\[ \int \cot (c+d x) (a+i a \tan (c+d x))^4 \, dx \]
Optimal antiderivative \[ 8 i a^{4} x +\frac {7 a^{4} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {a^{4} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {\left (a^{2}+i a^{2} \tan \left (d x +c \right )\right )^{2}}{2 d}-\frac {3 \left (a^{4}+i a^{4} \tan \left (d x +c \right )\right )}{d} \]
command
integrate(cot(d*x+c)*(a+I*a*tan(d*x+c))**4,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {a^{4} \left (\log {\left (e^{2 i d x} - e^{- 2 i c} \right )} + 7 \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}\right )}{d} + \frac {10 a^{4} e^{2 i c} e^{2 i d x} + 8 a^{4}}{d e^{4 i c} e^{4 i d x} + 2 d e^{2 i c} e^{2 i d x} + d} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Exception raised: NotInvertible} \]________________________________________________________________________________________