\[ \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx \]
Optimal antiderivative \[ \frac {8 \left (-1\right )^{\frac {1}{4}} a^{3} \left (i A +B \right ) \arctan \left (\left (-1\right )^{\frac {3}{4}} \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{d}+\frac {8 a^{3} \left (i A +B \right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{d}-\frac {8 a^{3} \left (-23 i B +21 A \right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{105 d}+\frac {2 i a B \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{2}}{7 d}-\frac {2 \left (-11 i B +7 A \right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (a^{3}+i a^{3} \tan \left (d x +c \right )\right )}{35 d} \]
command
integrate(tan(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^3*(A+B*tan(d*x+c)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\left (4 i - 4\right ) \, \sqrt {2} {\left (A a^{3} - i \, B a^{3}\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{d} - \frac {2 \, {\left (15 i \, B a^{3} d^{6} \tan \left (d x + c\right )^{\frac {7}{2}} + 21 i \, A a^{3} d^{6} \tan \left (d x + c\right )^{\frac {5}{2}} + 63 \, B a^{3} d^{6} \tan \left (d x + c\right )^{\frac {5}{2}} + 105 \, A a^{3} d^{6} \tan \left (d x + c\right )^{\frac {3}{2}} - 140 i \, B a^{3} d^{6} \tan \left (d x + c\right )^{\frac {3}{2}} - 420 i \, A a^{3} d^{6} \sqrt {\tan \left (d x + c\right )} - 420 \, B a^{3} d^{6} \sqrt {\tan \left (d x + c\right )}\right )}}{105 \, d^{7}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \sqrt {\tan \left (d x + c\right )}\,{d x} \]________________________________________________________________________________________