\[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {8 \left (-1\right )^{\frac {1}{4}} a^{3} \left (-i B +A \right ) \arctan \left (\left (-1\right )^{\frac {3}{4}} \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{d}-\frac {16 a^{3} \left (-6 i B +5 A \right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{15 d}+\frac {2 i a B \left (\sqrt {\tan }\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{2}}{5 d}-\frac {2 \left (-9 i B +5 A \right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \left (a^{3}+i a^{3} \tan \left (d x +c \right )\right )}{15 d} \]
command
integrate((a+I*a*tan(d*x+c))^3*(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
\[ \frac {\left (4 i - 4\right ) \, \sqrt {2} {\left (-i \, A a^{3} - B a^{3}\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{d} - \frac {2 \, {\left (3 i \, B a^{3} d^{4} \tan \left (d x + c\right )^{\frac {5}{2}} + 5 i \, A a^{3} d^{4} \tan \left (d x + c\right )^{\frac {3}{2}} + 15 \, B a^{3} d^{4} \tan \left (d x + c\right )^{\frac {3}{2}} + 45 \, A a^{3} d^{4} \sqrt {\tan \left (d x + c\right )} - 60 i \, B a^{3} d^{4} \sqrt {\tan \left (d x + c\right )}\right )}}{15 \, d^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________