\[ \int \frac {\sec ^3(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\sin \left (x \right )\right )}{2 a}+\frac {\left (-a c +b^{2}\right ) \arctanh \left (\sin \left (x \right )\right )}{a^{3}}-\frac {2 c \arctan \left (\frac {\sqrt {b -2 c -\sqrt {-4 a c +b^{2}}}\, \tan \left (\frac {x}{2}\right )}{\sqrt {b +2 c -\sqrt {-4 a c +b^{2}}}}\right ) \left (b^{3}-3 a b c +\left (-a c +b^{2}\right ) \sqrt {-4 a c +b^{2}}\right )}{a^{3} \sqrt {-4 a c +b^{2}}\, \sqrt {b -2 c -\sqrt {-4 a c +b^{2}}}\, \sqrt {b +2 c -\sqrt {-4 a c +b^{2}}}}+\frac {2 c \arctan \left (\frac {\sqrt {b -2 c +\sqrt {-4 a c +b^{2}}}\, \tan \left (\frac {x}{2}\right )}{\sqrt {b +2 c +\sqrt {-4 a c +b^{2}}}}\right ) \left (b^{3}-3 a b c -\left (-a c +b^{2}\right ) \sqrt {-4 a c +b^{2}}\right )}{a^{3} \sqrt {-4 a c +b^{2}}\, \sqrt {b -2 c +\sqrt {-4 a c +b^{2}}}\, \sqrt {b +2 c +\sqrt {-4 a c +b^{2}}}}-\frac {b \tan \left (x \right )}{a^{2}}+\frac {\sec \left (x \right ) \tan \left (x \right )}{2 a} \]
command
integrate(sec(x)^3/(a+b*cos(x)+c*cos(x)^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________