*(12+7*m+m^2)-2*a^3*B*(19+8*m+m^2)-a^2*A*b*(68+37*m+5*m^2))*tan(c+d*x)^(1+m)/(d*(4+m)*(3+4*m+m^2))+(a^4*A-6*a^2*A*b^2+A*b^4-4*a^3*b*B+4*a*b^3*B)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(c+d*x)^2)*tan(c+d*x)^(1+m)/(d*(1+m))+b^2*(2*a*A*b*(4+m)^2-b^2*B*(12+7*m+m^2)+a^2*B*(26+9*m+m^2))*tan(c+d*x)^(2+m)/(d*(2+m)*(3+m)*(4+m))+(4*a^3*A*b-4*a*A*b^3+a^4*B-6*a^2*b^2*B+b^4*B)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-tan(c+d*x)^2)*tan(c+d*x)^(2+m)/(d*(2+m))+b*(A*b*(4+m)+a*B*(7+m))*tan(c+d*x)^(1+m)*(a+b*tan(c+d*x))^2/(d*(3+m)*(4+m))+b*B*tan(c+d*x)^(1+m)*(a+b*tan(c+d*x))^3/(d*(4+m))],
[tan(c+d*x)^m*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,8,b*(3*a*A*b*(3+m)-b^2*B*(3+m)+2*a^2*B*(4+m))*tan(c+d*x)^(1+m)/(d*(1+m)*(3+m))+(a^3*A-3*a*A*b^2-3*a^2*b*B+b^3*B)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(c+d*x)^2)*tan(c+d*x)^(1+m)/(d*(1+m))+b^2*(A*b*(3+m)+a*B*(5+m))*tan(c+d*x)^(2+m)/(d*(2+m)*(3+m))+(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-tan(c+d*x)^2)*tan(c+d*x)^(2+m)/(d*(2+m))+b*B*tan(c+d*x)^(1+m)*(a+b*tan(c+d*x))^2/(d*(3+m))],
[tan(c+d*x)^m*(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,7,b*(A*b*(2+m)+a*B*(3+m))*tan(c+d*x)^(1+m)/(d*(1+m)*(2+m))+(a^2*A-A*b^2-2*a*b*B)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(c+d*x)^2)*tan(c+d*x)^(1+m)/(d*(1+m))+(2*a*A*b+a^2*B-b^2*B)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-tan(c+d*x)^2)*tan(c+d*x)^(2+m)/(d*(2+m))+b*B*tan(c+d*x)^(1+m)*(a+b*tan(c+d*x))/(d*(2+m))],
[tan(c+d*x)^m*(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,6,b*B*tan(c+d*x)^(1+m)/(d*(1+m))+(a*A-b*B)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(c+d*x)^2)*tan(c+d*x)^(1+m)/(d*(1+m))+(A*b+a*B)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-tan(c+d*x)^2)*tan(c+d*x)^(2+m)/(d*(2+m))],
[tan(c+d*x)^m*(A+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,8,(a*A+b*B)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(c+d*x)^2)*tan(c+d*x)^(1+m)/((a^2+b^2)*d*(1+m))+b*(A*b-a*B)*hypergeom([1,1+m],[2+m],-b*tan(c+d*x)/a)*tan(c+d*x)^(1+m)/(a*(a^2+b^2)*d*(1+m))-(A*b-a*B)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-tan(c+d*x)^2)*tan(c+d*x)^(2+m)/((a^2+b^2)*d*(2+m))],
[tan(c+d*x)^m*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,9,(a^2*A-A*b^2+2*a*b*B)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(c+d*x)^2)*tan(c+d*x)^(1+m)/((a^2+b^2)^2*d*(1+m))+b*(a^2*A*b*(2-m)-A*b^3*m+a*b^2*B*(1+m)-a^3*(B-B*m))*hypergeom([1,1+m],[2+m],-b*tan(c+d*x)/a)*tan(c+d*x)^(1+m)/(a^2*(a^2+b^2)^2*d*(1+m))-(2*a*A*b-a^2*B+b^2*B)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-tan(c+d*x)^2)*tan(c+d*x)^(2+m)/((a^2+b^2)^2*d*(2+m))+b*(A*b-a*B)*tan(c+d*x)^(1+m)/(a*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^m*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^3,x,10,(a^3*A-3*a*A*b^2+3*a^2*b*B-b^3*B)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(c+d*x)^2)*tan(c+d*x)^(1+m)/((a^2+b^2)^3*d*(1+m))-1/2*b*(A*b^5*(1-m)*m+a*b^4*B*m*(1+m)-2*a^3*b^2*B*(3+m-m^2)+2*a^2*A*b^3*(1+3*m-m^2)-a^4*A*b*(6-5*m+m^2)+a^5*B*(2-3*m+m^2))*hypergeom([1,1+m],[2+m],-b*tan(c+d*x)/a)*tan(c+d*x)^(1+m)/(a^3*(a^2+b^2)^3*d*(1+m))-(3*a^2*A*b-A*b^3-a^3*B+3*a*b^2*B)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-tan(c+d*x)^2)*tan(c+d*x)^(2+m)/((a^2+b^2)^3*d*(2+m))+1/2*b*(A*b-a*B)*tan(c+d*x)^(1+m)/(a*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+1/2*b*(A*b^3*(1-m)-a^3*B*(3-m)+a^2*A*b*(5-m)+a*b^2*B*(1+m))*tan(c+d*x)^(1+m)/(a^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^m*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^4,x,11,(a^4*A-6*a^2*A*b^2+A*b^4+4*a^3*b*B-4*a*b^3*B)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(c+d*x)^2)*tan(c+d*x)^(1+m)/((a^2+b^2)^4*d*(1+m))-1/6*b*(a*b^6*B*m*(1-m^2)+3*a^2*A*b^5*m*(2-5*m+m^2)+A*b^7*m*(2-3*m+m^2)+3*a^3*b^4*B*(2+5*m+2*m^2-m^3)+a^7*B*(6-11*m+6*m^2-m^3)-a^6*A*b*(24-26*m+9*m^2-m^3)+3*a^4*A*b^3*(8+10*m-7*m^2+m^3)-3*a^5*b^2*B*(12-m-4*m^2+m^3))*hypergeom([1,1+m],[2+m],-b*tan(c+d*x)/a)*tan(c+d*x)^(1+m)/(a^4*(a^2+b^2)^4*d*(1+m))-(4*a^3*A*b-4*a*A*b^3-a^4*B+6*a^2*b^2*B-b^4*B)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-tan(c+d*x)^2)*tan(c+d*x)^(2+m)/((a^2+b^2)^4*d*(2+m))+1/3*b*(A*b-a*B)*tan(c+d*x)^(1+m)/(a*(a^2+b^2)*d*(a+b*tan(c+d*x))^3)+1/6*b*(A*b^3*(2-m)-a^3*B*(5-m)+a^2*A*b*(8-m)+a*b^2*B*(1+m))*tan(c+d*x)^(1+m)/(a^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x))^2)+1/6*b*(a*b^4*B*(1-m^2)+2*a^3*b^2*B*(7+3*m-m^2)+a^4*A*b*(26-9*m+m^2)+2*a^2*A*b^3*(2-6*m+m^2)-a^5*B*(11-6*m+m^2)+A*b^5*(2-3*m+m^2))*tan(c+d*x)^(1+m)/(a^3*(a^2+b^2)^3*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^m*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,7,1/2*a^2*(A+I*B)*AppellF1(1+m,-5/2,1,2+m,-b*tan(c+d*x)/a,-I*tan(c+d*x))*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(1+b*tan(c+d*x)/a))+1/2*a^2*(A-I*B)*AppellF1(1+m,-5/2,1,2+m,-b*tan(c+d*x)/a,I*tan(c+d*x))*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(1+b*tan(c+d*x)/a))],
[tan(c+d*x)^m*(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,7,1/2*a*(A+I*B)*AppellF1(1+m,-3/2,1,2+m,-b*tan(c+d*x)/a,-I*tan(c+d*x))*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(1+b*tan(c+d*x)/a))+1/2*a*(A-I*B)*AppellF1(1+m,-3/2,1,2+m,-b*tan(c+d*x)/a,I*tan(c+d*x))*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(1+b*tan(c+d*x)/a))],
[tan(c+d*x)^m*(a+b*tan(c+d*x))^(1/2)*(A+B*tan(c+d*x)),x,7,1/2*(A+I*B)*AppellF1(1+m,-1/2,1,2+m,-b*tan(c+d*x)/a,-I*tan(c+d*x))*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(1+b*tan(c+d*x)/a))+1/2*(A-I*B)*AppellF1(1+m,-1/2,1,2+m,-b*tan(c+d*x)/a,I*tan(c+d*x))*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(1+b*tan(c+d*x)/a))],
[tan(c+d*x)^m*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(1/2),x,7,1/2*(A+I*B)*AppellF1(1+m,1/2,1,2+m,-b*tan(c+d*x)/a,-I*tan(c+d*x))*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(a+b*tan(c+d*x)))+1/2*(A-I*B)*AppellF1(1+m,1/2,1,2+m,-b*tan(c+d*x)/a,I*tan(c+d*x))*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(1+m)/(d*(1+m)*sqrt(a+b*tan(c+d*x)))],
[tan(c+d*x)^m*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,7,1/2*(A+I*B)*AppellF1(1+m,3/2,1,2+m,-b*tan(c+d*x)/a,-I*tan(c+d*x))*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(1+m)/(a*d*(1+m)*sqrt(a+b*tan(c+d*x)))+1/2*(A-I*B)*AppellF1(1+m,3/2,1,2+m,-b*tan(c+d*x)/a,I*tan(c+d*x))*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(1+m)/(a*d*(1+m)*sqrt(a+b*tan(c+d*x)))],
[tan(c+d*x)^m*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,7,1/2*(A+I*B)*AppellF1(1+m,5/2,1,2+m,-b*tan(c+d*x)/a,-I*tan(c+d*x))*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(1+m)/(a^2*d*(1+m)*sqrt(a+b*tan(c+d*x)))+1/2*(A-I*B)*AppellF1(1+m,5/2,1,2+m,-b*tan(c+d*x)/a,I*tan(c+d*x))*sqrt(1+b*tan(c+d*x)/a)*tan(c+d*x)^(1+m)/(a^2*d*(1+m)*sqrt(a+b*tan(c+d*x)))],

# Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+b Tan[c+d x])^n with n symbolic
[tan(c+d*x)^m*(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,7,1/2*(A+I*B)*AppellF1(1+m,-n,1,2+m,-b*tan(c+d*x)/a,-I*tan(c+d*x))*tan(c+d*x)^(1+m)*(a+b*tan(c+d*x))^n/(d*(1+m)*(1+b*tan(c+d*x)/a)^n)+1/2*(A-I*B)*AppellF1(1+m,-n,1,2+m,-b*tan(c+d*x)/a,I*tan(c+d*x))*tan(c+d*x)^(1+m)*(a+b*tan(c+d*x))^n/(d*(1+m)*(1+b*tan(c+d*x)/a)^n)],
[tan(c+d*x)^4*(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,9,-(A*b^3*(2+n)*(3+n)*(4+n)-a*(b^2*B*(3+n)*(4+n)-2*a*(3*a*B-A*b*(4+n))))*(a+b*tan(c+d*x))^(1+n)/(b^4*d*(1+n)*(2+n)*(3+n)*(4+n))+1/2*(A-I*B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-I*b))*(a+b*tan(c+d*x))^(1+n)/((I*a+b)*d*(1+n))-1/2*(A+I*B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+I*b))*(a+b*tan(c+d*x))^(1+n)/((I*a-b)*d*(1+n))-(b^2*B*(3+n)*(4+n)-2*a*(3*a*B-A*b*(4+n)))*tan(c+d*x)*(a+b*tan(c+d*x))^(1+n)/(b^3*d*(2+n)*(3+n)*(4+n))-(3*a*B-A*b*(4+n))*tan(c+d*x)^2*(a+b*tan(c+d*x))^(1+n)/(b^2*d*(3+n)*(4+n))+B*tan(c+d*x)^3*(a+b*tan(c+d*x))^(1+n)/(b*d*(4+n))],
[tan(c+d*x)^3*(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,8,(2*a^2*B-a*A*b*(3+n)-b^2*B*(6+5*n+n^2))*(a+b*tan(c+d*x))^(1+n)/(b^3*d*(1+n)*(2+n)*(3+n))+1/2*(I*A+B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-I*b))*(a+b*tan(c+d*x))^(1+n)/((I*a+b)*d*(1+n))+1/2*(A+I*B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+I*b))*(a+b*tan(c+d*x))^(1+n)/((a+I*b)*d*(1+n))-(2*a*B-A*b*(3+n))*tan(c+d*x)*(a+b*tan(c+d*x))^(1+n)/(b^2*d*(2+n)*(3+n))+B*tan(c+d*x)^2*(a+b*tan(c+d*x))^(1+n)/(b*d*(3+n))],
[tan(c+d*x)^2*(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,7,-(a*B-A*b*(2+n))*(a+b*tan(c+d*x))^(1+n)/(b^2*d*(1+n)*(2+n))+1/2*(I*A+B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-I*b))*(a+b*tan(c+d*x))^(1+n)/((a-I*b)*d*(1+n))+1/2*(A+I*B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+I*b))*(a+b*tan(c+d*x))^(1+n)/((I*a-b)*d*(1+n))+B*tan(c+d*x)*(a+b*tan(c+d*x))^(1+n)/(b*d*(2+n))],
[tan(c+d*x)*(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,6,B*(a+b*tan(c+d*x))^(1+n)/(b*d*(1+n))-1/2*(A-I*B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-I*b))*(a+b*tan(c+d*x))^(1+n)/((a-I*b)*d*(1+n))-1/2*(A+I*B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+I*b))*(a+b*tan(c+d*x))^(1+n)/((a+I*b)*d*(1+n))],
[(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,5,1/2*(A-I*B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-I*b))*(a+b*tan(c+d*x))^(1+n)/((I*a+b)*d*(1+n))+1/2*(I*A-B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+I*b))*(a+b*tan(c+d*x))^(1+n)/((a+I*b)*d*(1+n))],
[cot(c+d*x)*(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,8,1/2*(I*A+B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-I*b))*(a+b*tan(c+d*x))^(1+n)/((I*a+b)*d*(1+n))+1/2*(A+I*B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+I*b))*(a+b*tan(c+d*x))^(1+n)/((a+I*b)*d*(1+n))-A*hypergeom([1,1+n],[2+n],1+b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^(1+n)/(a*d*(1+n))],
[cot(c+d*x)^2*(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,9,-A*cot(c+d*x)*(a+b*tan(c+d*x))^(1+n)/(a*d)-1/2*(A-I*B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-I*b))*(a+b*tan(c+d*x))^(1+n)/((I*a+b)*d*(1+n))+1/2*(A+I*B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+I*b))*(a+b*tan(c+d*x))^(1+n)/((I*a-b)*d*(1+n))-(a*B+A*b*n)*hypergeom([1,1+n],[2+n],1+b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^(1+n)/(a^2*d*(1+n))],
[cot(c+d*x)^3*(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,10,-1/2*(2*a*B-A*b*(1-n))*cot(c+d*x)*(a+b*tan(c+d*x))^(1+n)/(a^2*d)-1/2*A*cot(c+d*x)^2*(a+b*tan(c+d*x))^(1+n)/(a*d)-1/2*(I*A+B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a-I*b))*(a+b*tan(c+d*x))^(1+n)/((I*a+b)*d*(1+n))-1/2*(A+I*B)*hypergeom([1,1+n],[2+n],(a+b*tan(c+d*x))/(a+I*b))*(a+b*tan(c+d*x))^(1+n)/((a+I*b)*d*(1+n))+1/2*(2*a^2*A-2*a*b*B*n+A*b^2*(1-n)*n)*hypergeom([1,1+n],[2+n],1+b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^(1+n)/(a^3*d*(1+n))],

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

# Integrands of the form (d Cot[e+f x])^m (a+a I Tan[e+f x])^n (A+B Tan[e+f x])

# Integrands of the form Cot[c+d x]^(m/2) (A+B Tan[c+d x]) (a+i a Tan[c+d x])^n

# n>0
[cot(c+d*x)^(7/2)*(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,6,2*(-1)^(1/4)*a*(A-I*B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-2/3*a*(I*A+B)*cot(c+d*x)^(3/2)/d-2/5*a*A*cot(c+d*x)^(5/2)/d+2*a*(A-I*B)*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,5,-2*(-1)^(1/4)*a*(I*A+B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-2/3*a*A*cot(c+d*x)^(3/2)/d-2*a*(I*A+B)*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,4,-2*(-1)^(1/4)*a*(A-I*B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-2*a*A*sqrt(cot(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,4,2*(-1)^(1/4)*a*(I*A+B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d+2*I*a*B/(d*sqrt(cot(c+d*x)))],
[(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x))/sqrt(cot(c+d*x)),x,5,2*(-1)^(1/4)*a*(A-I*B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d+2/3*I*a*B/(d*cot(c+d*x)^(3/2))+2*a*(I*A+B)/(d*sqrt(cot(c+d*x)))],
[(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x))/cot(c+d*x)^(3/2),x,6,-2*(-1)^(1/4)*a*(I*A+B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d+2/5*I*a*B/(d*cot(c+d*x)^(5/2))+2/3*a*(I*A+B)/(d*cot(c+d*x)^(3/2))+2*a*(A-I*B)/(d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(7/2)*(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,6,4*(-1)^(1/4)*a^2*(A-I*B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-2/15*a^2*(7*I*A+5*B)*cot(c+d*x)^(3/2)/d-2/5*A*cot(c+d*x)^(3/2)*(I*a^2+a^2*cot(c+d*x))/d+4*a^2*(A-I*B)*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,5,-4*(-1)^(1/4)*a^2*(I*A+B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-2/3*a^2*(5*I*A+3*B)*sqrt(cot(c+d*x))/d-2/3*A*(I*a^2+a^2*cot(c+d*x))*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,5,-4*(-1)^(1/4)*a^2*(A-I*B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d+2*I*B*(I*a^2+a^2*cot(c+d*x))/(d*sqrt(cot(c+d*x)))-2*a^2*(A+I*B)*sqrt(cot(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,5,4*(-1)^(1/4)*a^2*(I*A+B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d+2/3*I*B*(I*a^2+a^2*cot(c+d*x))/(d*cot(c+d*x)^(3/2))-2/3*a^2*(3*A-5*I*B)/(d*sqrt(cot(c+d*x)))],
[(a+I*a*tan(c+d*x))^2*(A+B*tan(c+d*x))/sqrt(cot(c+d*x)),x,6,4*(-1)^(1/4)*a^2*(A-I*B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-2/15*a^2*(5*A-7*I*B)/(d*cot(c+d*x)^(3/2))+2/5*I*B*(I*a^2+a^2*cot(c+d*x))/(d*cot(c+d*x)^(5/2))+4*a^2*(I*A+B)/(d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(9/2)*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,7,8*(-1)^(1/4)*a^3*(I*A+B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d+8/105*a^3*(23*A-21*I*B)*cot(c+d*x)^(3/2)/d-2/7*a*A*cot(c+d*x)^(3/2)*(I*a+a*cot(c+d*x))^2/d-2/35*(11*I*A+7*B)*cot(c+d*x)^(3/2)*(I*a^3+a^3*cot(c+d*x))/d+8*a^3*(I*A+B)*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(7/2)*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,6,8*(-1)^(1/4)*a^3*(A-I*B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d+16/15*a^3*(6*A-5*I*B)*sqrt(cot(c+d*x))/d-2/5*a*A*(I*a+a*cot(c+d*x))^2*sqrt(cot(c+d*x))/d-2/15*(9*I*A+5*B)*(I*a^3+a^3*cot(c+d*x))*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,6,-8*(-1)^(1/4)*a^3*(I*A+B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d+2*I*a*B*(I*a+a*cot(c+d*x))^2/(d*sqrt(cot(c+d*x)))-16/3*I*a^3*A*sqrt(cot(c+d*x))/d-2/3*(A+3*I*B)*(I*a^3+a^3*cot(c+d*x))*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,6,-8*(-1)^(1/4)*a^3*(A-I*B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d+2/3*I*a*B*(I*a+a*cot(c+d*x))^2/(d*cot(c+d*x)^(3/2))-2/3*(3*A-7*I*B)*(I*a^3+a^3*cot(c+d*x))/(d*sqrt(cot(c+d*x)))-16/3*I*a^3*B*sqrt(cot(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,6,8*(-1)^(1/4)*a^3*(I*A+B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d+2/5*I*a*B*(I*a+a*cot(c+d*x))^2/(d*cot(c+d*x)^(5/2))-2/15*(5*A-9*I*B)*(I*a^3+a^3*cot(c+d*x))/(d*cot(c+d*x)^(3/2))-16/15*a^3*(5*A-6*I*B)/(d*sqrt(cot(c+d*x)))],
[(a+I*a*tan(c+d*x))^3*(A+B*tan(c+d*x))/sqrt(cot(c+d*x)),x,7,8*(-1)^(1/4)*a^3*(A-I*B)*arctanh((-1)^(3/4)*sqrt(cot(c+d*x)))/d-8/105*a^3*(21*A-23*I*B)/(d*cot(c+d*x)^(3/2))+2/7*I*a*B*(I*a+a*cot(c+d*x))^2/(d*cot(c+d*x)^(7/2))-2/35*(7*A-11*I*B)*(I*a^3+a^3*cot(c+d*x))/(d*cot(c+d*x)^(5/2))+8*a^3*(I*A+B)/(d*sqrt(cot(c+d*x)))],

# n<0
[cot(c+d*x)^(5/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x)),x,14,-1/6*(7*A+3*I*B)*cot(c+d*x)^(3/2)/(a*d)+1/2*(A+I*B)*cot(c+d*x)^(5/2)/(d*(I*a+a*cot(c+d*x)))+(-1/4+1/4*I)*((6+I)*A+(1+4*I)*B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+1/4*((7-5*I)*A+(5+3*I)*B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+1/8*((7+5*I)*A+(-5+3*I)*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+1/8*((-7-5*I)*A+(5-3*I)*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+5/2*(I*A-B)*sqrt(cot(c+d*x))/(a*d)],
[cot(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x)),x,13,1/2*(A+I*B)*cot(c+d*x)^(3/2)/(d*(I*a+a*cot(c+d*x)))+1/4*((-5-3*I)*A+(3-I)*B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+1/4*((5+3*I)*A+(-3+I)*B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(-1/8+1/8*I)*((4+I)*A+(1+2*I)*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+1/8*((5-3*I)*A+(3+I)*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))-1/2*(5*A+I*B)*sqrt(cot(c+d*x))/(a*d)],
[sqrt(cot(c+d*x))*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x)),x,12,(1/4-1/4*I)*((2+I)*A+B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(-1/4+1/4*I)*((2+I)*A+B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))-1/8*((3+I)*A+(-1-I)*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+1/8*((3+I)*A+(-1-I)*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+1/2*(A+I*B)*sqrt(cot(c+d*x))/(d*(I*a+a*cot(c+d*x)))],
[(A+B*tan(c+d*x))/(sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))),x,12,(1/4-1/4*I)*(A+(2-I)*B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(-1/4+1/4*I)*(A+(2-I)*B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(1/8+1/8*I)*(A+(-2-I)*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(-1/8-1/8*I)*(A+(-2-I)*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+1/2*(I*A-B)*sqrt(cot(c+d*x))/(d*(I*a+a*cot(c+d*x)))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))),x,13,1/4*((1-3*I)*A+(3+5*I)*B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(1/4+1/4*I)*((1+2*I)*A+(-4-I)*B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(-1/8-1/8*I)*((2+I)*A+(1+4*I)*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(1/8+1/8*I)*((2+I)*A+(1+4*I)*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+1/2*(-A-5*I*B)/(a*d*sqrt(cot(c+d*x)))+1/2*(I*A-B)/(d*(I*a+a*cot(c+d*x))*sqrt(cot(c+d*x)))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))),x,14,1/6*(-3*A-7*I*B)/(a*d*cot(c+d*x)^(3/2))+1/2*(I*A-B)/(d*cot(c+d*x)^(3/2)*(I*a+a*cot(c+d*x)))+(1/4+1/4*I)*((4+I)*A+(1+6*I)*B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(-1/4-1/4*I)*((4+I)*A+(1+6*I)*B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+1/8*((3-5*I)*A+(5+7*I)*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))+(1/8+1/8*I)*((1+4*I)*A+(-6-I)*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a*d*sqrt(2))-5/2*(I*A-B)/(a*d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^2,x,14,1/8*(7*A+3*I*B)*cot(c+d*x)^(3/2)/(a^2*d*(I+cot(c+d*x)))+1/4*(A+I*B)*cot(c+d*x)^(5/2)/(d*(I*a+a*cot(c+d*x))^2)+(-1/16+1/16*I)*((2+23*I)*A+(-7-2*I)*B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+1/16*((25+21*I)*A+(-9+5*I)*B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(-1/32+1/32*I)*((23+2*I)*A+(2+7*I)*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(1/32-1/32*I)*((23+2*I)*A+(2+7*I)*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))-5/8*(5*A+I*B)*sqrt(cot(c+d*x))/(a^2*d)],
[sqrt(cot(c+d*x))*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^2,x,13,1/4*(A+I*B)*cot(c+d*x)^(3/2)/(d*(I*a+a*cot(c+d*x))^2)+1/16*((9-5*I)*A+(1-3*I)*B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(1/16+1/16*I)*((-2+7*I)*A+(1+2*I)*B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(1/32+1/32*I)*((-7+2*I)*A+(2+I)*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+1/32*((9+5*I)*A+(-1-3*I)*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+1/8*(5*A+I*B)*sqrt(cot(c+d*x))/(a^2*d*(I+cot(c+d*x)))],
[(A+B*tan(c+d*x))/(sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^2),x,13,-1/16*((-1+3*I)*A+(1+3*I)*B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+1/16*((-1+3*I)*A+(1+3*I)*B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+1/32*((1+3*I)*A+(1-3*I)*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))-1/32*((1+3*I)*A+(1-3*I)*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+1/8*(3*I*A+B)*sqrt(cot(c+d*x))/(a^2*d*(I+cot(c+d*x)))+1/4*(A+I*B)*sqrt(cot(c+d*x))/(d*(I*a+a*cot(c+d*x))^2)],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^2),x,13,(-1/16-1/16*I)*((2+I)*A+(7-2*I)*B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+1/16*((1+3*I)*A+(9+5*I)*B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+1/32*((1-3*I)*A+(-9+5*I)*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(1/32+1/32*I)*((1+2*I)*A+(2-7*I)*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+1/8*(A+5*I*B)*sqrt(cot(c+d*x))/(a^2*d*(I+cot(c+d*x)))+1/4*(I*A-B)*sqrt(cot(c+d*x))/(d*(I*a+a*cot(c+d*x))^2)],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^2),x,14,(-1/16+1/16*I)*((2+7*I)*A+(-23-2*I)*B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+1/16*((9+5*I)*A+(-25+21*I)*B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(-1/32+1/32*I)*((7+2*I)*A+(2+23*I)*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+(1/32-1/32*I)*((7+2*I)*A+(2+23*I)*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^2*d*sqrt(2))+5/8*(I*A-5*B)/(a^2*d*sqrt(cot(c+d*x)))+1/8*(3*A+7*I*B)/(a^2*d*(I+cot(c+d*x))*sqrt(cot(c+d*x)))+1/4*(I*A-B)/(d*(I*a+a*cot(c+d*x))^2*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,15,1/6*(A+I*B)*cot(c+d*x)^(7/2)/(d*(I*a+a*cot(c+d*x))^3)+1/12*(5*A+2*I*B)*cot(c+d*x)^(5/2)/(a*d*(I*a+a*cot(c+d*x))^2)+7/24*(4*A+I*B)*cot(c+d*x)^(3/2)/(d*(I*a^3+a^3*cot(c+d*x)))+(-1/16+1/16*I)*((1+29*I)*A+(-6-I)*B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/16*((30+28*I)*A+(-7+5*I)*B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+(-1/32+1/32*I)*((29+I)*A+(1+6*I)*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+(1/32-1/32*I)*((29+I)*A+(1+6*I)*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))-5/8*(6*A+I*B)*sqrt(cot(c+d*x))/(a^3*d)],
[sqrt(cot(c+d*x))*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^3,x,14,1/6*(A+I*B)*cot(c+d*x)^(5/2)/(d*(I*a+a*cot(c+d*x))^3)+1/12*(4*A+I*B)*cot(c+d*x)^(3/2)/(a*d*(I*a+a*cot(c+d*x))^2)-1/16*((-7+5*I)*A+2*I*B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/16*((-7+5*I)*A+2*I*B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))-1/32*((7+5*I)*A-2*I*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/32*((7+5*I)*A-2*I*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+5/8*A*sqrt(cot(c+d*x))/(d*(I*a^3+a^3*cot(c+d*x)))],
[(A+B*tan(c+d*x))/(sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^3),x,14,1/6*(A+I*B)*cot(c+d*x)^(3/2)/(d*(I*a+a*cot(c+d*x))^3)+(-1/16-1/16*I)*((1+I)*A+B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+(1/16+1/16*I)*((1+I)*A+B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/32*(2*I*A+(1-I)*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))-1/32*(2*I*A+(1-I)*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/4*A*sqrt(cot(c+d*x))/(a*d*(I*a+a*cot(c+d*x))^2)+1/8*(2*I*A+B)*sqrt(cot(c+d*x))/(d*(I*a^3+a^3*cot(c+d*x)))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^3),x,14,-1/16*((1+I)*A+2*B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/16*((1+I)*A+2*B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))-1/32*((-1+I)*A+2*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/32*((-1+I)*A+2*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/6*(A+I*B)*sqrt(cot(c+d*x))/(d*(I*a+a*cot(c+d*x))^3)+1/12*(2*I*A+B)*sqrt(cot(c+d*x))/(a*d*(I*a+a*cot(c+d*x))^2)+1/8*A*sqrt(cot(c+d*x))/(d*(I*a^3+a^3*cot(c+d*x)))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^3),x,14,-1/16*(2*A+(5-7*I)*B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/16*(2*A+(5-7*I)*B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))-1/32*(2*A+(-5-7*I)*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/32*(2*A+(-5-7*I)*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/6*(I*A-B)*sqrt(cot(c+d*x))/(d*(I*a+a*cot(c+d*x))^3)+1/12*(A+4*I*B)*sqrt(cot(c+d*x))/(a*d*(I*a+a*cot(c+d*x))^2)+5/8*B*sqrt(cot(c+d*x))/(d*(I*a^3+a^3*cot(c+d*x)))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(7/2)*(a+I*a*tan(c+d*x))^3),x,15,(1/16+1/16*I)*((1+6*I)*A+(-29-I)*B)*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+1/16*((5-7*I)*A+(28+30*I)*B)*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+(1/32+1/32*I)*((6+I)*A+(1+29*I)*B)*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+(-1/32-1/32*I)*((6+I)*A+(1+29*I)*B)*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(a^3*d*sqrt(2))+5/8*(A+6*I*B)/(a^3*d*sqrt(cot(c+d*x)))+1/6*(I*A-B)/(d*(I*a+a*cot(c+d*x))^3*sqrt(cot(c+d*x)))+1/12*(2*A+5*I*B)/(a*d*(I*a+a*cot(c+d*x))^2*sqrt(cot(c+d*x)))-7/24*(I*A-4*B)/(d*(I*a^3+a^3*cot(c+d*x))*sqrt(cot(c+d*x)))],

# Integrands of the form Cot[c+d x]^(m/2) (A+B Tan[c+d x]) (a+i a Tan[c+d x])^(n/2)

# n>0
[cot(c+d*x)^(7/2)*sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,7,(-1-I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/15*(I*A+5*B)*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/d-2/5*A*cot(c+d*x)^(5/2)*sqrt(a+I*a*tan(c+d*x))/d+2/15*(13*A-5*I*B)*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^(5/2)*sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,6,(1+I)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/3*A*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/d-2/3*(I*A+3*B)*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,5,(1+I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2*A*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x)),x,8,-2*(-1)^(3/4)*B*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(1-I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d],
[sqrt(a+I*a*tan(c+d*x))*(A+B*tan(c+d*x))/sqrt(cot(c+d*x)),x,9,-(-1)^(3/4)*(2*A-I*B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(-1-I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(a)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+B*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(9/2)*(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,8,(2-2*I)*a^(3/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+4/105*a*(19*A-21*I*B)*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/d-2/35*a*(8*I*A+7*B)*cot(c+d*x)^(5/2)*sqrt(a+I*a*tan(c+d*x))/d-2/7*a*A*cot(c+d*x)^(7/2)*sqrt(a+I*a*tan(c+d*x))/d+4/105*a*(67*I*A+63*B)*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^(7/2)*(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,7,(-2-2*I)*a^(3/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/15*a*(6*I*A+5*B)*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/d-2/5*a*A*cot(c+d*x)^(5/2)*sqrt(a+I*a*tan(c+d*x))/d+4/15*a*(9*A-10*I*B)*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,6,(2+2*I)*a^(3/2)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/3*a*A*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/d-2/3*a*(4*I*A+3*B)*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,9,2*(-1)^(1/4)*a^(3/2)*B*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(2+2*I)*a^(3/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2*a*A*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,9,-(-1)^(3/4)*a^(3/2)*(2*I*A+3*B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(2-2*I)*a^(3/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+I*a*B*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(cot(c+d*x)))],
[(a+I*a*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x))/sqrt(cot(c+d*x)),x,10,-1/4*(-1)^(3/4)*a^(3/2)*(12*A-11*I*B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(-2-2*I)*a^(3/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+1/2*I*a*B*sqrt(a+I*a*tan(c+d*x))/(d*cot(c+d*x)^(3/2))+1/4*a*(4*I*A+5*B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(11/2)*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,9,(4+4*I)*a^(5/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+8/315*a^2*(59*I*A+60*B)*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/d+2/105*a^2*(46*A-45*I*B)*cot(c+d*x)^(5/2)*sqrt(a+I*a*tan(c+d*x))/d-2/21*a^2*(4*I*A+3*B)*cot(c+d*x)^(7/2)*sqrt(a+I*a*tan(c+d*x))/d-8/315*a^2*(197*A-195*I*B)*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d-2/9*a*A*cot(c+d*x)^(9/2)*(a+I*a*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^(9/2)*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,8,(4-4*I)*a^(5/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+2/105*a^2*(80*A-77*I*B)*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/d-2/35*a^2*(10*I*A+7*B)*cot(c+d*x)^(5/2)*sqrt(a+I*a*tan(c+d*x))/d+4/105*a^2*(130*I*A+133*B)*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d-2/7*a*A*cot(c+d*x)^(7/2)*(a+I*a*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^(7/2)*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,7,(-4-4*I)*a^(5/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/15*a^2*(8*I*A+5*B)*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/d+2/15*a^2*(38*A-35*I*B)*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d-2/5*a*A*cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,10,2*(-1)^(3/4)*a^(5/2)*B*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(4+4*I)*a^(5/2)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2*a^2*(2*I*A+B)*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/d-2/3*a*A*cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,10,(-1)^(3/4)*a^(5/2)*(2*A-5*I*B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(4+4*I)*a^(5/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+a^2*(2*I*A-B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(cot(c+d*x)))-2*a*A*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(3/2)/d],
[sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,10,-1/4*(-1)^(3/4)*a^(5/2)*(20*I*A+23*B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(4-4*I)*a^(5/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-1/4*a^2*(4*A-7*I*B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(cot(c+d*x)))+1/2*I*a*B*(a+I*a*tan(c+d*x))^(3/2)/(d*sqrt(cot(c+d*x)))],
[(a+I*a*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/sqrt(cot(c+d*x)),x,11,-1/8*(-1)^(3/4)*a^(5/2)*(46*A-45*I*B)*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(-4-4*I)*a^(5/2)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-1/4*a^2*(2*A-3*I*B)*sqrt(a+I*a*tan(c+d*x))/(d*cot(c+d*x)^(3/2))+1/8*a^2*(18*I*A+19*B)*sqrt(a+I*a*tan(c+d*x))/(d*sqrt(cot(c+d*x)))+1/3*I*a*B*(a+I*a*tan(c+d*x))^(3/2)/(d*cot(c+d*x)^(3/2))],

# n<0
[cot(c+d*x)^(5/2)*(A+B*tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)),x,7,(1/2+1/2*I)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(a))+(A+I*B)*cot(c+d*x)^(3/2)/(d*sqrt(a+I*a*tan(c+d*x)))-1/3*(5*A+3*I*B)*cot(c+d*x)^(3/2)*sqrt(a+I*a*tan(c+d*x))/(a*d)+1/3*(7*I*A-9*B)*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[cot(c+d*x)^(3/2)*(A+B*tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)),x,6,(1/2+1/2*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(a))+(A+I*B)*sqrt(cot(c+d*x))/(d*sqrt(a+I*a*tan(c+d*x)))-(3*A+I*B)*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a*d)],
[sqrt(cot(c+d*x))*(A+B*tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)),x,5,(1/2-1/2*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(a))+(A+I*B)/(d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))),x,9,-2*(-1)^(1/4)*B*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(a))+(-1/2-1/2*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(a))+(I*A-B)/(d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))],
[cot(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(3/2),x,7,(1/4+1/4*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(3/2)*d)+1/6*(11*A+5*I*B)*sqrt(cot(c+d*x))/(a*d*sqrt(a+I*a*tan(c+d*x)))-1/6*(25*A+7*I*B)*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a^2*d)+1/3*(A+I*B)*sqrt(cot(c+d*x))/(d*(a+I*a*tan(c+d*x))^(3/2))],
[sqrt(cot(c+d*x))*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(3/2),x,6,(1/4-1/4*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(3/2)*d)+1/6*(7*A+I*B)/(a*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/3*(A+I*B)/(d*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(3/2)),x,6,(-1/4-1/4*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(3/2)*d)+1/6*(I*A+5*B)/(a*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/3*(I*A-B)/(d*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2)),x,10,2*(-1)^(3/4)*B*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(3/2)*d)+(1/4+1/4*I)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(3/2)*d)+1/2*(A+3*I*B)/(a*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/3*(I*A-B)/(d*cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2))],
[cot(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,8,(1/8+1/8*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(5/2)*d)+1/60*(151*A+41*I*B)*sqrt(cot(c+d*x))/(a^2*d*sqrt(a+I*a*tan(c+d*x)))-1/60*(317*A+67*I*B)*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x))/(a^3*d)+1/5*(A+I*B)*sqrt(cot(c+d*x))/(d*(a+I*a*tan(c+d*x))^(5/2))+1/30*(17*A+7*I*B)*sqrt(cot(c+d*x))/(a*d*(a+I*a*tan(c+d*x))^(3/2))],
[sqrt(cot(c+d*x))*(A+B*tan(c+d*x))/(a+I*a*tan(c+d*x))^(5/2),x,7,(1/8-1/8*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(5/2)*d)+1/60*(67*A-3*I*B)/(a^2*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/5*(A+I*B)/(d*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(5/2))+1/30*(13*A+3*I*B)/(a*d*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(5/2)),x,7,(-1/8-1/8*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(5/2)*d)+1/60*(-3*I*A+13*B)/(a^2*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/5*(I*A-B)/(d*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(5/2))+1/30*(3*I*A+7*B)/(a*d*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(5/2)),x,7,(1/8+1/8*I)*(I*A+B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(5/2)*d)+1/60*(13*A-37*I*B)/(a^2*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/5*(I*A-B)/(d*cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(5/2))+1/30*(A+11*I*B)/(a*d*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(5/2)),x,11,2*(-1)^(1/4)*B*arctan((-1)^(3/4)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(5/2)*d)+(1/8+1/8*I)*(A-I*B)*arctanh((1+I)*sqrt(a)*sqrt(tan(c+d*x))/sqrt(a+I*a*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(a^(5/2)*d)+1/4*(-I*A+7*B)/(a^2*d*sqrt(cot(c+d*x))*sqrt(a+I*a*tan(c+d*x)))+1/5*(I*A-B)/(d*cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^(5/2))+1/6*(A+3*I*B)/(a*d*cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^(3/2))],

# Integrands of the form Cot[c+d x]^(m/2) (A+B Tan[c+d x]) (a+i a Tan[c+d x])^n with n symbolic
[cot(c+d*x)^m*(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,8,(A-I*B)*AppellF1(1-m,1-n,1,2-m,-I*tan(c+d*x),I*tan(c+d*x))*cot(c+d*x)^(-1+m)*(a+I*a*tan(c+d*x))^n/(d*(1-m)*(1+I*tan(c+d*x))^n)+I*B*cot(c+d*x)^(-1+m)*hypergeom([1-m,1-n],[2-m],-I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(1-m)*(1+I*tan(c+d*x))^n)],
[cot(c+d*x)^(5/2)*(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,11,-2/3*A*cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^n/d-2/3*(3*B+2*I*A*n)*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^n/d-2*(A-I*B)*AppellF1(1/2,1-n,1,3/2,-I*tan(c+d*x),I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*sqrt(cot(c+d*x))*(1+I*tan(c+d*x))^n)-2/3*(1-2*n)*(3*I*B-2*A*n)*hypergeom([1/2,1-n],[3/2],-I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*sqrt(cot(c+d*x))*(1+I*tan(c+d*x))^n)],
[cot(c+d*x)^(3/2)*(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,10,-2*A*sqrt(cot(c+d*x))*(a+I*a*tan(c+d*x))^n/d+2*(I*A+B)*AppellF1(1/2,1-n,1,3/2,-I*tan(c+d*x),I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*sqrt(cot(c+d*x))*(1+I*tan(c+d*x))^n)-2*I*A*(1-2*n)*hypergeom([1/2,1-n],[3/2],-I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*sqrt(cot(c+d*x))*(1+I*tan(c+d*x))^n)],
[cot(c+d*x)^(1/2)*(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,9,2*(A-I*B)*AppellF1(1/2,1-n,1,3/2,-I*tan(c+d*x),I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*sqrt(cot(c+d*x))*(1+I*tan(c+d*x))^n)+2*I*B*hypergeom([1/2,1-n],[3/2],-I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*sqrt(cot(c+d*x))*(1+I*tan(c+d*x))^n)],
[(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x))/cot(c+d*x)^(1/2),x,10,2*B*(a+I*a*tan(c+d*x))^n/(d*(1+2*n)*sqrt(cot(c+d*x)))-2*(I*A+B)*AppellF1(1/2,1-n,1,3/2,-I*tan(c+d*x),I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*sqrt(cot(c+d*x))*(1+I*tan(c+d*x))^n)+2*(2*B*n+I*A*(1+2*n))*hypergeom([1/2,1-n],[3/2],-I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(1+2*n)*sqrt(cot(c+d*x))*(1+I*tan(c+d*x))^n)],
[(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x))/cot(c+d*x)^(3/2),x,11,2*B*(a+I*a*tan(c+d*x))^n/(d*(3+2*n)*cot(c+d*x)^(3/2))-2*(2*I*B*n-A*(3+2*n))*(a+I*a*tan(c+d*x))^n/(d*(3+8*n+4*n^2)*sqrt(cot(c+d*x)))-2*(A-I*B)*AppellF1(1/2,1-n,1,3/2,-I*tan(c+d*x),I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*sqrt(cot(c+d*x))*(1+I*tan(c+d*x))^n)+2*(2*A*n*(3+2*n)-I*B*(3+6*n+4*n^2))*hypergeom([1/2,1-n],[3/2],-I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(3+8*n+4*n^2)*sqrt(cot(c+d*x))*(1+I*tan(c+d*x))^n)],
[(a+I*a*tan(c+d*x))^n*(A+B*tan(c+d*x))/cot(c+d*x)^(5/2),x,12,2*B*(a+I*a*tan(c+d*x))^n/(d*(5+2*n)*cot(c+d*x)^(5/2))-2*(2*I*B*n-A*(5+2*n))*(a+I*a*tan(c+d*x))^n/(d*(3+2*n)*(5+2*n)*cot(c+d*x)^(3/2))-2*(2*I*A*n*(5+2*n)+B*(15+10*n+4*n^2))*(a+I*a*tan(c+d*x))^n/(d*(5+2*n)*(3+8*n+4*n^2)*sqrt(cot(c+d*x)))+2*(I*A+B)*AppellF1(1/2,1-n,1,3/2,-I*tan(c+d*x),I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*sqrt(cot(c+d*x))*(1+I*tan(c+d*x))^n)-2*(4*B*n*(9+8*n+2*n^2)+I*A*(15+36*n+32*n^2+8*n^3))*hypergeom([1/2,1-n],[3/2],-I*tan(c+d*x))*(a+I*a*tan(c+d*x))^n/(d*(5+2*n)*(3+8*n+4*n^2)*sqrt(cot(c+d*x))*(1+I*tan(c+d*x))^n)],

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

# Integrands of the form Cot[c+d x]^(m/2) (A+B Tan[c+d x]) (a+b Tan[c+d x])^n

# n>0
[cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,13,-2/3*a*A*cot(c+d*x)^(3/2)/d-(b*(A-B)+a*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(b*(A-B)+a*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a*(A-B)-b*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a*(A-B)-b*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-2*(A*b+a*B)*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,12,-(a*(A-B)-b*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(a*(A-B)-b*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(b*(A-B)+a*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(b*(A-B)+a*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-2*a*A*sqrt(cot(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,12,(b*(A-B)+a*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(b*(A-B)+a*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a*(A-B)-b*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a*(A-B)-b*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*b*B/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))*(A+B*tan(c+d*x))/sqrt(cot(c+d*x)),x,13,2/3*b*B/(d*cot(c+d*x)^(3/2))+(a*(A-B)-b*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a*(A-B)-b*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(b*(A-B)+a*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(b*(A-B)+a*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*(A*b+a*B)/(d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(7/2)*(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,14,-2/15*a*(7*A*b+5*a*B)*cot(c+d*x)^(3/2)/d-2/5*a*A*cot(c+d*x)^(3/2)*(b+a*cot(c+d*x))/d+(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*(a^2*A-A*b^2-2*a*b*B)*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,13,-(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-2/3*a*(5*A*b+3*a*B)*sqrt(cot(c+d*x))/d-2/3*a*A*(b+a*cot(c+d*x))*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,13,-(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*b^2*B/(d*sqrt(cot(c+d*x)))-2*a^2*A*sqrt(cot(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x)),x,13,2/3*b^2*B/(d*cot(c+d*x)^(3/2))+(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*b*(A*b+2*a*B)/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))^2*(A+B*tan(c+d*x))/sqrt(cot(c+d*x)),x,14,2/5*b^2*B/(d*cot(c+d*x)^(5/2))+2/3*b*(A*b+2*a*B)/(d*cot(c+d*x)^(3/2))+(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*(2*a*A*b+a^2*B-b^2*B)/(d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(9/2)*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,15,2/21*a*(7*a^2*A-18*A*b^2-21*a*b*B)*cot(c+d*x)^(3/2)/d-2/35*a^2*(11*A*b+7*a*B)*cot(c+d*x)^(5/2)/d-2/7*a*A*cot(c+d*x)^(3/2)*(b+a*cot(c+d*x))^2/d+(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(7/2)*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,14,-2/15*a^2*(9*A*b+5*a*B)*cot(c+d*x)^(3/2)/d+(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2/5*a*(5*a^2*A-14*A*b^2-15*a*b*B)*sqrt(cot(c+d*x))/d-2/5*a*A*(b+a*cot(c+d*x))^2*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,14,-2/3*a^2*(a*A+3*b*B)*cot(c+d*x)^(3/2)/d-(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*b*B*(b+a*cot(c+d*x))^2/(d*sqrt(cot(c+d*x)))-2*a*(3*a*A*b+a^2*B+2*b^2*B)*sqrt(cot(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,14,2/3*b*B*(b+a*cot(c+d*x))^2/(d*cot(c+d*x)^(3/2))-(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2/3*b^2*(3*A*b+7*a*B)/(d*sqrt(cot(c+d*x)))-2/3*a^2*(3*a*A+b*B)*sqrt(cot(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x)),x,14,2/15*b^2*(5*A*b+9*a*B)/(d*cot(c+d*x)^(3/2))+2/5*b*B*(b+a*cot(c+d*x))^2/(d*cot(c+d*x)^(5/2))+(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2/5*b*(15*a*A*b+14*a^2*B-5*b^2*B)/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))^3*(A+B*tan(c+d*x))/sqrt(cot(c+d*x)),x,15,2/35*b^2*(7*A*b+11*a*B)/(d*cot(c+d*x)^(5/2))+2/21*b*(21*a*A*b+18*a^2*B-7*b^2*B)/(d*cot(c+d*x)^(3/2))+2/7*b*B*(b+a*cot(c+d*x))^2/(d*cot(c+d*x)^(7/2))+(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*(3*a^2*A*b-A*b^3+a^3*B-3*a*b^2*B)/(d*sqrt(cot(c+d*x)))],

# n<0
[cot(c+d*x)^(5/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,17,-2*b^(5/2)*(A*b-a*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(a^(5/2)*(a^2+b^2)*d)-2/3*A*cot(c+d*x)^(3/2)/(a*d)+(b*(A-B)-a*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-(b*(A-B)-a*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a*(A-B)+b*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a*(A-B)+b*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+2*(A*b-a*B)*sqrt(cot(c+d*x))/(a^2*d)],
[cot(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,16,2*b^(3/2)*(A*b-a*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(a^(3/2)*(a^2+b^2)*d)-(a*(A-B)+b*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(a*(A-B)+b*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(b*(A-B)-a*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(b*(A-B)-a*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-2*A*sqrt(cot(c+d*x))/(a*d)],
[sqrt(cot(c+d*x))*(A+B*tan(c+d*x))/(a+b*tan(c+d*x)),x,15,-(b*(A-B)-a*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(b*(A-B)-a*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a*(A-B)+b*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a*(A-B)+b*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-2*(A*b-a*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))*sqrt(b)/((a^2+b^2)*d*sqrt(a))],
[(A+B*tan(c+d*x))/(sqrt(cot(c+d*x))*(a+b*tan(c+d*x))),x,15,(a*(A-B)+b*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-(a*(A-B)+b*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(b*(A-B)-a*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(b*(A-B)-a*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+2*(A*b-a*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))*sqrt(a)/((a^2+b^2)*d*sqrt(b))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))),x,16,-2*a^(3/2)*(A*b-a*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(b^(3/2)*(a^2+b^2)*d)+(b*(A-B)-a*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-(b*(A-B)-a*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(a*(A-B)+b*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(a*(A-B)+b*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+2*B/(b*d*sqrt(cot(c+d*x)))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))),x,17,2*a^(5/2)*(A*b-a*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(b^(5/2)*(a^2+b^2)*d)+2/3*B/(b*d*cot(c+d*x)^(3/2))-(a*(A-B)+b*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+(a*(A-B)+b*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+1/2*(b*(A-B)-a*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))-1/2*(b*(A-B)-a*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)*d*sqrt(2))+2*(A*b-a*B)/(b^2*d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,17,b^(3/2)*(7*a^2*A*b+3*A*b^3-5*a^3*B-a*b^2*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(a^(5/2)*(a^2+b^2)^2*d)+b*(A*b-a*B)*cot(c+d*x)^(3/2)/(a*(a^2+b^2)*d*(b+a*cot(c+d*x)))-(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(2*a^2*A+3*A*b^2-a*b*B)*sqrt(cot(c+d*x))/(a^2*(a^2+b^2)*d)],
[sqrt(cot(c+d*x))*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^2,x,16,-(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(5*a^2*A*b+A*b^3-3*a^3*B+a*b^2*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))*sqrt(b)/(a^(3/2)*(a^2+b^2)^2*d)+b*(A*b-a*B)*sqrt(cot(c+d*x))/(a*(a^2+b^2)*d*(b+a*cot(c+d*x)))],
[(A+B*tan(c+d*x))/(sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^2),x,16,(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(3*a^2*A*b-A*b^3-a^3*B+3*a*b^2*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/((a^2+b^2)^2*d*sqrt(a)*sqrt(b))-(A*b-a*B)*sqrt(cot(c+d*x))/((a^2+b^2)*d*(b+a*cot(c+d*x)))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^2),x,16,(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-(a^2*A*b-3*A*b^3+a^3*B+5*a*b^2*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))*sqrt(a)/(b^(3/2)*(a^2+b^2)^2*d)+a*(A*b-a*B)*sqrt(cot(c+d*x))/(b*(a^2+b^2)*d*(b+a*cot(c+d*x)))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^2),x,17,-a^(3/2)*(a^2*A*b+5*A*b^3-3*a^3*B-7*a*b^2*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(b^(5/2)*(a^2+b^2)^2*d)-(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+1/2*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))-1/2*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^2*d*sqrt(2))+(-a*A*b+3*a^2*B+2*b^2*B)/(b^2*(a^2+b^2)*d*sqrt(cot(c+d*x)))+a*(A*b-a*B)/(b*(a^2+b^2)*d*(b+a*cot(c+d*x))*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^3,x,18,1/4*b^(3/2)*(63*a^4*A*b+46*a^2*A*b^3+15*A*b^5-35*a^5*B-6*a^3*b^2*B-3*a*b^4*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(a^(7/2)*(a^2+b^2)^3*d)+1/2*b*(A*b-a*B)*cot(c+d*x)^(5/2)/(a*(a^2+b^2)*d*(b+a*cot(c+d*x))^2)+1/4*b*(13*a^2*A*b+5*A*b^3-9*a^3*B-a*b^2*B)*cot(c+d*x)^(3/2)/(a^2*(a^2+b^2)^2*d*(b+a*cot(c+d*x)))-(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/4*(8*a^4*A+31*a^2*A*b^2+15*A*b^4-11*a^3*b*B-3*a*b^3*B)*sqrt(cot(c+d*x))/(a^3*(a^2+b^2)^2*d)],
[sqrt(cot(c+d*x))*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^3,x,17,1/2*b*(A*b-a*B)*cot(c+d*x)^(3/2)/(a*(a^2+b^2)*d*(b+a*cot(c+d*x))^2)-(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/4*(35*a^4*A*b+6*a^2*A*b^3+3*A*b^5-15*a^5*B+18*a^3*b^2*B+a*b^4*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))*sqrt(b)/(a^(5/2)*(a^2+b^2)^3*d)+1/4*b*(11*a^2*A*b+3*A*b^3-7*a^3*B+a*b^2*B)*sqrt(cot(c+d*x))/(a^2*(a^2+b^2)^2*d*(b+a*cot(c+d*x)))],
[(A+B*tan(c+d*x))/(sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^3),x,17,(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/4*(15*a^4*A*b-18*a^2*A*b^3-A*b^5-3*a^5*B+26*a^3*b^2*B-3*a*b^4*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(a^(3/2)*(a^2+b^2)^3*d*sqrt(b))+1/2*b*(A*b-a*B)*sqrt(cot(c+d*x))/(a*(a^2+b^2)*d*(b+a*cot(c+d*x))^2)-1/4*(9*a^2*A*b+A*b^3-5*a^3*B+3*a*b^2*B)*sqrt(cot(c+d*x))/(a*(a^2+b^2)^2*d*(b+a*cot(c+d*x)))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^3),x,17,(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/4*(3*a^4*A*b-26*a^2*A*b^3+3*A*b^5+a^5*B+18*a^3*b^2*B-15*a*b^4*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(b^(3/2)*(a^2+b^2)^3*d*sqrt(a))-1/2*(A*b-a*B)*sqrt(cot(c+d*x))/((a^2+b^2)*d*(b+a*cot(c+d*x))^2)+1/4*(5*a^2*A*b-3*A*b^3-a^3*B+7*a*b^2*B)*sqrt(cot(c+d*x))/(b*(a^2+b^2)^2*d*(b+a*cot(c+d*x)))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^3),x,17,-(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/4*(a^4*A*b+18*a^2*A*b^3-15*A*b^5+3*a^5*B+6*a^3*b^2*B+35*a*b^4*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))*sqrt(a)/(b^(5/2)*(a^2+b^2)^3*d)+1/2*a*(A*b-a*B)*sqrt(cot(c+d*x))/(b*(a^2+b^2)*d*(b+a*cot(c+d*x))^2)-1/4*a*(a^2*A*b-7*A*b^3+3*a^3*B+11*a*b^2*B)*sqrt(cot(c+d*x))/(b^2*(a^2+b^2)^2*d*(b+a*cot(c+d*x)))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(7/2)*(a+b*tan(c+d*x))^3),x,18,-1/4*a^(3/2)*(3*a^4*A*b+6*a^2*A*b^3+35*A*b^5-15*a^5*B-46*a^3*b^2*B-63*a*b^4*B)*arctan(sqrt(a)*sqrt(cot(c+d*x))/sqrt(b))/(b^(7/2)*(a^2+b^2)^3*d)-(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))-1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/((a^2+b^2)^3*d*sqrt(2))+1/4*(-3*a^3*A*b-11*a*A*b^3+15*a^4*B+31*a^2*b^2*B+8*b^4*B)/(b^3*(a^2+b^2)^2*d*sqrt(cot(c+d*x)))+1/2*a*(A*b-a*B)/(b*(a^2+b^2)*d*(b+a*cot(c+d*x))^2*sqrt(cot(c+d*x)))+1/4*a*(a^2*A*b+9*A*b^3-5*a^3*B-13*a*b^2*B)/(b^2*(a^2+b^2)^2*d*(b+a*cot(c+d*x))*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(5/2)*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x)),x,13,-2/3*B*cot(c+d*x)^(3/2)/d-B*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+B*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*B*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*B*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))],
[cot(c+d*x)^(3/2)*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x)),x,13,-B*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+B*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*B*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*B*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-2*B*sqrt(cot(c+d*x))/d],
[sqrt(cot(c+d*x))*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x)),x,12,B*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-B*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*B*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*B*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))],
[(a*B+b*B*tan(c+d*x))/(sqrt(cot(c+d*x))*(a+b*tan(c+d*x))),x,12,B*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-B*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*B*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*B*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))],
[(a*B+b*B*tan(c+d*x))/(cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))),x,13,-B*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+B*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*B*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*B*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+2*B/(d*sqrt(cot(c+d*x)))],
[(a*B+b*B*tan(c+d*x))/(cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))),x,13,2/3*B/(d*cot(c+d*x)^(3/2))-B*arctan(1-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+B*arctan(1+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))-1/2*B*log(1+cot(c+d*x)-sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))+1/2*B*log(1+cot(c+d*x)+sqrt(2)*sqrt(cot(c+d*x)))/(d*sqrt(2))],

# Integrands of the form Cot[c+d x]^(m/2) (A+B Tan[c+d x]) (a+b Tan[c+d x])^(n/2)

# n>0
[cot(c+d*x)^(9/2)*sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,12,-(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+2/105*(35*a^2*A+4*A*b^2-7*a*b*B)*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/(a^2*d)-2/35*(A*b+7*a*B)*cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))/(a*d)-2/7*A*cot(c+d*x)^(7/2)*sqrt(a+b*tan(c+d*x))/d+2/105*(35*a^2*A*b-8*A*b^3+105*a^3*B+14*a*b^2*B)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a^3*d)],
[cot(c+d*x)^(7/2)*sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,11,(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/15*(A*b+5*a*B)*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/(a*d)-2/5*A*cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))/d+2/15*(15*a^2*A+2*A*b^2-5*a*b*B)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a^2*d)],
[cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,10,(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/3*A*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/d-2/3*(A*b+3*a*B)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a*d)],
[cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,9,-(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2*A*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/d],
[sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x)),x,13,-(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+2*B*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d],
[sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x))/sqrt(cot(c+d*x)),x,14,(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(2*A*b+a*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(b))-(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+B*sqrt(a+b*tan(c+d*x))/(d*sqrt(cot(c+d*x)))],
[sqrt(a+b*tan(c+d*x))*(A+B*tan(c+d*x))/cot(c+d*x)^(3/2),x,15,1/4*(4*a*A*b-a^2*B-8*b^2*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(b^(3/2)*d)+(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a-b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(I*a+b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+1/4*(4*A*b-a*B)*sqrt(a+b*tan(c+d*x))/(b*d*sqrt(cot(c+d*x)))+1/2*B*(a+b*tan(c+d*x))^(3/2)/(b*d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(11/2)*(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,13,(I*a-b)^(3/2)*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(I*a+b)^(3/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+2/315*(126*a^2*A*b+4*A*b^3+105*a^3*B-9*a*b^2*B)*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/(a^2*d)+2/105*(21*a^2*A-A*b^2-24*a*b*B)*cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))/(a*d)-2/63*(10*A*b+9*a*B)*cot(c+d*x)^(7/2)*sqrt(a+b*tan(c+d*x))/d-2/9*a*A*cot(c+d*x)^(9/2)*sqrt(a+b*tan(c+d*x))/d-2/315*(315*a^4*A-63*a^2*A*b^2+8*A*b^4-420*a^3*b*B-18*a*b^3*B)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a^3*d)],
[cot(c+d*x)^(9/2)*(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,12,-(I*a-b)^(3/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(I*a+b)^(3/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+2/105*(35*a^2*A-3*A*b^2-42*a*b*B)*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/(a*d)-2/35*(8*A*b+7*a*B)*cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))/d-2/7*a*A*cot(c+d*x)^(7/2)*sqrt(a+b*tan(c+d*x))/d+2/105*(140*a^2*A*b+6*A*b^3+105*a^3*B-21*a*b^2*B)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a^2*d)],
[cot(c+d*x)^(7/2)*(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,11,(I*a+b)^(3/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(a+I*b)^2*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))-2/15*(6*A*b+5*a*B)*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/d-2/5*a*A*cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))/d+2/15*(15*a^2*A-3*A*b^2-20*a*b*B)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a*d)],
[cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,10,(I*a-b)^(3/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(I*a+b)^(3/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/3*a*A*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/d-2/3*(4*A*b+3*a*B)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/d],
[cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,14,2*b^(3/2)*B*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(I*a+b)^(3/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(a+I*b)^2*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))-2*a*A*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/d],
[sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x)),x,14,-(I*a-b)^(3/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(I*a+b)^(3/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(2*A*b+3*a*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+b*B*sqrt(a+b*tan(c+d*x))/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x))/sqrt(cot(c+d*x)),x,15,(I*a+b)^(3/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(a+I*b)^2*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))+1/4*(12*a*A*b+3*a^2*B-8*b^2*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(b))+1/2*b*B*sqrt(a+b*tan(c+d*x))/(d*cot(c+d*x)^(3/2))+1/4*(4*A*b+5*a*B)*sqrt(a+b*tan(c+d*x))/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))^(3/2)*(A+B*tan(c+d*x))/cot(c+d*x)^(3/2),x,16,(I*a-b)^(3/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+1/8*(6*a^2*A*b-16*A*b^3-a^3*B-24*a*b^2*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(b^(3/2)*d)+(I*a+b)^(3/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+1/8*(6*a*A*b-a^2*B-8*b^2*B)*sqrt(a+b*tan(c+d*x))/(b*d*sqrt(cot(c+d*x)))+1/12*(6*A*b-a*B)*(a+b*tan(c+d*x))^(3/2)/(b*d*sqrt(cot(c+d*x)))+1/3*B*(a+b*tan(c+d*x))^(5/2)/(b*d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(13/2)*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,14,-(I*a-b)^(5/2)*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(I*a+b)^(5/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/3465*(1155*a^4*A-1485*a^2*A*b^2-20*A*b^4-2541*a^3*b*B+55*a*b^3*B)*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/(a^2*d)+2/1155*(495*a^2*A*b-5*A*b^3+231*a^3*B-275*a*b^2*B)*cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))/(a*d)+2/693*(99*a^2*A-113*A*b^2-209*a*b*B)*cot(c+d*x)^(7/2)*sqrt(a+b*tan(c+d*x))/d-2/99*a*(14*A*b+11*a*B)*cot(c+d*x)^(9/2)*sqrt(a+b*tan(c+d*x))/d-2/3465*(8085*a^4*A*b-495*a^2*A*b^3+40*A*b^5+3465*a^5*B-5313*a^3*b^2*B-110*a*b^4*B)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a^3*d)-2/11*a*A*cot(c+d*x)^(11/2)*(a+b*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^(11/2)*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,13,(I*a-b)^(5/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(I*a+b)^(5/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+2/315*(231*a^2*A*b-5*A*b^3+105*a^3*B-135*a*b^2*B)*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/(a*d)+2/105*(21*a^2*A-25*A*b^2-45*a*b*B)*cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))/d-2/21*a*(4*A*b+3*a*B)*cot(c+d*x)^(7/2)*sqrt(a+b*tan(c+d*x))/d-2/315*(315*a^4*A-483*a^2*A*b^2-10*A*b^4-735*a^3*b*B+45*a*b^3*B)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a^2*d)-2/9*a*A*cot(c+d*x)^(9/2)*(a+b*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^(9/2)*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,12,(I*a-b)^(5/2)*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(I*a+b)^(5/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+2/105*(35*a^2*A-45*A*b^2-77*a*b*B)*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/d-2/35*a*(10*A*b+7*a*B)*cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))/d+2/105*(245*a^2*A*b-15*A*b^3+105*a^3*B-161*a*b^2*B)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a*d)-2/7*a*A*cot(c+d*x)^(7/2)*(a+b*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^(7/2)*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,11,-(I*a-b)^(5/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(I*a+b)^(5/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2/15*a*(8*A*b+5*a*B)*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/d+2/15*(15*a^2*A-23*A*b^2-35*a*b*B)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/d-2/5*a*A*cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,15,-(I*a-b)^(5/2)*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+2*b^(5/2)*B*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(I*a+b)^(5/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-2*a*(2*A*b+a*B)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/d-2/3*a*A*cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(3/2)/d],
[cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,15,(I*a-b)^(5/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+b^(3/2)*(2*A*b+5*a*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d-(I*a+b)^(5/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+b*(2*a*A+b*B)*sqrt(a+b*tan(c+d*x))/(d*sqrt(cot(c+d*x)))-2*a*A*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2)/d],
[sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x)),x,15,(I*a-b)^(5/2)*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(I*a+b)^(5/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+1/4*(20*a*A*b+15*a^2*B-8*b^2*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(b)*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+1/4*b*(4*A*b+7*a*B)*sqrt(a+b*tan(c+d*x))/(d*sqrt(cot(c+d*x)))+1/2*b*B*(a+b*tan(c+d*x))^(3/2)/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/sqrt(cot(c+d*x)),x,16,-(I*a-b)^(5/2)*(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+(I*a+b)^(5/2)*(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+1/8*(30*a^2*A*b-16*A*b^3+5*a^3*B-40*a*b^2*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(b))+1/8*(14*a*A*b+5*a^2*B-8*b^2*B)*sqrt(a+b*tan(c+d*x))/(d*sqrt(cot(c+d*x)))+1/3*b*B*(a+b*tan(c+d*x))^(3/2)/(d*cot(c+d*x)^(3/2))+1/4*(2*A*b+3*a*B)*(a+b*tan(c+d*x))^(3/2)/(d*sqrt(cot(c+d*x)))],
[(a+b*tan(c+d*x))^(5/2)*(A+B*tan(c+d*x))/cot(c+d*x)^(3/2),x,17,-(I*a-b)^(5/2)*(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+1/64*(40*a^3*A*b-320*a*A*b^3-5*a^4*B-240*a^2*b^2*B+128*b^4*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(b^(3/2)*d)-(I*a+b)^(5/2)*(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/d+1/64*(40*a^2*A*b-64*A*b^3-5*a^3*B-112*a*b^2*B)*sqrt(a+b*tan(c+d*x))/(b*d*sqrt(cot(c+d*x)))+1/96*(40*a*A*b-5*a^2*B-48*b^2*B)*(a+b*tan(c+d*x))^(3/2)/(b*d*sqrt(cot(c+d*x)))+1/24*(8*A*b-a*B)*(a+b*tan(c+d*x))^(5/2)/(b*d*sqrt(cot(c+d*x)))+1/4*B*(a+b*tan(c+d*x))^(7/2)/(b*d*sqrt(cot(c+d*x)))],

# n<0
[cot(c+d*x)^(7/2)*(A+B*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,11,(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a+b))+2/15*(4*A*b-5*a*B)*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/(a^2*d)-2/5*A*cot(c+d*x)^(5/2)*sqrt(a+b*tan(c+d*x))/(a*d)+2/15*(15*a^2*A-8*A*b^2+10*a*b*B)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a^3*d)],
[cot(c+d*x)^(5/2)*(A+B*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,10,-(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))-(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a+b))-2/3*A*cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))/(a*d)+2/3*(2*A*b-3*a*B)*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a^2*d)],
[cot(c+d*x)^(3/2)*(A+B*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,9,-(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))+(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a+b))-2*A*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))/(a*d)],
[sqrt(cot(c+d*x))*(A+B*tan(c+d*x))/sqrt(a+b*tan(c+d*x)),x,8,(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))+(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a+b))],
[(A+B*tan(c+d*x))/(sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x))),x,13,(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))+2*B*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(b))-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a+b))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(3/2)*sqrt(a+b*tan(c+d*x))),x,14,(2*A*b-a*B)*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(b^(3/2)*d)-(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))-(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a+b))+B*sqrt(a+b*tan(c+d*x))/(b*d*sqrt(cot(c+d*x)))],
[cot(c+d*x)^(5/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,11,-(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(3/2)*d)-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(3/2)*d)-2/3*A*cot(c+d*x)^(3/2)/(a*d*sqrt(a+b*tan(c+d*x)))+2/3*b*(5*a^2*A*b+8*A*b^3-3*a^3*B-6*a*b^2*B)/(a^3*(a^2+b^2)*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))+2/3*(4*A*b-3*a*B)*sqrt(cot(c+d*x))/(a^2*d*sqrt(a+b*tan(c+d*x)))],
[cot(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,10,(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(3/2)*d)-(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(3/2)*d)-2*b*(a^2*A+2*A*b^2-a*b*B)/(a^2*(a^2+b^2)*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))-2*A*sqrt(cot(c+d*x))/(a*d*sqrt(a+b*tan(c+d*x)))],
[cot(c+d*x)^(1/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,9,(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(3/2)*d)+(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(3/2)*d)+2*b*(A*b-a*B)/(a*(a^2+b^2)*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(1/2)*(a+b*tan(c+d*x))^(3/2)),x,9,-(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(3/2)*d)+(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(3/2)*d)-2*(A*b-a*B)/((a^2+b^2)*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(3/2)),x,14,-(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(3/2)*d)+2*B*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(b^(3/2)*d)-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(3/2)*d)+2*a*(A*b-a*B)/(b*(a^2+b^2)*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))],
[cot(c+d*x)^(5/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,12,(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(5/2)*d)+(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(5/2)*d)+2/3*b*(8*a^4*A*b+30*a^2*A*b^3+16*A*b^5-3*a^5*B-17*a^3*b^2*B-8*a*b^4*B)/(a^4*(a^2+b^2)^2*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))-2/3*A*cot(c+d*x)^(3/2)/(a*d*(a+b*tan(c+d*x))^(3/2))+2/3*b*(7*a^2*A*b+8*A*b^3-3*a^3*B-4*a*b^2*B)/(a^3*(a^2+b^2)*d*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2))+2*(2*A*b-a*B)*sqrt(cot(c+d*x))/(a^2*d*(a+b*tan(c+d*x))^(3/2))],
[cot(c+d*x)^(3/2)*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,11,(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(5/2)*d)-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(5/2)*d)-2/3*b*(3*a^4*A+17*a^2*A*b^2+8*A*b^4-8*a^3*b*B-2*a*b^3*B)/(a^3*(a^2+b^2)^2*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))-2/3*b*(3*a^2*A+4*A*b^2-a*b*B)/(a^2*(a^2+b^2)*d*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2))-2*A*sqrt(cot(c+d*x))/(a*d*(a+b*tan(c+d*x))^(3/2))],
[sqrt(cot(c+d*x))*(A+B*tan(c+d*x))/(a+b*tan(c+d*x))^(5/2),x,10,-(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(5/2)*d)-(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(5/2)*d)+2/3*b*(8*a^2*A*b+2*A*b^3-5*a^3*B+a*b^2*B)/(a^2*(a^2+b^2)^2*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))+2/3*b*(A*b-a*B)/(a*(a^2+b^2)*d*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(5/2)),x,10,-(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(5/2)*d)+(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(5/2)*d)-2/3*(5*a^2*A*b-A*b^3-2*a^3*B+4*a*b^2*B)/(a*(a^2+b^2)^2*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))-2/3*(A*b-a*B)/((a^2+b^2)*d*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(5/2)),x,10,(A+I*B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(5/2)*d)+(A-I*B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(5/2)*d)+2/3*(2*a^2*A*b-4*A*b^3+a^3*B+7*a*b^2*B)/(b*(a^2+b^2)^2*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))+2/3*a*(A*b-a*B)/(b*(a^2+b^2)*d*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2))],
[(A+B*tan(c+d*x))/(cot(c+d*x)^(5/2)*(a+b*tan(c+d*x))^(5/2)),x,15,(I*A-B)*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a-b)^(5/2)*d)+2*B*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(b^(5/2)*d)-(I*A+B)*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/((I*a+b)^(5/2)*d)+2*a*(2*A*b^3-a*(a^2+3*b^2)*B)/(b^2*(a^2+b^2)^2*d*sqrt(cot(c+d*x))*sqrt(a+b*tan(c+d*x)))+2/3*a*(A*b-a*B)/(b*(a^2+b^2)*d*cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(3/2))],
[sqrt(cot(c+d*x))*(a*B+b*B*tan(c+d*x))/(a+b*tan(c+d*x))^(3/2),x,9,B*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))+B*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a+b))],
[(a*B+b*B*tan(c+d*x))/(sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^(3/2)),x,9,I*B*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))-I*B*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a+b))],
[(a*B+b*B*tan(c+d*x))/(cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^(3/2)),x,14,-B*arctan(sqrt(I*a-b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a-b))+2*B*arctanh(sqrt(b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(b))-B*arctanh(sqrt(I*a+b)*sqrt(tan(c+d*x))/sqrt(a+b*tan(c+d*x)))*sqrt(cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(I*a+b))],

# Integrands of the form Cot[c+d x]^(m/2) (A+B Tan[c+d x]) (a+b Tan[c+d x])^n with n symbolic
[cot(c+d*x)^m*(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,8,1/2*(A+I*B)*AppellF1(1-m,-n,1,2-m,-b*tan(c+d*x)/a,-I*tan(c+d*x))*cot(c+d*x)^(-1+m)*(a+b*tan(c+d*x))^n/(d*(1-m)*(1+b*tan(c+d*x)/a)^n)+1/2*(A-I*B)*AppellF1(1-m,-n,1,2-m,-b*tan(c+d*x)/a,I*tan(c+d*x))*cot(c+d*x)^(-1+m)*(a+b*tan(c+d*x))^n/(d*(1-m)*(1+b*tan(c+d*x)/a)^n)],
[cot(c+d*x)^(3/2)*(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,10,-(A+I*B)*AppellF1(-1/2,1,-n,1/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)-(A-I*B)*AppellF1(-1/2,1,-n,1/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)],
[sqrt(cot(c+d*x))*(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,10,(A+I*B)*AppellF1(1/2,1,-n,3/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*sqrt(cot(c+d*x))*(1+b*tan(c+d*x)/a)^n)+(A-I*B)*AppellF1(1/2,1,-n,3/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*sqrt(cot(c+d*x))*(1+b*tan(c+d*x)/a)^n)],
[(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x))/sqrt(cot(c+d*x)),x,10,1/3*(A+I*B)*AppellF1(3/2,1,-n,5/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*cot(c+d*x)^(3/2)*(1+b*tan(c+d*x)/a)^n)+1/3*(A-I*B)*AppellF1(3/2,1,-n,5/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*cot(c+d*x)^(3/2)*(1+b*tan(c+d*x)/a)^n)],
[(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x))/cot(c+d*x)^(3/2),x,10,1/5*(A+I*B)*AppellF1(5/2,1,-n,7/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*cot(c+d*x)^(5/2)*(1+b*tan(c+d*x)/a)^n)+1/5*(A-I*B)*AppellF1(5/2,1,-n,7/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*cot(c+d*x)^(5/2)*(1+b*tan(c+d*x)/a)^n)],
[tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,9,1/5*(A+I*B)*AppellF1(5/2,1,-n,7/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)+1/5*(A-I*B)*AppellF1(5/2,1,-n,7/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*tan(c+d*x)^(5/2)*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)],
[tan(c+d*x)^(1/2)*(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x)),x,9,1/3*(A+I*B)*AppellF1(3/2,1,-n,5/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)+1/3*(A-I*B)*AppellF1(3/2,1,-n,5/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*tan(c+d*x)^(3/2)*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)],
[(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x))/tan(c+d*x)^(1/2),x,9,(A+I*B)*AppellF1(1/2,1,-n,3/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)+(A-I*B)*AppellF1(1/2,1,-n,3/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*sqrt(tan(c+d*x))*(a+b*tan(c+d*x))^n/(d*(1+b*tan(c+d*x)/a)^n)],
[(a+b*tan(c+d*x))^n*(A+B*tan(c+d*x))/tan(c+d*x)^(3/2),x,9,-(A+I*B)*AppellF1(-1/2,1,-n,1/2,-I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*sqrt(tan(c+d*x))*(1+b*tan(c+d*x)/a)^n)-(A-I*B)*AppellF1(-1/2,1,-n,1/2,I*tan(c+d*x),-b*tan(c+d*x)/a)*(a+b*tan(c+d*x))^n/(d*sqrt(tan(c+d*x))*(1+b*tan(c+d*x)/a)^n)],

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

# Integrands of the form (a+a I Tan[e+f x])^m (c-c I Tan[e+f x])^n (A+B Tan[e+f x])

# Integrands of the form (a+a I Tan[e+f x])^m (c-c I Tan[e+f x])^n (A+B Tan[e+f x])

# m>0
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^n,x,3,a*(I*A+B)*(c-I*c*tan(e+f*x))^n/(f*n)-a*B*(c-I*c*tan(e+f*x))^(1+n)/(c*f*(1+n))],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^4,x,3,1/4*a*(I*A+B)*c^4*(1-I*tan(e+f*x))^4/f-1/5*a*B*c^4*(1-I*tan(e+f*x))^5/f],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^3,x,3,1/3*a*(I*A+B)*c^3*(1-I*tan(e+f*x))^3/f-1/4*a*B*c^3*(1-I*tan(e+f*x))^4/f],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^2,x,3,a*A*c^2*tan(e+f*x)/f-1/2*a*(I*A-B)*c^2*tan(e+f*x)^2/f-1/3*I*a*B*c^2*tan(e+f*x)^3/f],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x)),x,2,a*A*c*tan(e+f*x)/f+1/2*a*B*c*tan(e+f*x)^2/f],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x)),x,2,a*(A-I*B)*x-a*(I*A+B)*log(cos(e+f*x))/f+I*a*B*tan(e+f*x)/f],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x)),x,3,I*a*B*x/c+a*B*log(cos(e+f*x))/(c*f)+a*(A-I*B)/(c*f*(I+tan(e+f*x)))],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^2,x,2,1/2*a*(A+B*tan(e+f*x))^2/((I*A+B)*c^2*f*(1-I*tan(e+f*x))^2)],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^3,x,3,-1/3*a*(A-I*B)/(c^3*f*(I+tan(e+f*x))^3)-1/2*a*B/(c^3*f*(I+tan(e+f*x))^2)],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^4,x,3,-1/4*a*(I*A+B)/(c^4*f*(I+tan(e+f*x))^4)-1/3*I*a*B/(c^4*f*(I+tan(e+f*x))^3)],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^5,x,3,1/5*a*(A-I*B)/(c^5*f*(I+tan(e+f*x))^5)+1/4*a*B/(c^5*f*(I+tan(e+f*x))^4)],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^n,x,3,2*a^2*(I*A+B)*(c-I*c*tan(e+f*x))^n/(f*n)-a^2*(I*A+3*B)*(c-I*c*tan(e+f*x))^(1+n)/(c*f*(1+n))+a^2*B*(c-I*c*tan(e+f*x))^(2+n)/(c^2*f*(2+n))],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^5,x,3,2/5*a^2*(I*A+B)*c^5*(1-I*tan(e+f*x))^5/f-1/6*a^2*(I*A+3*B)*c^5*(1-I*tan(e+f*x))^6/f+1/7*a^2*B*c^5*(1-I*tan(e+f*x))^7/f],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^4,x,3,1/2*a^2*(I*A+B)*c^4*(1-I*tan(e+f*x))^4/f-1/5*a^2*(I*A+3*B)*c^4*(1-I*tan(e+f*x))^5/f+1/6*a^2*B*c^4*(1-I*tan(e+f*x))^6/f],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^3,x,3,2/3*a^2*(I*A+B)*c^3*(1-I*tan(e+f*x))^3/f-1/4*a^2*(I*A+3*B)*c^3*(1-I*tan(e+f*x))^4/f+1/5*a^2*B*c^3*(1-I*tan(e+f*x))^5/f],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^2,x,4,1/4*a^2*B*c^2*sec(e+f*x)^4/f+a^2*A*c^2*tan(e+f*x)/f+1/3*a^2*A*c^2*tan(e+f*x)^3/f],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x)),x,3,a^2*A*c*tan(e+f*x)/f+1/2*a^2*(I*A+B)*c*tan(e+f*x)^2/f+1/3*I*a^2*B*c*tan(e+f*x)^3/f],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x)),x,3,2*a^2*(A-I*B)*x-2*a^2*(I*A+B)*log(cos(e+f*x))/f-a^2*(A-I*B)*tan(e+f*x)/f+1/2*B*(a+I*a*tan(e+f*x))^2/f],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x)),x,3,-a^2*(A-3*I*B)*x/c+a^2*(I*A+3*B)*log(cos(e+f*x))/(c*f)-I*a^2*B*tan(e+f*x)/(c*f)+2*a^2*(A-I*B)/(c*f*(I+tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^2,x,3,-I*a^2*B*x/c^2-a^2*B*log(cos(e+f*x))/(c^2*f)+a^2*(I*A+B)/(c^2*f*(I+tan(e+f*x))^2)-a^2*(A-3*I*B)/(c^2*f*(I+tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^3,x,3,-2/3*a^2*(A-I*B)/(c^3*f*(I+tan(e+f*x))^3)-1/2*a^2*(I*A+3*B)/(c^3*f*(I+tan(e+f*x))^2)-I*a^2*B/(c^3*f*(I+tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^4,x,3,-1/2*a^2*(I*A+B)/(c^4*f*(I+tan(e+f*x))^4)+1/3*a^2*(A-3*I*B)/(c^4*f*(I+tan(e+f*x))^3)+1/2*a^2*B/(c^4*f*(I+tan(e+f*x))^2)],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^5,x,3,2/5*a^2*(A-I*B)/(c^5*f*(I+tan(e+f*x))^5)+1/4*a^2*(I*A+3*B)/(c^5*f*(I+tan(e+f*x))^4)+1/3*I*a^2*B/(c^5*f*(I+tan(e+f*x))^3)],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^6,x,3,1/3*a^2*(I*A+B)/(c^6*f*(I+tan(e+f*x))^6)-1/5*a^2*(A-3*I*B)/(c^6*f*(I+tan(e+f*x))^5)-1/4*a^2*B/(c^6*f*(I+tan(e+f*x))^4)],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^n,x,3,4*a^3*(I*A+B)*(c-I*c*tan(e+f*x))^n/(f*n)-4*a^3*(I*A+2*B)*(c-I*c*tan(e+f*x))^(1+n)/(c*f*(1+n))+a^3*(I*A+5*B)*(c-I*c*tan(e+f*x))^(2+n)/(c^2*f*(2+n))-a^3*B*(c-I*c*tan(e+f*x))^(3+n)/(c^3*f*(3+n))],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^6,x,3,2/3*a^3*(I*A+B)*c^6*(1-I*tan(e+f*x))^6/f-4/7*a^3*(I*A+2*B)*c^6*(1-I*tan(e+f*x))^7/f+1/8*a^3*(I*A+5*B)*c^6*(1-I*tan(e+f*x))^8/f-1/9*a^3*B*c^6*(1-I*tan(e+f*x))^9/f],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^5,x,3,4/5*a^3*(I*A+B)*c^5*(1-I*tan(e+f*x))^5/f-2/3*a^3*(I*A+2*B)*c^5*(1-I*tan(e+f*x))^6/f+1/7*a^3*(I*A+5*B)*c^5*(1-I*tan(e+f*x))^7/f-1/8*a^3*B*c^5*(1-I*tan(e+f*x))^8/f],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^4,x,3,a^3*(I*A+B)*c^4*(1-I*tan(e+f*x))^4/f-4/5*a^3*(I*A+2*B)*c^4*(1-I*tan(e+f*x))^5/f+1/6*a^3*(I*A+5*B)*c^4*(1-I*tan(e+f*x))^6/f-1/7*a^3*B*c^4*(1-I*tan(e+f*x))^7/f],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^3,x,5,1/6*a^3*B*c^3*sec(e+f*x)^6/f+a^3*A*c^3*tan(e+f*x)/f+2/3*a^3*A*c^3*tan(e+f*x)^3/f+1/5*a^3*A*c^3*tan(e+f*x)^5/f],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^2,x,3,-2/3*a^3*(I*A-B)*c^2*(1+I*tan(e+f*x))^3/f+1/4*a^3*(I*A-3*B)*c^2*(1+I*tan(e+f*x))^4/f+1/5*a^3*B*c^2*(1+I*tan(e+f*x))^5/f],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x)),x,3,-1/3*a^3*(I*A-B)*c*(1+I*tan(e+f*x))^3/f-1/4*a^3*B*c*(1+I*tan(e+f*x))^4/f],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x)),x,4,4*a^3*(A-I*B)*x-4*a^3*(I*A+B)*log(cos(e+f*x))/f-2*a^3*(A-I*B)*tan(e+f*x)/f+1/2*a*(I*A+B)*(a+I*a*tan(e+f*x))^2/f+1/3*B*(a+I*a*tan(e+f*x))^3/f],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x)),x,3,-4*a^3*(A-2*I*B)*x/c+4*a^3*(I*A+2*B)*log(cos(e+f*x))/(c*f)+a^3*(A-4*I*B)*tan(e+f*x)/(c*f)+1/2*a^3*B*tan(e+f*x)^2/(c*f)+4*a^3*(A-I*B)/(c*f*(I+tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^2,x,3,a^3*(A-5*I*B)*x/c^2-a^3*(I*A+5*B)*log(cos(e+f*x))/(c^2*f)+I*a^3*B*tan(e+f*x)/(c^2*f)+2*a^3*(I*A+B)/(c^2*f*(I+tan(e+f*x))^2)-4*a^3*(A-2*I*B)/(c^2*f*(I+tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^3,x,4,I*a^3*B*x/c^3+a^3*B*log(cos(e+f*x))/(c^3*f)-1/6*a^3*(I*A+B)*(1+I*tan(e+f*x))^3/(c^3*f*(1-I*tan(e+f*x))^3)-2*a^3*B/(c^3*f*(I+tan(e+f*x))^2)-4*I*a^3*B/(c^3*f*(I+tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^4,x,3,-1/8*a^3*(I*A+B)*(1+I*tan(e+f*x))^3/(c^4*f*(1-I*tan(e+f*x))^4)-1/48*a^3*(I*A-7*B)*(1+I*tan(e+f*x))^3/(c^4*f*(1-I*tan(e+f*x))^3)],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^5,x,3,4/5*a^3*(A-I*B)/(c^5*f*(I+tan(e+f*x))^5)+a^3*(I*A+2*B)/(c^5*f*(I+tan(e+f*x))^4)-1/3*a^3*(A-5*I*B)/(c^5*f*(I+tan(e+f*x))^3)-1/2*a^3*B/(c^5*f*(I+tan(e+f*x))^2)],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^6,x,3,2/3*a^3*(I*A+B)/(c^6*f*(I+tan(e+f*x))^6)-4/5*a^3*(A-2*I*B)/(c^6*f*(I+tan(e+f*x))^5)-1/4*a^3*(I*A+5*B)/(c^6*f*(I+tan(e+f*x))^4)-1/3*I*a^3*B/(c^6*f*(I+tan(e+f*x))^3)],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^7,x,3,-4/7*a^3*(A-I*B)/(c^7*f*(I+tan(e+f*x))^7)-2/3*a^3*(I*A+2*B)/(c^7*f*(I+tan(e+f*x))^6)+1/5*a^3*(A-5*I*B)/(c^7*f*(I+tan(e+f*x))^5)+1/4*a^3*B/(c^7*f*(I+tan(e+f*x))^4)],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^8,x,3,-1/2*a^3*(I*A+B)/(c^8*f*(I+tan(e+f*x))^8)+4/7*a^3*(A-2*I*B)/(c^8*f*(I+tan(e+f*x))^7)+1/6*a^3*(I*A+5*B)/(c^8*f*(I+tan(e+f*x))^6)+1/5*I*a^3*B/(c^8*f*(I+tan(e+f*x))^5)],

# m<0
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^n/(a+I*a*tan(e+f*x)),x,3,1/4*(I*A*(1-n)+B*(1+n))*hypergeom([1,n],[1+n],1/2*(1-I*tan(e+f*x)))*(c-I*c*tan(e+f*x))^n/(a*f*n)+1/2*(I*A-B)*(c-I*c*tan(e+f*x))^n/(a*f*(1+I*tan(e+f*x)))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^4/(a+I*a*tan(e+f*x)),x,3,-4*(3*A+5*I*B)*c^4*x/a-4*(3*I*A-5*B)*c^4*log(cos(e+f*x))/(a*f)-8*(A+I*B)*c^4/(a*f*(I-tan(e+f*x)))+(5*A+12*I*B)*c^4*tan(e+f*x)/(a*f)-1/2*(I*A-5*B)*c^4*tan(e+f*x)^2/(a*f)-1/3*I*B*c^4*tan(e+f*x)^3/(a*f)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^3/(a+I*a*tan(e+f*x)),x,3,-4*(A+2*I*B)*c^3*x/a-4*(I*A-2*B)*c^3*log(cos(e+f*x))/(a*f)-4*(A+I*B)*c^3/(a*f*(I-tan(e+f*x)))+(A+4*I*B)*c^3*tan(e+f*x)/(a*f)+1/2*B*c^3*tan(e+f*x)^2/(a*f)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^2/(a+I*a*tan(e+f*x)),x,3,-(A+3*I*B)*c^2*x/a-(I*A-3*B)*c^2*log(cos(e+f*x))/(a*f)-2*(A+I*B)*c^2/(a*f*(I-tan(e+f*x)))+I*B*c^2*tan(e+f*x)/(a*f)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))/(a+I*a*tan(e+f*x)),x,3,-I*B*c*x/a+B*c*log(cos(e+f*x))/(a*f)-(A+I*B)*c/(a*f*(I-tan(e+f*x)))],
[(A+B*tan(e+f*x))/(a+I*a*tan(e+f*x)),x,2,1/2*(A-I*B)*x/a+1/2*(I*A-B)/(f*(a+I*a*tan(e+f*x)))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))),x,4,1/2*A*x/(a*c)-1/2*cos(e+f*x)^2*(B-A*tan(e+f*x))/(a*c*f)],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^2),x,4,1/8*(3*A+I*B)*x/(a*c^2)+1/8*(-A-I*B)/(a*c^2*f*(I-tan(e+f*x)))+1/8*(I*A+B)/(a*c^2*f*(I+tan(e+f*x))^2)+1/4*A/(a*c^2*f*(I+tan(e+f*x)))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^3),x,4,1/8*(2*A+I*B)*x/(a*c^3)+1/16*(-A-I*B)/(a*c^3*f*(I-tan(e+f*x)))+1/12*(-A+I*B)/(a*c^3*f*(I+tan(e+f*x))^3)+1/8*I*A/(a*c^3*f*(I+tan(e+f*x))^2)+1/16*(3*A+I*B)/(a*c^3*f*(I+tan(e+f*x)))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^4),x,4,1/32*(5*A+3*I*B)*x/(a*c^4)+1/32*(-A-I*B)/(a*c^4*f*(I-tan(e+f*x)))+1/16*(-I*A-B)/(a*c^4*f*(I+tan(e+f*x))^4)-1/12*A/(a*c^4*f*(I+tan(e+f*x))^3)+1/32*(3*I*A-B)/(a*c^4*f*(I+tan(e+f*x))^2)+1/16*(2*A+I*B)/(a*c^4*f*(I+tan(e+f*x)))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^n/(a+I*a*tan(e+f*x))^2,x,3,1/16*(I*A*(2-n)+B*(2+n))*hypergeom([2,n],[1+n],1/2*(1-I*tan(e+f*x)))*(c-I*c*tan(e+f*x))^n/(a^2*f*n)+1/4*(I*A-B)*(c-I*c*tan(e+f*x))^n/(a^2*f*(1+I*tan(e+f*x))^2)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^5/(a+I*a*tan(e+f*x))^2,x,3,8*(3*A+7*I*B)*c^5*x/a^2+8*(3*I*A-7*B)*c^5*log(cos(e+f*x))/(a^2*f)-8*(I*A-B)*c^5/(a^2*f*(I-tan(e+f*x))^2)+16*(2*A+3*I*B)*c^5/(a^2*f*(I-tan(e+f*x)))-(7*A+24*I*B)*c^5*tan(e+f*x)/(a^2*f)+1/2*(I*A-7*B)*c^5*tan(e+f*x)^2/(a^2*f)+1/3*I*B*c^5*tan(e+f*x)^3/(a^2*f)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^4/(a+I*a*tan(e+f*x))^2,x,3,6*(A+3*I*B)*c^4*x/a^2+6*(I*A-3*B)*c^4*log(cos(e+f*x))/(a^2*f)-4*(I*A-B)*c^4/(a^2*f*(I-tan(e+f*x))^2)+4*(3*A+5*I*B)*c^4/(a^2*f*(I-tan(e+f*x)))-(A+6*I*B)*c^4*tan(e+f*x)/(a^2*f)-1/2*B*c^4*tan(e+f*x)^2/(a^2*f)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^3/(a+I*a*tan(e+f*x))^2,x,3,(A+5*I*B)*c^3*x/a^2+(I*A-5*B)*c^3*log(cos(e+f*x))/(a^2*f)-2*(I*A-B)*c^3/(a^2*f*(I-tan(e+f*x))^2)+4*(A+2*I*B)*c^3/(a^2*f*(I-tan(e+f*x)))-I*B*c^3*tan(e+f*x)/(a^2*f)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^2/(a+I*a*tan(e+f*x))^2,x,3,I*B*c^2*x/a^2-B*c^2*log(cos(e+f*x))/(a^2*f)-(I*A-B)*c^2/(a^2*f*(I-tan(e+f*x))^2)+(A+3*I*B)*c^2/(a^2*f*(I-tan(e+f*x)))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))/(a+I*a*tan(e+f*x))^2,x,2,-1/2*c*(A+B*tan(e+f*x))^2/(a^2*(I*A-B)*f*(1+I*tan(e+f*x))^2)],
[(A+B*tan(e+f*x))/(a+I*a*tan(e+f*x))^2,x,3,1/4*(A-I*B)*x/a^2+1/4*(I*A-B)/(f*(a+I*a*tan(e+f*x))^2)+1/4*(I*A+B)/(f*(a^2+I*a^2*tan(e+f*x)))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))),x,4,1/8*(3*A-I*B)*x/(a^2*c)+1/8*(-I*A+B)/(a^2*c*f*(I-tan(e+f*x))^2)-1/4*A/(a^2*c*f*(I-tan(e+f*x)))+1/8*(A-I*B)/(a^2*c*f*(I+tan(e+f*x)))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^2),x,5,3/8*A*x/(a^2*c^2)+3/8*A*cos(e+f*x)*sin(e+f*x)/(a^2*c^2*f)-1/4*cos(e+f*x)^4*(B-A*tan(e+f*x))/(a^2*c^2*f)],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^3),x,4,1/16*(5*A+I*B)*x/(a^2*c^3)+1/32*(-I*A+B)/(a^2*c^3*f*(I-tan(e+f*x))^2)+1/16*(-2*A-I*B)/(a^2*c^3*f*(I-tan(e+f*x)))+1/24*(-A+I*B)/(a^2*c^3*f*(I+tan(e+f*x))^3)+1/32*(3*I*A+B)/(a^2*c^3*f*(I+tan(e+f*x))^2)+3/16*A/(a^2*c^3*f*(I+tan(e+f*x)))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^4),x,4,5/64*(3*A+I*B)*x/(a^2*c^4)+1/64*(-I*A+B)/(a^2*c^4*f*(I-tan(e+f*x))^2)+1/64*(-5*A-3*I*B)/(a^2*c^4*f*(I-tan(e+f*x)))+1/32*(-I*A-B)/(a^2*c^4*f*(I+tan(e+f*x))^4)+1/48*(-3*A+I*B)/(a^2*c^4*f*(I+tan(e+f*x))^3)+3/32*I*A/(a^2*c^4*f*(I+tan(e+f*x))^2)+1/32*(5*A+I*B)/(a^2*c^4*f*(I+tan(e+f*x)))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^5),x,4,3/128*(7*A+3*I*B)*x/(a^2*c^5)+1/128*(-I*A+B)/(a^2*c^5*f*(I-tan(e+f*x))^2)+1/64*(-3*A-2*I*B)/(a^2*c^5*f*(I-tan(e+f*x)))+1/40*(A-I*B)/(a^2*c^5*f*(I+tan(e+f*x))^5)+1/64*(-3*I*A-B)/(a^2*c^5*f*(I+tan(e+f*x))^4)-1/16*A/(a^2*c^5*f*(I+tan(e+f*x))^3)+1/64*(5*I*A-B)/(a^2*c^5*f*(I+tan(e+f*x))^2)+5/128*(3*A+I*B)/(a^2*c^5*f*(I+tan(e+f*x)))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^n/(a+I*a*tan(e+f*x))^3,x,3,1/48*(I*A*(3-n)+B*(3+n))*hypergeom([3,n],[1+n],1/2*(1-I*tan(e+f*x)))*(c-I*c*tan(e+f*x))^n/(a^3*f*n)+1/6*(I*A-B)*(c-I*c*tan(e+f*x))^n/(a^3*f*(1+I*tan(e+f*x))^3)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^5/(a+I*a*tan(e+f*x))^3,x,3,-8*(A+4*I*B)*c^5*x/a^3-8*(I*A-4*B)*c^5*log(cos(e+f*x))/(a^3*f)+16/3*(A+I*B)*c^5/(a^3*f*(I-tan(e+f*x))^3)+8*(2*I*A-3*B)*c^5/(a^3*f*(I-tan(e+f*x))^2)-8*(3*A+7*I*B)*c^5/(a^3*f*(I-tan(e+f*x)))+(A+8*I*B)*c^5*tan(e+f*x)/(a^3*f)+1/2*B*c^5*tan(e+f*x)^2/(a^3*f)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^4/(a+I*a*tan(e+f*x))^3,x,3,-(A+7*I*B)*c^4*x/a^3-(I*A-7*B)*c^4*log(cos(e+f*x))/(a^3*f)+8/3*(A+I*B)*c^4/(a^3*f*(I-tan(e+f*x))^3)+2*(3*I*A-5*B)*c^4/(a^3*f*(I-tan(e+f*x))^2)-6*(A+3*I*B)*c^4/(a^3*f*(I-tan(e+f*x)))+I*B*c^4*tan(e+f*x)/(a^3*f)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^3/(a+I*a*tan(e+f*x))^3,x,4,-I*B*c^3*x/a^3+B*c^3*log(cos(e+f*x))/(a^3*f)-2*B*c^3/(a^3*f*(I-tan(e+f*x))^2)-4*I*B*c^3/(a^3*f*(I-tan(e+f*x)))+1/6*(I*A-B)*c^3*(1-I*tan(e+f*x))^3/(a^3*f*(1+I*tan(e+f*x))^3)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^2/(a+I*a*tan(e+f*x))^3,x,3,2/3*(A+I*B)*c^2/(a^3*f*(I-tan(e+f*x))^3)+1/2*(I*A-3*B)*c^2/(a^3*f*(I-tan(e+f*x))^2)-I*B*c^2/(a^3*f*(I-tan(e+f*x)))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))/(a+I*a*tan(e+f*x))^3,x,3,1/3*(A+I*B)*c/(a^3*f*(I-tan(e+f*x))^3)-1/2*B*c/(a^3*f*(I-tan(e+f*x))^2)],
[(A+B*tan(e+f*x))/(a+I*a*tan(e+f*x))^3,x,4,1/8*(A-I*B)*x/a^3+1/6*(I*A-B)/(f*(a+I*a*tan(e+f*x))^3)+1/8*(I*A+B)/(a*f*(a+I*a*tan(e+f*x))^2)+1/8*(I*A+B)/(f*(a^3+I*a^3*tan(e+f*x)))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))),x,4,1/8*(2*A-I*B)*x/(a^3*c)+1/12*(A+I*B)/(a^3*c*f*(I-tan(e+f*x))^3)-1/8*I*A/(a^3*c*f*(I-tan(e+f*x))^2)+1/16*(-3*A+I*B)/(a^3*c*f*(I-tan(e+f*x)))+1/16*(A-I*B)/(a^3*c*f*(I+tan(e+f*x)))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^2),x,4,1/16*(5*A-I*B)*x/(a^3*c^2)+1/24*(A+I*B)/(a^3*c^2*f*(I-tan(e+f*x))^3)+1/32*(-3*I*A+B)/(a^3*c^2*f*(I-tan(e+f*x))^2)-3/16*A/(a^3*c^2*f*(I-tan(e+f*x)))+1/32*(I*A+B)/(a^3*c^2*f*(I+tan(e+f*x))^2)+1/16*(2*A-I*B)/(a^3*c^2*f*(I+tan(e+f*x)))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^3),x,6,5/16*A*x/(a^3*c^3)+5/16*A*cos(e+f*x)*sin(e+f*x)/(a^3*c^3*f)+5/24*A*cos(e+f*x)^3*sin(e+f*x)/(a^3*c^3*f)-1/6*cos(e+f*x)^6*(B-A*tan(e+f*x))/(a^3*c^3*f)],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^4),x,4,5/128*(7*A+I*B)*x/(a^3*c^4)+1/96*(A+I*B)/(a^3*c^4*f*(I-tan(e+f*x))^3)+1/128*(-5*I*A+3*B)/(a^3*c^4*f*(I-tan(e+f*x))^2)-5/128*(3*A+I*B)/(a^3*c^4*f*(I-tan(e+f*x)))+1/64*(-I*A-B)/(a^3*c^4*f*(I+tan(e+f*x))^4)+1/48*(-2*A+I*B)/(a^3*c^4*f*(I+tan(e+f*x))^3)+1/64*(5*I*A+B)/(a^3*c^4*f*(I+tan(e+f*x))^2)+5/32*A/(a^3*c^4*f*(I+tan(e+f*x)))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^5),x,4,7/128*(4*A+I*B)*x/(a^3*c^5)+1/192*(A+I*B)/(a^3*c^5*f*(I-tan(e+f*x))^3)+1/128*(-3*I*A+2*B)/(a^3*c^5*f*(I-tan(e+f*x))^2)-3/256*(7*A+3*I*B)/(a^3*c^5*f*(I-tan(e+f*x)))+1/80*(A-I*B)/(a^3*c^5*f*(I+tan(e+f*x))^5)+1/64*(-2*I*A-B)/(a^3*c^5*f*(I+tan(e+f*x))^4)+1/96*(-5*A+I*B)/(a^3*c^5*f*(I+tan(e+f*x))^3)+5/64*I*A/(a^3*c^5*f*(I+tan(e+f*x))^2)+5/256*(7*A+I*B)/(a^3*c^5*f*(I+tan(e+f*x)))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^6),x,4,7/128*(3*A+I*B)*x/(a^3*c^6)+1/384*(A+I*B)/(a^3*c^6*f*(I-tan(e+f*x))^3)+1/512*(-7*I*A+5*B)/(a^3*c^6*f*(I-tan(e+f*x))^2)-7/256*(2*A+I*B)/(a^3*c^6*f*(I-tan(e+f*x)))+1/96*(I*A+B)/(a^3*c^6*f*(I+tan(e+f*x))^6)+1/80*(2*A-I*B)/(a^3*c^6*f*(I+tan(e+f*x))^5)+1/128*(-5*I*A-B)/(a^3*c^6*f*(I+tan(e+f*x))^4)-5/96*A/(a^3*c^6*f*(I+tan(e+f*x))^3)+5/512*(7*I*A-B)/(a^3*c^6*f*(I+tan(e+f*x))^2)+7/256*(4*A+I*B)/(a^3*c^6*f*(I+tan(e+f*x)))],

# Integrands of the form (a+a I Tan[e+f x])^m (c-c I Tan[e+f x])^(n/2) (A+B Tan[e+f x])

# m>0
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(7/2),x,3,2/7*a*(I*A+B)*(c-I*c*tan(e+f*x))^(7/2)/f-2/9*a*B*(c-I*c*tan(e+f*x))^(9/2)/(c*f)],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2),x,3,2/5*a*(I*A+B)*(c-I*c*tan(e+f*x))^(5/2)/f-2/7*a*B*(c-I*c*tan(e+f*x))^(7/2)/(c*f)],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2),x,3,2/3*a*(I*A+B)*(c-I*c*tan(e+f*x))^(3/2)/f-2/5*a*B*(c-I*c*tan(e+f*x))^(5/2)/(c*f)],
[sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x)),x,3,2*a*(I*A+B)*sqrt(c-I*c*tan(e+f*x))/f-2/3*a*B*(c-I*c*tan(e+f*x))^(3/2)/(c*f)],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))/sqrt(c-I*c*tan(e+f*x)),x,3,-2*a*(I*A+B)/(f*sqrt(c-I*c*tan(e+f*x)))-2*a*B*sqrt(c-I*c*tan(e+f*x))/(c*f)],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(3/2),x,3,2*a*B/(c*f*sqrt(c-I*c*tan(e+f*x)))-2/3*a*(I*A+B)/(f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(5/2),x,3,-2/5*a*(I*A+B)/(f*(c-I*c*tan(e+f*x))^(5/2))+2/3*a*B/(c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(7/2),x,3,-2/7*a*(I*A+B)/(f*(c-I*c*tan(e+f*x))^(7/2))+2/5*a*B/(c*f*(c-I*c*tan(e+f*x))^(5/2))],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(7/2),x,3,4/7*a^2*(I*A+B)*(c-I*c*tan(e+f*x))^(7/2)/f-2/9*a^2*(I*A+3*B)*(c-I*c*tan(e+f*x))^(9/2)/(c*f)+2/11*a^2*B*(c-I*c*tan(e+f*x))^(11/2)/(c^2*f)],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2),x,3,4/5*a^2*(I*A+B)*(c-I*c*tan(e+f*x))^(5/2)/f-2/7*a^2*(I*A+3*B)*(c-I*c*tan(e+f*x))^(7/2)/(c*f)+2/9*a^2*B*(c-I*c*tan(e+f*x))^(9/2)/(c^2*f)],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2),x,3,4/3*a^2*(I*A+B)*(c-I*c*tan(e+f*x))^(3/2)/f-2/5*a^2*(I*A+3*B)*(c-I*c*tan(e+f*x))^(5/2)/(c*f)+2/7*a^2*B*(c-I*c*tan(e+f*x))^(7/2)/(c^2*f)],
[sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x)),x,3,4*a^2*(I*A+B)*sqrt(c-I*c*tan(e+f*x))/f-2/3*a^2*(I*A+3*B)*(c-I*c*tan(e+f*x))^(3/2)/(c*f)+2/5*a^2*B*(c-I*c*tan(e+f*x))^(5/2)/(c^2*f)],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))/sqrt(c-I*c*tan(e+f*x)),x,3,-4*a^2*(I*A+B)/(f*sqrt(c-I*c*tan(e+f*x)))-2*a^2*(I*A+3*B)*sqrt(c-I*c*tan(e+f*x))/(c*f)+2/3*a^2*B*(c-I*c*tan(e+f*x))^(3/2)/(c^2*f)],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(3/2),x,3,2*a^2*(I*A+3*B)/(c*f*sqrt(c-I*c*tan(e+f*x)))+2*a^2*B*sqrt(c-I*c*tan(e+f*x))/(c^2*f)-4/3*a^2*(I*A+B)/(f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(5/2),x,3,-2*a^2*B/(c^2*f*sqrt(c-I*c*tan(e+f*x)))-4/5*a^2*(I*A+B)/(f*(c-I*c*tan(e+f*x))^(5/2))+2/3*a^2*(I*A+3*B)/(c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^2*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(7/2),x,3,-4/7*a^2*(I*A+B)/(f*(c-I*c*tan(e+f*x))^(7/2))+2/5*a^2*(I*A+3*B)/(c*f*(c-I*c*tan(e+f*x))^(5/2))-2/3*a^2*B/(c^2*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(7/2),x,3,8/7*a^3*(I*A+B)*(c-I*c*tan(e+f*x))^(7/2)/f-8/9*a^3*(I*A+2*B)*(c-I*c*tan(e+f*x))^(9/2)/(c*f)+2/11*a^3*(I*A+5*B)*(c-I*c*tan(e+f*x))^(11/2)/(c^2*f)-2/13*a^3*B*(c-I*c*tan(e+f*x))^(13/2)/(c^3*f)],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2),x,3,8/5*a^3*(I*A+B)*(c-I*c*tan(e+f*x))^(5/2)/f-8/7*a^3*(I*A+2*B)*(c-I*c*tan(e+f*x))^(7/2)/(c*f)+2/9*a^3*(I*A+5*B)*(c-I*c*tan(e+f*x))^(9/2)/(c^2*f)-2/11*a^3*B*(c-I*c*tan(e+f*x))^(11/2)/(c^3*f)],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2),x,3,8/3*a^3*(I*A+B)*(c-I*c*tan(e+f*x))^(3/2)/f-8/5*a^3*(I*A+2*B)*(c-I*c*tan(e+f*x))^(5/2)/(c*f)+2/7*a^3*(I*A+5*B)*(c-I*c*tan(e+f*x))^(7/2)/(c^2*f)-2/9*a^3*B*(c-I*c*tan(e+f*x))^(9/2)/(c^3*f)],
[sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x)),x,3,8*a^3*(I*A+B)*sqrt(c-I*c*tan(e+f*x))/f-8/3*a^3*(I*A+2*B)*(c-I*c*tan(e+f*x))^(3/2)/(c*f)+2/5*a^3*(I*A+5*B)*(c-I*c*tan(e+f*x))^(5/2)/(c^2*f)-2/7*a^3*B*(c-I*c*tan(e+f*x))^(7/2)/(c^3*f)],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))/sqrt(c-I*c*tan(e+f*x)),x,3,-8*a^3*(I*A+B)/(f*sqrt(c-I*c*tan(e+f*x)))-8*a^3*(I*A+2*B)*sqrt(c-I*c*tan(e+f*x))/(c*f)+2/3*a^3*(I*A+5*B)*(c-I*c*tan(e+f*x))^(3/2)/(c^2*f)-2/5*a^3*B*(c-I*c*tan(e+f*x))^(5/2)/(c^3*f)],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(3/2),x,3,8*a^3*(I*A+2*B)/(c*f*sqrt(c-I*c*tan(e+f*x)))+2*a^3*(I*A+5*B)*sqrt(c-I*c*tan(e+f*x))/(c^2*f)-8/3*a^3*(I*A+B)/(f*(c-I*c*tan(e+f*x))^(3/2))-2/3*a^3*B*(c-I*c*tan(e+f*x))^(3/2)/(c^3*f)],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(5/2),x,3,-2*a^3*(I*A+5*B)/(c^2*f*sqrt(c-I*c*tan(e+f*x)))-2*a^3*B*sqrt(c-I*c*tan(e+f*x))/(c^3*f)-8/5*a^3*(I*A+B)/(f*(c-I*c*tan(e+f*x))^(5/2))+8/3*a^3*(I*A+2*B)/(c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^3*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(7/2),x,3,2*a^3*B/(c^3*f*sqrt(c-I*c*tan(e+f*x)))-8/7*a^3*(I*A+B)/(f*(c-I*c*tan(e+f*x))^(7/2))+8/5*a^3*(I*A+2*B)/(c*f*(c-I*c*tan(e+f*x))^(5/2))-2/3*a^3*(I*A+5*B)/(c^2*f*(c-I*c*tan(e+f*x))^(3/2))],

# m<0
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(7/2)/(a+I*a*tan(e+f*x)),x,7,-2*(5*I*A-9*B)*c^(7/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))*sqrt(2)/(a*f)+2*(5*I*A-9*B)*c^3*sqrt(c-I*c*tan(e+f*x))/(a*f)+1/3*(5*I*A-9*B)*c^2*(c-I*c*tan(e+f*x))^(3/2)/(a*f)+1/10*(5*I*A-9*B)*c*(c-I*c*tan(e+f*x))^(5/2)/(a*f)+1/2*(I*A-B)*(c-I*c*tan(e+f*x))^(7/2)/(a*f*(1+I*tan(e+f*x)))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x)),x,6,-(3*I*A-7*B)*c^(5/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))*sqrt(2)/(a*f)+(3*I*A-7*B)*c^2*sqrt(c-I*c*tan(e+f*x))/(a*f)+1/6*(3*I*A-7*B)*c*(c-I*c*tan(e+f*x))^(3/2)/(a*f)+1/2*(I*A-B)*(c-I*c*tan(e+f*x))^(5/2)/(a*f*(1+I*tan(e+f*x)))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x)),x,5,-(I*A-5*B)*c^(3/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a*f*sqrt(2))+1/2*(I*A-5*B)*c*sqrt(c-I*c*tan(e+f*x))/(a*f)+1/2*(I*A-B)*(c-I*c*tan(e+f*x))^(3/2)/(a*f*(1+I*tan(e+f*x)))],
[sqrt(c-I*c*tan(e+f*x))*(A+B*tan(e+f*x))/(a+I*a*tan(e+f*x)),x,4,1/2*(I*A+3*B)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))*sqrt(c)/(a*f*sqrt(2))+1/2*(I*A-B)*sqrt(c-I*c*tan(e+f*x))/(a*f*(1+I*tan(e+f*x)))],
[(A+B*tan(e+f*x))/(sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))),x,5,1/4*(3*I*A+B)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a*f*sqrt(2)*sqrt(c))+1/4*(-3*I*A-B)/(a*f*sqrt(c-I*c*tan(e+f*x)))+1/2*(I*A-B)/(a*f*sqrt(c-I*c*tan(e+f*x))*(1+I*tan(e+f*x)))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)),x,6,1/8*(5*I*A-B)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a*c^(3/2)*f*sqrt(2))+1/8*(-5*I*A+B)/(a*c*f*sqrt(c-I*c*tan(e+f*x)))+1/12*(-5*I*A+B)/(a*f*(c-I*c*tan(e+f*x))^(3/2))+1/2*(I*A-B)/(a*f*(1+I*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2)),x,7,1/16*(7*I*A-3*B)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a*c^(5/2)*f*sqrt(2))+1/16*(-7*I*A+3*B)/(a*c^2*f*sqrt(c-I*c*tan(e+f*x)))+1/20*(-7*I*A+3*B)/(a*f*(c-I*c*tan(e+f*x))^(5/2))+1/2*(I*A-B)/(a*f*(1+I*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2))+1/24*(-7*I*A+3*B)/(a*c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(9/2)/(a+I*a*tan(e+f*x))^2,x,8,7*(5*I*A-13*B)*c^(9/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^2*f*sqrt(2))-7/2*(5*I*A-13*B)*c^4*sqrt(c-I*c*tan(e+f*x))/(a^2*f)-7/12*(5*I*A-13*B)*c^3*(c-I*c*tan(e+f*x))^(3/2)/(a^2*f)-7/40*(5*I*A-13*B)*c^2*(c-I*c*tan(e+f*x))^(5/2)/(a^2*f)-1/8*(5*I*A-13*B)*c*(c-I*c*tan(e+f*x))^(7/2)/(a^2*f*(1+I*tan(e+f*x)))+1/4*(I*A-B)*(c-I*c*tan(e+f*x))^(9/2)/(a^2*f*(1+I*tan(e+f*x))^2)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(7/2)/(a+I*a*tan(e+f*x))^2,x,7,5/2*(3*I*A-11*B)*c^(7/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^2*f*sqrt(2))-5/4*(3*I*A-11*B)*c^3*sqrt(c-I*c*tan(e+f*x))/(a^2*f)-5/24*(3*I*A-11*B)*c^2*(c-I*c*tan(e+f*x))^(3/2)/(a^2*f)-1/8*(3*I*A-11*B)*c*(c-I*c*tan(e+f*x))^(5/2)/(a^2*f*(1+I*tan(e+f*x)))+1/4*(I*A-B)*(c-I*c*tan(e+f*x))^(7/2)/(a^2*f*(1+I*tan(e+f*x))^2)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^2,x,6,3/4*(I*A-9*B)*c^(5/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^2*f*sqrt(2))-3/8*(I*A-9*B)*c^2*sqrt(c-I*c*tan(e+f*x))/(a^2*f)-1/8*(I*A-9*B)*c*(c-I*c*tan(e+f*x))^(3/2)/(a^2*f*(1+I*tan(e+f*x)))+1/4*(I*A-B)*(c-I*c*tan(e+f*x))^(5/2)/(a^2*f*(1+I*tan(e+f*x))^2)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^2,x,5,-1/8*(I*A+7*B)*c^(3/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^2*f*sqrt(2))+1/8*(I*A+7*B)*c*sqrt(c-I*c*tan(e+f*x))/(a^2*f*(1+I*tan(e+f*x)))+1/4*(I*A-B)*(c-I*c*tan(e+f*x))^(3/2)/(a^2*f*(1+I*tan(e+f*x))^2)],
[sqrt(c-I*c*tan(e+f*x))*(A+B*tan(e+f*x))/(a+I*a*tan(e+f*x))^2,x,5,1/16*(3*I*A+5*B)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))*sqrt(c)/(a^2*f*sqrt(2))+1/4*(I*A-B)*sqrt(c-I*c*tan(e+f*x))/(a^2*f*(1+I*tan(e+f*x))^2)+1/16*(3*I*A+5*B)*sqrt(c-I*c*tan(e+f*x))/(a^2*f*(1+I*tan(e+f*x)))],
[(A+B*tan(e+f*x))/(sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^2),x,6,3/32*(5*I*A+3*B)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^2*f*sqrt(2)*sqrt(c))-3/32*(5*I*A+3*B)/(a^2*f*sqrt(c-I*c*tan(e+f*x)))+1/4*(I*A-B)/(a^2*f*sqrt(c-I*c*tan(e+f*x))*(1+I*tan(e+f*x))^2)+1/16*(5*I*A+3*B)/(a^2*f*sqrt(c-I*c*tan(e+f*x))*(1+I*tan(e+f*x)))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^(3/2)),x,7,5/64*(7*I*A+B)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^2*c^(3/2)*f*sqrt(2))-5/64*(7*I*A+B)/(a^2*c*f*sqrt(c-I*c*tan(e+f*x)))-5/96*(7*I*A+B)/(a^2*f*(c-I*c*tan(e+f*x))^(3/2))+1/4*(I*A-B)/(a^2*f*(1+I*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^(3/2))+1/16*(7*I*A+B)/(a^2*f*(1+I*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^(5/2)),x,8,7/128*(9*I*A-B)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^2*c^(5/2)*f*sqrt(2))-7/128*(9*I*A-B)/(a^2*c^2*f*sqrt(c-I*c*tan(e+f*x)))-7/160*(9*I*A-B)/(a^2*f*(c-I*c*tan(e+f*x))^(5/2))+1/4*(I*A-B)/(a^2*f*(1+I*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^(5/2))+1/16*(9*I*A-B)/(a^2*f*(1+I*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2))-7/192*(9*I*A-B)/(a^2*c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(9/2)/(a+I*a*tan(e+f*x))^3,x,8,-35/4*(I*A-5*B)*c^(9/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^3*f*sqrt(2))+35/8*(I*A-5*B)*c^4*sqrt(c-I*c*tan(e+f*x))/(a^3*f)+35/48*(I*A-5*B)*c^3*(c-I*c*tan(e+f*x))^(3/2)/(a^3*f)+7/16*(I*A-5*B)*c^2*(c-I*c*tan(e+f*x))^(5/2)/(a^3*f*(1+I*tan(e+f*x)))-1/8*(I*A-5*B)*c*(c-I*c*tan(e+f*x))^(7/2)/(a^3*f*(1+I*tan(e+f*x))^2)+1/6*(I*A-B)*(c-I*c*tan(e+f*x))^(9/2)/(a^3*f*(1+I*tan(e+f*x))^3)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(7/2)/(a+I*a*tan(e+f*x))^3,x,7,-5/8*(I*A-13*B)*c^(7/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^3*f*sqrt(2))+5/16*(I*A-13*B)*c^3*sqrt(c-I*c*tan(e+f*x))/(a^3*f)+5/48*(I*A-13*B)*c^2*(c-I*c*tan(e+f*x))^(3/2)/(a^3*f*(1+I*tan(e+f*x)))-1/24*(I*A-13*B)*c*(c-I*c*tan(e+f*x))^(5/2)/(a^3*f*(1+I*tan(e+f*x))^2)+1/6*(I*A-B)*(c-I*c*tan(e+f*x))^(7/2)/(a^3*f*(1+I*tan(e+f*x))^3)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^3,x,6,1/16*(I*A+11*B)*c^(5/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^3*f*sqrt(2))-1/16*(I*A+11*B)*c^2*sqrt(c-I*c*tan(e+f*x))/(a^3*f*(1+I*tan(e+f*x)))+1/24*(I*A+11*B)*c*(c-I*c*tan(e+f*x))^(3/2)/(a^3*f*(1+I*tan(e+f*x))^2)+1/6*(I*A-B)*(c-I*c*tan(e+f*x))^(5/2)/(a^3*f*(1+I*tan(e+f*x))^3)],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^3,x,6,-1/32*(I*A+3*B)*c^(3/2)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^3*f*sqrt(2))+1/8*(I*A+3*B)*c*sqrt(c-I*c*tan(e+f*x))/(a^3*f*(1+I*tan(e+f*x))^2)-1/32*(I*A+3*B)*c*sqrt(c-I*c*tan(e+f*x))/(a^3*f*(1+I*tan(e+f*x)))+1/6*(I*A-B)*(c-I*c*tan(e+f*x))^(3/2)/(a^3*f*(1+I*tan(e+f*x))^3)],
[sqrt(c-I*c*tan(e+f*x))*(A+B*tan(e+f*x))/(a+I*a*tan(e+f*x))^3,x,6,1/64*(5*I*A+7*B)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))*sqrt(c)/(a^3*f*sqrt(2))+1/6*(I*A-B)*sqrt(c-I*c*tan(e+f*x))/(a^3*f*(1+I*tan(e+f*x))^3)+1/48*(5*I*A+7*B)*sqrt(c-I*c*tan(e+f*x))/(a^3*f*(1+I*tan(e+f*x))^2)+1/64*(5*I*A+7*B)*sqrt(c-I*c*tan(e+f*x))/(a^3*f*(1+I*tan(e+f*x)))],
[(A+B*tan(e+f*x))/(sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^3),x,7,5/128*(7*I*A+5*B)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^3*f*sqrt(2)*sqrt(c))-5/128*(7*I*A+5*B)/(a^3*f*sqrt(c-I*c*tan(e+f*x)))+1/6*(I*A-B)/(a^3*f*sqrt(c-I*c*tan(e+f*x))*(1+I*tan(e+f*x))^3)+1/48*(7*I*A+5*B)/(a^3*f*sqrt(c-I*c*tan(e+f*x))*(1+I*tan(e+f*x))^2)+5/192*(7*I*A+5*B)/(a^3*f*sqrt(c-I*c*tan(e+f*x))*(1+I*tan(e+f*x)))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^(3/2)),x,8,35/256*(3*I*A+B)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^3*c^(3/2)*f*sqrt(2))-35/256*(3*I*A+B)/(a^3*c*f*sqrt(c-I*c*tan(e+f*x)))-35/384*(3*I*A+B)/(a^3*f*(c-I*c*tan(e+f*x))^(3/2))+1/6*(I*A-B)/(a^3*f*(1+I*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^(3/2))+1/16*(3*I*A+B)/(a^3*f*(1+I*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^(3/2))+7/64*(3*I*A+B)/(a^3*f*(1+I*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^(5/2)),x,9,21/512*(11*I*A+B)*arctanh(sqrt(c-I*c*tan(e+f*x))/(sqrt(2)*sqrt(c)))/(a^3*c^(5/2)*f*sqrt(2))-21/512*(11*I*A+B)/(a^3*c^2*f*sqrt(c-I*c*tan(e+f*x)))-21/640*(11*I*A+B)/(a^3*f*(c-I*c*tan(e+f*x))^(5/2))+1/6*(I*A-B)/(a^3*f*(1+I*tan(e+f*x))^3*(c-I*c*tan(e+f*x))^(5/2))+1/48*(11*I*A+B)/(a^3*f*(1+I*tan(e+f*x))^2*(c-I*c*tan(e+f*x))^(5/2))+3/64*(11*I*A+B)/(a^3*f*(1+I*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2))-7/256*(11*I*A+B)/(a^3*c*f*(c-I*c*tan(e+f*x))^(3/2))],

# Integrands of the form (a+a I Tan[e+f x])^(m/2) (c-c I Tan[e+f x])^(n/2) (A+B Tan[e+f x])

# m>0
[sqrt(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(7/2),x,8,-5/4*(4*I*A-3*B)*c^(7/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))*sqrt(a)/f-5/8*(4*I*A-3*B)*c^3*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/f-5/24*(4*I*A-3*B)*c^2*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)/f-1/12*(4*I*A-3*B)*c*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2)/f+1/4*B*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(7/2)/f],
[sqrt(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2),x,7,-(3*I*A-2*B)*c^(5/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))*sqrt(a)/f-1/2*(3*I*A-2*B)*c^2*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/f-1/6*(3*I*A-2*B)*c*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)/f+1/3*B*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2)/f],
[sqrt(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2),x,6,-(2*I*A-B)*c^(3/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))*sqrt(a)/f-1/2*(2*I*A-B)*c*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/f+1/2*B*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)/f],
[sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))*(A+B*tan(e+f*x)),x,5,-2*I*A*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))*sqrt(a)*sqrt(c)/f+B*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/f],
[sqrt(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))/sqrt(c-I*c*tan(e+f*x)),x,5,2*B*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))*sqrt(a)/(f*sqrt(c))-(I*A+B)*sqrt(a+I*a*tan(e+f*x))/(f*sqrt(c-I*c*tan(e+f*x)))],
[sqrt(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(3/2),x,3,-1/3*(I*A-2*B)*sqrt(a+I*a*tan(e+f*x))/(c*f*sqrt(c-I*c*tan(e+f*x)))-1/3*(I*A+B)*sqrt(a+I*a*tan(e+f*x))/(f*(c-I*c*tan(e+f*x))^(3/2))],
[sqrt(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(5/2),x,4,-1/15*(2*I*A-3*B)*sqrt(a+I*a*tan(e+f*x))/(c^2*f*sqrt(c-I*c*tan(e+f*x)))-1/5*(I*A+B)*sqrt(a+I*a*tan(e+f*x))/(f*(c-I*c*tan(e+f*x))^(5/2))-1/15*(2*I*A-3*B)*sqrt(a+I*a*tan(e+f*x))/(c*f*(c-I*c*tan(e+f*x))^(3/2))],
[sqrt(a+I*a*tan(e+f*x))*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(7/2),x,5,-2/105*(3*I*A-4*B)*sqrt(a+I*a*tan(e+f*x))/(c^3*f*sqrt(c-I*c*tan(e+f*x)))-1/7*(I*A+B)*sqrt(a+I*a*tan(e+f*x))/(f*(c-I*c*tan(e+f*x))^(7/2))-1/35*(3*I*A-4*B)*sqrt(a+I*a*tan(e+f*x))/(c*f*(c-I*c*tan(e+f*x))^(5/2))-2/105*(3*I*A-4*B)*sqrt(a+I*a*tan(e+f*x))/(c^2*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(7/2),x,8,-1/4*a^(3/2)*(5*I*A-2*B)*c^(7/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/f+1/8*a*(5*A+2*I*B)*c^3*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))*tan(e+f*x)/f-1/12*(5*I*A-2*B)*c^2*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2)/f-1/20*(5*I*A-2*B)*c*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(5/2)/f+1/5*B*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(7/2)/f],
[(a+I*a*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2),x,7,-1/4*a^(3/2)*(4*I*A-B)*c^(5/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/f+1/8*a*(4*A+I*B)*c^2*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))*tan(e+f*x)/f-1/12*(4*I*A-B)*c*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2)/f+1/4*B*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(5/2)/f],
[(a+I*a*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2),x,6,-I*a^(3/2)*A*c^(3/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/f+1/2*a*A*c*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))*tan(e+f*x)/f+1/3*B*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2)/f],
[sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x)),x,6,-a^(3/2)*(2*I*A+B)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))*sqrt(c)/f+1/2*a*(2*I*A+B)*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/f+1/2*B*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2)/f],
[(a+I*a*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x))/sqrt(c-I*c*tan(e+f*x)),x,6,2*a^(3/2)*(I*A+2*B)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(f*sqrt(c))-a*(I*A+2*B)*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(c*f)-(I*A+B)*(a+I*a*tan(e+f*x))^(3/2)/(f*sqrt(c-I*c*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(3/2),x,6,-2*a^(3/2)*B*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(c^(3/2)*f)+2*a*B*sqrt(a+I*a*tan(e+f*x))/(c*f*sqrt(c-I*c*tan(e+f*x)))-1/3*(I*A+B)*(a+I*a*tan(e+f*x))^(3/2)/(f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(5/2),x,3,-1/5*(I*A+B)*(a+I*a*tan(e+f*x))^(3/2)/(f*(c-I*c*tan(e+f*x))^(5/2))-1/15*(I*A-4*B)*(a+I*a*tan(e+f*x))^(3/2)/(c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(7/2),x,4,-1/7*(I*A+B)*(a+I*a*tan(e+f*x))^(3/2)/(f*(c-I*c*tan(e+f*x))^(7/2))-1/35*(2*I*A-5*B)*(a+I*a*tan(e+f*x))^(3/2)/(c*f*(c-I*c*tan(e+f*x))^(5/2))-1/105*(2*I*A-5*B)*(a+I*a*tan(e+f*x))^(3/2)/(c^2*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(9/2),x,5,-1/9*(I*A+B)*(a+I*a*tan(e+f*x))^(3/2)/(f*(c-I*c*tan(e+f*x))^(9/2))-1/21*(I*A-2*B)*(a+I*a*tan(e+f*x))^(3/2)/(c*f*(c-I*c*tan(e+f*x))^(7/2))-2/105*(I*A-2*B)*(a+I*a*tan(e+f*x))^(3/2)/(c^2*f*(c-I*c*tan(e+f*x))^(5/2))-2/315*(I*A-2*B)*(a+I*a*tan(e+f*x))^(3/2)/(c^3*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(11/2),x,6,-1/11*(I*A+B)*(a+I*a*tan(e+f*x))^(3/2)/(f*(c-I*c*tan(e+f*x))^(11/2))-1/99*(4*I*A-7*B)*(a+I*a*tan(e+f*x))^(3/2)/(c*f*(c-I*c*tan(e+f*x))^(9/2))-1/231*(4*I*A-7*B)*(a+I*a*tan(e+f*x))^(3/2)/(c^2*f*(c-I*c*tan(e+f*x))^(7/2))-2/1155*(4*I*A-7*B)*(a+I*a*tan(e+f*x))^(3/2)/(c^3*f*(c-I*c*tan(e+f*x))^(5/2))-2/3465*(4*I*A-7*B)*(a+I*a*tan(e+f*x))^(3/2)/(c^4*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(7/2),x,8,-1/8*a^(5/2)*(6*I*A-B)*c^(7/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/f+1/16*a^2*(6*A+I*B)*c^3*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))*tan(e+f*x)/f+1/24*a*(6*A+I*B)*c^2*tan(e+f*x)*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2)/f-1/30*(6*I*A-B)*c*(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(5/2)/f+1/6*B*(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(7/2)/f],
[(a+I*a*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2),x,7,-3/4*I*a^(5/2)*A*c^(5/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/f+3/8*a^2*A*c^2*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))*tan(e+f*x)/f+1/4*a*A*c*tan(e+f*x)*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2)/f+1/5*B*(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(5/2)/f],
[(a+I*a*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2),x,7,-1/4*a^(5/2)*(4*I*A+B)*c^(3/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/f+1/8*a^2*(4*A-I*B)*c*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))*tan(e+f*x)/f+1/12*a*(4*I*A+B)*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2)/f+1/4*B*(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(3/2)/f],
[sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x)),x,7,-a^(5/2)*(3*I*A+2*B)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))*sqrt(c)/f+1/2*a^2*(3*I*A+2*B)*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/f+1/6*a*(3*I*A+2*B)*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2)/f+1/3*B*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(5/2)/f],
[(a+I*a*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x))/sqrt(c-I*c*tan(e+f*x)),x,7,3*a^(5/2)*(2*I*A+3*B)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(f*sqrt(c))-3/2*a^2*(2*I*A+3*B)*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(c*f)-1/2*a*(2*I*A+3*B)*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2)/(c*f)-(I*A+B)*(a+I*a*tan(e+f*x))^(5/2)/(f*sqrt(c-I*c*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(3/2),x,7,-2*a^(5/2)*(I*A+4*B)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(c^(3/2)*f)+a^2*(I*A+4*B)*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(c^2*f)+2/3*a*(I*A+4*B)*(a+I*a*tan(e+f*x))^(3/2)/(c*f*sqrt(c-I*c*tan(e+f*x)))-1/3*(I*A+B)*(a+I*a*tan(e+f*x))^(5/2)/(f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(5/2),x,7,2*a^(5/2)*B*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(c^(5/2)*f)-2*a^2*B*sqrt(a+I*a*tan(e+f*x))/(c^2*f*sqrt(c-I*c*tan(e+f*x)))-1/5*(I*A+B)*(a+I*a*tan(e+f*x))^(5/2)/(f*(c-I*c*tan(e+f*x))^(5/2))+2/3*a*B*(a+I*a*tan(e+f*x))^(3/2)/(c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(7/2),x,3,-1/7*(I*A+B)*(a+I*a*tan(e+f*x))^(5/2)/(f*(c-I*c*tan(e+f*x))^(7/2))-1/35*(I*A-6*B)*(a+I*a*tan(e+f*x))^(5/2)/(c*f*(c-I*c*tan(e+f*x))^(5/2))],
[(a+I*a*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(9/2),x,4,-1/9*(I*A+B)*(a+I*a*tan(e+f*x))^(5/2)/(f*(c-I*c*tan(e+f*x))^(9/2))-1/63*(2*I*A-7*B)*(a+I*a*tan(e+f*x))^(5/2)/(c*f*(c-I*c*tan(e+f*x))^(7/2))-1/315*(2*I*A-7*B)*(a+I*a*tan(e+f*x))^(5/2)/(c^2*f*(c-I*c*tan(e+f*x))^(5/2))],
[(a+I*a*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(11/2),x,5,-1/11*(I*A+B)*(a+I*a*tan(e+f*x))^(5/2)/(f*(c-I*c*tan(e+f*x))^(11/2))-1/99*(3*I*A-8*B)*(a+I*a*tan(e+f*x))^(5/2)/(c*f*(c-I*c*tan(e+f*x))^(9/2))-2/693*(3*I*A-8*B)*(a+I*a*tan(e+f*x))^(5/2)/(c^2*f*(c-I*c*tan(e+f*x))^(7/2))-2/3465*(3*I*A-8*B)*(a+I*a*tan(e+f*x))^(5/2)/(c^3*f*(c-I*c*tan(e+f*x))^(5/2))],
[(a+I*a*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(13/2),x,6,-1/13*(I*A+B)*(a+I*a*tan(e+f*x))^(5/2)/(f*(c-I*c*tan(e+f*x))^(13/2))-1/143*(4*I*A-9*B)*(a+I*a*tan(e+f*x))^(5/2)/(c*f*(c-I*c*tan(e+f*x))^(11/2))-1/429*(4*I*A-9*B)*(a+I*a*tan(e+f*x))^(5/2)/(c^2*f*(c-I*c*tan(e+f*x))^(9/2))-2/3003*(4*I*A-9*B)*(a+I*a*tan(e+f*x))^(5/2)/(c^3*f*(c-I*c*tan(e+f*x))^(7/2))-2/15015*(4*I*A-9*B)*(a+I*a*tan(e+f*x))^(5/2)/(c^4*f*(c-I*c*tan(e+f*x))^(5/2))],
[(a+I*a*tan(e+f*x))^(7/2)*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(9/2),x,9,-5/64*a^(7/2)*(8*I*A-B)*c^(9/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/f+5/128*a^3*(8*A+I*B)*c^4*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))*tan(e+f*x)/f+5/192*a^2*(8*A+I*B)*c^3*tan(e+f*x)*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2)/f+1/48*a*(8*A+I*B)*c^2*tan(e+f*x)*(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(5/2)/f-1/56*(8*I*A-B)*c*(a+I*a*tan(e+f*x))^(7/2)*(c-I*c*tan(e+f*x))^(7/2)/f+1/8*B*(a+I*a*tan(e+f*x))^(7/2)*(c-I*c*tan(e+f*x))^(9/2)/f],
[(a+I*a*tan(e+f*x))^(7/2)*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(7/2),x,8,-5/8*I*a^(7/2)*A*c^(7/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/f+5/16*a^3*A*c^3*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))*tan(e+f*x)/f+5/24*a^2*A*c^2*tan(e+f*x)*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2)/f+1/6*a*A*c*tan(e+f*x)*(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(5/2)/f+1/7*B*(a+I*a*tan(e+f*x))^(7/2)*(c-I*c*tan(e+f*x))^(7/2)/f],
[(a+I*a*tan(e+f*x))^(7/2)*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2),x,8,-1/8*a^(7/2)*(6*I*A+B)*c^(5/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/f+1/16*a^3*(6*A-I*B)*c^2*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))*tan(e+f*x)/f+1/24*a^2*(6*A-I*B)*c*tan(e+f*x)*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2)/f+1/30*a*(6*I*A+B)*(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(5/2)/f+1/6*B*(a+I*a*tan(e+f*x))^(7/2)*(c-I*c*tan(e+f*x))^(5/2)/f],
[(a+I*a*tan(e+f*x))^(7/2)*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2),x,8,-1/4*a^(7/2)*(5*I*A+2*B)*c^(3/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/f+1/8*a^3*(5*A-2*I*B)*c*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))*tan(e+f*x)/f+1/12*a^2*(5*I*A+2*B)*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2)/f+1/20*a*(5*I*A+2*B)*(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(3/2)/f+1/5*B*(a+I*a*tan(e+f*x))^(7/2)*(c-I*c*tan(e+f*x))^(3/2)/f],
[sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(7/2)*(A+B*tan(e+f*x)),x,8,-5/4*a^(7/2)*(4*I*A+3*B)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))*sqrt(c)/f+5/8*a^3*(4*I*A+3*B)*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/f+5/24*a^2*(4*I*A+3*B)*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2)/f+1/12*a*(4*I*A+3*B)*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(5/2)/f+1/4*B*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(7/2)/f],
[(a+I*a*tan(e+f*x))^(7/2)*(A+B*tan(e+f*x))/sqrt(c-I*c*tan(e+f*x)),x,8,5*a^(7/2)*(3*I*A+4*B)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(f*sqrt(c))-5/2*a^3*(3*I*A+4*B)*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(c*f)-5/6*a^2*(3*I*A+4*B)*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2)/(c*f)-1/3*a*(3*I*A+4*B)*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(5/2)/(c*f)-(I*A+B)*(a+I*a*tan(e+f*x))^(7/2)/(f*sqrt(c-I*c*tan(e+f*x)))],
[(a+I*a*tan(e+f*x))^(7/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(3/2),x,8,-5*a^(7/2)*(2*I*A+5*B)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(c^(3/2)*f)+5/2*a^3*(2*I*A+5*B)*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(c^2*f)+5/6*a^2*(2*I*A+5*B)*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2)/(c^2*f)+2/3*a*(2*I*A+5*B)*(a+I*a*tan(e+f*x))^(5/2)/(c*f*sqrt(c-I*c*tan(e+f*x)))-1/3*(I*A+B)*(a+I*a*tan(e+f*x))^(7/2)/(f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(7/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(5/2),x,8,2*a^(7/2)*(I*A+6*B)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(c^(5/2)*f)-a^3*(I*A+6*B)*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(c^3*f)-2/3*a^2*(I*A+6*B)*(a+I*a*tan(e+f*x))^(3/2)/(c^2*f*sqrt(c-I*c*tan(e+f*x)))-1/5*(I*A+B)*(a+I*a*tan(e+f*x))^(7/2)/(f*(c-I*c*tan(e+f*x))^(5/2))+2/15*a*(I*A+6*B)*(a+I*a*tan(e+f*x))^(5/2)/(c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(7/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(7/2),x,8,-2*a^(7/2)*B*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(c^(7/2)*f)+2*a^3*B*sqrt(a+I*a*tan(e+f*x))/(c^3*f*sqrt(c-I*c*tan(e+f*x)))-1/7*(I*A+B)*(a+I*a*tan(e+f*x))^(7/2)/(f*(c-I*c*tan(e+f*x))^(7/2))+2/5*a*B*(a+I*a*tan(e+f*x))^(5/2)/(c*f*(c-I*c*tan(e+f*x))^(5/2))-2/3*a^2*B*(a+I*a*tan(e+f*x))^(3/2)/(c^2*f*(c-I*c*tan(e+f*x))^(3/2))],
[(a+I*a*tan(e+f*x))^(7/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(9/2),x,3,-1/9*(I*A+B)*(a+I*a*tan(e+f*x))^(7/2)/(f*(c-I*c*tan(e+f*x))^(9/2))-1/63*(I*A-8*B)*(a+I*a*tan(e+f*x))^(7/2)/(c*f*(c-I*c*tan(e+f*x))^(7/2))],
[(a+I*a*tan(e+f*x))^(7/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(11/2),x,4,-1/11*(I*A+B)*(a+I*a*tan(e+f*x))^(7/2)/(f*(c-I*c*tan(e+f*x))^(11/2))-1/99*(2*I*A-9*B)*(a+I*a*tan(e+f*x))^(7/2)/(c*f*(c-I*c*tan(e+f*x))^(9/2))-1/693*(2*I*A-9*B)*(a+I*a*tan(e+f*x))^(7/2)/(c^2*f*(c-I*c*tan(e+f*x))^(7/2))],
[(a+I*a*tan(e+f*x))^(7/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(13/2),x,5,-1/13*(I*A+B)*(a+I*a*tan(e+f*x))^(7/2)/(f*(c-I*c*tan(e+f*x))^(13/2))-1/143*(3*I*A-10*B)*(a+I*a*tan(e+f*x))^(7/2)/(c*f*(c-I*c*tan(e+f*x))^(11/2))-2/1287*(3*I*A-10*B)*(a+I*a*tan(e+f*x))^(7/2)/(c^2*f*(c-I*c*tan(e+f*x))^(9/2))-2/9009*(3*I*A-10*B)*(a+I*a*tan(e+f*x))^(7/2)/(c^3*f*(c-I*c*tan(e+f*x))^(7/2))],
[(a+I*a*tan(e+f*x))^(7/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(15/2),x,6,-1/15*(I*A+B)*(a+I*a*tan(e+f*x))^(7/2)/(f*(c-I*c*tan(e+f*x))^(15/2))-1/195*(4*I*A-11*B)*(a+I*a*tan(e+f*x))^(7/2)/(c*f*(c-I*c*tan(e+f*x))^(13/2))-1/715*(4*I*A-11*B)*(a+I*a*tan(e+f*x))^(7/2)/(c^2*f*(c-I*c*tan(e+f*x))^(11/2))-2/6435*(4*I*A-11*B)*(a+I*a*tan(e+f*x))^(7/2)/(c^3*f*(c-I*c*tan(e+f*x))^(9/2))-2/45045*(4*I*A-11*B)*(a+I*a*tan(e+f*x))^(7/2)/(c^4*f*(c-I*c*tan(e+f*x))^(7/2))],
[(a+I*a*tan(e+f*x))^(7/2)*(A+B*tan(e+f*x))/(c-I*c*tan(e+f*x))^(17/2),x,7,-1/17*(I*A+B)*(a+I*a*tan(e+f*x))^(7/2)/(f*(c-I*c*tan(e+f*x))^(17/2))-1/255*(5*I*A-12*B)*(a+I*a*tan(e+f*x))^(7/2)/(c*f*(c-I*c*tan(e+f*x))^(15/2))-4/3315*(5*I*A-12*B)*(a+I*a*tan(e+f*x))^(7/2)/(c^2*f*(c-I*c*tan(e+f*x))^(13/2))-4/12155*(5*I*A-12*B)*(a+I*a*tan(e+f*x))^(7/2)/(c^3*f*(c-I*c*tan(e+f*x))^(11/2))-8/109395*(5*I*A-12*B)*(a+I*a*tan(e+f*x))^(7/2)/(c^4*f*(c-I*c*tan(e+f*x))^(9/2))-8/765765*(5*I*A-12*B)*(a+I*a*tan(e+f*x))^(7/2)/(c^5*f*(c-I*c*tan(e+f*x))^(7/2))],

# m<0
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2)/sqrt(a+I*a*tan(e+f*x)),x,7,3*(2*I*A-3*B)*c^(5/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(f*sqrt(a))+3/2*(2*I*A-3*B)*c^2*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(a*f)+1/2*(2*I*A-3*B)*c*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)/(a*f)+(I*A-B)*(c-I*c*tan(e+f*x))^(5/2)/(f*sqrt(a+I*a*tan(e+f*x)))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)/sqrt(a+I*a*tan(e+f*x)),x,6,2*(I*A-2*B)*c^(3/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(f*sqrt(a))+(I*A-2*B)*c*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(a*f)+(I*A-B)*(c-I*c*tan(e+f*x))^(3/2)/(f*sqrt(a+I*a*tan(e+f*x)))],
[sqrt(c-I*c*tan(e+f*x))*(A+B*tan(e+f*x))/sqrt(a+I*a*tan(e+f*x)),x,5,-2*B*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))*sqrt(c)/(f*sqrt(a))+(I*A-B)*sqrt(c-I*c*tan(e+f*x))/(f*sqrt(a+I*a*tan(e+f*x)))],
[(A+B*tan(e+f*x))/(sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))),x,3,(-I*A-B)/(f*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x)))+I*A*sqrt(c-I*c*tan(e+f*x))/(c*f*sqrt(a+I*a*tan(e+f*x)))],
[(A+B*tan(e+f*x))/(sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)),x,4,-1/3*(2*I*A-B)*sqrt(a+I*a*tan(e+f*x))/(a*c*f*sqrt(c-I*c*tan(e+f*x)))+(I*A-B)/(f*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2))-1/3*(2*I*A-B)*sqrt(a+I*a*tan(e+f*x))/(a*f*(c-I*c*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))/(sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2)),x,5,-2/15*(3*I*A-2*B)*sqrt(a+I*a*tan(e+f*x))/(a*c^2*f*sqrt(c-I*c*tan(e+f*x)))+(I*A-B)/(f*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2))-1/5*(3*I*A-2*B)*sqrt(a+I*a*tan(e+f*x))/(a*f*(c-I*c*tan(e+f*x))^(5/2))-2/15*(3*I*A-2*B)*sqrt(a+I*a*tan(e+f*x))/(a*c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(7/2)/(a+I*a*tan(e+f*x))^(3/2),x,8,-5*(2*I*A-5*B)*c^(7/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(a^(3/2)*f)-5/2*(2*I*A-5*B)*c^3*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(a^2*f)-5/6*(2*I*A-5*B)*c^2*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)/(a^2*f)-2/3*(2*I*A-5*B)*c*(c-I*c*tan(e+f*x))^(5/2)/(a*f*sqrt(a+I*a*tan(e+f*x)))+1/3*(I*A-B)*(c-I*c*tan(e+f*x))^(7/2)/(f*(a+I*a*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^(3/2),x,7,-2*(I*A-4*B)*c^(5/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(a^(3/2)*f)-(I*A-4*B)*c^2*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(a^2*f)-2/3*(I*A-4*B)*c*(c-I*c*tan(e+f*x))^(3/2)/(a*f*sqrt(a+I*a*tan(e+f*x)))+1/3*(I*A-B)*(c-I*c*tan(e+f*x))^(5/2)/(f*(a+I*a*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^(3/2),x,6,2*B*c^(3/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(a^(3/2)*f)+2*B*c*sqrt(c-I*c*tan(e+f*x))/(a*f*sqrt(a+I*a*tan(e+f*x)))+1/3*(I*A-B)*(c-I*c*tan(e+f*x))^(3/2)/(f*(a+I*a*tan(e+f*x))^(3/2))],
[sqrt(c-I*c*tan(e+f*x))*(A+B*tan(e+f*x))/(a+I*a*tan(e+f*x))^(3/2),x,3,1/3*(I*A+2*B)*sqrt(c-I*c*tan(e+f*x))/(a*f*sqrt(a+I*a*tan(e+f*x)))+1/3*(I*A-B)*sqrt(c-I*c*tan(e+f*x))/(f*(a+I*a*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))/(sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2)),x,4,1/3*(2*I*A+B)*sqrt(c-I*c*tan(e+f*x))/(a*c*f*sqrt(a+I*a*tan(e+f*x)))+(-I*A-B)/(f*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2))+1/3*(2*I*A+B)*sqrt(c-I*c*tan(e+f*x))/(c*f*(a+I*a*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2)),x,4,2/3*A*tan(e+f*x)/(a*c*f*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x)))+1/3*I*A/(c*f*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2))+1/3*(-I*A-B)/(f*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(5/2)),x,6,-2/15*(4*I*A-B)*sqrt(a+I*a*tan(e+f*x))/(a^2*c^2*f*sqrt(c-I*c*tan(e+f*x)))+1/3*(4*I*A-B)/(a*f*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2))-1/5*(4*I*A-B)*sqrt(a+I*a*tan(e+f*x))/(a^2*f*(c-I*c*tan(e+f*x))^(5/2))+1/3*(I*A-B)/(f*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(5/2))-2/15*(4*I*A-B)*sqrt(a+I*a*tan(e+f*x))/(a^2*c*f*(c-I*c*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(9/2)/(a+I*a*tan(e+f*x))^(5/2),x,9,7*(2*I*A-7*B)*c^(9/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(a^(5/2)*f)+7/2*(2*I*A-7*B)*c^4*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(a^3*f)+7/6*(2*I*A-7*B)*c^3*sqrt(a+I*a*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)/(a^3*f)+14/15*(2*I*A-7*B)*c^2*(c-I*c*tan(e+f*x))^(5/2)/(a^2*f*sqrt(a+I*a*tan(e+f*x)))-2/15*(2*I*A-7*B)*c*(c-I*c*tan(e+f*x))^(7/2)/(a*f*(a+I*a*tan(e+f*x))^(3/2))+1/5*(I*A-B)*(c-I*c*tan(e+f*x))^(9/2)/(f*(a+I*a*tan(e+f*x))^(5/2))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(7/2)/(a+I*a*tan(e+f*x))^(5/2),x,8,2*(I*A-6*B)*c^(7/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(a^(5/2)*f)+(I*A-6*B)*c^3*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x))/(a^3*f)+2/3*(I*A-6*B)*c^2*(c-I*c*tan(e+f*x))^(3/2)/(a^2*f*sqrt(a+I*a*tan(e+f*x)))-2/15*(I*A-6*B)*c*(c-I*c*tan(e+f*x))^(5/2)/(a*f*(a+I*a*tan(e+f*x))^(3/2))+1/5*(I*A-B)*(c-I*c*tan(e+f*x))^(7/2)/(f*(a+I*a*tan(e+f*x))^(5/2))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(5/2)/(a+I*a*tan(e+f*x))^(5/2),x,7,-2*B*c^(5/2)*arctan(sqrt(c)*sqrt(a+I*a*tan(e+f*x))/(sqrt(a)*sqrt(c-I*c*tan(e+f*x))))/(a^(5/2)*f)-2*B*c^2*sqrt(c-I*c*tan(e+f*x))/(a^2*f*sqrt(a+I*a*tan(e+f*x)))+2/3*B*c*(c-I*c*tan(e+f*x))^(3/2)/(a*f*(a+I*a*tan(e+f*x))^(3/2))+1/5*(I*A-B)*(c-I*c*tan(e+f*x))^(5/2)/(f*(a+I*a*tan(e+f*x))^(5/2))],
[(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(3/2)/(a+I*a*tan(e+f*x))^(5/2),x,3,1/5*(I*A-B)*(c-I*c*tan(e+f*x))^(3/2)/(f*(a+I*a*tan(e+f*x))^(5/2))+1/15*(I*A+4*B)*(c-I*c*tan(e+f*x))^(3/2)/(a*f*(a+I*a*tan(e+f*x))^(3/2))],
[sqrt(c-I*c*tan(e+f*x))*(A+B*tan(e+f*x))/(a+I*a*tan(e+f*x))^(5/2),x,4,1/15*(2*I*A+3*B)*sqrt(c-I*c*tan(e+f*x))/(a^2*f*sqrt(a+I*a*tan(e+f*x)))+1/5*(I*A-B)*sqrt(c-I*c*tan(e+f*x))/(f*(a+I*a*tan(e+f*x))^(5/2))+1/15*(2*I*A+3*B)*sqrt(c-I*c*tan(e+f*x))/(a*f*(a+I*a*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))/(sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(5/2)),x,5,2/15*(3*I*A+2*B)*sqrt(c-I*c*tan(e+f*x))/(a^2*c*f*sqrt(a+I*a*tan(e+f*x)))+(-I*A-B)/(f*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(5/2))+1/5*(3*I*A+2*B)*sqrt(c-I*c*tan(e+f*x))/(c*f*(a+I*a*tan(e+f*x))^(5/2))+2/15*(3*I*A+2*B)*sqrt(c-I*c*tan(e+f*x))/(a*c*f*(a+I*a*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(3/2)),x,5,2/15*(4*A-I*B)*tan(e+f*x)/(a^2*c*f*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x)))+1/15*(4*I*A+B)/(c*f*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(5/2))+1/15*(4*I*A+B)/(a*c*f*sqrt(c-I*c*tan(e+f*x))*(a+I*a*tan(e+f*x))^(3/2))+1/3*(-I*A-B)/(f*(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x))/((a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(5/2)),x,5,8/15*A*tan(e+f*x)/(a^2*c^2*f*sqrt(a+I*a*tan(e+f*x))*sqrt(c-I*c*tan(e+f*x)))+1/5*(-I*A-B)/(f*(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(5/2))+1/5*I*A/(c*f*(a+I*a*tan(e+f*x))^(5/2)*(c-I*c*tan(e+f*x))^(3/2))+4/15*A*tan(e+f*x)/(a*c*f*(a+I*a*tan(e+f*x))^(3/2)*(c-I*c*tan(e+f*x))^(3/2))],

# Integrands of the form (a+a I Tan[e+f x])^m (c-c I Tan[e+f x])^n (A+B Tan[e+f x]) when m and n symbolic
[(a+I*a*tan(e+f*x))^m*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^n,x,4,1/2*(I*A+B)*(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^n/(f*n)-2^(-1+n)*(B*(m-n)+I*A*(m+n))*hypergeom([m,-n],[1+m],1/2*(1+I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^n/(f*m*n*(1-I*tan(e+f*x))^n)],
[(a+I*a*tan(e+f*x))^(1+m)*(A+B*tan(e+f*x))*(c-I*c*tan(e+f*x))^(-1-m),x,4,-1/2*(I*A+B)*(a+I*a*tan(e+f*x))^(1+m)*(c-I*c*tan(e+f*x))^(-1-m)/(f*(1+m))+2^m*a*B*hypergeom([-m,-m],[1-m],1/2*(1-I*tan(e+f*x)))*(a+I*a*tan(e+f*x))^m/(c*f*m*(1+I*tan(e+f*x))^m*(c-I*c*tan(e+f*x))^m)],
[(c-I*c*tan(e+f*x))^n*(-I*(2+n)+(-2+n)*tan(e+f*x))/(-I+tan(e+f*x))^2,x,2,(c-I*c*tan(e+f*x))^n/(f*(I-tan(e+f*x))^2)],

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

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

# Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^n (A+B Tan[e+f x])
[(A+B*tan(e+f*x))*(c+d*tan(e+f*x))/(a+I*a*tan(e+f*x))^2,x,3,1/4*(A-I*B)*(c-I*d)*x/a^2+1/4*(B*(c+3*I*d)+A*(I*c+d))/(a^2*f*(1+I*tan(e+f*x)))+1/4*(I*A-B)*(c+I*d)/(f*(a+I*a*tan(e+f*x))^2)],

# Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^(n/2) (A+B Tan[e+f x])
[(A+B*tan(e+f*x))*(c+d*tan(e+f*x))/(a+I*a*tan(e+f*x))^(3/2),x,4,-1/2*(I*A+B)*(c-I*d)*arctanh(sqrt(a+I*a*tan(e+f*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*f*sqrt(2))+1/2*(B*(c+3*I*d)+A*(I*c+d))/(a*f*sqrt(a+I*a*tan(e+f*x)))+1/3*(I*A-B)*(c+I*d)/(f*(a+I*a*tan(e+f*x))^(3/2))]]:

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