\[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {154 i \left (\cosh ^{\frac {3}{2}}\left (x \right )\right ) \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticE \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\right )}{195 \cosh \left (\frac {x}{2}\right ) a^{2} \sqrt {a \left (\cosh ^{3}\left (x \right )\right )}}+\frac {154 \cosh \left (x \right ) \sinh \left (x \right )}{195 a^{2} \sqrt {a \left (\cosh ^{3}\left (x \right )\right )}}+\frac {154 \tanh \left (x \right )}{585 a^{2} \sqrt {a \left (\cosh ^{3}\left (x \right )\right )}}+\frac {22 \mathrm {sech}\left (x \right )^{2} \tanh \left (x \right )}{117 a^{2} \sqrt {a \left (\cosh ^{3}\left (x \right )\right )}}+\frac {2 \mathrm {sech}\left (x \right )^{4} \tanh \left (x \right )}{13 a^{2} \sqrt {a \left (\cosh ^{3}\left (x \right )\right )}} \]
command
integrate(1/(a*cosh(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 \[ {\rm integral}\left (\frac {\sqrt {a \cosh \left (x\right )^{3}}}{a^{3} \cosh \left (x\right )^{9}}, x\right ) \]