\[ \int \frac {1}{\sqrt {a+b \cos ^2(x)}} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \EllipticF \left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) \sqrt {1+\frac {b \left (\cos ^{2}\left (x \right )\right )}{a}}}{\sin \left (x \right ) \sqrt {a +b \left (\cos ^{2}\left (x \right )\right )}} \]
command
integrate(1/(a+b*cos(x)^2)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (-2 i \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} - 2 i \, a - i \, b\right )} \sqrt {b} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} {\rm ellipticF}\left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} {\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right )}, \frac {8 \, a^{2} + 8 \, a b + b^{2} + 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}\right ) + {\left (2 i \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 i \, a + i \, b\right )} \sqrt {b} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} {\rm ellipticF}\left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} {\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right )}, \frac {8 \, a^{2} + 8 \, a b + b^{2} + 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}\right )}{b^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {1}{\sqrt {b \cos \left (x\right )^{2} + a}}, x\right ) \]