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