\[ \int \frac {1}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \arctanh \left (\frac {\left (1+i\right ) \sqrt {a}\, \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a +i a \tan \left (d x +c \right )}}\right )}{a^{\frac {3}{2}} d}+\frac {7 \left (\sqrt {\tan }\left (d x +c \right )\right )}{6 a d \sqrt {a +i a \tan \left (d x +c \right )}}+\frac {\sqrt {\tan }\left (d x +c \right )}{3 d \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}} \]
command
integrate(1/tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(3/2),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 \[ \int \frac {1}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sqrt {\tan \left (d x + c\right )}}\,{d x} \]_______________________________________________________