\[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {154 \coth \left (x \right )}{585 a^{2} \sqrt {a \left (\sinh ^{3}\left (x \right )\right )}}+\frac {22 \coth \left (x \right ) \mathrm {csch}\left (x \right )^{2}}{117 a^{2} \sqrt {a \left (\sinh ^{3}\left (x \right )\right )}}-\frac {2 \coth \left (x \right ) \mathrm {csch}\left (x \right )^{4}}{13 a^{2} \sqrt {a \left (\sinh ^{3}\left (x \right )\right )}}+\frac {154 \cosh \left (x \right ) \sinh \left (x \right )}{195 a^{2} \sqrt {a \left (\sinh ^{3}\left (x \right )\right )}}-\frac {154 i \sqrt {\frac {1}{2}+\frac {i \sinh \left (x \right )}{2}}\, \EllipticE \left (\cos \left (\frac {\pi }{4}+\frac {i x}{2}\right ), \sqrt {2}\right ) \left (\sinh ^{2}\left (x \right )\right )}{195 \sin \left (\frac {\pi }{4}+\frac {i x}{2}\right ) a^{2} \sqrt {i \sinh \left (x \right )}\, \sqrt {a \left (\sinh ^{3}\left (x \right )\right )}} \]
command
integrate(1/(a*sinh(x)^3)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {\sqrt {a \sinh \left (x\right )^{3}}}{a^{3} \sinh \left (x\right )^{9}}, x\right ) \]