\[ \int \frac {1}{(c+d x)^2 (a+i a \tan (e+f x))^3} \, dx \]
Optimal antiderivative \[ -\frac {1}{8 a^{3} d \left (d x +c \right )}-\frac {3 i f \sinIntegral \left (\frac {6 c f}{d}+6 f x \right ) \sin \left (-6 e +\frac {6 c f}{d}\right )}{4 a^{3} d^{2}}-\frac {3 i f \sinIntegral \left (\frac {4 c f}{d}+4 f x \right ) \sin \left (-4 e +\frac {4 c f}{d}\right )}{2 a^{3} d^{2}}-\frac {3 i f \cosineIntegral \left (\frac {4 c f}{d}+4 f x \right ) \cos \left (-4 e +\frac {4 c f}{d}\right )}{2 a^{3} d^{2}}-\frac {9 \cos \left (2 f x +2 e \right )}{32 a^{3} d \left (d x +c \right )}-\frac {3 \left (\cos ^{2}\left (2 f x +2 e \right )\right )}{8 a^{3} d \left (d x +c \right )}-\frac {\cos ^{3}\left (2 f x +2 e \right )}{8 a^{3} d \left (d x +c \right )}-\frac {3 \cos \left (6 f x +6 e \right )}{32 a^{3} d \left (d x +c \right )}-\frac {3 f \cos \left (-2 e +\frac {2 c f}{d}\right ) \sinIntegral \left (\frac {2 c f}{d}+2 f x \right )}{4 a^{3} d^{2}}-\frac {3 f \cos \left (-4 e +\frac {4 c f}{d}\right ) \sinIntegral \left (\frac {4 c f}{d}+4 f x \right )}{2 a^{3} d^{2}}-\frac {3 f \cos \left (-6 e +\frac {6 c f}{d}\right ) \sinIntegral \left (\frac {6 c f}{d}+6 f x \right )}{4 a^{3} d^{2}}+\frac {3 f \cosineIntegral \left (\frac {6 c f}{d}+6 f x \right ) \sin \left (-6 e +\frac {6 c f}{d}\right )}{4 a^{3} d^{2}}-\frac {3 i f \sinIntegral \left (\frac {2 c f}{d}+2 f x \right ) \sin \left (-2 e +\frac {2 c f}{d}\right )}{4 a^{3} d^{2}}+\frac {3 f \cosineIntegral \left (\frac {4 c f}{d}+4 f x \right ) \sin \left (-4 e +\frac {4 c f}{d}\right )}{2 a^{3} d^{2}}-\frac {3 i f \cosineIntegral \left (\frac {6 c f}{d}+6 f x \right ) \cos \left (-6 e +\frac {6 c f}{d}\right )}{4 a^{3} d^{2}}+\frac {3 f \cosineIntegral \left (\frac {2 c f}{d}+2 f x \right ) \sin \left (-2 e +\frac {2 c f}{d}\right )}{4 a^{3} d^{2}}+\frac {3 i \sin \left (4 f x +4 e \right )}{8 a^{3} d \left (d x +c \right )}+\frac {3 i \sin \left (6 f x +6 e \right )}{32 a^{3} d \left (d x +c \right )}+\frac {3 \left (\sin ^{2}\left (2 f x +2 e \right )\right )}{8 a^{3} d \left (d x +c \right )}+\frac {15 i \sin \left (2 f x +2 e \right )}{32 a^{3} d \left (d x +c \right )}-\frac {i \left (\sin ^{3}\left (2 f x +2 e \right )\right )}{8 a^{3} d \left (d x +c \right )}-\frac {3 i f \cosineIntegral \left (\frac {2 c f}{d}+2 f x \right ) \cos \left (-2 e +\frac {2 c f}{d}\right )}{4 a^{3} d^{2}} \]
command
integrate(1/(d*x+c)^2/(a+I*a*tan(f*x+e))^3,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 \[ \text {Timed out} \]_______________________________________________________