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