Test folder name
test_cases/4_Trig_functions/4.3_Tangent/4.3.2.1-a+b_tan-^m-c+d_tan-^n
\[ \int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^3 \, dx \] Optimal antiderivative
\[ -\frac{2 d \left (c^2 (-(m+3))+i c d m+d^2\right ) (a+i a \tan (e+f x))^m}{f m (m+2)}-\frac{d^2 (d m+i c (m+4)) (a+i a \tan (e+f x))^{m+1}}{a f (m+1) (m+2)}+\frac{(d+i c)^3 (a+i a \tan (e+f x))^m \, _2F_1\left (1,m;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right )}{2 f m}+\frac{d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (m+2)} \] command
Integrate[(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^3,x]
Mathematica 12.1 output
\[ \int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^3 \, dx \] Mathematica 12 output
\[ \frac{2^{m-1} \left (e^{i f x}\right )^m e^{-i m (e+2 f x)} \left (\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m \left (\frac{(d+i c)^3 e^{i (e (m+6)+2 f (m+3) x)} \, _2F_1\left (1,1;m+4;-e^{2 i (e+f x)}\right )}{m+3}+\frac{(-d+i c)^3 e^{i m (e+2 f x)} \left (2 (m+2) e^{2 i (e+f x)}+2 e^{4 i (e+f x)}+m^2+3 m+2\right )}{m (m+1) (m+2)}-\frac{3 i (c-i d) (c+i d)^2 e^{i (e (m+2)+2 f (m+1) x)} \left (e^{2 i (e+f x)}+m+2\right )}{(m+1) (m+2)}+\frac{3 (c-i d)^2 (d-i c) e^{i (e (m+4)+2 f (m+2) x)}}{m+2}\right )}{f \left (1+e^{2 i (e+f x)}\right )^2} \]
Test folder name
test_cases/4_Trig_functions/4.3_Tangent/4.3.3.1-a+b_tan-^m-c+d_tan-^n-A+B_tan-
\[ \int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx \] Optimal antiderivative
\[ \frac{(B (n+2)+i A (2-n)) (c-i c \tan (e+f x))^n \, _2F_1\left (2,n;n+1;\frac{1}{2} (1-i \tan (e+f x))\right )}{16 a^2 f n}+\frac{(-B+i A) (c-i c \tan (e+f x))^n}{4 a^2 f (1+i \tan (e+f x))^2} \] command
Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n)/(a + I*a*Tan[e + f*x])^2,x]
Mathematica 12.1 output
\[ \int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx \] Mathematica 12 output
\[ \frac{2^{n-2} \left (\frac{c}{1+e^{2 i (e+f x)}}\right )^n \left (i (A (n-2)+i B (n+2)) e^{4 i (e+f x)} \, _2F_1\left (2,2-n;3-n;1+e^{2 i (e+f x)}\right )+(n-2) (B-i A)\right )}{a^2 f (n-2) (\tan (e+f x)-i)^2} \]