\[ \int \left (a \sec ^3(x)\right )^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {154 a^{2} \left (\cos ^{\frac {3}{2}}\left (x \right )\right ) \sqrt {\frac {\cos \left (x \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {x}{2}\right ), \sqrt {2}\right ) \sqrt {a \left (\sec ^{3}\left (x \right )\right )}}{195 \cos \left (\frac {x}{2}\right )}+\frac {154 a^{2} \cos \left (x \right ) \sin \left (x \right ) \sqrt {a \left (\sec ^{3}\left (x \right )\right )}}{195}+\frac {154 a^{2} \sqrt {a \left (\sec ^{3}\left (x \right )\right )}\, \tan \left (x \right )}{585}+\frac {22 a^{2} \left (\sec ^{2}\left (x \right )\right ) \sqrt {a \left (\sec ^{3}\left (x \right )\right )}\, \tan \left (x \right )}{117}+\frac {2 a^{2} \left (\sec ^{4}\left (x \right )\right ) \sqrt {a \left (\sec ^{3}\left (x \right )\right )}\, \tan \left (x \right )}{13} \]
command
integrate((a*sec(x)^3)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {231 i \, \sqrt {2} a^{\frac {5}{2}} \cos \left (x\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) - 231 i \, \sqrt {2} a^{\frac {5}{2}} \cos \left (x\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) + 2 \, {\left (231 \, a^{2} \cos \left (x\right )^{6} + 77 \, a^{2} \cos \left (x\right )^{4} + 55 \, a^{2} \cos \left (x\right )^{2} + 45 \, a^{2}\right )} \sqrt {\frac {a}{\cos \left (x\right )^{3}}} \sin \left (x\right )}{585 \, \cos \left (x\right )^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {a \sec \left (x\right )^{3}} a^{2} \sec \left (x\right )^{6}, x\right ) \]