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