\[ \int \frac {1}{\sqrt {-1-\cos ^2(x)}} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \EllipticF \left (\cos \left (x \right ), i\right ) \sqrt {1+\cos ^{2}\left (x \right )}}{\sin \left (x \right ) \sqrt {-1-\left (\cos ^{2}\left (x \right )\right )}} \]
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
\[ 2 \, {\left (2 \, \sqrt {2} + 3\right )} \sqrt {2 \, \sqrt {2} - 3} {\rm ellipticF}\left (\sqrt {2 \, \sqrt {2} - 3} e^{\left (i \, x\right )}, 12 \, \sqrt {2} + 17\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {2}{\sqrt {e^{\left (4 i \, x\right )} + 6 \, e^{\left (2 i \, x\right )} + 1}}, x\right ) \]