\[ \int \frac {(a+i a \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac {5}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-1\right )^{\frac {1}{4}} a \left (-i B +A \right ) \arctan \left (\left (-1\right )^{\frac {3}{4}} \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{d}-\frac {2 a \left (i A +B \right )}{d \sqrt {\tan \left (d x +c \right )}}-\frac {2 a A}{3 d \tan \left (d x +c \right )^{\frac {3}{2}}} \]
command
integrate((a+I*a*tan(d*x+c))*(A+B*tan(d*x+c))/tan(d*x+c)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\left (i - 1\right ) \, \sqrt {2} {\left (-i \, A a - B a\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{d} - \frac {2 \, {\left (3 i \, A a \tan \left (d x + c\right ) + 3 \, B a \tan \left (d x + c\right ) + A a\right )}}{3 \, d \tan \left (d x + c\right )^{\frac {3}{2}}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}}{\tan \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]________________________________________________________________________________________