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