\[ \int \frac {1}{(5+3 i \sinh (c+d x))^4} \, dx \]
Optimal antiderivative \[ \frac {385 x}{32768}-\frac {385 i \arctan \left (\frac {\cosh \left (d x +c \right )}{3+i \sinh \left (d x +c \right )}\right )}{16384 d}-\frac {i \cosh \left (d x +c \right )}{16 d \left (5+3 i \sinh \left (d x +c \right )\right )^{3}}-\frac {25 i \cosh \left (d x +c \right )}{512 d \left (5+3 i \sinh \left (d x +c \right )\right )^{2}}-\frac {311 i \cosh \left (d x +c \right )}{8192 d \left (5+3 i \sinh \left (d x +c \right )\right )} \]
command
integrate(1/(5+3*I*sinh(d*x+c))**4,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {- 10395 i e^{5 c} e^{5 d x} - 86625 e^{4 c} e^{4 d x} + 239470 i e^{3 c} e^{3 d x} + 218466 e^{2 c} e^{2 d x} - 73575 i e^{c} e^{d x} - 8397}{331776 d e^{6 c} e^{6 d x} - 3317760 i d e^{5 c} e^{5 d x} - 12054528 d e^{4 c} e^{4 d x} + 18923520 i d e^{3 c} e^{3 d x} + 12054528 d e^{2 c} e^{2 d x} - 3317760 i d e^{c} e^{d x} - 331776 d} + \frac {- \frac {385 \log {\left (e^{d x} - 3 i e^{- c} \right )}}{32768} + \frac {385 \log {\left (e^{d x} - \frac {i e^{- c}}{3} \right )}}{32768}}{d} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Exception raised: NotInvertible} \]________________________________________________________________________________________