\[ \int \frac {1}{(i \sinh (c+d x))^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {6 i \sqrt {\frac {1}{2}+\frac {\sin \left (i d x +i c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {1}{2} i c +\frac {1}{4} \pi +\frac {1}{2} i d x \right ), \sqrt {2}\right )}{5 \sin \left (\frac {1}{2} i c +\frac {1}{4} \pi +\frac {1}{2} i d x \right ) d}+\frac {2 i \cosh \left (d x +c \right )}{5 d \left (i \sinh \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {6 i \cosh \left (d x +c \right )}{5 d \sqrt {i \sinh \left (d x +c \right )}} \]
command
integrate(1/(I*sinh(d*x+c))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, \sqrt {\frac {1}{2}} {\left (3 \, e^{\left (6 \, d x + 6 \, c\right )} - 8 \, e^{\left (4 \, d x + 4 \, c\right )} + e^{\left (2 \, d x + 2 \, c\right )}\right )} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} + 3 \, {\left (\sqrt {2} \sqrt {i} e^{\left (6 \, d x + 6 \, c\right )} - 3 \, \sqrt {2} \sqrt {i} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, \sqrt {2} \sqrt {i} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {2} \sqrt {i}\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, e^{\left (d x + c\right )}\right )\right )\right )}}{5 \, {\left (d e^{\left (6 \, d x + 6 \, c\right )} - 3 \, d e^{\left (4 \, d x + 4 \, c\right )} + 3 \, d e^{\left (2 \, d x + 2 \, c\right )} - d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {4 \, \sqrt {\frac {1}{2}} {\left (3 \, e^{\left (6 \, d x + 6 \, c\right )} - 8 \, e^{\left (4 \, d x + 4 \, c\right )} + e^{\left (2 \, d x + 2 \, c\right )}\right )} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} + 5 \, {\left (d e^{\left (6 \, d x + 6 \, c\right )} - 3 \, d e^{\left (4 \, d x + 4 \, c\right )} + 3 \, d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} {\rm integral}\left (-\frac {6 \, \sqrt {\frac {1}{2}} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{5 \, {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )}}, x\right )}{5 \, {\left (d e^{\left (6 \, d x + 6 \, c\right )} - 3 \, d e^{\left (4 \, d x + 4 \, c\right )} + 3 \, d e^{\left (2 \, d x + 2 \, c\right )} - d\right )}} \]