\[ \int \frac {1}{\left (a \cos ^3(x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {10 \left (\cos ^{\frac {3}{2}}\left (x \right )\right ) \sqrt {\frac {\cos \left (x \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {x}{2}\right ), \sqrt {2}\right )}{21 \cos \left (\frac {x}{2}\right ) a \sqrt {a \left (\cos ^{3}\left (x \right )\right )}}+\frac {10 \sin \left (x \right )}{21 a \sqrt {a \left (\cos ^{3}\left (x \right )\right )}}+\frac {2 \sec \left (x \right ) \tan \left (x \right )}{7 a \sqrt {a \left (\cos ^{3}\left (x \right )\right )}} \]
command
integrate(1/(a*cos(x)^3)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {5 i \, \sqrt {2} \sqrt {a} \cos \left (x\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 5 i \, \sqrt {2} \sqrt {a} \cos \left (x\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, \sqrt {a \cos \left (x\right )^{3}} {\left (5 \, \cos \left (x\right )^{2} + 3\right )} \sin \left (x\right )}{21 \, a^{2} \cos \left (x\right )^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {a \cos \left (x\right )^{3}}}{a^{2} \cos \left (x\right )^{6}}, x\right ) \]