\[ \int \left (a \text {csch}^3(x)\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {10 a \cosh \left (x \right ) \sqrt {a \mathrm {csch}\left (x \right )^{3}}}{21}-\frac {2 a \coth \left (x \right ) \mathrm {csch}\left (x \right ) \sqrt {a \mathrm {csch}\left (x \right )^{3}}}{7}+\frac {10 i a \sqrt {\frac {1}{2}+\frac {i \sinh \left (x \right )}{2}}\, \EllipticF \left (\cos \left (\frac {\pi }{4}+\frac {i x}{2}\right ), \sqrt {2}\right ) \sinh \left (x \right ) \sqrt {a \mathrm {csch}\left (x \right )^{3}}\, \sqrt {i \sinh \left (x \right )}}{21 \sin \left (\frac {\pi }{4}+\frac {i x}{2}\right )} \]
command
integrate((a*csch(x)^3)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (5 \, \sqrt {2} {\left (a \cosh \left (x\right )^{6} + 6 \, a \cosh \left (x\right ) \sinh \left (x\right )^{5} + a \sinh \left (x\right )^{6} - 3 \, a \cosh \left (x\right )^{4} + 3 \, {\left (5 \, a \cosh \left (x\right )^{2} - a\right )} \sinh \left (x\right )^{4} + 4 \, {\left (5 \, a \cosh \left (x\right )^{3} - 3 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )^{2} + 3 \, {\left (5 \, a \cosh \left (x\right )^{4} - 6 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{2} + 6 \, {\left (a \cosh \left (x\right )^{5} - 2 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) - a\right )} \sqrt {a} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \sqrt {2} {\left (5 \, a \cosh \left (x\right )^{6} + 30 \, a \cosh \left (x\right ) \sinh \left (x\right )^{5} + 5 \, a \sinh \left (x\right )^{6} - 17 \, a \cosh \left (x\right )^{4} + {\left (75 \, a \cosh \left (x\right )^{2} - 17 \, a\right )} \sinh \left (x\right )^{4} + 4 \, {\left (25 \, a \cosh \left (x\right )^{3} - 17 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 17 \, a \cosh \left (x\right )^{2} + {\left (75 \, a \cosh \left (x\right )^{4} - 102 \, a \cosh \left (x\right )^{2} - 17 \, a\right )} \sinh \left (x\right )^{2} + 2 \, {\left (15 \, a \cosh \left (x\right )^{5} - 34 \, a \cosh \left (x\right )^{3} - 17 \, a \cosh \left (x\right )\right )} \sinh \left (x\right ) + 5 \, a\right )} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}}\right )}}{21 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} - 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \, {\left (\cosh \left (x\right )^{5} - 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {a \operatorname {csch}\left (x\right )^{3}} a \operatorname {csch}\left (x\right )^{3}, x\right ) \]