\[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx \]
Optimal antiderivative \[ \frac {\left (-\frac {1}{32}+\frac {i}{32}\right ) \left (\left (76+i\right ) A +\left (1+29 i\right ) B \right ) \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{a^{3} d}+\frac {\left (-\frac {1}{32}+\frac {i}{32}\right ) \left (\left (76+i\right ) A +\left (1+29 i\right ) B \right ) \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}}{a^{3} d}+\frac {\left (\left (77+75 i\right ) A +\left (-30+28 i\right ) B \right ) \ln \left (1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{64 a^{3} d}+\frac {\left (-\frac {1}{64}+\frac {i}{64}\right ) \left (\left (1+76 i\right ) A +\left (-29-i\right ) B \right ) \ln \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )\right ) \sqrt {2}}{a^{3} d}+\frac {\frac {75 i A}{8}-\frac {15 B}{4}}{a^{3} d \sqrt {\tan \left (d x +c \right )}}-\frac {7 \left (4 i B +11 A \right )}{24 a^{3} d \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {i B +A}{6 d \tan \left (d x +c \right )^{\frac {3}{2}} \left (a +i a \tan \left (d x +c \right )\right )^{3}}+\frac {i B +2 A}{4 a d \tan \left (d x +c \right )^{\frac {3}{2}} \left (a +i a \tan \left (d x +c \right )\right )^{2}}+\frac {\frac {3 i B}{4}+\frac {15 A}{8}}{d \tan \left (d x +c \right )^{\frac {3}{2}} \left (a^{3}+i a^{3} \tan \left (d x +c \right )\right )} \]
command
integrate((A+B*tan(d*x+c))/tan(d*x+c)^(5/2)/(a+I*a*tan(d*x+c))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\left (i - 1\right ) \, \sqrt {2} {\left (i \, A + B\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{16 \, a^{3} d} - \frac {\left (i - 1\right ) \, \sqrt {2} {\left (76 \, A + 29 i \, B\right )} \arctan \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{16 \, a^{3} d} - \frac {225 \, A \tan \left (d x + c\right )^{4} + 90 i \, B \tan \left (d x + c\right )^{4} - 598 i \, A \tan \left (d x + c\right )^{3} + 242 \, B \tan \left (d x + c\right )^{3} - 489 \, A \tan \left (d x + c\right )^{2} - 204 i \, B \tan \left (d x + c\right )^{2} + 96 i \, A \tan \left (d x + c\right ) - 48 \, B \tan \left (d x + c\right ) - 16 \, A}{24 \, {\left (-i \, \tan \left (d x + c\right )^{\frac {3}{2}} - \sqrt {\tan \left (d x + c\right )}\right )}^{3} a^{3} d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {B \tan \left (d x + c\right ) + A}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \tan \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]________________________________________________________________________________________