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