\[ \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (-\frac {1}{8}+\frac {i}{8}\right ) \arctanh \left (\frac {\left (1+i\right ) \sqrt {a}\, \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a +i a \tan \left (d x +c \right )}}\right )}{a^{\frac {5}{2}} d}+\frac {707 i \sqrt {a +i a \tan \left (d x +c \right )}}{60 a^{3} d \sqrt {\tan \left (d x +c \right )}}+\frac {89}{20 a^{2} d \sqrt {a +i a \tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {361 \sqrt {a +i a \tan \left (d x +c \right )}}{60 a^{3} d \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {1}{5 d \tan \left (d x +c \right )^{\frac {3}{2}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {7}{10 a d \tan \left (d x +c \right )^{\frac {3}{2}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}} \]
command
integrate(1/tan(d*x+c)^(5/2)/(a+I*a*tan(d*x+c))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \int \frac {1}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \tan \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]_______________________________________________________