\[ \int \frac {1}{(i \sinh (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 i \sqrt {\frac {1}{2}+\frac {\sin \left (i d x +i c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {1}{2} i c +\frac {1}{4} \pi +\frac {1}{2} i d x \right ), \sqrt {2}\right )}{\sin \left (\frac {1}{2} i c +\frac {1}{4} \pi +\frac {1}{2} i d x \right ) d}+\frac {2 i \cosh \left (d x +c \right )}{d \sqrt {i \sinh \left (d x +c \right )}} \]
command
integrate(1/(I*sinh(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )} + {\left (\sqrt {2} \sqrt {i} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {2} \sqrt {i}\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, e^{\left (d x + c\right )}\right )\right )\right )}}{d e^{\left (2 \, d x + 2 \, c\right )} - d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {4 \, \sqrt {\frac {1}{2}} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )} + {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} {\rm integral}\left (-\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{d e^{\left (2 \, d x + 2 \, c\right )} - d}, x\right )}{d e^{\left (2 \, d x + 2 \, c\right )} - d} \]