\[ \int \frac {1}{(b \text {csch}(c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \cosh \left (d x +c \right )}{5 b d \left (b \,\mathrm {csch}\left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {6 i \sqrt {\frac {1}{2}+\frac {\sin \left (i d x +i c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {1}{2} i c +\frac {1}{4} \pi +\frac {1}{2} i d x \right ), \sqrt {2}\right )}{5 \sin \left (\frac {1}{2} i c +\frac {1}{4} \pi +\frac {1}{2} i d x \right ) b^{2} d \sqrt {b \,\mathrm {csch}\left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}} \]
command
integrate(1/(b*csch(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {24 \, \sqrt {2} {\left (\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )\right ) + \sqrt {2} {\left (\cosh \left (d x + c\right )^{6} + 6 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + \sinh \left (d x + c\right )^{6} + {\left (15 \, \cosh \left (d x + c\right )^{2} + 11\right )} \sinh \left (d x + c\right )^{4} + 11 \, \cosh \left (d x + c\right )^{4} + 4 \, {\left (5 \, \cosh \left (d x + c\right )^{3} + 11 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + {\left (15 \, \cosh \left (d x + c\right )^{4} + 66 \, \cosh \left (d x + c\right )^{2} - 13\right )} \sinh \left (d x + c\right )^{2} - 13 \, \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, \cosh \left (d x + c\right )^{5} + 22 \, \cosh \left (d x + c\right )^{3} - 13 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 1\right )} \sqrt {\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1}}}{20 \, {\left (b^{3} d \cosh \left (d x + c\right )^{3} + 3 \, b^{3} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, b^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b^{3} d \sinh \left (d x + c\right )^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b \operatorname {csch}\left (d x + c\right )}}{b^{3} \operatorname {csch}\left (d x + c\right )^{3}}, x\right ) \]