\[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {10 \cosh \left (x \right )}{21 a \sqrt {a \left (\sinh ^{3}\left (x \right )\right )}}-\frac {2 \coth \left (x \right ) \mathrm {csch}\left (x \right )}{7 a \sqrt {a \left (\sinh ^{3}\left (x \right )\right )}}+\frac {10 i \sqrt {\frac {1}{2}+\frac {i \sinh \left (x \right )}{2}}\, \EllipticF \left (\cos \left (\frac {\pi }{4}+\frac {i x}{2}\right ), \sqrt {2}\right ) \sinh \left (x \right ) \sqrt {i \sinh \left (x \right )}}{21 \sin \left (\frac {\pi }{4}+\frac {i x}{2}\right ) a \sqrt {a \left (\sinh ^{3}\left (x \right )\right )}} \]
command
integrate(1/(a*sinh(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 \, {\left (\sqrt {2} \cosh \left (x\right )^{8} + 8 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sqrt {2} \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right )^{6} - 4 \, \sqrt {2} \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{3} - 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \sqrt {2} \cosh \left (x\right )^{4} - 30 \, \sqrt {2} \cosh \left (x\right )^{2} + 3 \, \sqrt {2}\right )} \sinh \left (x\right )^{4} + 6 \, \sqrt {2} \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{5} - 10 \, \sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{6} - 15 \, \sqrt {2} \cosh \left (x\right )^{4} + 9 \, \sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right )^{2} - 4 \, \sqrt {2} \cosh \left (x\right )^{2} + 8 \, {\left (\sqrt {2} \cosh \left (x\right )^{7} - 3 \, \sqrt {2} \cosh \left (x\right )^{5} + 3 \, \sqrt {2} \cosh \left (x\right )^{3} - \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + 2 \, {\left (5 \, \cosh \left (x\right )^{7} + 35 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + 5 \, \sinh \left (x\right )^{7} + {\left (105 \, \cosh \left (x\right )^{2} - 17\right )} \sinh \left (x\right )^{5} - 17 \, \cosh \left (x\right )^{5} + 5 \, {\left (35 \, \cosh \left (x\right )^{3} - 17 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + {\left (175 \, \cosh \left (x\right )^{4} - 170 \, \cosh \left (x\right )^{2} - 17\right )} \sinh \left (x\right )^{3} - 17 \, \cosh \left (x\right )^{3} + {\left (105 \, \cosh \left (x\right )^{5} - 170 \, \cosh \left (x\right )^{3} - 51 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (35 \, \cosh \left (x\right )^{6} - 85 \, \cosh \left (x\right )^{4} - 51 \, \cosh \left (x\right )^{2} + 5\right )} \sinh \left (x\right ) + 5 \, \cosh \left (x\right )\right )} \sqrt {a \sinh \left (x\right )}\right )}}{21 \, {\left (a^{2} \cosh \left (x\right )^{8} + 8 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right )^{7} + a^{2} \sinh \left (x\right )^{8} - 4 \, a^{2} \cosh \left (x\right )^{6} + 4 \, {\left (7 \, a^{2} \cosh \left (x\right )^{2} - a^{2}\right )} \sinh \left (x\right )^{6} + 6 \, a^{2} \cosh \left (x\right )^{4} + 8 \, {\left (7 \, a^{2} \cosh \left (x\right )^{3} - 3 \, a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, a^{2} \cosh \left (x\right )^{4} - 30 \, a^{2} \cosh \left (x\right )^{2} + 3 \, a^{2}\right )} \sinh \left (x\right )^{4} - 4 \, a^{2} \cosh \left (x\right )^{2} + 8 \, {\left (7 \, a^{2} \cosh \left (x\right )^{5} - 10 \, a^{2} \cosh \left (x\right )^{3} + 3 \, a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, a^{2} \cosh \left (x\right )^{6} - 15 \, a^{2} \cosh \left (x\right )^{4} + 9 \, a^{2} \cosh \left (x\right )^{2} - a^{2}\right )} \sinh \left (x\right )^{2} + a^{2} + 8 \, {\left (a^{2} \cosh \left (x\right )^{7} - 3 \, a^{2} \cosh \left (x\right )^{5} + 3 \, a^{2} \cosh \left (x\right )^{3} - a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {a \sinh \left (x\right )^{3}}}{a^{2} \sinh \left (x\right )^{6}}, x\right ) \]