\[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^n \, dx \]
Optimal antiderivative \[ -\frac {i \left (a +i a \tan \left (d x +c \right )\right )^{1+n}}{a d \left (1+n \right )} \]
command
integrate(sec(d*x+c)^2*(a+I*a*tan(d*x+c))^n,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {i \, \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}\right )^{n + 1}}{a d {\left (n + 1\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right )^{2}\,{d x} \]________________________________________________________________________________________