\[ \int \left (a \text {sech}^3(x)\right )^{5/2} \, dx \]
Optimal antiderivative \[ \frac {154 i a^{2} \left (\cosh ^{\frac {3}{2}}\left (x \right )\right ) \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticE \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\right ) \sqrt {a \mathrm {sech}\left (x \right )^{3}}}{195 \cosh \left (\frac {x}{2}\right )}+\frac {154 a^{2} \cosh \left (x \right ) \sinh \left (x \right ) \sqrt {a \mathrm {sech}\left (x \right )^{3}}}{195}+\frac {154 a^{2} \sqrt {a \mathrm {sech}\left (x \right )^{3}}\, \tanh \left (x \right )}{585}+\frac {22 a^{2} \mathrm {sech}\left (x \right )^{2} \sqrt {a \mathrm {sech}\left (x \right )^{3}}\, \tanh \left (x \right )}{117}+\frac {2 a^{2} \mathrm {sech}\left (x \right )^{4} \sqrt {a \mathrm {sech}\left (x \right )^{3}}\, \tanh \left (x \right )}{13} \]
command
integrate((a*sech(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 \[ {\rm integral}\left (\sqrt {a \operatorname {sech}\left (x\right )^{3}} a^{2} \operatorname {sech}\left (x\right )^{6}, x\right ) \]