\[ \int \frac {\tan ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (\left (1+3 i\right ) A +\left (9+5 i\right ) B \right ) \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{32 a^{2} d}-\frac {\left (\left (1+3 i\right ) A +\left (9+5 i\right ) B \right ) \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{32 a^{2} d}+\frac {\left (\left (1-3 i\right ) A +\left (-9+5 i\right ) B \right ) \ln \left (1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{64 a^{2} d}-\frac {\left (\left (1-3 i\right ) A +\left (-9+5 i\right ) B \right ) \ln \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{64 a^{2} d}+\frac {\left (5 i B +A \right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{8 a^{2} d \left (1+i \tan \left (d x +c \right )\right )}+\frac {\left (i A -B \right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{4 d \left (a +i a \tan \left (d x +c \right )\right )^{2}} \]
command
integrate(tan(d*x+c)^(3/2)*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\left (i + 1\right ) \, \sqrt {2} {\left (A - i \, B\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{8 \, a^{2} d} + \frac {\left (i - 1\right ) \, \sqrt {2} {\left (A - 7 i \, B\right )} \arctan \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{16 \, a^{2} d} - \frac {3 i \, A \tan \left (d x + c\right )^{\frac {3}{2}} - 7 \, B \tan \left (d x + c\right )^{\frac {3}{2}} + A \sqrt {\tan \left (d x + c\right )} + 5 i \, B \sqrt {\tan \left (d x + c\right )}}{8 \, a^{2} d {\left (\tan \left (d x + c\right ) - i\right )}^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{\frac {3}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]________________________________________________________________________________________