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