\[ \int \frac {1}{\left (a \sec ^3(x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {14 \sqrt {\frac {\cos \left (x \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {x}{2}\right ), \sqrt {2}\right )}{15 \cos \left (\frac {x}{2}\right ) a \cos \left (x \right )^{\frac {3}{2}} \sqrt {a \left (\sec ^{3}\left (x \right )\right )}}+\frac {14 \sin \left (x \right )}{45 a \sqrt {a \left (\sec ^{3}\left (x \right )\right )}}+\frac {2 \left (\cos ^{2}\left (x \right )\right ) \sin \left (x \right )}{9 a \sqrt {a \left (\sec ^{3}\left (x \right )\right )}} \]
command
integrate(1/(a*sec(x)^3)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (5 \, \cos \left (x\right )^{5} + 7 \, \cos \left (x\right )^{3}\right )} \sqrt {\frac {a}{\cos \left (x\right )^{3}}} \sin \left (x\right ) - 21 i \, \sqrt {2} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) + 21 i \, \sqrt {2} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right )}{45 \, a^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {a \sec \left (x\right )^{3}}}{a^{2} \sec \left (x\right )^{6}}, x\right ) \]