\[ \int \frac {1}{\sqrt {1+\cos ^2(x)}} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \EllipticF \left (\cos \left (x \right ), i\right )}{\sin \left (x \right )} \]
command
integrate(1/(1+cos(x)^2)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \sqrt {2 \, \sqrt {2} - 3} {\left (2 i \, \sqrt {2} + 3 i\right )} {\rm ellipticF}\left (\sqrt {2 \, \sqrt {2} - 3} {\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right )}, 12 \, \sqrt {2} + 17\right ) + \sqrt {2 \, \sqrt {2} - 3} {\left (-2 i \, \sqrt {2} - 3 i\right )} {\rm ellipticF}\left (\sqrt {2 \, \sqrt {2} - 3} {\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right )}, 12 \, \sqrt {2} + 17\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {1}{\sqrt {\cos \left (x\right )^{2} + 1}}, x\right ) \]