\[ \int \frac {\sqrt {\tan (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx \]
Optimal antiderivative \[ \frac {\left (-\frac {1}{32}-\frac {i}{32}\right ) \left (\left (1+i\right ) A +B \right ) \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{a^{3} d}+\frac {\left (-\frac {1}{32}-\frac {i}{32}\right ) \left (\left (1+i\right ) A +B \right ) \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{a^{3} d}+\frac {\left (2 i A +\left (1-i\right ) B \right ) \ln \left (1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{64 a^{3} d}-\frac {\left (2 i A +\left (1-i\right ) B \right ) \ln \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{64 a^{3} d}+\frac {\left (i A -B \right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{6 d \left (a +i a \tan \left (d x +c \right )\right )^{3}}+\frac {\left (i A +2 B \right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{12 a d \left (a +i a \tan \left (d x +c \right )\right )^{2}}+\frac {B \left (\sqrt {\tan }\left (d x +c \right )\right )}{8 d \left (a^{3}+i a^{3} \tan \left (d x +c \right )\right )} \]
command
integrate(tan(d*x+c)^(1/2)*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\left (i + 1\right ) \, \sqrt {2} A \arctan \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{16 \, a^{3} d} - \frac {\left (i - 1\right ) \, \sqrt {2} {\left (A - i \, B\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{16 \, a^{3} d} - \frac {3 i \, B \tan \left (d x + c\right )^{\frac {5}{2}} + 2 i \, A \tan \left (d x + c\right )^{\frac {3}{2}} + 10 \, B \tan \left (d x + c\right )^{\frac {3}{2}} + 6 \, A \sqrt {\tan \left (d x + c\right )} - 3 i \, B \sqrt {\tan \left (d x + c\right )}}{24 \, a^{3} d {\left (\tan \left (d x + c\right ) - i\right )}^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {\tan \left (d x + c\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]________________________________________________________________________________________