\[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {4 \left (-1\right )^{\frac {1}{4}} a^{2} \left (i A +B \right ) \arctan \left (\left (-1\right )^{\frac {3}{4}} \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{d}+\frac {2 a^{2} \left (i A -B \right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{d}-\frac {2 A \left (a^{2}+i a^{2} \tan \left (d x +c \right )\right )}{d \sqrt {\tan \left (d x +c \right )}} \]
command
integrate((a+I*a*tan(d*x+c))^2*(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
\[ -\frac {2 \, B a^{2} \sqrt {\tan \left (d x + c\right )}}{d} - \frac {\left (2 i + 2\right ) \, \sqrt {2} {\left (-i \, A a^{2} - B a^{2}\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{d} - \frac {2 \, A a^{2}}{d \sqrt {\tan \left (d x + c\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}{\tan \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]________________________________________________________________________________________