\[ \int \frac {1}{(c+d x)^2 (a+i a \tan (e+f x))^2} \, dx \]
Optimal antiderivative \[ -\frac {1}{4 a^{2} d \left (d x +c \right )}-\frac {i f \cosineIntegral \left (\frac {4 c f}{d}+4 f x \right ) \cos \left (-4 e +\frac {4 c f}{d}\right )}{a^{2} d^{2}}-\frac {i f \cosineIntegral \left (\frac {2 c f}{d}+2 f x \right ) \cos \left (-2 e +\frac {2 c f}{d}\right )}{a^{2} d^{2}}-\frac {\cos \left (2 f x +2 e \right )}{2 a^{2} d \left (d x +c \right )}-\frac {\cos ^{2}\left (2 f x +2 e \right )}{4 a^{2} d \left (d x +c \right )}-\frac {f \cos \left (-2 e +\frac {2 c f}{d}\right ) \sinIntegral \left (\frac {2 c f}{d}+2 f x \right )}{a^{2} d^{2}}-\frac {f \cos \left (-4 e +\frac {4 c f}{d}\right ) \sinIntegral \left (\frac {4 c f}{d}+4 f x \right )}{a^{2} d^{2}}+\frac {f \cosineIntegral \left (\frac {4 c f}{d}+4 f x \right ) \sin \left (-4 e +\frac {4 c f}{d}\right )}{a^{2} d^{2}}-\frac {i f \sinIntegral \left (\frac {4 c f}{d}+4 f x \right ) \sin \left (-4 e +\frac {4 c f}{d}\right )}{a^{2} d^{2}}+\frac {f \cosineIntegral \left (\frac {2 c f}{d}+2 f x \right ) \sin \left (-2 e +\frac {2 c f}{d}\right )}{a^{2} d^{2}}-\frac {i f \sinIntegral \left (\frac {2 c f}{d}+2 f x \right ) \sin \left (-2 e +\frac {2 c f}{d}\right )}{a^{2} d^{2}}+\frac {i \sin \left (2 f x +2 e \right )}{2 a^{2} d \left (d x +c \right )}+\frac {\sin ^{2}\left (2 f x +2 e \right )}{4 a^{2} d \left (d x +c \right )}+\frac {i \sin \left (4 f x +4 e \right )}{4 a^{2} d \left (d x +c \right )} \]
command
integrate(1/(d*x+c)^2/(a+I*a*tan(f*x+e))^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 \[ \text {Timed out} \]_______________________________________________________