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