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