\[ \int \left (a \sec ^3(x)\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {10 a \left (\cos ^{\frac {3}{2}}\left (x \right )\right ) \sqrt {\frac {\cos \left (x \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {x}{2}\right ), \sqrt {2}\right ) \sqrt {a \left (\sec ^{3}\left (x \right )\right )}}{21 \cos \left (\frac {x}{2}\right )}+\frac {10 a \sin \left (x \right ) \sqrt {a \left (\sec ^{3}\left (x \right )\right )}}{21}+\frac {2 a \sec \left (x \right ) \sqrt {a \left (\sec ^{3}\left (x \right )\right )}\, \tan \left (x \right )}{7} \]
command
integrate((a*sec(x)^3)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {5 i \, \sqrt {2} a^{\frac {3}{2}} \cos \left (x\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 5 i \, \sqrt {2} a^{\frac {3}{2}} \cos \left (x\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (5 \, a \cos \left (x\right )^{2} + 3 \, a\right )} \sqrt {\frac {a}{\cos \left (x\right )^{3}}} \sin \left (x\right )}{21 \, \cos \left (x\right )^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {a \sec \left (x\right )^{3}} a \sec \left (x\right )^{3}, x\right ) \]