\[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx \]
Optimal antiderivative \[ -\frac {\left (\left (30+28 i\right ) A +\left (-7+5 i\right ) B \right ) \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{32 a^{3} d}+\frac {\left (-\frac {1}{32}+\frac {i}{32}\right ) \left (\left (1+29 i\right ) A +\left (-6-i\right ) B \right ) \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{a^{3} d}+\frac {\left (-\frac {1}{64}+\frac {i}{64}\right ) \left (\left (29+i\right ) A +\left (1+6 i\right ) B \right ) \ln \left (1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{a^{3} d}+\frac {\left (\frac {1}{64}-\frac {i}{64}\right ) \left (\left (29+i\right ) A +\left (1+6 i\right ) B \right ) \ln \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{a^{3} d}-\frac {5 \left (i B +6 A \right )}{8 a^{3} d \sqrt {\tan \left (d x +c \right )}}+\frac {i B +A}{6 d \sqrt {\tan \left (d x +c \right )}\, \left (a +i a \tan \left (d x +c \right )\right )^{3}}+\frac {2 i B +5 A}{12 a d \sqrt {\tan \left (d x +c \right )}\, \left (a +i a \tan \left (d x +c \right )\right )^{2}}+\frac {\frac {7 i B}{24}+\frac {7 A}{6}}{d \sqrt {\tan \left (d x +c \right )}\, \left (a^{3}+i a^{3} \tan \left (d x +c \right )\right )} \]
command
integrate((A+B*tan(d*x+c))/tan(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\left (i + 1\right ) \, \sqrt {2} {\left (29 \, A + 6 i \, B\right )} \arctan \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{16 \, a^{3} d} - \frac {\left (i + 1\right ) \, \sqrt {2} {\left (i \, A + B\right )} \arctan \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{16 \, a^{3} d} - \frac {2 \, A}{a^{3} d \sqrt {\tan \left (d x + c\right )}} - \frac {42 i \, A \tan \left (d x + c\right )^{\frac {5}{2}} - 15 \, B \tan \left (d x + c\right )^{\frac {5}{2}} + 98 \, A \tan \left (d x + c\right )^{\frac {3}{2}} + 38 i \, B \tan \left (d x + c\right )^{\frac {3}{2}} - 60 i \, A \sqrt {\tan \left (d x + c\right )} + 27 \, B \sqrt {\tan \left (d x + c\right )}}{24 \, a^{3} d {\left (-i \, \tan \left (d x + c\right ) - 1\right )}^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {B \tan \left (d x + c\right ) + A}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \tan \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]________________________________________________________________________________________