\[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {8 \left (-1\right )^{\frac {1}{4}} a^{3} \left (i A +B \right ) \arctan \left (\left (-1\right )^{\frac {3}{4}} \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{d}-\frac {16 a^{3} B \left (\sqrt {\tan }\left (d x +c \right )\right )}{3 d}-\frac {2 a A \left (a +i a \tan \left (d x +c \right )\right )^{2}}{d \sqrt {\tan \left (d x +c \right )}}+\frac {2 \left (3 i A -B \right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \left (a^{3}+i a^{3} \tan \left (d x +c \right )\right )}{3 d} \]
command
integrate((a+I*a*tan(d*x+c))^3*(A+B*tan(d*x+c))/tan(d*x+c)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, A a^{3}}{d \sqrt {\tan \left (d x + c\right )}} - \frac {\left (4 i + 4\right ) \, \sqrt {2} {\left (-i \, A a^{3} - B a^{3}\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{d} - \frac {2 \, {\left (i \, B a^{3} d^{2} \tan \left (d x + c\right )^{\frac {3}{2}} + 3 i \, A a^{3} d^{2} \sqrt {\tan \left (d x + c\right )} + 9 \, B a^{3} d^{2} \sqrt {\tan \left (d x + c\right )}\right )}}{3 \, d^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]________________________________________________________________________________________