# Maple integration test file: "4 Trig functions\4.3 Tangent\4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).txt"

lst:=[

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

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

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

# n>0
[tan(c+d*x)*(a+b*tan(c+d*x))*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,5,-(a*B-b*C)*x+(b*B+a*C)*log(cos(c+d*x))/d+(a*B-b*C)*tan(c+d*x)/d+1/2*(b*B+a*C)*tan(c+d*x)^2/d+1/3*b*C*tan(c+d*x)^3/d],
[(a+b*tan(c+d*x))*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,3,-(b*B+a*C)*x-(a*B-b*C)*log(cos(c+d*x))/d+b*B*tan(c+d*x)/d+1/2*C*(a+b*tan(c+d*x))^2/(b*d)],
[cot(c+d*x)*(a+b*tan(c+d*x))*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,3,(a*B-b*C)*x-(b*B+a*C)*log(cos(c+d*x))/d+b*C*tan(c+d*x)/d],
[cot(c+d*x)^2*(a+b*tan(c+d*x))*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,5,(b*B+a*C)*x-b*C*log(cos(c+d*x))/d+a*B*log(sin(c+d*x))/d],
[cot(c+d*x)^3*(a+b*tan(c+d*x))*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,4,-(a*B-b*C)*x-a*B*cot(c+d*x)/d+(b*B+a*C)*log(sin(c+d*x))/d],
[cot(c+d*x)^4*(a+b*tan(c+d*x))*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,5,-(b*B+a*C)*x-(b*B+a*C)*cot(c+d*x)/d-1/2*a*B*cot(c+d*x)^2/d-(a*B-b*C)*log(sin(c+d*x))/d],
[cot(c+d*x)^5*(a+b*tan(c+d*x))*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,6,(a*B-b*C)*x+(a*B-b*C)*cot(c+d*x)/d-1/2*(b*B+a*C)*cot(c+d*x)^2/d-1/3*a*B*cot(c+d*x)^3/d-(b*B+a*C)*log(sin(c+d*x))/d],
[cot(c+d*x)^6*(a+b*tan(c+d*x))*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,7,(b*B+a*C)*x+(b*B+a*C)*cot(c+d*x)/d+1/2*(a*B-b*C)*cot(c+d*x)^2/d-1/3*(b*B+a*C)*cot(c+d*x)^3/d-1/4*a*B*cot(c+d*x)^4/d+(a*B-b*C)*log(sin(c+d*x))/d],
[tan(c+d*x)*(a+b*tan(c+d*x))^2*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,6,-(a^2*B-b^2*B-2*a*b*C)*x+(2*a*b*B+a^2*C-b^2*C)*log(cos(c+d*x))/d-b*(b*B+a*C)*tan(c+d*x)/d-1/2*C*(a+b*tan(c+d*x))^2/d+1/12*(4*b*B-a*C)*(a+b*tan(c+d*x))^3/(b^2*d)+1/4*C*tan(c+d*x)*(a+b*tan(c+d*x))^3/(b*d)],
[(a+b*tan(c+d*x))^2*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,4,-(2*a*b*B+a^2*C-b^2*C)*x-(a^2*B-b^2*B-2*a*b*C)*log(cos(c+d*x))/d+b*(a*B-b*C)*tan(c+d*x)/d+1/2*B*(a+b*tan(c+d*x))^2/d+1/3*C*(a+b*tan(c+d*x))^3/(b*d)],
[cot(c+d*x)*(a+b*tan(c+d*x))^2*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,4,(a^2*B-b^2*B-2*a*b*C)*x-(2*a*b*B+a^2*C-b^2*C)*log(cos(c+d*x))/d+b*(b*B+a*C)*tan(c+d*x)/d+1/2*C*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^2*(a+b*tan(c+d*x))^2*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,5,(2*a*b*B+a^2*C-b^2*C)*x-b*(b*B+2*a*C)*log(cos(c+d*x))/d+a^2*B*log(sin(c+d*x))/d+b^2*C*tan(c+d*x)/d],
[cot(c+d*x)^3*(a+b*tan(c+d*x))^2*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,5,-(a^2*B-b^2*B-2*a*b*C)*x-a^2*B*cot(c+d*x)/d-b^2*C*log(cos(c+d*x))/d+a*(2*b*B+a*C)*log(sin(c+d*x))/d],
[cot(c+d*x)^4*(a+b*tan(c+d*x))^2*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,5,(b^2*C-a*(2*b*B+a*C))*x-a*(2*b*B+a*C)*cot(c+d*x)/d-1/2*a^2*B*cot(c+d*x)^2/d-(a^2*B-b^2*B-2*a*b*C)*log(sin(c+d*x))/d],
[cot(c+d*x)^5*(a+b*tan(c+d*x))^2*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,6,(a^2*B-b^2*B-2*a*b*C)*x+(a^2*B-b^2*B-2*a*b*C)*cot(c+d*x)/d-1/2*a*(2*b*B+a*C)*cot(c+d*x)^2/d-1/3*a^2*B*cot(c+d*x)^3/d+(b^2*C-a*(2*b*B+a*C))*log(sin(c+d*x))/d],
[cot(c+d*x)^6*(a+b*tan(c+d*x))^2*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,7,(2*a*b*B+a^2*C-b^2*C)*x-(b^2*C-a*(2*b*B+a*C))*cot(c+d*x)/d+1/2*(a^2*B-b^2*B-2*a*b*C)*cot(c+d*x)^2/d-1/3*a*(2*b*B+a*C)*cot(c+d*x)^3/d-1/4*a^2*B*cot(c+d*x)^4/d+(a^2*B-b^2*B-2*a*b*C)*log(sin(c+d*x))/d],
[(a+b*tan(c+d*x))^3*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,5,-(3*a^2*b*B-b^3*B+a^3*C-3*a*b^2*C)*x-(a^3*B-3*a*b^2*B-3*a^2*b*C+b^3*C)*log(cos(c+d*x))/d+b*(a^2*B-b^2*B-2*a*b*C)*tan(c+d*x)/d+1/2*(a*B-b*C)*(a+b*tan(c+d*x))^2/d+1/3*B*(a+b*tan(c+d*x))^3/d+1/4*C*(a+b*tan(c+d*x))^4/(b*d)],
[cot(c+d*x)*(a+b*tan(c+d*x))^3*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,5,(a^3*B-3*a*b^2*B-3*a^2*b*C+b^3*C)*x-(3*a^2*b*B-b^3*B+a^3*C-3*a*b^2*C)*log(cos(c+d*x))/d+b*(2*a*b*B+a^2*C-b^2*C)*tan(c+d*x)/d+1/2*(b*B+a*C)*(a+b*tan(c+d*x))^2/d+1/3*C*(a+b*tan(c+d*x))^3/d],
[cot(c+d*x)^2*(a+b*tan(c+d*x))^3*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,6,(3*a^2*b*B-b^3*B+a^3*C-3*a*b^2*C)*x-b*(3*a*b*B+3*a^2*C-b^2*C)*log(cos(c+d*x))/d+a^3*B*log(sin(c+d*x))/d+b^2*(b*B+2*a*C)*tan(c+d*x)/d+1/2*b*C*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^3*(a+b*tan(c+d*x))^3*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,6,-(a^3*B-3*a*b^2*B-3*a^2*b*C+b^3*C)*x-b^2*(b*B+3*a*C)*log(cos(c+d*x))/d+a^2*(3*b*B+a*C)*log(sin(c+d*x))/d+b^2*(a*B+b*C)*tan(c+d*x)/d-a*B*cot(c+d*x)*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^4*(a+b*tan(c+d*x))^3*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,6,-(3*a^2*b*B-b^3*B+a^3*C-3*a*b^2*C)*x-a^2*(2*b*B+a*C)*cot(c+d*x)/d-b^3*C*log(cos(c+d*x))/d-a*(a^2*B-3*b^2*B-3*a*b*C)*log(sin(c+d*x))/d-1/2*a*B*cot(c+d*x)^2*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^5*(a+b*tan(c+d*x))^3*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,6,(a^3*B-3*a*b^2*B-3*a^2*b*C+b^3*C)*x+1/3*a*(3*a^2*B-8*b^2*B-9*a*b*C)*cot(c+d*x)/d-1/6*a^2*(5*b*B+3*a*C)*cot(c+d*x)^2/d-(3*a^2*b*B-b^3*B+a^3*C-3*a*b^2*C)*log(sin(c+d*x))/d-1/3*a*B*cot(c+d*x)^3*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^6*(a+b*tan(c+d*x))^3*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,7,(3*a^2*b*B-b^3*B+a^3*C-3*a*b^2*C)*x+(3*a^2*b*B-b^3*B+a^3*C-3*a*b^2*C)*cot(c+d*x)/d+1/4*a*(2*a^2*B-5*b^2*B-6*a*b*C)*cot(c+d*x)^2/d-1/6*a^2*(3*b*B+2*a*C)*cot(c+d*x)^3/d+(a^3*B-3*a*b^2*B-3*a^2*b*C+b^3*C)*log(sin(c+d*x))/d-1/4*a*B*cot(c+d*x)^4*(a+b*tan(c+d*x))^2/d],
[cot(c+d*x)^7*(a+b*tan(c+d*x))^3*(B*tan(c+d*x)+C*tan(c+d*x)^2),x,8,-(a^3*B-3*a*b^2*B-3*a^2*b*C+b^3*C)*x-(a^3*B-3*a*b^2*B-3*a^2*b*C+b^3*C)*cot(c+d*x)/d+1/2*(3*a^2*b*B-b^3*B+a^3*C-3*a*b^2*C)*cot(c+d*x)^2/d+1/15*a*(5*a^2*B-12*b^2*B-15*a*b*C)*cot(c+d*x)^3/d-1/20*a^2*(7*b*B+5*a*C)*cot(c+d*x)^4/d+(3*a^2*b*B-b^3*B+a^3*C-3*a*b^2*C)*log(sin(c+d*x))/d-1/5*a*B*cot(c+d*x)^5*(a+b*tan(c+d*x))^2/d],

# n<0
[tan(c+d*x)^2*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x)),x,7,-(b*B-a*C)*x/(a^2+b^2)+(a*B+b*C)*log(cos(c+d*x))/((a^2+b^2)*d)-a^3*(b*B-a*C)*log(a+b*tan(c+d*x))/(b^3*(a^2+b^2)*d)+(b*B-a*C)*tan(c+d*x)/(b^2*d)+1/2*C*tan(c+d*x)^2/(b*d)],
[tan(c+d*x)*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x)),x,6,-(a*B+b*C)*x/(a^2+b^2)-(b*B-a*C)*log(cos(c+d*x))/((a^2+b^2)*d)+a^2*(b*B-a*C)*log(a+b*tan(c+d*x))/(b^2*(a^2+b^2)*d)+C*tan(c+d*x)/(b*d)],
[(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x)),x,6,(b*B-a*C)*x/(a^2+b^2)-(a*B+b*C)*log(cos(c+d*x))/((a^2+b^2)*d)-a*(b*B-a*C)*log(a+b*tan(c+d*x))/(b*(a^2+b^2)*d)],
[cot(c+d*x)*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x)),x,3,(a*B+b*C)*x/(a^2+b^2)+(b*B-a*C)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)*d)],
[cot(c+d*x)^2*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x)),x,4,-(b*B-a*C)*x/(a^2+b^2)+B*log(sin(c+d*x))/(a*d)-b*(b*B-a*C)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a*(a^2+b^2)*d)],
[cot(c+d*x)^3*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x)),x,5,-(a*B+b*C)*x/(a^2+b^2)-B*cot(c+d*x)/(a*d)-(b*B-a*C)*log(sin(c+d*x))/(a^2*d)+b^2*(b*B-a*C)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^2*(a^2+b^2)*d)],
[cot(c+d*x)^4*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x)),x,6,(b*B-a*C)*x/(a^2+b^2)+(b*B-a*C)*cot(c+d*x)/(a^2*d)-1/2*B*cot(c+d*x)^2/(a*d)-(a^2*B-b^2*B+a*b*C)*log(sin(c+d*x))/(a^3*d)-b^3*(b*B-a*C)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^3*(a^2+b^2)*d)],
[tan(c+d*x)^2*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x))^2,x,7,-(2*a*b*B-a^2*C+b^2*C)*x/(a^2+b^2)^2+(a^2*B-b^2*B+2*a*b*C)*log(cos(c+d*x))/((a^2+b^2)^2*d)+a^2*(a^2*b*B+3*b^3*B-2*a^3*C-4*a*b^2*C)*log(a+b*tan(c+d*x))/(b^3*(a^2+b^2)^2*d)-(a*b*B-2*a^2*C-b^2*C)*tan(c+d*x)/(b^2*(a^2+b^2)*d)+a*(b*B-a*C)*tan(c+d*x)^2/(b*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x))^2,x,6,-(a^2*B-b^2*B+2*a*b*C)*x/(a^2+b^2)^2-(2*a*b*B-a^2*C+b^2*C)*log(cos(c+d*x))/((a^2+b^2)^2*d)-a*(2*b^3*B-a^3*C-3*a*b^2*C)*log(a+b*tan(c+d*x))/(b^2*(a^2+b^2)^2*d)-a^2*(b*B-a*C)/(b^2*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x))^2,x,3,(2*a*b*B-a^2*C+b^2*C)*x/(a^2+b^2)^2-(a^2*B-b^2*B+2*a*b*C)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^2*d)+a*(b*B-a*C)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x))^2,x,4,(a^2*B-b^2*B+2*a*b*C)*x/(a^2+b^2)^2+(2*a*b*B-a^2*C+b^2*C)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^2*d)+(-b*B+a*C)/((a^2+b^2)*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)^2*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x))^2,x,5,-(2*a*b*B-a^2*C+b^2*C)*x/(a^2+b^2)^2+B*log(sin(c+d*x))/(a^2*d)-b*(3*a^2*b*B+b^3*B-2*a^3*C)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^2*(a^2+b^2)^2*d)+b*(b*B-a*C)/(a*(a^2+b^2)*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)^3*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x))^2,x,6,-(a^2*B-b^2*B+2*a*b*C)*x/(a^2+b^2)^2-(2*b*B-a*C)*log(sin(c+d*x))/(a^3*d)+b^2*(4*a^2*b*B+2*b^3*B-3*a^3*C-a*b^2*C)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^3*(a^2+b^2)^2*d)-b*(a^2*B+2*b^2*B-a*b*C)/(a^2*(a^2+b^2)*d*(a+b*tan(c+d*x)))-B*cot(c+d*x)/(a*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^3*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x))^3,x,8,(a^3*B-3*a*b^2*B+3*a^2*b*C-b^3*C)*x/(a^2+b^2)^3+(3*a^2*b*B-b^3*B-a^3*C+3*a*b^2*C)*log(cos(c+d*x))/((a^2+b^2)^3*d)+a^2*(a^4*b*B+3*a^2*b^3*B+6*b^5*B-3*a^5*C-9*a^3*b^2*C-10*a*b^4*C)*log(a+b*tan(c+d*x))/(b^4*(a^2+b^2)^3*d)-(a^3*b*B+3*a*b^3*B-3*a^4*C-6*a^2*b^2*C-b^4*C)*tan(c+d*x)/(b^3*(a^2+b^2)^2*d)+1/2*a*(b*B-a*C)*tan(c+d*x)^3/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+1/2*a*(a^2*b*B+5*b^3*B-3*a^3*C-7*a*b^2*C)*tan(c+d*x)^2/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)^2*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x))^3,x,7,-(3*a^2*b*B-b^3*B-a^3*C+3*a*b^2*C)*x/(a^2+b^2)^3+(a^3*B-3*a*b^2*B+3*a^2*b*C-b^3*C)*log(cos(c+d*x))/((a^2+b^2)^3*d)+a*(a^2*b^3*B-3*b^5*B+a^5*C+3*a^3*b^2*C+6*a*b^4*C)*log(a+b*tan(c+d*x))/(b^3*(a^2+b^2)^3*d)+1/2*a*(b*B-a*C)*tan(c+d*x)^2/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)-a^2*(2*b^3*B-a^3*C-3*a*b^2*C)/(b^3*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[tan(c+d*x)*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x))^3,x,5,-(a^3*B-3*a*b^2*B+3*a^2*b*C-b^3*C)*x/(a^2+b^2)^3-(3*a^2*b*B-b^3*B-a^3*C+3*a*b^2*C)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^3*d)-1/2*a^2*(b*B-a*C)/(b^2*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+a*(2*b^3*B-a^3*C-3*a*b^2*C)/(b^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x))^3,x,4,(3*a^2*b*B-b^3*B-a^3*C+3*a*b^2*C)*x/(a^2+b^2)^3-(a^3*B-3*a*b^2*B+3*a^2*b*C-b^3*C)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^3*d)+1/2*a*(b*B-a*C)/(b*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+(a^2*B-b^2*B+2*a*b*C)/((a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x))^3,x,5,(a^3*B-3*a*b^2*B+3*a^2*b*C-b^3*C)*x/(a^2+b^2)^3+(3*a^2*b*B-b^3*B-a^3*C+3*a*b^2*C)*log(a*cos(c+d*x)+b*sin(c+d*x))/((a^2+b^2)^3*d)+1/2*(-b*B+a*C)/((a^2+b^2)*d*(a+b*tan(c+d*x))^2)+(-2*a*b*B+a^2*C-b^2*C)/((a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)^2*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x))^3,x,6,-(3*a^2*b*B-b^3*B-a^3*C+3*a*b^2*C)*x/(a^2+b^2)^3+B*log(sin(c+d*x))/(a^3*d)-b*(6*a^4*b*B+3*a^2*b^3*B+b^5*B-3*a^5*C+a^3*b^2*C)*log(a*cos(c+d*x)+b*sin(c+d*x))/(a^3*(a^2+b^2)^3*d)+1/2*b*(b*B-a*C)/(a*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)+b*(3*a^2*b*B+b^3*B-2*a^3*C)/(a^2*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],
[cot(c+d*x)^3*(B*tan(c+d*x)+C*tan(c+d*x)^2)/(a+b*tan(c+d*x))^3,x,7,-(a^3*B-3*a*b^2*B+3*a^2*b*C-b^3*C)*x/(a^2+b^2)^3-(3*b*B-a*C)*log(sin(c+d*x))/(a^4*d)+b^2*(10*a^4*b*B+9*a^2*b^3*B+3*b^5*B-6*a^5*C-3*a^3*b^2*C-a*b^4*C)*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*B+3*b^2*B-a*b*C)/(a^2*(a^2+b^2)*d*(a+b*tan(c+d*x))^2)-B*cot(c+d*x)/(a*d*(a+b*tan(c+d*x))^2)-b*(a^4*B+6*a^2*b^2*B+3*b^4*B-3*a^3*b*C-a*b^3*C)/(a^3*(a^2+b^2)^2*d*(a+b*tan(c+d*x)))],

# Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]+C Tan[c+d x]^2) (b Tan[c+d x])^n
[tan(c+d*x)^2*(b*tan(c+d*x))^n*(A+B*tan(c+d*x)+C*tan(c+d*x)^2),x,7,C*(b*tan(c+d*x))^(3+n)/(b^3*d*(3+n))+(A-C)*hypergeom([1,1/2*(3+n)],[1/2*(5+n)],-tan(c+d*x)^2)*(b*tan(c+d*x))^(3+n)/(b^3*d*(3+n))+B*hypergeom([1,1/2*(4+n)],[1/2*(6+n)],-tan(c+d*x)^2)*(b*tan(c+d*x))^(4+n)/(b^4*d*(4+n))],
[tan(c+d*x)^m*(b*tan(c+d*x))^n*(A+B*tan(c+d*x)+C*tan(c+d*x)^2),x,7,C*tan(c+d*x)^(1+m)*(b*tan(c+d*x))^n/(d*(1+m+n))+(A-C)*hypergeom([1,1/2*(1+m+n)],[1/2*(3+m+n)],-tan(c+d*x)^2)*tan(c+d*x)^(1+m)*(b*tan(c+d*x))^n/(d*(1+m+n))+B*hypergeom([1,1/2*(2+m+n)],[1/2*(4+m+n)],-tan(c+d*x)^2)*tan(c+d*x)^(2+m)*(b*tan(c+d*x))^n/(d*(2+m+n))],
[tan(c+d*x)^m*(b*tan(c+d*x))^(1/2)*(A+B*tan(c+d*x)+C*tan(c+d*x)^2),x,7,2*C*sqrt(b*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(3+2*m))+2*(A-C)*hypergeom([1,1/4*(3+2*m)],[1/4*(7+2*m)],-tan(c+d*x)^2)*sqrt(b*tan(c+d*x))*tan(c+d*x)^(1+m)/(d*(3+2*m))+2*B*hypergeom([1,1/4*(5+2*m)],[1/4*(9+2*m)],-tan(c+d*x)^2)*sqrt(b*tan(c+d*x))*tan(c+d*x)^(2+m)/(d*(5+2*m))],
[tan(c+d*x)^m*(A+B*tan(c+d*x)+C*tan(c+d*x)^2)/(b*tan(c+d*x))^(1/2),x,7,2*C*tan(c+d*x)^(1+m)/(d*(1+2*m)*sqrt(b*tan(c+d*x)))+2*(A-C)*hypergeom([1,1/4*(1+2*m)],[1/4*(5+2*m)],-tan(c+d*x)^2)*tan(c+d*x)^(1+m)/(d*(1+2*m)*sqrt(b*tan(c+d*x)))+2*B*hypergeom([1,1/4*(3+2*m)],[1/4*(7+2*m)],-tan(c+d*x)^2)*tan(c+d*x)^(2+m)/(d*(3+2*m)*sqrt(b*tan(c+d*x)))],

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

# n>0

# n<0
[tan(c+d*x)^m*(A+B*tan(c+d*x)+C*tan(c+d*x)^2)/sqrt(a+b*tan(c+d*x)),x,13,2*C*hypergeom([1/2,-m],[3/2],1+b*tan(c+d*x)/a)*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^m/(b*d*(-b*tan(c+d*x)/a)^m)-AppellF1(1/2,1,-m,3/2,(a+b*tan(c+d*x))/(a+sqrt(-b^2)),1+b*tan(c+d*x)/a)*(b*B-(A-C)*sqrt(-b^2))*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^m/(b*d*(a+sqrt(-b^2))*(-b*tan(c+d*x)/a)^m)-AppellF1(1/2,1,-m,3/2,(a+b*tan(c+d*x))/(a-sqrt(-b^2)),1+b*tan(c+d*x)/a)*(b*B+(A-C)*sqrt(-b^2))*sqrt(a+b*tan(c+d*x))*tan(c+d*x)^m/(b*d*(a-sqrt(-b^2))*(-b*tan(c+d*x)/a)^m)],

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

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

# n > 0
[(a+b*tan(e+f*x))^3*(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,6,(a^3*(A*c-c*C-B*d)-3*a*b^2*(A*c-c*C-B*d)-3*a^2*b*(B*c+(A-C)*d)+b^3*(B*c+(A-C)*d))*x-(3*a^2*b*(A*c-c*C-B*d)-b^3*(A*c-c*C-B*d)+a^3*(B*c+(A-C)*d)-3*a*b^2*(B*c+(A-C)*d))*log(cos(e+f*x))/f+b*(2*a*b*(A*c-c*C-B*d)+a^2*(B*c+(A-C)*d)-b^2*(B*c+(A-C)*d))*tan(e+f*x)/f+1/2*(A*b*c+a*B*c-b*c*C+a*A*d-b*B*d-a*C*d)*(a+b*tan(e+f*x))^2/f+1/3*(B*c+(A-C)*d)*(a+b*tan(e+f*x))^3/f-1/20*(a*C*d-5*b*(c*C+B*d))*(a+b*tan(e+f*x))^4/(b^2*f)+1/5*C*d*tan(e+f*x)*(a+b*tan(e+f*x))^4/(b*f)],
[(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,5,(a^2*(A*c-c*C-B*d)-b^2*(A*c-c*C-B*d)-2*a*b*(B*c+(A-C)*d))*x-(2*a*b*(A*c-c*C-B*d)+a^2*(B*c+(A-C)*d)-b^2*(B*c+(A-C)*d))*log(cos(e+f*x))/f+b*(A*b*c+a*B*c-b*c*C+a*A*d-b*B*d-a*C*d)*tan(e+f*x)/f+1/2*(B*c+(A-C)*d)*(a+b*tan(e+f*x))^2/f-1/12*(a*C*d-4*b*(c*C+B*d))*(a+b*tan(e+f*x))^3/(b^2*f)+1/4*C*d*tan(e+f*x)*(a+b*tan(e+f*x))^3/(b*f)],
[(a+b*tan(e+f*x))*(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,4,(a*(A*c-c*C-B*d)-b*(B*c+(A-C)*d))*x-(A*b*c+a*B*c-b*c*C+a*A*d-b*B*d-a*C*d)*log(cos(e+f*x))/f+(A*b+a*B-b*C)*d*tan(e+f*x)/f-1/6*(b*c*C-3*b*B*d-3*a*C*d)*(c+d*tan(e+f*x))^2/(d^2*f)+1/3*b*C*tan(e+f*x)*(c+d*tan(e+f*x))^2/(d*f)],
[(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,3,(A*c-c*C-B*d)*x-(B*c+(A-C)*d)*log(cos(e+f*x))/f+B*d*tan(e+f*x)/f+1/2*C*(c+d*tan(e+f*x))^2/(d*f)],
[(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x)),x,5,(a*(A*c-c*C-B*d)+b*(B*c+(A-C)*d))*x/(a^2+b^2)+(A*b*c-a*B*c-b*c*C-a*A*d-b*B*d+a*C*d)*log(cos(e+f*x))/((a^2+b^2)*f)+(A*b^2-a*(b*B-a*C))*(b*c-a*d)*log(a+b*tan(e+f*x))/(b^2*(a^2+b^2)*f)+C*d*tan(e+f*x)/(b*f)],
[(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^2,x,5,(a^2*(A*c-c*C-B*d)-b^2*(A*c-c*C-B*d)+2*a*b*(B*c+(A-C)*d))*x/(a^2+b^2)^2+(2*a*b*(A*c-c*C-B*d)-a^2*(B*c+(A-C)*d)+b^2*(B*c+(A-C)*d))*log(cos(e+f*x))/((a^2+b^2)^2*f)+(a^4*C*d+b^4*(B*c+A*d)+2*a*b^3*(A*c-c*C-B*d)-a^2*b^2*(B*c+(A-3*C)*d))*log(a+b*tan(e+f*x))/(b^2*(a^2+b^2)^2*f)-(A*b^2-a*(b*B-a*C))*(b*c-a*d)/(b^2*(a^2+b^2)*f*(a+b*tan(e+f*x)))],
[(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^3,x,4,(a^3*(A*c-c*C-B*d)-3*a*b^2*(A*c-c*C-B*d)+3*a^2*b*(B*c+(A-C)*d)-b^3*(B*c+(A-C)*d))*x/(a^2+b^2)^3+(3*a^2*b*(A*c-c*C-B*d)-b^3*(A*c-c*C-B*d)-a^3*(B*c+(A-C)*d)+3*a*b^2*(B*c+(A-C)*d))*log(a*cos(e+f*x)+b*sin(e+f*x))/((a^2+b^2)^3*f)-1/2*(A*b^2-a*(b*B-a*C))*(b*c-a*d)/(b^2*(a^2+b^2)*f*(a+b*tan(e+f*x))^2)+(-a^4*C*d-b^4*(B*c+A*d)-2*a*b^3*(A*c-c*C-B*d)+a^2*b^2*(B*c+(A-3*C)*d))/(b^2*(a^2+b^2)^2*f*(a+b*tan(e+f*x)))],
[(a+b*tan(e+f*x))^3*(c+d*tan(e+f*x))^2*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,7,-(a^3*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-3*a*b^2*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))+3*a^2*b*(2*c*(A-C)*d+B*(c^2-d^2))-b^3*(2*c*(A-C)*d+B*(c^2-d^2)))*x+(3*a^2*b*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-b^3*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-a^3*(2*c*(A-C)*d+B*(c^2-d^2))+3*a*b^2*(2*c*(A-C)*d+B*(c^2-d^2)))*log(cos(e+f*x))/f+d*(3*a^2*b*(A*c-c*C-B*d)-b^3*(A*c-c*C-B*d)+a^3*(B*c+(A-C)*d)-3*a*b^2*(B*c+(A-C)*d))*tan(e+f*x)/f+1/2*(a^3*B-3*a*b^2*B+3*a^2*b*(A-C)-b^3*(A-C))*(c+d*tan(e+f*x))^2/f+1/60*(4*a^3*C*d^3-3*a^2*b*d^2*(3*c*C-16*B*d)+3*a*b^2*d*(2*c^2*C-5*B*c*d+20*(A-C)*d^2)-b^3*(c^3*C-2*B*c^2*d+5*c*(A-C)*d^2+20*B*d^3))*(c+d*tan(e+f*x))^3/(d^4*f)+1/20*b*(5*b*(A*b+a*B-b*C)*d^2+(b*c-a*d)*(b*c*C-2*b*B*d-a*C*d))*tan(e+f*x)*(c+d*tan(e+f*x))^3/(d^3*f)-1/10*(b*c*C-2*b*B*d-a*C*d)*(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^3/(d^2*f)+1/6*C*(a+b*tan(e+f*x))^3*(c+d*tan(e+f*x))^3/(d*f)],
[(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^2*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,6,-(a^2*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-b^2*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))+2*a*b*(2*c*(A-C)*d+B*(c^2-d^2)))*x+(2*a*b*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-a^2*(2*c*(A-C)*d+B*(c^2-d^2))+b^2*(2*c*(A-C)*d+B*(c^2-d^2)))*log(cos(e+f*x))/f+d*(2*a*b*(A*c-c*C-B*d)+a^2*(B*c+(A-C)*d)-b^2*(B*c+(A-C)*d))*tan(e+f*x)/f+1/2*(a^2*B-b^2*B+2*a*b*(A-C))*(c+d*tan(e+f*x))^2/f+1/60*(8*a^2*C*d^2-10*a*b*d*(c*C-4*B*d)+b^2*(2*c^2*C-5*B*c*d+20*(A-C)*d^2))*(c+d*tan(e+f*x))^3/(d^3*f)-1/20*b*(2*b*c*C-5*b*B*d-2*a*C*d)*tan(e+f*x)*(c+d*tan(e+f*x))^3/(d^2*f)+1/5*C*(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^3/(d*f)],
[(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^2*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,5,-(a*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))+b*(2*c*(A-C)*d+B*(c^2-d^2)))*x-(a*(B*c^2-2*c*C*d-B*d^2)-b*(c^2*C+2*B*c*d-C*d^2)+A*(2*a*c*d+b*(c^2-d^2)))*log(cos(e+f*x))/f+d*(A*b*c+a*B*c-b*c*C+a*A*d-b*B*d-a*C*d)*tan(e+f*x)/f+1/2*(A*b+a*B-b*C)*(c+d*tan(e+f*x))^2/f-1/12*(b*c*C-4*b*B*d-4*a*C*d)*(c+d*tan(e+f*x))^3/(d^2*f)+1/4*b*C*tan(e+f*x)*(c+d*tan(e+f*x))^3/(d*f)],
[(c+d*tan(e+f*x))^2*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,4,-(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))*x-(2*c*(A-C)*d+B*(c^2-d^2))*log(cos(e+f*x))/f+d*(B*c+(A-C)*d)*tan(e+f*x)/f+1/2*B*(c+d*tan(e+f*x))^2/f+1/3*C*(c+d*tan(e+f*x))^3/(d*f)],
[(c+d*tan(e+f*x))^2*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x)),x,6,-(a*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-b*(2*c*(A-C)*d+B*(c^2-d^2)))*x/(a^2+b^2)-(a*(B*c^2-2*c*C*d-B*d^2)+b*(c^2*C+2*B*c*d-C*d^2)+A*(2*a*c*d-b*(c^2-d^2)))*log(cos(e+f*x))/((a^2+b^2)*f)+(A*b^2-a*(b*B-a*C))*(b*c-a*d)^2*log(a+b*tan(e+f*x))/(b^3*(a^2+b^2)*f)+d*(b*c*C+b*B*d-a*C*d)*tan(e+f*x)/(b^2*f)+1/2*C*(c+d*tan(e+f*x))^2/(b*f)],
[(c+d*tan(e+f*x))^2*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^2,x,6,-(a^2*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-b^2*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-2*a*b*(2*c*(A-C)*d+B*(c^2-d^2)))*x/(a^2+b^2)^2-(2*a*b*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))+a^2*(2*c*(A-C)*d+B*(c^2-d^2))-b^2*(2*c*(A-C)*d+B*(c^2-d^2)))*log(cos(e+f*x))/((a^2+b^2)^2*f)-(b*c-a*d)*(a^3*b*B*d-2*a^4*C*d-b^4*(B*c+2*A*d)-a*b^3*(2*A*c-2*c*C-3*B*d)+a^2*b^2*(B*c-4*C*d))*log(a+b*tan(e+f*x))/(b^3*(a^2+b^2)^2*f)+(A*b^2-a*b*B+2*a^2*C+b^2*C)*d^2*tan(e+f*x)/(b^2*(a^2+b^2)*f)-(A*b^2-a*(b*B-a*C))*(c+d*tan(e+f*x))^2/(b*(a^2+b^2)*f*(a+b*tan(e+f*x)))],
[(c+d*tan(e+f*x))^2*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^3,x,6,-(a^3*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-3*a*b^2*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-3*a^2*b*(2*c*(A-C)*d+B*(c^2-d^2))+b^3*(2*c*(A-C)*d+B*(c^2-d^2)))*x/(a^2+b^2)^3-(3*a^2*b*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-b^3*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))+a^3*(2*c*(A-C)*d+B*(c^2-d^2))-3*a*b^2*(2*c*(A-C)*d+B*(c^2-d^2)))*log(cos(e+f*x))/((a^2+b^2)^3*f)+(a^6*C*d^2+3*a^4*b^2*C*d^2-3*a^2*b^4*(c^2*C+2*B*c*d-2*C*d^2-A*(c^2-d^2))+b^6*(c*(c*C+2*B*d)-A*(c^2-d^2))-a^3*b^3*(2*c*(A-C)*d+B*(c^2-d^2))+3*a*b^5*(2*c*(A-C)*d+B*(c^2-d^2)))*log(a+b*tan(e+f*x))/(b^3*(a^2+b^2)^3*f)-(b*c-a*d)*(a^4*C*d+b^4*(B*c+A*d)+2*a*b^3*(A*c-c*C-B*d)-a^2*b^2*(B*c+(A-3*C)*d))/(b^3*(a^2+b^2)^2*f*(a+b*tan(e+f*x)))-1/2*(A*b^2-a*(b*B-a*C))*(c+d*tan(e+f*x))^2/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^2)],
[(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^3*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,7,(a^2*(A*c^3-c^3*C-3*B*c^2*d-3*A*c*d^2+3*c*C*d^2+B*d^3)+b^2*(c^3*C+3*B*c^2*d-3*c*C*d^2-B*d^3-A*(c^3-3*c*d^2))-2*a*b*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2)))*x+(2*a*b*(c^3*C+3*B*c^2*d-3*c*C*d^2-B*d^3-A*(c^3-3*c*d^2))-a^2*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2))+b^2*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2)))*log(cos(e+f*x))/f-d*(2*a*b*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-a^2*(2*c*(A-C)*d+B*(c^2-d^2))+b^2*(2*c*(A-C)*d+B*(c^2-d^2)))*tan(e+f*x)/f+1/2*(2*a*b*(A*c-c*C-B*d)+a^2*(B*c+(A-C)*d)-b^2*(B*c+(A-C)*d))*(c+d*tan(e+f*x))^2/f+1/3*(a^2*B-b^2*B+2*a*b*(A-C))*(c+d*tan(e+f*x))^3/f+1/60*(5*a^2*C*d^2-6*a*b*d*(c*C-5*B*d)+b^2*(c^2*C-3*B*c*d+15*(A-C)*d^2))*(c+d*tan(e+f*x))^4/(d^3*f)-1/15*b*(b*c*C-3*b*B*d-a*C*d)*tan(e+f*x)*(c+d*tan(e+f*x))^4/(d^2*f)+1/6*C*(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^4/(d*f)],
[(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^3*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,6,(a*(A*c^3-c^3*C-3*B*c^2*d-3*A*c*d^2+3*c*C*d^2+B*d^3)-b*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2)))*x-(A*(b*c^3+3*a*c^2*d-3*b*c*d^2-a*d^3)-b*(c^3*C+3*B*c^2*d-3*c*C*d^2-B*d^3)+a*(B*c^3-3*c^2*C*d-3*B*c*d^2+C*d^3))*log(cos(e+f*x))/f+d*(a*(B*c^2-2*c*C*d-B*d^2)-b*(c^2*C+2*B*c*d-C*d^2)+A*(2*a*c*d+b*(c^2-d^2)))*tan(e+f*x)/f+1/2*(A*b*c+a*B*c-b*c*C+a*A*d-b*B*d-a*C*d)*(c+d*tan(e+f*x))^2/f+1/3*(A*b+a*B-b*C)*(c+d*tan(e+f*x))^3/f-1/20*(b*c*C-5*b*B*d-5*a*C*d)*(c+d*tan(e+f*x))^4/(d^2*f)+1/5*b*C*tan(e+f*x)*(c+d*tan(e+f*x))^4/(d*f)],
[(c+d*tan(e+f*x))^3*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,5,-(c^3*C+3*B*c^2*d-3*c*C*d^2-B*d^3-A*(c^3-3*c*d^2))*x-((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2))*log(cos(e+f*x))/f+d*(2*c*(A-C)*d+B*(c^2-d^2))*tan(e+f*x)/f+1/2*(B*c+(A-C)*d)*(c+d*tan(e+f*x))^2/f+1/3*B*(c+d*tan(e+f*x))^3/f+1/4*C*(c+d*tan(e+f*x))^4/(d*f)],
[(c+d*tan(e+f*x))^3*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x)),x,7,-(a*(c^3*C+3*B*c^2*d-3*c*C*d^2-B*d^3-A*(c^3-3*c*d^2))-b*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2)))*x/(a^2+b^2)-(b*(c^3*C+3*B*c^2*d-3*c*C*d^2-B*d^3)+a*(B*c^3-3*c^2*C*d-3*B*c*d^2+C*d^3)+A*(a*d*(3*c^2-d^2)-b*(c^3-3*c*d^2)))*log(cos(e+f*x))/((a^2+b^2)*f)+(A*b^2-a*(b*B-a*C))*(b*c-a*d)^3*log(a+b*tan(e+f*x))/(b^4*(a^2+b^2)*f)+d*(b^2*d*(B*c+(A-C)*d)+(b*c-a*d)*(b*c*C+b*B*d-a*C*d))*tan(e+f*x)/(b^3*f)+1/2*(b*c*C+b*B*d-a*C*d)*(c+d*tan(e+f*x))^2/(b^2*f)+1/3*C*(c+d*tan(e+f*x))^3/(b*f)],
[(c+d*tan(e+f*x))^3*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^2,x,7,-(b^2*(A*c^3-c^3*C-3*B*c^2*d-3*A*c*d^2+3*c*C*d^2+B*d^3)+a^2*(c^3*C+3*B*c^2*d-3*c*C*d^2-B*d^3-A*(c^3-3*c*d^2))-2*a*b*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2)))*x/(a^2+b^2)^2+(2*a*b*(A*c^3-c^3*C-3*B*c^2*d-3*A*c*d^2+3*c*C*d^2+B*d^3)-a^2*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2))+b^2*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2)))*log(cos(e+f*x))/((a^2+b^2)^2*f)-(b*c-a*d)^2*(2*a^3*b*B*d-3*a^4*C*d-b^4*(B*c+3*A*d)-2*a*b^3*(A*c-c*C-2*B*d)+a^2*b^2*(B*c-(A+5*C)*d))*log(a+b*tan(e+f*x))/(b^4*(a^2+b^2)^2*f)-d^2*(3*a^3*C*d-A*b^2*(b*c-a*d)-b^3*(2*c*C+B*d)-a^2*b*(3*c*C+2*B*d)+a*b^2*(B*c+2*C*d))*tan(e+f*x)/(b^3*(a^2+b^2)*f)+1/2*(2*A*b^2-2*a*b*B+3*a^2*C+b^2*C)*d*(c+d*tan(e+f*x))^2/(b^2*(a^2+b^2)*f)-(A*b^2-a*(b*B-a*C))*(c+d*tan(e+f*x))^3/(b*(a^2+b^2)*f*(a+b*tan(e+f*x)))],
[(c+d*tan(e+f*x))^3*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^3,x,7,-(3*a*b^2*(A*c^3-c^3*C-3*B*c^2*d-3*A*c*d^2+3*c*C*d^2+B*d^3)+a^3*(c^3*C+3*B*c^2*d-3*c*C*d^2-B*d^3-A*(c^3-3*c*d^2))-3*a^2*b*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2))+b^3*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2)))*x/(a^2+b^2)^3-(b^3*(A*c^3-c^3*C-3*B*c^2*d-3*A*c*d^2+3*c*C*d^2+B*d^3)+3*a^2*b*(c^3*C+3*B*c^2*d-3*c*C*d^2-B*d^3-A*(c^3-3*c*d^2))+a^3*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2))-3*a*b^2*((A-C)*d*(3*c^2-d^2)+B*(c^3-3*c*d^2)))*log(cos(e+f*x))/((a^2+b^2)^3*f)-(b*c-a*d)*(a^5*b*B*d^2-3*a^6*C*d^2+a^4*b^2*d*(B*c-9*C*d)+a^3*b^3*B*(c^2+3*d^2)-b^6*(c*(c*C+3*B*d)-A*(c^2-3*d^2))-a*b^5*(8*c*(A-C)*d+3*B*(c^2-2*d^2))+a^2*b^4*(3*c^2*C+6*B*c*d-10*C*d^2-A*(3*c^2-d^2)))*log(a+b*tan(e+f*x))/(b^4*(a^2+b^2)^3*f)-d^2*(a^3*b*B*d-3*a^4*C*d-a*b^3*(2*A*c-2*c*C-3*B*d)+a^2*b^2*(B*c-6*C*d)-b^4*(B*c+(2*A+C)*d))*tan(e+f*x)/(b^3*(a^2+b^2)^2*f)+1/2*(a^3*b*B*d-3*a^4*C*d-b^4*(2*B*c+3*A*d)-a*b^3*(4*A*c-4*c*C-5*B*d)+a^2*b^2*(2*B*c+(A-7*C)*d))*(c+d*tan(e+f*x))^2/(b^2*(a^2+b^2)^2*f*(a+b*tan(e+f*x)))-1/2*(A*b^2-a*(b*B-a*C))*(c+d*tan(e+f*x))^3/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^2)],

# n < 0
[(a+b*tan(e+f*x))^3*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x)),x,7,(a^3*(A*c-c*C+B*d)-3*a*b^2*(A*c-c*C+B*d)-3*a^2*b*(B*c-(A-C)*d)+b^3*(B*c-(A-C)*d))*x/(c^2+d^2)-(3*a^2*b*(A*c-c*C+B*d)-b^3*(A*c-c*C+B*d)+a^3*(B*c-(A-C)*d)-3*a*b^2*(B*c-(A-C)*d))*log(cos(e+f*x))/((c^2+d^2)*f)-(b*c-a*d)^3*(c^2*C-B*c*d+A*d^2)*log(c+d*tan(e+f*x))/(d^4*(c^2+d^2)*f)+b*(b*(A*b+a*B-b*C)*d^2+(b*c-a*d)*(b*c*C-b*B*d-a*C*d))*tan(e+f*x)/(d^3*f)-1/2*(b*c*C-b*B*d-a*C*d)*(a+b*tan(e+f*x))^2/(d^2*f)+1/3*C*(a+b*tan(e+f*x))^3/(d*f)],
[(a+b*tan(e+f*x))^2*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x)),x,6,(a^2*(A*c-c*C+B*d)-b^2*(A*c-c*C+B*d)-2*a*b*(B*c-(A-C)*d))*x/(c^2+d^2)-(2*a*b*(A*c-c*C+B*d)+a^2*(B*c-(A-C)*d)-b^2*(B*c-(A-C)*d))*log(cos(e+f*x))/((c^2+d^2)*f)+(b*c-a*d)^2*(c^2*C-B*c*d+A*d^2)*log(c+d*tan(e+f*x))/(d^3*(c^2+d^2)*f)-b*(b*c*C-b*B*d-a*C*d)*tan(e+f*x)/(d^2*f)+1/2*C*(a+b*tan(e+f*x))^2/(d*f)],
[(a+b*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x)),x,5,(a*(A*c-c*C+B*d)-b*(B*c-(A-C)*d))*x/(c^2+d^2)-(A*b*c+a*B*c-b*c*C-a*A*d+b*B*d+a*C*d)*log(cos(e+f*x))/((c^2+d^2)*f)-(b*c-a*d)*(c^2*C-B*c*d+A*d^2)*log(c+d*tan(e+f*x))/(d^2*(c^2+d^2)*f)+b*C*tan(e+f*x)/(d*f)],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x)),x,4,(A*c-c*C+B*d)*x/(c^2+d^2)-(B*c-(A-C)*d)*log(cos(e+f*x))/((c^2+d^2)*f)+(c^2*C-B*c*d+A*d^2)*log(c+d*tan(e+f*x))/(d*(c^2+d^2)*f)],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/((a+b*tan(e+f*x))*(c+d*tan(e+f*x))),x,3,(a*(A*c-c*C+B*d)+b*(B*c-(A-C)*d))*x/((a^2+b^2)*(c^2+d^2))+(A*b^2-a*(b*B-a*C))*log(a*cos(e+f*x)+b*sin(e+f*x))/((a^2+b^2)*(b*c-a*d)*f)-(c^2*C-B*c*d+A*d^2)*log(c*cos(e+f*x)+d*sin(e+f*x))/((b*c-a*d)*(c^2+d^2)*f)],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/((a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))),x,4,(a^2*(A*c-c*C+B*d)-b^2*(A*c-c*C+B*d)+2*a*b*(B*c-(A-C)*d))*x/((a^2+b^2)^2*(c^2+d^2))+(2*a*b^3*c*(A-C)+2*a^3*b*B*d-a^4*C*d+b^4*(B*c-A*d)-a^2*b^2*(B*c+3*A*d-C*d))*log(a*cos(e+f*x)+b*sin(e+f*x))/((a^2+b^2)^2*(b*c-a*d)^2*f)+d*(c^2*C-B*c*d+A*d^2)*log(c*cos(e+f*x)+d*sin(e+f*x))/((b*c-a*d)^2*(c^2+d^2)*f)+(-A*b^2+a*(b*B-a*C))/((a^2+b^2)*(b*c-a*d)*f*(a+b*tan(e+f*x)))],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/((a+b*tan(e+f*x))^3*(c+d*tan(e+f*x))),x,5,(a^3*(A*c-c*C+B*d)-3*a*b^2*(A*c-c*C+B*d)+3*a^2*b*(B*c-(A-C)*d)-b^3*(B*c-(A-C)*d))*x/((a^2+b^2)^3*(c^2+d^2))+(3*a*b^5*B*c^2-3*a^5*b*B*d^2+a^6*C*d^2+3*a^4*b^2*d*(B*c+2*A*d-C*d)+b^6*(c*(c*C-B*d)-A*(c^2-d^2))-a^3*b^3*(8*c*(A-C)*d+B*(c^2-d^2))-3*a^2*b^4*(c*(c*C+2*B*d)-A*(c^2+d^2)))*log(a*cos(e+f*x)+b*sin(e+f*x))/((a^2+b^2)^3*(b*c-a*d)^3*f)-d^2*(c^2*C-B*c*d+A*d^2)*log(c*cos(e+f*x)+d*sin(e+f*x))/((b*c-a*d)^3*(c^2+d^2)*f)+1/2*(-A*b^2+a*(b*B-a*C))/((a^2+b^2)*(b*c-a*d)*f*(a+b*tan(e+f*x))^2)+(-2*a*b^3*c*(A-C)-2*a^3*b*B*d+a^4*C*d-b^4*(B*c-A*d)+a^2*b^2*(B*c+3*A*d-C*d))/((a^2+b^2)^2*(b*c-a*d)^2*f*(a+b*tan(e+f*x)))],
[(a+b*tan(e+f*x))^3*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^2,x,7,-(a^3*(c^2*C-2*B*c*d-C*d^2-A*(c^2-d^2))-3*a*b^2*(c^2*C-2*B*c*d-C*d^2-A*(c^2-d^2))-3*a^2*b*(2*c*(A-C)*d-B*(c^2-d^2))+b^3*(2*c*(A-C)*d-B*(c^2-d^2)))*x/(c^2+d^2)^2+(3*a^2*b*(c^2*C-2*B*c*d-C*d^2-A*(c^2-d^2))-b^3*(c^2*C-2*B*c*d-C*d^2-A*(c^2-d^2))+a^3*(2*c*(A-C)*d-B*(c^2-d^2))-3*a*b^2*(2*c*(A-C)*d-B*(c^2-d^2)))*log(cos(e+f*x))/((c^2+d^2)^2*f)+(b*c-a*d)^2*(b*(3*c^4*C-2*B*c^3*d+c^2*(A+5*C)*d^2-4*B*c*d^3+3*A*d^4)+a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))*log(c+d*tan(e+f*x))/(d^4*(c^2+d^2)^2*f)+b^2*(a*d*(3*c^2*C-B*c*d+(A+2*C)*d^2)-b*(3*c^3*C-2*B*c^2*d+c*(A+2*C)*d^2-B*d^3))*tan(e+f*x)/(d^3*(c^2+d^2)*f)+1/2*b*(3*c^2*C-2*B*c*d+(2*A+C)*d^2)*(a+b*tan(e+f*x))^2/(d^2*(c^2+d^2)*f)-(c^2*C-B*c*d+A*d^2)*(a+b*tan(e+f*x))^3/(d*(c^2+d^2)*f*(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))^2*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^2,x,6,-(a^2*(c^2*C-2*B*c*d-C*d^2-A*(c^2-d^2))-b^2*(c^2*C-2*B*c*d-C*d^2-A*(c^2-d^2))-2*a*b*(2*c*(A-C)*d-B*(c^2-d^2)))*x/(c^2+d^2)^2+(2*a*b*(c^2*C-2*B*c*d-C*d^2-A*(c^2-d^2))+a^2*(2*c*(A-C)*d-B*(c^2-d^2))-b^2*(2*c*(A-C)*d-B*(c^2-d^2)))*log(cos(e+f*x))/((c^2+d^2)^2*f)-(b*c-a*d)*(b*(2*c^4*C-B*c^3*d+4*c^2*C*d^2-3*B*c*d^3+2*A*d^4)+a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))*log(c+d*tan(e+f*x))/(d^3*(c^2+d^2)^2*f)+b^2*(2*c^2*C-B*c*d+(A+C)*d^2)*tan(e+f*x)/(d^2*(c^2+d^2)*f)-(c^2*C-B*c*d+A*d^2)*(a+b*tan(e+f*x))^2/(d*(c^2+d^2)*f*(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^2,x,5,-(a*(c^2*C-2*B*c*d-C*d^2-A*(c^2-d^2))-b*(2*c*(A-C)*d-B*(c^2-d^2)))*x/(c^2+d^2)^2-(a*(B*c^2+2*c*C*d-B*d^2)-b*(c^2*C-2*B*c*d-C*d^2)-A*(2*a*c*d-b*(c^2-d^2)))*log(cos(e+f*x))/((c^2+d^2)^2*f)+(b*(c^4*C-c^2*(A-3*C)*d^2-2*B*c*d^3+A*d^4)+a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))*log(c+d*tan(e+f*x))/(d^2*(c^2+d^2)^2*f)+(b*c-a*d)*(c^2*C-B*c*d+A*d^2)/(d^2*(c^2+d^2)*f*(c+d*tan(e+f*x)))],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^2,x,3,-(c^2*C-2*B*c*d-C*d^2-A*(c^2-d^2))*x/(c^2+d^2)^2+(2*c*(A-C)*d-B*(c^2-d^2))*log(c*cos(e+f*x)+d*sin(e+f*x))/((c^2+d^2)^2*f)+(-c^2*C+B*c*d-A*d^2)/(d*(c^2+d^2)*f*(c+d*tan(e+f*x)))],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/((a+b*tan(e+f*x))*(c+d*tan(e+f*x))^2),x,4,-(a*(c^2*C-2*B*c*d-C*d^2-A*(c^2-d^2))+b*(2*c*(A-C)*d-B*(c^2-d^2)))*x/((a^2+b^2)*(c^2+d^2)^2)+b*(A*b^2-a*(b*B-a*C))*log(a*cos(e+f*x)+b*sin(e+f*x))/((a^2+b^2)*(b*c-a*d)^2*f)-(b*(c^4*C-2*B*c^3*d+c^2*(3*A-C)*d^2+A*d^4)-a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))*log(c*cos(e+f*x)+d*sin(e+f*x))/((b*c-a*d)^2*(c^2+d^2)^2*f)+(c^2*C-B*c*d+A*d^2)/((b*c-a*d)*(c^2+d^2)*f*(c+d*tan(e+f*x)))],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/((a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^2),x,5,-(a^2*(c^2*C-2*B*c*d-C*d^2-A*(c^2-d^2))-b^2*(c^2*C-2*B*c*d-C*d^2-A*(c^2-d^2))+2*a*b*(2*c*(A-C)*d-B*(c^2-d^2)))*x/((a^2+b^2)^2*(c^2+d^2)^2)+b*(3*a^3*b*B*d-2*a^4*C*d+b^4*(B*c-2*A*d)-a^2*b^2*(B*c+4*A*d)+a*b^3*(2*A*c-2*c*C+B*d))*log(a*cos(e+f*x)+b*sin(e+f*x))/((a^2+b^2)^2*(b*c-a*d)^3*f)+d*(b*(2*c^4*C-3*B*c^3*d+4*A*c^2*d^2-B*c*d^3+2*A*d^4)-a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))*log(c*cos(e+f*x)+d*sin(e+f*x))/((b*c-a*d)^3*(c^2+d^2)^2*f)-d*(b^2*c*(c*C-B*d)-a*b*B*(c^2+d^2)+a^2*(2*c^2*C-B*c*d+C*d^2)+A*(a^2*d^2+b^2*(c^2+2*d^2)))/((a^2+b^2)*(b*c-a*d)^2*(c^2+d^2)*f*(c+d*tan(e+f*x)))+(-A*b^2+a*(b*B-a*C))/((a^2+b^2)*(b*c-a*d)*f*(a+b*tan(e+f*x))*(c+d*tan(e+f*x)))],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/((a+b*tan(e+f*x))^3*(c+d*tan(e+f*x))^2),x,6,-(a^3*(c^2*C-2*B*c*d-C*d^2-A*(c^2-d^2))-3*a*b^2*(c^2*C-2*B*c*d-C*d^2-A*(c^2-d^2))+3*a^2*b*(2*c*(A-C)*d-B*(c^2-d^2))-b^3*(2*c*(A-C)*d-B*(c^2-d^2)))*x/((a^2+b^2)^3*(c^2+d^2)^2)-b*(6*a^5*b*B*d^2-3*a^6*C*d^2-a^4*b^2*d*(4*B*c+(10*A-C)*d)-b^6*(c*(c*C-2*B*d)-A*(c^2-3*d^2))+a*b^5*(2*c*(A-C)*d-B*(3*c^2-d^2))+3*a^2*b^4*(c*(c*C+2*B*d)-A*(c^2+3*d^2))+a^3*b^3*(10*c*(A-C)*d+B*(c^2+3*d^2)))*log(a*cos(e+f*x)+b*sin(e+f*x))/((a^2+b^2)^3*(b*c-a*d)^4*f)-d^2*(b*(3*c^4*C-4*B*c^3*d+c^2*(5*A+C)*d^2-2*B*c*d^3+3*A*d^4)-a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))*log(c*cos(e+f*x)+d*sin(e+f*x))/((b*c-a*d)^4*(c^2+d^2)^2*f)-d*(3*a^3*b*B*d*(c^2+d^2)+a*b^3*(2*A*c-2*c*C+B*d)*(c^2+d^2)-a^4*d*(3*c^2*C-B*c*d+(A+2*C)*d^2)-a^2*b^2*(B*c^3+4*A*c^2*d+2*c^2*C*d-B*c*d^2+6*A*d^3)-b^4*(d*(2*A*c^2+c^2*C+3*A*d^2)-B*(c^3+2*c*d^2)))/((a^2+b^2)^2*(b*c-a*d)^3*(c^2+d^2)*f*(c+d*tan(e+f*x)))+1/2*(-A*b^2+a*(b*B-a*C))/((a^2+b^2)*(b*c-a*d)*f*(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x)))+1/2*(-5*a^3*b*B*d+3*a^4*C*d-b^4*(2*B*c-3*A*d)-a*b^3*(4*A*c-4*c*C+B*d)+a^2*b^2*(2*B*c+(7*A-C)*d))/((a^2+b^2)^2*(b*c-a*d)^2*f*(a+b*tan(e+f*x))*(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))^3*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^3,x,7,-(3*a*b^2*(A*c^3-c^3*C+3*B*c^2*d-3*A*c*d^2+3*c*C*d^2-B*d^3)+a^3*(c^3*C-3*B*c^2*d-3*c*C*d^2+B*d^3-A*(c^3-3*c*d^2))-3*a^2*b*((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2))+b^3*((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2)))*x/(c^2+d^2)^3-(3*a^2*b*(A*c^3-c^3*C+3*B*c^2*d-3*A*c*d^2+3*c*C*d^2-B*d^3)-b^3*(A*c^3-c^3*C+3*B*c^2*d-3*A*c*d^2+3*c*C*d^2-B*d^3)-a^3*((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2))+3*a*b^2*((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2)))*log(cos(e+f*x))/((c^2+d^2)^3*f)-(b*c-a*d)*(b^2*(3*c^6*C-B*c^5*d+9*c^4*C*d^2-3*B*c^3*d^3-c^2*(A-10*C)*d^4-6*B*c*d^5+3*A*d^6)+a^2*d^3*((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2))+a*b*d^2*(8*c*(A-C)*d^3-B*(c^4+6*c^2*d^2-3*d^4)))*log(c+d*tan(e+f*x))/(d^4*(c^2+d^2)^3*f)+b^2*(b*(3*c^4*C-B*c^3*d+6*c^2*C*d^2-3*B*c*d^3+(2*A+C)*d^4)+a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))*tan(e+f*x)/(d^3*(c^2+d^2)^2*f)-1/2*(c^2*C-B*c*d+A*d^2)*(a+b*tan(e+f*x))^3/(d*(c^2+d^2)*f*(c+d*tan(e+f*x))^2)-1/2*(b*(3*c^4*C-B*c^3*d-c^2*(A-7*C)*d^2-5*B*c*d^3+3*A*d^4)+2*a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))*(a+b*tan(e+f*x))^2/(d^2*(c^2+d^2)^2*f*(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))^2*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^3,x,6,-(b^2*(A*c^3-c^3*C+3*B*c^2*d-3*A*c*d^2+3*c*C*d^2-B*d^3)+a^2*(c^3*C-3*B*c^2*d-3*c*C*d^2+B*d^3-A*(c^3-3*c*d^2))-2*a*b*((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2)))*x/(c^2+d^2)^3-(2*a*b*(A*c^3-c^3*C+3*B*c^2*d-3*A*c*d^2+3*c*C*d^2-B*d^3)-a^2*((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2))+b^2*((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2)))*log(cos(e+f*x))/((c^2+d^2)^3*f)-(2*a*b*d^3*(A*c^3-c^3*C+3*B*c^2*d-3*A*c*d^2+3*c*C*d^2-B*d^3)-b^2*(c^6*C+3*c^4*C*d^2+B*c^3*d^3-3*c^2*(A-2*C)*d^4-3*B*c*d^5+A*d^6)-a^2*d^3*((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2)))*log(c+d*tan(e+f*x))/(d^3*(c^2+d^2)^3*f)-1/2*(c^2*C-B*c*d+A*d^2)*(a+b*tan(e+f*x))^2/(d*(c^2+d^2)*f*(c+d*tan(e+f*x))^2)+(b*c-a*d)*(b*(c^4*C-c^2*(A-3*C)*d^2-2*B*c*d^3+A*d^4)+a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))/(d^3*(c^2+d^2)^2*f*(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^3,x,4,-(a*(c^3*C-3*B*c^2*d-3*c*C*d^2+B*d^3-A*(c^3-3*c*d^2))-b*((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2)))*x/(c^2+d^2)^3+(b*(c^3*C-3*B*c^2*d-3*c*C*d^2+B*d^3)-a*(B*c^3+3*c^2*C*d-3*B*c*d^2-C*d^3)+A*(a*d*(3*c^2-d^2)-b*(c^3-3*c*d^2)))*log(c*cos(e+f*x)+d*sin(e+f*x))/((c^2+d^2)^3*f)+1/2*(b*c-a*d)*(c^2*C-B*c*d+A*d^2)/(d^2*(c^2+d^2)*f*(c+d*tan(e+f*x))^2)+(-b*(c^4*C-c^2*(A-3*C)*d^2-2*B*c*d^3+A*d^4)-a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))/(d^2*(c^2+d^2)^2*f*(c+d*tan(e+f*x)))],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^3,x,4,-(c^3*C-3*B*c^2*d-3*c*C*d^2+B*d^3-A*(c^3-3*c*d^2))*x/(c^2+d^2)^3+((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2))*log(c*cos(e+f*x)+d*sin(e+f*x))/((c^2+d^2)^3*f)+1/2*(-c^2*C+B*c*d-A*d^2)/(d*(c^2+d^2)*f*(c+d*tan(e+f*x))^2)+(-2*c*(A-C)*d+B*(c^2-d^2))/((c^2+d^2)^2*f*(c+d*tan(e+f*x)))],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/((a+b*tan(e+f*x))*(c+d*tan(e+f*x))^3),x,5,-(a*(c^3*C-3*B*c^2*d-3*c*C*d^2+B*d^3-A*(c^3-3*c*d^2))+b*((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2)))*x/((a^2+b^2)*(c^2+d^2)^3)+b^2*(A*b^2-a*(b*B-a*C))*log(a*cos(e+f*x)+b*sin(e+f*x))/((a^2+b^2)*(b*c-a*d)^3*f)-(b^2*(c^6*C-3*B*c^5*d+3*c^4*(2*A-C)*d^2+B*c^3*d^3+3*A*c^2*d^4+A*d^6)+a^2*d^3*((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2))-a*b*d^2*(8*c^3*(A-C)*d-B*(3*c^4-6*c^2*d^2-d^4)))*log(c*cos(e+f*x)+d*sin(e+f*x))/((b*c-a*d)^3*(c^2+d^2)^3*f)+1/2*(c^2*C-B*c*d+A*d^2)/((b*c-a*d)*(c^2+d^2)*f*(c+d*tan(e+f*x))^2)+(b*(c^4*C-2*B*c^3*d+c^2*(3*A-C)*d^2+A*d^4)-a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))/((b*c-a*d)^2*(c^2+d^2)^2*f*(c+d*tan(e+f*x)))],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/((a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^3),x,6,-(b^2*(A*c^3-c^3*C+3*B*c^2*d-3*A*c*d^2+3*c*C*d^2-B*d^3)+a^2*(c^3*C-3*B*c^2*d-3*c*C*d^2+B*d^3-A*(c^3-3*c*d^2))+2*a*b*((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2)))*x/((a^2+b^2)^2*(c^2+d^2)^3)+b^2*(4*a^3*b*B*d-3*a^4*C*d+b^4*(B*c-3*A*d)+2*a*b^3*(A*c-c*C+B*d)-a^2*b^2*(B*c+(5*A+C)*d))*log(a*cos(e+f*x)+b*sin(e+f*x))/((a^2+b^2)^2*(b*c-a*d)^4*f)+d*(b^2*(3*c^6*C-6*B*c^5*d+c^4*(10*A-C)*d^2-3*B*c^3*d^3+9*A*c^2*d^4-B*c*d^5+3*A*d^6)+a^2*d^3*((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2))-2*a*b*d^2*(c*(A-C)*d*(5*c^2+d^2)-B*(2*c^4-3*c^2*d^2-d^4)))*log(c*cos(e+f*x)+d*sin(e+f*x))/((b*c-a*d)^4*(c^2+d^2)^3*f)-1/2*d*(b^2*c*(c*C-B*d)-2*a*b*B*(c^2+d^2)+a^2*(3*c^2*C-B*c*d+2*C*d^2)+A*(a^2*d^2+b^2*(2*c^2+3*d^2)))/((a^2+b^2)*(b*c-a*d)^2*(c^2+d^2)*f*(c+d*tan(e+f*x))^2)+(-A*b^2+a*(b*B-a*C))/((a^2+b^2)*(b*c-a*d)*f*(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^2)-d*(b^3*c*(2*c^3*C-3*B*c^2*d-B*d^3)+a^2*b*(3*c^4*C-3*B*c^3*d+2*c^2*C*d^2-B*c*d^3+C*d^4)+a^3*d^2*(2*c*C*d+B*(c^2-d^2))+a*b^2*(2*c*C*d^3-B*(c^4+c^2*d^2+2*d^4))-A*(2*a^3*c*d^3+2*a*b^2*c*d^3-2*a^2*b*d^2*(2*c^2+d^2)-b^3*(c^4+6*c^2*d^2+3*d^4)))/((a^2+b^2)*(b*c-a*d)^3*(c^2+d^2)^2*f*(c+d*tan(e+f*x)))],

# 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]+C Tan[e+f x]^2)

# n > 0
[sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^3*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,12,-(a-I*b)^3*(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/f+(a+I*b)^3*(I*A-B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))*sqrt(c+I*d)/f+2*(a^3*B-3*a*b^2*B+3*a^2*b*(A-C)-b^3*(A-C))*sqrt(c+d*tan(e+f*x))/f+2/315*(40*a^3*C*d^3-6*a^2*b*d^2*(16*c*C-45*B*d)+9*a*b^2*d*(8*c^2*C-14*B*c*d+35*(A-C)*d^2)-b^3*(16*c^3*C-24*B*c^2*d+42*c*(A-C)*d^2+105*B*d^3))*(c+d*tan(e+f*x))^(3/2)/(d^4*f)+2/105*b*(21*b*(A*b+a*B-b*C)*d^2+4*(b*c-a*d)*(2*b*c*C-3*b*B*d-2*a*C*d))*tan(e+f*x)*(c+d*tan(e+f*x))^(3/2)/(d^3*f)-2/21*(2*b*c*C-3*b*B*d-2*a*C*d)*(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^(3/2)/(d^2*f)+2/9*C*(a+b*tan(e+f*x))^3*(c+d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^2*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,11,-(a-I*b)^2*(B+I*(A-C))*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/f-(a+I*b)^2*(B-I*(A-C))*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))*sqrt(c+I*d)/f+2*(a^2*B-b^2*B+2*a*b*(A-C))*sqrt(c+d*tan(e+f*x))/f+2/105*(20*a^2*C*d^2-14*a*b*d*(2*c*C-5*B*d)+b^2*(8*c^2*C-14*B*c*d+35*(A-C)*d^2))*(c+d*tan(e+f*x))^(3/2)/(d^3*f)-2/35*b*(4*b*c*C-7*b*B*d-4*a*C*d)*tan(e+f*x)*(c+d*tan(e+f*x))^(3/2)/(d^2*f)+2/7*C*(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,10,-(I*a+b)*(A-I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/f+(I*a-b)*(A+I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))*sqrt(c+I*d)/f+2*(A*b+a*B-b*C)*sqrt(c+d*tan(e+f*x))/f-2/15*(2*b*c*C-5*b*B*d-5*a*C*d)*(c+d*tan(e+f*x))^(3/2)/(d^2*f)+2/5*b*C*tan(e+f*x)*(c+d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,9,-(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/f-(B-I*(A-C))*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))*sqrt(c+I*d)/f+2*B*sqrt(c+d*tan(e+f*x))/f+2/3*C*(c+d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x)),x,12,-(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/((a-I*b)*f)+(I*A-B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))*sqrt(c+I*d)/((a+I*b)*f)-2*(A*b^2-a*(b*B-a*C))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))*sqrt(b*c-a*d)/(b^(3/2)*(a^2+b^2)*f)+2*C*sqrt(c+d*tan(e+f*x))/(b*f)],
[sqrt(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^2,x,12,-(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/((a-I*b)^2*f)-(B-I*(A-C))*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))*sqrt(c+I*d)/((a+I*b)^2*f)-(a^3*b*B*d+a^4*C*d+b^4*(2*B*c+A*d)+a*b^3*(4*A*c-4*c*C-3*B*d)-a^2*b^2*(2*B*c+3*A*d-5*C*d))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/(b^(3/2)*(a^2+b^2)^2*f*sqrt(b*c-a*d))-(A*b^2-a*(b*B-a*C))*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)*f*(a+b*tan(e+f*x)))],
[sqrt(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^3,x,13,1/4*(3*a^5*b*B*d^2+a^6*C*d^2-3*a^4*b^2*d*(4*B*c+5*A*d-6*C*d)-3*a^2*b^4*(8*A*c^2-8*c^2*C-16*B*c*d-6*A*d^2+5*C*d^2)+2*a^3*b^3*(20*c*(A-C)*d+B*(4*c^2-13*d^2))-3*a*b^5*(8*c*(A-C)*d+B*(8*c^2-d^2))-b^6*(4*c*(2*c*C+B*d)-A*(8*c^2+d^2)))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/(b^(3/2)*(a^2+b^2)^3*(b*c-a*d)^(3/2)*f)-(A-I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))*sqrt(c-I*d)/((I*a+b)^3*f)+(A+I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))*sqrt(c+I*d)/((I*a-b)^3*f)-1/2*(A*b^2-a*(b*B-a*C))*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^2)-1/4*(3*a^3*b*B*d+a^4*C*d+b^4*(4*B*c+A*d)+a*b^3*(8*A*c-8*c*C-5*B*d)-a^2*b^2*(4*B*c+7*A*d-9*C*d))*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)^2*(b*c-a*d)*f*(a+b*tan(e+f*x)))],
[(a+b*tan(e+f*x))^3*(c+d*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,13,(I*a+b)^3*(A-I*B-C)*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f+(a+I*b)^3*(I*A-B-I*C)*(c+I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/f+2*(3*a^2*b*(A*c-c*C-B*d)-b^3*(A*c-c*C-B*d)+a^3*(B*c+(A-C)*d)-3*a*b^2*(B*c+(A-C)*d))*sqrt(c+d*tan(e+f*x))/f+2/3*(a^3*B-3*a*b^2*B+3*a^2*b*(A-C)-b^3*(A-C))*(c+d*tan(e+f*x))^(3/2)/f+2/3465*(168*a^3*C*d^3-2*a^2*b*d^2*(192*c*C-847*B*d)+33*a*b^2*d*(8*c^2*C-18*B*c*d+63*(A-C)*d^2)-b^3*(48*c^3*C-88*B*c^2*d+198*c*(A-C)*d^2+693*B*d^3))*(c+d*tan(e+f*x))^(5/2)/(d^4*f)+2/693*b*(99*b*(A*b+a*B-b*C)*d^2+4*(b*c-a*d)*(6*b*c*C-11*b*B*d-6*a*C*d))*tan(e+f*x)*(c+d*tan(e+f*x))^(5/2)/(d^3*f)-2/99*(6*b*c*C-11*b*B*d-6*a*C*d)*(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^(5/2)/(d^2*f)+2/11*C*(a+b*tan(e+f*x))^3*(c+d*tan(e+f*x))^(5/2)/(d*f)],
[(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,12,-(a-I*b)^2*(B+I*(A-C))*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f+(a+I*b)^2*(I*A-B-I*C)*(c+I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/f+2*(2*a*b*(A*c-c*C-B*d)+a^2*(B*c+(A-C)*d)-b^2*(B*c+(A-C)*d))*sqrt(c+d*tan(e+f*x))/f+2/3*(a^2*B-b^2*B+2*a*b*(A-C))*(c+d*tan(e+f*x))^(3/2)/f+2/315*(28*a^2*C*d^2-18*a*b*d*(2*c*C-7*B*d)+b^2*(8*c^2*C-18*B*c*d+63*(A-C)*d^2))*(c+d*tan(e+f*x))^(5/2)/(d^3*f)-2/63*b*(4*b*c*C-9*b*B*d-4*a*C*d)*tan(e+f*x)*(c+d*tan(e+f*x))^(5/2)/(d^2*f)+2/9*C*(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^(5/2)/(d*f)],
[(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,11,-(I*a+b)*(A-I*B-C)*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f+(I*a-b)*(A+I*B-C)*(c+I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/f+2*(A*b*c+a*B*c-b*c*C+a*A*d-b*B*d-a*C*d)*sqrt(c+d*tan(e+f*x))/f+2/3*(A*b+a*B-b*C)*(c+d*tan(e+f*x))^(3/2)/f-2/35*(2*b*c*C-7*b*B*d-7*a*C*d)*(c+d*tan(e+f*x))^(5/2)/(d^2*f)+2/7*b*C*tan(e+f*x)*(c+d*tan(e+f*x))^(5/2)/(d*f)],
[(c+d*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,10,-(I*A+B-I*C)*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f-(B-I*(A-C))*(c+I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/f+2*(B*c+(A-C)*d)*sqrt(c+d*tan(e+f*x))/f+2/3*B*(c+d*tan(e+f*x))^(3/2)/f+2/5*C*(c+d*tan(e+f*x))^(5/2)/(d*f)],
[(c+d*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x)),x,13,-(I*A+B-I*C)*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((a-I*b)*f)-(A+I*B-C)*(c+I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((I*a-b)*f)-2*(A*b^2-a*(b*B-a*C))*(b*c-a*d)^(3/2)*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/(b^(5/2)*(a^2+b^2)*f)+2*(b*c*C+b*B*d-a*C*d)*sqrt(c+d*tan(e+f*x))/(b^2*f)+2/3*C*(c+d*tan(e+f*x))^(3/2)/(b*f)],
[(c+d*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^2,x,13,-(I*A+B-I*C)*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((a-I*b)^2*f)-(B-I*(A-C))*(c+I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((a+I*b)^2*f)+(a^3*b*B*d-3*a^4*C*d-b^4*(2*B*c+3*A*d)-a*b^3*(4*A*c-4*c*C-5*B*d)+a^2*b^2*(2*B*c+(A-7*C)*d))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))*sqrt(b*c-a*d)/(b^(5/2)*(a^2+b^2)^2*f)+(A*b^2-a*b*B+3*a^2*C+2*b^2*C)*d*sqrt(c+d*tan(e+f*x))/(b^2*(a^2+b^2)*f)-(A*b^2-a*(b*B-a*C))*(c+d*tan(e+f*x))^(3/2)/(b*(a^2+b^2)*f*(a+b*tan(e+f*x)))],
[(c+d*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^3,x,13,-(A-I*B-C)*(c-I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((I*a+b)^3*f)+(A+I*B-C)*(c+I*d)^(3/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((I*a-b)^3*f)-1/4*(a^5*b*B*d^2+3*a^6*C*d^2+a^4*b^2*d*(4*B*c+3*(A+2*C)*d)-b^6*(8*A*c^2-8*c^2*C-12*B*c*d-3*A*d^2)+a^2*b^4*(24*A*c^2-24*c^2*C-48*B*c*d-26*A*d^2+35*C*d^2)-2*a^3*b^3*(12*c*(A-C)*d+B*(4*c^2-9*d^2))+a*b^5*(40*c*(A-C)*d+3*B*(8*c^2-5*d^2)))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/(b^(5/2)*(a^2+b^2)^3*f*sqrt(b*c-a*d))-1/4*(a^3*b*B*d+3*a^4*C*d+b^4*(4*B*c+3*A*d)+a*b^3*(8*A*c-8*c*C-7*B*d)-a^2*b^2*(4*B*c+5*A*d-11*C*d))*sqrt(c+d*tan(e+f*x))/(b^2*(a^2+b^2)^2*f*(a+b*tan(e+f*x)))-1/2*(A*b^2-a*(b*B-a*C))*(c+d*tan(e+f*x))^(3/2)/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^2)],
[(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,13,-(a-I*b)^2*(I*A+B-I*C)*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f+(a+I*b)^2*(I*A-B-I*C)*(c+I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/f-2*(2*a*b*(c^2*C+2*B*c*d-C*d^2-A*(c^2-d^2))-a^2*(2*c*(A-C)*d+B*(c^2-d^2))+b^2*(2*c*(A-C)*d+B*(c^2-d^2)))*sqrt(c+d*tan(e+f*x))/f+2/3*(2*a*b*(A*c-c*C-B*d)+a^2*(B*c+(A-C)*d)-b^2*(B*c+(A-C)*d))*(c+d*tan(e+f*x))^(3/2)/f+2/5*(a^2*B-b^2*B+2*a*b*(A-C))*(c+d*tan(e+f*x))^(5/2)/f+2/693*(36*a^2*C*d^2-22*a*b*d*(2*c*C-9*B*d)+b^2*(8*c^2*C-22*B*c*d+99*(A-C)*d^2))*(c+d*tan(e+f*x))^(7/2)/(d^3*f)-2/99*b*(4*b*c*C-11*b*B*d-4*a*C*d)*tan(e+f*x)*(c+d*tan(e+f*x))^(7/2)/(d^2*f)+2/11*C*(a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^(7/2)/(d*f)],
[(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,12,-(I*a+b)*(A-I*B-C)*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f+(I*a-b)*(A+I*B-C)*(c+I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/f+2*(a*(B*c^2-2*c*C*d-B*d^2)-b*(c^2*C+2*B*c*d-C*d^2)+A*(2*a*c*d+b*(c^2-d^2)))*sqrt(c+d*tan(e+f*x))/f+2/3*(A*b*c+a*B*c-b*c*C+a*A*d-b*B*d-a*C*d)*(c+d*tan(e+f*x))^(3/2)/f+2/5*(A*b+a*B-b*C)*(c+d*tan(e+f*x))^(5/2)/f-2/63*(2*b*c*C-9*b*B*d-9*a*C*d)*(c+d*tan(e+f*x))^(7/2)/(d^2*f)+2/9*b*C*tan(e+f*x)*(c+d*tan(e+f*x))^(7/2)/(d*f)],
[(c+d*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,11,-(I*A+B-I*C)*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/f-(B-I*(A-C))*(c+I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/f+2*(2*c*(A-C)*d+B*(c^2-d^2))*sqrt(c+d*tan(e+f*x))/f+2/3*(B*c+(A-C)*d)*(c+d*tan(e+f*x))^(3/2)/f+2/5*B*(c+d*tan(e+f*x))^(5/2)/f+2/7*C*(c+d*tan(e+f*x))^(7/2)/(d*f)],
[(c+d*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x)),x,14,-(I*A+B-I*C)*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((a-I*b)*f)+(I*A-B-I*C)*(c+I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((a+I*b)*f)-2*(A*b^2-a*(b*B-a*C))*(b*c-a*d)^(5/2)*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/(b^(7/2)*(a^2+b^2)*f)+2*(b^2*d*(B*c+(A-C)*d)+(b*c-a*d)*(b*c*C+b*B*d-a*C*d))*sqrt(c+d*tan(e+f*x))/(b^3*f)+2/3*(b*c*C+b*B*d-a*C*d)*(c+d*tan(e+f*x))^(3/2)/(b^2*f)+2/5*C*(c+d*tan(e+f*x))^(5/2)/(b*f)],
[(c+d*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^2,x,14,-(I*A+B-I*C)*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((a-I*b)^2*f)-(B-I*(A-C))*(c+I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((a+I*b)^2*f)+(b*c-a*d)^(3/2)*(3*a^3*b*B*d-5*a^4*C*d-b^4*(2*B*c+5*A*d)-a*b^3*(4*A*c-4*c*C-7*B*d)+a^2*b^2*(2*B*c-(A+9*C)*d))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/(b^(7/2)*(a^2+b^2)^2*f)-d*(5*a^3*C*d-A*b^2*(b*c-a*d)-2*b^3*(2*c*C+B*d)-a^2*b*(5*c*C+3*B*d)+a*b^2*(B*c+4*C*d))*sqrt(c+d*tan(e+f*x))/(b^3*(a^2+b^2)*f)+1/3*(3*A*b^2-3*a*b*B+5*a^2*C+2*b^2*C)*d*(c+d*tan(e+f*x))^(3/2)/(b^2*(a^2+b^2)*f)-(A*b^2-a*(b*B-a*C))*(c+d*tan(e+f*x))^(5/2)/(b*(a^2+b^2)*f*(a+b*tan(e+f*x)))],
[(c+d*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^3,x,14,-(A-I*B-C)*(c-I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((I*a+b)^3*f)+(A+I*B-C)*(c+I*d)^(5/2)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((I*a-b)^3*f)+1/4*(3*a^5*b*B*d^2-15*a^6*C*d^2+a^4*b^2*d*(4*B*c+(A-46*C)*d)-3*a^2*b^4*(8*A*c^2-8*c^2*C-16*B*c*d-6*A*d^2+21*C*d^2)-a*b^5*(56*c*(A-C)*d+B*(24*c^2-35*d^2))-b^6*(4*c*(2*c*C+5*B*d)-A*(8*c^2-15*d^2))+2*a^3*b^3*(4*c*(A-C)*d+B*(4*c^2+3*d^2)))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))*sqrt(b*c-a*d)/(b^(7/2)*(a^2+b^2)^3*f)-1/4*d*(3*a^3*b*B*d-15*a^4*C*d-a*b^3*(8*A*c-8*c*C-11*B*d)+a^2*b^2*(4*B*c+(A-31*C)*d)-b^4*(4*B*c+7*A*d+8*C*d))*sqrt(c+d*tan(e+f*x))/(b^3*(a^2+b^2)^2*f)+1/4*(a^3*b*B*d-5*a^4*C*d-b^4*(4*B*c+5*A*d)-a*b^3*(8*A*c-8*c*C-9*B*d)+a^2*b^2*(4*B*c+3*A*d-13*C*d))*(c+d*tan(e+f*x))^(3/2)/(b^2*(a^2+b^2)^2*f*(a+b*tan(e+f*x)))-1/2*(A*b^2-a*(b*B-a*C))*(c+d*tan(e+f*x))^(5/2)/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^2)],

# n < 0
[(a+b*tan(e+f*x))^3*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/sqrt(c+d*tan(e+f*x)),x,11,(I*a+b)^3*(A-I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(f*sqrt(c-I*d))-(I*a-b)^3*(A+I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(f*sqrt(c+I*d))+2/105*(72*a^3*C*d^3-6*a^2*b*d^2*(32*c*C-49*B*d)+21*a*b^2*d*(8*c^2*C-10*B*c*d+15*(A-C)*d^2)-b^3*(48*c^3*C-56*B*c^2*d+70*c*(A-C)*d^2+105*B*d^3))*sqrt(c+d*tan(e+f*x))/(d^4*f)+2/105*b*(35*b*(A*b+a*B-b*C)*d^2+4*(b*c-a*d)*(6*b*c*C-7*b*B*d-6*a*C*d))*sqrt(c+d*tan(e+f*x))*tan(e+f*x)/(d^3*f)-2/35*(6*b*c*C-7*b*B*d-6*a*C*d)*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^2/(d^2*f)+2/7*C*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^3/(d*f)],
[(a+b*tan(e+f*x))^2*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/sqrt(c+d*tan(e+f*x)),x,10,-(a-I*b)^2*(B+I*(A-C))*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(f*sqrt(c-I*d))+(a+I*b)^2*(I*A-B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(f*sqrt(c+I*d))+2/15*(12*a^2*C*d^2-10*a*b*d*(2*c*C-3*B*d)+b^2*(8*c^2*C-10*B*c*d+15*(A-C)*d^2))*sqrt(c+d*tan(e+f*x))/(d^3*f)-2/15*b*(4*b*c*C-5*b*B*d-4*a*C*d)*sqrt(c+d*tan(e+f*x))*tan(e+f*x)/(d^2*f)+2/5*C*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^2/(d*f)],
[(a+b*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/sqrt(c+d*tan(e+f*x)),x,9,-(I*a+b)*(A-I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(f*sqrt(c-I*d))+(I*a-b)*(A+I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(f*sqrt(c+I*d))-2/3*(2*b*c*C-3*b*B*d-3*a*C*d)*sqrt(c+d*tan(e+f*x))/(d^2*f)+2/3*b*C*sqrt(c+d*tan(e+f*x))*tan(e+f*x)/(d*f)],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/sqrt(c+d*tan(e+f*x)),x,8,-(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/(f*sqrt(c-I*d))-(B-I*(A-C))*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/(f*sqrt(c+I*d))+2*C*sqrt(c+d*tan(e+f*x))/(d*f)],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))),x,11,-(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((a-I*b)*f*sqrt(c-I*d))-(A+I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((I*a-b)*f*sqrt(c+I*d))-2*(A*b^2-a*(b*B-a*C))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/((a^2+b^2)*f*sqrt(b)*sqrt(b*c-a*d))],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^2),x,12,-(3*a^3*b*B*d-a^4*C*d+b^4*(2*B*c-A*d)+a*b^3*(4*A*c-4*c*C-B*d)-a^2*b^2*(2*B*c+5*A*d-3*C*d))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/((a^2+b^2)^2*(b*c-a*d)^(3/2)*f*sqrt(b))-(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((a-I*b)^2*f*sqrt(c-I*d))-(B-I*(A-C))*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((a+I*b)^2*f*sqrt(c+I*d))-(A*b^2-a*(b*B-a*C))*sqrt(c+d*tan(e+f*x))/((a^2+b^2)*(b*c-a*d)*f*(a+b*tan(e+f*x)))],
[(a+b*tan(e+f*x))^3*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^(3/2),x,11,-(a-I*b)^3*(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(3/2)*f)-(I*a-b)^3*(A+I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(3/2)*f)+2/15*b*(6*a^2*d^2*(12*c^2*C-5*B*c*d+(5*A+7*C)*d^2)-15*a*b*d*(8*c^3*C-6*B*c^2*d+c*(3*A+5*C)*d^2-3*B*d^3)+b^2*(48*c^4*C-40*B*c^3*d+6*c^2*(5*A+3*C)*d^2-25*B*c*d^3+15*(A-C)*d^4))*sqrt(c+d*tan(e+f*x))/(d^4*(c^2+d^2)*f)-2/15*b^2*(4*(b*c-a*d)*(6*c^2*C-5*B*c*d+(5*A+C)*d^2)-5*d^2*((A-C)*(b*c-a*d)+B*(a*c+b*d)))*sqrt(c+d*tan(e+f*x))*tan(e+f*x)/(d^3*(c^2+d^2)*f)+2/5*b*(6*c^2*C-5*B*c*d+(5*A+C)*d^2)*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^2/(d^2*(c^2+d^2)*f)-2*(c^2*C-B*c*d+A*d^2)*(a+b*tan(e+f*x))^3/(d*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))^2*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^(3/2),x,10,-(a-I*b)^2*(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(3/2)*f)-(a+I*b)^2*(B-I*(A-C))*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(3/2)*f)+2/3*b*(6*a*d*(2*c^2*C-B*c*d+(A+C)*d^2)-b*(8*c^3*C-6*B*c^2*d+c*(3*A+5*C)*d^2-3*B*d^3))*sqrt(c+d*tan(e+f*x))/(d^3*(c^2+d^2)*f)+2/3*b^2*(4*c^2*C-3*B*c*d+(3*A+C)*d^2)*sqrt(c+d*tan(e+f*x))*tan(e+f*x)/(d^2*(c^2+d^2)*f)-2*(c^2*C-B*c*d+A*d^2)*(a+b*tan(e+f*x))^2/(d*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^(3/2),x,9,-(I*a+b)*(A-I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(3/2)*f)+(I*a-b)*(A+I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(3/2)*f)+2*(b*c-a*d)*(c^2*C-B*c*d+A*d^2)/(d^2*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))+2*b*C*sqrt(c+d*tan(e+f*x))/(d^2*f)],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^(3/2),x,8,-(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(3/2)*f)-(B-I*(A-C))*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(3/2)*f)-2*(c^2*C-B*c*d+A*d^2)/(d*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/((a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)),x,12,(A-I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((I*a+b)*(c-I*d)^(3/2)*f)+(I*A-B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((a+I*b)*(c+I*d)^(3/2)*f)-2*(A*b^2-a*(b*B-a*C))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))*sqrt(b)/((a^2+b^2)*(b*c-a*d)^(3/2)*f)+2*(c^2*C-B*c*d+A*d^2)/((b*c-a*d)*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/((a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^(3/2)),x,13,-(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((a-I*b)^2*(c-I*d)^(3/2)*f)-(B-I*(A-C))*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((a+I*b)^2*(c+I*d)^(3/2)*f)-(5*a^3*b*B*d-3*a^4*C*d+b^4*(2*B*c-3*A*d)+a*b^3*(4*A*c-4*c*C+B*d)-a^2*b^2*(2*B*c+(7*A-C)*d))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))*sqrt(b)/((a^2+b^2)^2*(b*c-a*d)^(5/2)*f)-d*(2*b^2*c*(c*C-B*d)-a*b*B*(c^2+d^2)+a^2*(3*c^2*C-2*B*c*d+C*d^2)+A*(2*a^2*d^2+b^2*(c^2+3*d^2)))/((a^2+b^2)*(b*c-a*d)^2*(c^2+d^2)*f*sqrt(c+d*tan(e+f*x)))+(-A*b^2+a*(b*B-a*C))/((a^2+b^2)*(b*c-a*d)*f*sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x)))],
[(a+b*tan(e+f*x))^3*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^(5/2),x,11,-(a-I*b)^3*(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(5/2)*f)-(I*a-b)^3*(A+I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(5/2)*f)+2/3*b*(3*a*b*d*(8*c^4*C-2*B*c^3*d-c^2*(A-17*C)*d^2-8*B*c*d^3+(5*A+3*C)*d^4)-b^2*(16*c^5*C-8*B*c^4*d+2*c^3*(A+15*C)*d^2-17*B*c^2*d^3+8*c*(A+C)*d^4-3*B*d^5)+6*a^2*d^3*(2*c*(A-C)*d-B*(c^2-d^2)))*sqrt(c+d*tan(e+f*x))/(d^4*(c^2+d^2)^2*f)+2/3*b^2*(b*(8*c^4*C-4*B*c^3*d+c^2*(A+15*C)*d^2-10*B*c*d^3+(7*A+C)*d^4)+3*a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))*sqrt(c+d*tan(e+f*x))*tan(e+f*x)/(d^3*(c^2+d^2)^2*f)-2*(b*(2*c^4*C-B*c^3*d+4*c^2*C*d^2-3*B*c*d^3+2*A*d^4)+a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))*(a+b*tan(e+f*x))^2/(d^2*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))-2/3*(c^2*C-B*c*d+A*d^2)*(a+b*tan(e+f*x))^3/(d*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))^2*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^(5/2),x,10,-(a-I*b)^2*(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(5/2)*f)-(a+I*b)^2*(B-I*(A-C))*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(5/2)*f)+2/3*(b*c-a*d)*(b*(4*c^4*C-B*c^3*d-2*c^2*(A-5*C)*d^2-7*B*c*d^3+4*A*d^4)+3*a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))/(d^3*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))+2/3*b^2*(4*c^2*C-B*c*d+(A+3*C)*d^2)*sqrt(c+d*tan(e+f*x))/(d^3*(c^2+d^2)*f)-2/3*(c^2*C-B*c*d+A*d^2)*(a+b*tan(e+f*x))^2/(d*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^(5/2),x,9,-(a-I*b)*(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(5/2)*f)+(I*a-b)*(A+I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(5/2)*f)-2*(b*(c^4*C-c^2*(A-3*C)*d^2-2*B*c*d^3+A*d^4)+a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))/(d^2*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))+2/3*(b*c-a*d)*(c^2*C-B*c*d+A*d^2)/(d^2*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^(5/2),x,9,-(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((c-I*d)^(5/2)*f)-(B-I*(A-C))*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((c+I*d)^(5/2)*f)-2*(2*c*(A-C)*d-B*(c^2-d^2))/((c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))-2/3*(c^2*C-B*c*d+A*d^2)/(d*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/((a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2)),x,13,(A-I*B-C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((I*a+b)*(c-I*d)^(5/2)*f)+(I*A-B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((a+I*b)*(c+I*d)^(5/2)*f)-2*b^(3/2)*(A*b^2-a*(b*B-a*C))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/((a^2+b^2)*(b*c-a*d)^(5/2)*f)+2*(b*(c^4*C-2*B*c^3*d+c^2*(3*A-C)*d^2+A*d^4)-a*d^2*(2*c*(A-C)*d-B*(c^2-d^2)))/((b*c-a*d)^2*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))+2/3*(c^2*C-B*c*d+A*d^2)/((b*c-a*d)*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))],
[(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/((a+b*tan(e+f*x))^2*(c+d*tan(e+f*x))^(5/2)),x,14,-(I*A+B-I*C)*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c-I*d))/((a-I*b)^2*(c-I*d)^(5/2)*f)-(B-I*(A-C))*arctanh(sqrt(c+d*tan(e+f*x))/sqrt(c+I*d))/((a+I*b)^2*(c+I*d)^(5/2)*f)-b^(3/2)*(7*a^3*b*B*d-5*a^4*C*d+b^4*(2*B*c-5*A*d)+a*b^3*(4*A*c-4*c*C+3*B*d)-a^2*b^2*(2*B*c+(9*A+C)*d))*arctanh(sqrt(b)*sqrt(c+d*tan(e+f*x))/sqrt(b*c-a*d))/((a^2+b^2)^2*(b*c-a*d)^(7/2)*f)-d*(2*a^3*d^2*(B*c^2+2*c*C*d-B*d^2)+2*b^3*c*(2*c^3*C-3*B*c^2*d-B*d^3)-a*b^2*(B*c^4-4*c*C*d^3+3*B*d^4)+a^2*b*(5*c^4*C-6*B*c^3*d+2*c^2*C*d^2-2*B*c*d^3+C*d^4)-A*(4*a^3*c*d^3+4*a*b^2*c*d^3-4*a^2*b*d^2*(2*c^2+d^2)-b^3*(c^4+10*c^2*d^2+5*d^4)))/((a^2+b^2)*(b*c-a*d)^3*(c^2+d^2)^2*f*sqrt(c+d*tan(e+f*x)))-1/3*d*(2*b^2*c*(c*C-B*d)-3*a*b*B*(c^2+d^2)+a^2*(5*c^2*C-2*B*c*d+3*C*d^2)+A*(2*a^2*d^2+b^2*(3*c^2+5*d^2)))/((a^2+b^2)*(b*c-a*d)^2*(c^2+d^2)*f*(c+d*tan(e+f*x))^(3/2))+(-A*b^2+a*(b*B-a*C))/((a^2+b^2)*(b*c-a*d)*f*(a+b*tan(e+f*x))*(c+d*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]+C Tan[e+f x]^2)

# n > 0
[sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,16,-1/64*(5*a^4*C*d^4-20*a^3*b*d^3*(c*C+2*B*d)+30*a^2*b^2*d^2*(c^2*C-4*B*c*d-8*(A-C)*d^2)-20*a*b^3*d*(c^3*C-2*B*c^2*d+8*c*(A-C)*d^2-16*B*d^3)+b^4*(5*c^4*C-8*B*c^3*d+16*c^2*(A-C)*d^2+64*B*c*d^3+128*(A-C)*d^4))*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(b^(3/2)*d^(7/2)*f)-(a-I*b)^(5/2)*(I*A+B-I*C)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c-I*d)/f-(a+I*b)^(5/2)*(B-I*(A-C))*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c+I*d)/f+1/64*(64*b*(a^2*B-b^2*B+2*a*b*(A-C))*d^3-(b*c-a*d)*(16*b*(A*b+a*B-b*C)*d^2+(b*c-a*d)*(5*b*c*C-8*b*B*d-5*a*C*d)))*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(b*d^3*f)+1/32*(16*b*(A*b+a*B-b*C)*d^2+(b*c-a*d)*(5*b*c*C-8*b*B*d-5*a*C*d))*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/(d^3*f)-1/24*(5*b*c*C-8*b*B*d-5*a*C*d)*(a+b*tan(e+f*x))^(3/2)*(c+d*tan(e+f*x))^(3/2)/(d^2*f)+1/4*C*(a+b*tan(e+f*x))^(5/2)*(c+d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(c+d*tan(e+f*x))*(a+b*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,15,-1/8*(a^3*C*d^3-3*a^2*b*d^2*(c*C+2*B*d)+3*a*b^2*d*(c^2*C-4*B*c*d-8*(A-C)*d^2)-b^3*(c^3*C-2*B*c^2*d+8*c*(A-C)*d^2-16*B*d^3))*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(b^(3/2)*d^(5/2)*f)-(a-I*b)^(3/2)*(I*A+B-I*C)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c-I*d)/f+(a+I*b)^(3/2)*(I*A-B-I*C)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c+I*d)/f+1/8*(8*b*(A*b+a*B-b*C)*d^2+(b*c-a*d)*(b*c*C-2*b*B*d-a*C*d))*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(b*d^2*f)-1/4*(b*c*C-2*b*B*d-a*C*d)*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/(d^2*f)+1/3*C*(a+b*tan(e+f*x))^(3/2)*(c+d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,14,-1/4*(a^2*C*d^2-2*a*b*d*(c*C+2*B*d)+b^2*(c^2*C-4*B*c*d-8*(A-C)*d^2))*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(b^(3/2)*d^(3/2)*f)-(I*A+B-I*C)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a-I*b)*sqrt(c-I*d)/f-(B-I*(A-C))*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a+I*b)*sqrt(c+I*d)/f-1/4*(b*c*C-4*b*B*d-a*C*d)*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(b*d*f)+1/2*C*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/(d*f)],
[sqrt(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/sqrt(a+b*tan(e+f*x)),x,13,-(I*A+B-I*C)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c-I*d)/(f*sqrt(a-I*b))-(B-I*(A-C))*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c+I*d)/(f*sqrt(a+I*b))+(b*c*C+2*b*B*d-a*C*d)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(b^(3/2)*f*sqrt(d))+C*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(b*f)],
[sqrt(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^(3/2),x,13,-(I*A+B-I*C)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c-I*d)/((a-I*b)^(3/2)*f)-(B-I*(A-C))*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c+I*d)/((a+I*b)^(3/2)*f)+2*C*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))*sqrt(d)/(b^(3/2)*f)-2*(A*b^2-a*(b*B-a*C))*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)*f*sqrt(a+b*tan(e+f*x)))],
[sqrt(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^(5/2),x,9,-(I*A+B-I*C)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c-I*d)/((a-I*b)^(5/2)*f)-(B-I*(A-C))*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c+I*d)/((a+I*b)^(5/2)*f)-2/3*(2*a^3*b*B*d+a^4*C*d+b^4*(3*B*c+A*d)+2*a*b^3*(3*A*c-3*c*C-2*B*d)-a^2*b^2*(3*B*c+5*A*d-7*C*d))*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)^2*(b*c-a*d)*f*sqrt(a+b*tan(e+f*x)))-2/3*(A*b^2-a*(b*B-a*C))*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^(3/2))],
[sqrt(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^(7/2),x,10,-(I*A+B-I*C)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c-I*d)/((a-I*b)^(7/2)*f)-(B-I*(A-C))*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(c+I*d)/((a+I*b)^(7/2)*f)+2/15*(8*a^5*b*B*d^2+2*a^6*C*d^2-a^4*b^2*d*(25*B*c+33*A*d-39*C*d)-a^2*b^4*(45*A*c^2-45*c^2*C-90*B*c*d-29*A*d^2+23*C*d^2)+a^3*b^3*(80*c*(A-C)*d+B*(15*c^2-49*d^2))-a*b^5*(40*c*(A-C)*d+B*(45*c^2-3*d^2))-b^6*(5*c*(3*c*C+B*d)-A*(15*c^2+2*d^2)))*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)^3*(b*c-a*d)^2*f*sqrt(a+b*tan(e+f*x)))-2/5*(A*b^2-a*(b*B-a*C))*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^(5/2))-2/15*(4*a^3*b*B*d+a^4*C*d+b^4*(5*B*c+A*d)+2*a*b^3*(5*A*c-5*c*C-3*B*d)-a^2*b^2*(5*B*c+9*A*d-11*C*d))*sqrt(c+d*tan(e+f*x))/(b*(a^2+b^2)^2*(b*c-a*d)*f*(a+b*tan(e+f*x))^(3/2))],
[(a+b*tan(e+f*x))^(3/2)*(c+d*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,16,-(a-I*b)^(3/2)*(B+I*(A-C))*(c-I*d)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/f-(a+I*b)^(3/2)*(B-I*(A-C))*(c+I*d)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/f+1/64*(3*a^4*C*d^4-4*a^3*b*d^3*(3*c*C+2*B*d)+6*a^2*b^2*d^2*(3*c^2*C+12*B*c*d+8*(A-C)*d^2)-12*a*b^3*d*(c^3*C-6*B*c^2*d-24*c*(A-C)*d^2+16*B*d^3)+b^4*(3*c^4*C-8*B*c^3*d+48*c^2*(A-C)*d^2-192*B*c*d^3-128*(A-C)*d^4))*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(b^(5/2)*d^(5/2)*f)+1/64*(64*b*(a^2*B-b^2*B+2*a*b*(A-C))*d^3+(b*c-a*d)*(48*b*(A*b+a*B-b*C)*d^2+(b*c-a*d)*(3*b*c*C-8*b*B*d-3*a*C*d)))*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(b^2*d^2*f)+1/96*(48*b*(A*b+a*B-b*C)*d^2+(b*c-a*d)*(3*b*c*C-8*b*B*d-3*a*C*d))*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/(b*d^2*f)-1/24*(3*b*c*C-8*b*B*d-3*a*C*d)*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2)/(d^2*f)+1/4*C*(a+b*tan(e+f*x))^(3/2)*(c+d*tan(e+f*x))^(5/2)/(d*f)],
[sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,15,1/8*(a^3*C*d^3-a^2*b*d^2*(3*c*C+2*B*d)+a*b^2*d*(3*c^2*C+12*B*c*d+8*(A-C)*d^2)-b^3*(c^3*C-6*B*c^2*d-24*c*(A-C)*d^2+16*B*d^3))*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(b^(5/2)*d^(3/2)*f)-(I*A+B-I*C)*(c-I*d)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a-I*b)/f-(B-I*(A-C))*(c+I*d)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a+I*b)/f+1/8*(8*b*(A*b+a*B-b*C)*d^2-(b*c-a*d)*(b*c*C-6*b*B*d-a*C*d))*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(b^2*d*f)-1/12*(b*c*C-6*b*B*d-a*C*d)*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/(b*d*f)+1/3*C*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2)/(d*f)],
[(c+d*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/sqrt(a+b*tan(e+f*x)),x,14,-(I*A+B-I*C)*(c-I*d)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(a-I*b))+(I*A-B-I*C)*(c+I*d)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(a+I*b))+1/4*(3*a^2*C*d^2-2*a*b*d*(3*c*C+2*B*d)+b^2*(3*c^2*C+12*B*c*d+8*(A-C)*d^2))*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(b^(5/2)*f*sqrt(d))+1/4*(3*b*c*C+4*b*B*d-3*a*C*d)*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(b^2*f)+1/2*C*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/(b*f)],
[(c+d*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^(3/2),x,14,-(I*A+B-I*C)*(c-I*d)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(3/2)*f)-(B-I*(A-C))*(c+I*d)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(3/2)*f)+(3*b*c*C+2*b*B*d-3*a*C*d)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))*sqrt(d)/(b^(5/2)*f)+(2*A*b^2-2*a*b*B+3*a^2*C+b^2*C)*d*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(b^2*(a^2+b^2)*f)-2*(A*b^2-a*(b*B-a*C))*(c+d*tan(e+f*x))^(3/2)/(b*(a^2+b^2)*f*sqrt(a+b*tan(e+f*x)))],
[(c+d*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^(5/2),x,14,-(I*A+B-I*C)*(c-I*d)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(5/2)*f)-(B-I*(A-C))*(c+I*d)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(5/2)*f)+2*C*d^(3/2)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(b^(5/2)*f)-2*(a^4*C*d+b^4*(B*c+A*d)+2*a*b^3*(A*c-c*C-B*d)-a^2*b^2*(B*c+(A-3*C)*d))*sqrt(c+d*tan(e+f*x))/(b^2*(a^2+b^2)^2*f*sqrt(a+b*tan(e+f*x)))-2/3*(A*b^2-a*(b*B-a*C))*(c+d*tan(e+f*x))^(3/2)/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^(3/2))],
[(c+d*tan(e+f*x))^(3/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^(7/2),x,10,-(I*A+B-I*C)*(c-I*d)^(3/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(7/2)*f)-(B-I*(A-C))*(c+I*d)^(3/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(7/2)*f)-2/15*(2*a^5*b*B*d^2+3*a^6*C*d^2+a^4*b^2*d*(10*B*c+(8*A+C)*d)+a^2*b^4*(45*A*c^2-45*c^2*C-90*B*c*d-49*A*d^2+58*C*d^2)-a^3*b^3*(50*c*(A-C)*d+B*(15*c^2-39*d^2))+a*b^5*(70*c*(A-C)*d+B*(45*c^2-23*d^2))+b^6*(5*c*(3*c*C+4*B*d)-3*A*(5*c^2-d^2)))*sqrt(c+d*tan(e+f*x))/(b^2*(a^2+b^2)^3*(b*c-a*d)*f*sqrt(a+b*tan(e+f*x)))-2/15*(2*a^3*b*B*d+3*a^4*C*d+b^4*(5*B*c+3*A*d)+2*a*b^3*(5*A*c-5*c*C-4*B*d)-a^2*b^2*(5*B*c+7*A*d-13*C*d))*sqrt(c+d*tan(e+f*x))/(b^2*(a^2+b^2)^2*f*(a+b*tan(e+f*x))^(3/2))-2/5*(A*b^2-a*(b*B-a*C))*(c+d*tan(e+f*x))^(3/2)/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^(5/2))],
[sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,16,-1/64*(5*a^4*C*d^4-4*a^3*b*d^3*(5*c*C+2*B*d)+2*a^2*b^2*d^2*(15*c^2*C+20*B*c*d+8*(A-C)*d^2)-4*a*b^3*d*(5*c^3*C+30*B*c^2*d+40*c*(A-C)*d^2-16*B*d^3)+b^4*(5*c^4*C-40*B*c^3*d-240*c^2*(A-C)*d^2+320*B*c*d^3+128*(A-C)*d^4))*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(b^(7/2)*d^(3/2)*f)-(I*A+B-I*C)*(c-I*d)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a-I*b)/f+(I*A-B-I*C)*(c+I*d)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))*sqrt(a+I*b)/f+1/64*(64*b^2*d^2*(A*b*c+a*B*c-b*c*C+a*A*d-b*B*d-a*C*d)+(b*c-a*d)*(48*b*(A*b+a*B-b*C)*d^2-5*(b*c-a*d)*(b*c*C-8*b*B*d-a*C*d)))*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(b^3*d*f)+1/96*(48*b*(A*b+a*B-b*C)*d^2-5*(b*c-a*d)*(b*c*C-8*b*B*d-a*C*d))*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/(b^2*d*f)-1/24*(b*c*C-8*b*B*d-a*C*d)*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2)/(b*d*f)+1/4*C*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(7/2)/(d*f)],
[(c+d*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/sqrt(a+b*tan(e+f*x)),x,15,-(I*A+B-I*C)*(c-I*d)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(a-I*b))-(B-I*(A-C))*(c+I*d)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/(f*sqrt(a+I*b))-1/8*(5*a^3*C*d^3-3*a^2*b*d^2*(5*c*C+2*B*d)+a*b^2*d*(15*c^2*C+20*B*c*d+8*(A-C)*d^2)-b^3*(5*c^3*C+30*B*c^2*d+40*c*(A-C)*d^2-16*B*d^3))*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(b^(7/2)*f*sqrt(d))+1/8*(8*b^2*d*(B*c+(A-C)*d)+(b*c-a*d)*(5*b*c*C+6*b*B*d-5*a*C*d))*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(b^3*f)+1/12*(5*b*c*C+6*b*B*d-5*a*C*d)*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/(b^2*f)+1/3*C*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(5/2)/(b*f)],
[(c+d*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^(3/2),x,15,-(I*A+B-I*C)*(c-I*d)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(3/2)*f)-(B-I*(A-C))*(c+I*d)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(3/2)*f)+1/4*(15*a^2*C*d^2-6*a*b*d*(5*c*C+2*B*d)+b^2*(15*c^2*C+20*B*c*d+8*(A-C)*d^2))*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))*sqrt(d)/(b^(7/2)*f)-1/4*d*(15*a^3*C*d-8*A*b^2*(b*c-a*d)-3*a^2*b*(5*c*C+4*B*d)-b^3*(7*c*C+4*B*d)+a*b^2*(8*B*c+7*C*d))*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(b^3*(a^2+b^2)*f)+1/2*(4*A*b^2-4*a*b*B+5*a^2*C+b^2*C)*d*sqrt(a+b*tan(e+f*x))*(c+d*tan(e+f*x))^(3/2)/(b^2*(a^2+b^2)*f)-2*(A*b^2-a*(b*B-a*C))*(c+d*tan(e+f*x))^(5/2)/(b*(a^2+b^2)*f*sqrt(a+b*tan(e+f*x)))],
[(c+d*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^(5/2),x,15,-(I*A+B-I*C)*(c-I*d)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(5/2)*f)-(B-I*(A-C))*(c+I*d)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(5/2)*f)+d^(3/2)*(5*b*c*C+2*b*B*d-5*a*C*d)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(b^(7/2)*f)-d*(2*a^3*b*B*d-5*a^4*C*d-2*a*b^3*(2*A*c-2*c*C-3*B*d)+2*a^2*b^2*(B*c-5*C*d)-b^4*(2*B*c+(4*A+C)*d))*sqrt(a+b*tan(e+f*x))*sqrt(c+d*tan(e+f*x))/(b^3*(a^2+b^2)^2*f)+2/3*(2*a^3*b*B*d-5*a^4*C*d-b^4*(3*B*c+5*A*d)-2*a*b^3*(3*A*c-3*c*C-4*B*d)+a^2*b^2*(3*B*c+(A-11*C)*d))*(c+d*tan(e+f*x))^(3/2)/(b^2*(a^2+b^2)^2*f*sqrt(a+b*tan(e+f*x)))-2/3*(A*b^2-a*(b*B-a*C))*(c+d*tan(e+f*x))^(5/2)/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^(3/2))],
[(c+d*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^(7/2),x,15,-(I*A+B-I*C)*(c-I*d)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(7/2)*f)-(B-I*(A-C))*(c+I*d)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(7/2)*f)+2*C*d^(5/2)*arctanh(sqrt(d)*sqrt(a+b*tan(e+f*x))/(sqrt(b)*sqrt(c+d*tan(e+f*x))))/(b^(7/2)*f)-2*(a^6*C*d^2+3*a^4*b^2*C*d^2-3*a^2*b^4*(c^2*C+2*B*c*d-2*C*d^2-A*(c^2-d^2))+b^6*(c*(c*C+2*B*d)-A*(c^2-d^2))-a^3*b^3*(2*c*(A-C)*d+B*(c^2-d^2))+3*a*b^5*(2*c*(A-C)*d+B*(c^2-d^2)))*sqrt(c+d*tan(e+f*x))/(b^3*(a^2+b^2)^3*f*sqrt(a+b*tan(e+f*x)))-2/3*(a^4*C*d+b^4*(B*c+A*d)+2*a*b^3*(A*c-c*C-B*d)-a^2*b^2*(B*c+(A-3*C)*d))*(c+d*tan(e+f*x))^(3/2)/(b^2*(a^2+b^2)^2*f*(a+b*tan(e+f*x))^(3/2))-2/5*(A*b^2-a*(b*B-a*C))*(c+d*tan(e+f*x))^(5/2)/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^(5/2))],
[(c+d*tan(e+f*x))^(5/2)*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(a+b*tan(e+f*x))^(9/2),x,11,-(I*A+B-I*C)*(c-I*d)^(5/2)*arctanh(sqrt(c-I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a-I*b)*sqrt(c+d*tan(e+f*x))))/((a-I*b)^(9/2)*f)-(B-I*(A-C))*(c+I*d)^(5/2)*arctanh(sqrt(c+I*d)*sqrt(a+b*tan(e+f*x))/(sqrt(a+I*b)*sqrt(c+d*tan(e+f*x))))/((a+I*b)^(9/2)*f)-2/105*(6*a^7*b*B*d^3+15*a^8*C*d^3+2*a^6*b^2*d^2*(7*B*c+4*A*d+26*C*d)-2*a*b^7*(210*A*c^3-210*c^3*C-525*B*c^2*d-406*A*c*d^2+406*c*C*d^2+88*B*d^3)-a^4*b^4*(105*B*c^3+525*A*c^2*d-525*c^2*C*d-749*B*c*d^2-311*A*d^3+221*C*d^3)+2*a^2*b^6*(315*B*c^3+875*A*c^2*d-875*c^2*C*d-812*B*c*d^2-261*A*d^3+291*C*d^3)+2*a^5*b^3*d*(56*c*(A-C)*d+B*(35*c^2-12*d^2))-b^8*(5*d*(49*A*c^2-49*c^2*C-3*A*d^2)+7*B*(15*c^3-23*c*d^2))-2*a^3*b^5*(210*c^3*C+700*B*c^2*d-798*c*C*d^2-317*B*d^3-42*A*(5*c^3-19*c*d^2)))*sqrt(c+d*tan(e+f*x))/(b^3*(a^2+b^2)^4*(b*c-a*d)*f*sqrt(a+b*tan(e+f*x)))-2/105*(6*a^5*b*B*d^2+15*a^6*C*d^2+a^4*b^2*d*(14*B*c+8*A*d+37*C*d)+3*a^2*b^4*(35*A*c^2-35*c^2*C-70*B*c*d-39*A*d^2+54*C*d^2)-a^3*b^3*(98*c*(A-C)*d+B*(35*c^2-75*d^2))+a*b^5*(182*c*(A-C)*d+B*(105*c^2-71*d^2))+b^6*(7*c*(5*c*C+8*B*d)-5*A*(7*c^2-3*d^2)))*sqrt(c+d*tan(e+f*x))/(b^3*(a^2+b^2)^3*f*(a+b*tan(e+f*x))^(3/2))-2/35*(2*a^3*b*B*d+5*a^4*C*d+b^4*(7*B*c+5*A*d)+2*a*b^3*(7*A*c-7*c*C-6*B*d)-a^2*b^2*(7*B*c+9*A*d-19*C*d))*(c+d*tan(e+f*x))^(3/2)/(b^2*(a^2+b^2)^2*f*(a+b*tan(e+f*x))^(5/2))-2/7*(A*b^2-a*(b*B-a*C))*(c+d*tan(e+f*x))^(5/2)/(b*(a^2+b^2)*f*(a+b*tan(e+f*x))^(7/2))],

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

# Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^n (A+B Tan[e+f x]+C Tan[e+f x]^2) with m and/or n symbolic
[(a+b*tan(e+f*x))^m*(c+d*tan(e+f*x))^n*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,9,-1/2*(B+I*(A-C))*AppellF1(1+m,-n,1,2+m,-d*(a+b*tan(e+f*x))/(b*c-a*d),(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)*(c+d*tan(e+f*x))^n/((a-I*b)*f*(1+m)*(b*(c+d*tan(e+f*x))/(b*c-a*d))^n)-1/2*(A+I*B-C)*AppellF1(1+m,-n,1,2+m,-d*(a+b*tan(e+f*x))/(b*c-a*d),(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)*(c+d*tan(e+f*x))^n/((I*a-b)*f*(1+m)*(b*(c+d*tan(e+f*x))/(b*c-a*d))^n)+C*hypergeom([1+m,-n],[2+m],-d*(a+b*tan(e+f*x))/(b*c-a*d))*(a+b*tan(e+f*x))^(1+m)*(c+d*tan(e+f*x))^n/(b*f*(1+m)*(b*(c+d*tan(e+f*x))/(b*c-a*d))^n)],
[(a+b*tan(e+f*x))^m*(c+d*tan(e+f*x))^3*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,9,(b*c*(2+m)*(b^2*d*(B*c+(A-C)*d)*(3+m)*(4+m)-2*(b*c-a*d)*(3*a*C*d-b*(3*c*C+B*d*(4+m))))+d*(b^3*(2*c*(A-C)*d+B*(c^2-d^2))*(2+m)*(3+m)*(4+m)-a*(b^2*d*(B*c+(A-C)*d)*(3+m)*(4+m)-2*(b*c-a*d)*(3*a*C*d-b*(3*c*C+B*d*(4+m))))))*(a+b*tan(e+f*x))^(1+m)/(b^4*f*(1+m)*(2+m)*(3+m)*(4+m))+1/2*(A-I*B-C)*(c-I*d)^3*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*f*(1+m))-1/2*(A+I*B-C)*(c+I*d)^3*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a-b)*f*(1+m))+d*(b^2*d*(B*c+(A-C)*d)*(3+m)*(4+m)-2*(b*c-a*d)*(3*a*C*d-b*(3*c*C+B*d*(4+m))))*tan(e+f*x)*(a+b*tan(e+f*x))^(1+m)/(b^3*f*(2+m)*(3+m)*(4+m))-(3*a*C*d-b*(3*c*C+B*d*(4+m)))*(a+b*tan(e+f*x))^(1+m)*(c+d*tan(e+f*x))^2/(b^2*f*(3+m)*(4+m))+C*(a+b*tan(e+f*x))^(1+m)*(c+d*tan(e+f*x))^3/(b*f*(4+m))],
[(a+b*tan(e+f*x))^m*(c+d*tan(e+f*x))^2*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,8,(2*a^2*C*d^2-a*b*d*(2*c*C+B*d)*(3+m)+b^2*(2+m)*(2*c^2*C+2*B*c*d*(3+m)+(A-C)*d^2*(3+m)))*(a+b*tan(e+f*x))^(1+m)/(b^3*f*(1+m)*(2+m)*(3+m))+1/2*(A-I*B-C)*(c-I*d)^2*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*f*(1+m))+1/2*(I*A-B-I*C)*(c+I*d)^2*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)/((a+I*b)*f*(1+m))-d*(2*a*C*d-b*(2*c*C+B*d*(3+m)))*tan(e+f*x)*(a+b*tan(e+f*x))^(1+m)/(b^2*f*(2+m)*(3+m))+C*(a+b*tan(e+f*x))^(1+m)*(c+d*tan(e+f*x))^2/(b*f*(3+m))],
[(a+b*tan(e+f*x))^m*(c+d*tan(e+f*x))*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,7,-(a*C*d-b*(c*C+B*d)*(2+m))*(a+b*tan(e+f*x))^(1+m)/(b^2*f*(1+m)*(2+m))+1/2*(A-I*B-C)*(c-I*d)*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*f*(1+m))-1/2*(A+I*B-C)*(c+I*d)*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a-b)*f*(1+m))+C*d*tan(e+f*x)*(a+b*tan(e+f*x))^(1+m)/(b*f*(2+m))],
[(a+b*tan(e+f*x))^m*(A+B*tan(e+f*x)+C*tan(e+f*x)^2),x,6,C*(a+b*tan(e+f*x))^(1+m)/(b*f*(1+m))+1/2*(A-I*B-C)*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*f*(1+m))+1/2*(I*A-B-I*C)*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)/((a+I*b)*f*(1+m))],
[(a+b*tan(e+f*x))^m*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x)),x,8,-1/2*(I*A+B-I*C)*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)/((a-I*b)*(c-I*d)*f*(1+m))-1/2*(A+I*B-C)*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a-b)*(c+I*d)*f*(1+m))+(c^2*C-B*c*d+A*d^2)*hypergeom([1,1+m],[2+m],-d*(a+b*tan(e+f*x))/(b*c-a*d))*(a+b*tan(e+f*x))^(1+m)/((b*c-a*d)*(c^2+d^2)*f*(1+m))],
[(a+b*tan(e+f*x))^m*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^2,x,9,1/2*(A-I*B-C)*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*(c-I*d)^2*f*(1+m))+1/2*(I*A-B-I*C)*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)/((a+I*b)*(c+I*d)^2*f*(1+m))-(a*d^2*(2*c*(A-C)*d-B*(c^2-d^2))-b*(A*d^2*(c^2*(2-m)-d^2*m)-B*c*d*(c^2*(1-m)-d^2*(1+m))-c^2*C*(c^2*m+d^2*(2+m))))*hypergeom([1,1+m],[2+m],-d*(a+b*tan(e+f*x))/(b*c-a*d))*(a+b*tan(e+f*x))^(1+m)/((b*c-a*d)^2*(c^2+d^2)^2*f*(1+m))+(c^2*C-B*c*d+A*d^2)*(a+b*tan(e+f*x))^(1+m)/((b*c-a*d)*(c^2+d^2)*f*(c+d*tan(e+f*x)))],
[(a+b*tan(e+f*x))^m*(A+B*tan(e+f*x)+C*tan(e+f*x)^2)/(c+d*tan(e+f*x))^3,x,10,1/2*(A-I*B-C)*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a-I*b))*(a+b*tan(e+f*x))^(1+m)/((I*a+b)*(c-I*d)^3*f*(1+m))+1/2*(A+I*B-C)*hypergeom([1,1+m],[2+m],(a+b*tan(e+f*x))/(a+I*b))*(a+b*tan(e+f*x))^(1+m)/((a+I*b)*(I*c-d)^3*f*(1+m))+1/2*(2*a^2*d^3*((A-C)*d*(3*c^2-d^2)-B*(c^3-3*c*d^2))-2*a*b*d^2*(B*(6*c^2*d^2-c^4*(2-m)-d^4*m)+2*c*(A-C)*d*(c^2*(3-m)-d^2*(1+m)))-b^2*(A*d^2*(d^4*(1-m)*m+2*c^2*d^2*(1+3*m-m^2)-c^4*(6-5*m+m^2))+B*c*d*(d^4*m*(1+m)-2*c^2*d^2*(3+m-m^2)+c^4*(2-3*m+m^2))+c^2*C*(c^4*(1-m)*m+2*c^2*d^2*(3-m-m^2)-d^4*(2+3*m+m^2))))*hypergeom([1,1+m],[2+m],-d*(a+b*tan(e+f*x))/(b*c-a*d))*(a+b*tan(e+f*x))^(1+m)/((b*c-a*d)^3*(c^2+d^2)^3*f*(1+m))+1/2*(c^2*C-B*c*d+A*d^2)*(a+b*tan(e+f*x))^(1+m)/((b*c-a*d)*(c^2+d^2)*f*(c+d*tan(e+f*x))^2)-1/2*(2*a*d^2*(2*c*(A-C)*d-B*(c^2-d^2))-b*(c^4*C*(1-m)+A*d^4*(1-m)-B*c^3*d*(3-m)+B*c*d^3*(1+m)+c^2*d^2*(A*(5-m)-C*(3+m))))*(a+b*tan(e+f*x))^(1+m)/((b*c-a*d)^2*(c^2+d^2)^2*f*(c+d*tan(e+f*x)))]]:
#  {(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 18, -((2*(3*a*C*d - b*(3*c*C + B*d*(5 + 2*m)))*(a + b*Tan[e + f*x])^(1 + m)*Sqrt[c + d*Tan[e + f*x]])/(b^2*f*(3 + 2*m)*(5 + 2*m))) + (2*C*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x])^(3/2))/(b*f*(5 + 2*m)) - ((B*d*(c^2 - d^2 - 2*c*Sqrt[-d^2]) + (A - C)*(2*c*d^2 + c^2*Sqrt[-d^2] + (-d^2)^(3/2)))*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c - Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c - Sqrt[-d^2])*f)) - ((B*d*(c^2 - d^2 + 2*c*Sqrt[-d^2]) + (A - C)*(2*c*d^2 - c^2*Sqrt[-d^2] - (-d^2)^(3/2)))*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c + Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c + Sqrt[-d^2])*f)) + (1/(b^2*d*f*(3 + 2*m)*(5 + 2*m)))*((2*(3*a^2*C*d^2 - a*b*d*(6*c*C + B*d*(5 + 2*m)) + b^2*(3*c^2*C + 2*B*c*d*(10 + 9*m + 2*m^2) + (A - C)*d^2*(15 + 16*m + 4*m^2)))*Hypergeometric2F1[1/2, -m, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d)]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/(1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m)}
# {(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(1/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 16, (2*C*(a + b*Tan[e + f*x])^(1 + m)*Sqrt[c + d*Tan[e + f*x]])/(b*f*(3 + 2*m)) - ((B*d*(c - Sqrt[-d^2]) + (A - C)*(d^2 + c*Sqrt[-d^2]))*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c - Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c - Sqrt[-d^2])*f)) - ((B*d*(c + Sqrt[-d^2]) + (A - C)*(d^2 - c*Sqrt[-d^2]))*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c + Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c + Sqrt[-d^2])*f)) - (2*(a*C*d - b*(c*C + B*d*(3 + 2*m)))*Hypergeometric2F1[1/2, -m, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d)]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(b*d*f*(3 + 2*m)))}
# {((a + b*Tan[e + f*x])^m*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(1/2), x, 15, -(((B*d + (A - C)*Sqrt[-d^2])*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c - Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c - Sqrt[-d^2])*f))) - ((B*d - (A - C)*Sqrt[-d^2])*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c + Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c + Sqrt[-d^2])*f)) + (2*C*Hypergeometric2F1[1/2, -m, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d)]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*f))}

# {((a + b*Tan[e + f*x])^m*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2), x, 16, If[$VersionNumber<9, (2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) - ((B*c - (A - C)*d - (d*(A*c - c*C + B*d))/Sqrt[-d^2])*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c - Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*((c^2 + d^2)*(c - Sqrt[-d^2])*f)) - ((B*c - (A - C)*d + (d*(A*c - c*C + B*d))/Sqrt[-d^2])*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c + Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*((c^2 + d^2)*(c + Sqrt[-d^2])*f)) - (2*b*(c^2*C - B*c*d + A*d^2)*(1 + 2*m)*Hypergeometric2F1[1/2, -m, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d)]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(b*c - a*d)*(c^2 + d^2)*f)), (2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) - ((B*d*(c + Sqrt[-d^2]) - (A - C)*(d^2 - c*Sqrt[-d^2]))*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c - Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c^2 + d^2)*(c - Sqrt[-d^2])*f)) - ((B*d*(c - Sqrt[-d^2]) - (A - C)*(d^2 + c*Sqrt[-d^2]))*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c + Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c^2 + d^2)*(c + Sqrt[-d^2])*f)) - (2*b*(c^2*C - B*c*d + A*d^2)*(1 + 2*m)*Hypergeometric2F1[1/2, -m, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d)]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(b*c - a*d)*(c^2 + d^2)*f))]} 
