\[ \int \frac {\sec ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx \]
Optimal antiderivative \[ -\frac {b \arctanh \left (\sin \left (x \right )\right )}{a^{2}}+\frac {2 b c \arctan \left (\frac {\sqrt {b -2 c -\sqrt {-4 a c +b^{2}}}\, \tan \left (\frac {x}{2}\right )}{\sqrt {b +2 c -\sqrt {-4 a c +b^{2}}}}\right ) \left (1+\frac {-2 a c +b^{2}}{b \sqrt {-4 a c +b^{2}}}\right )}{a^{2} \sqrt {b -2 c -\sqrt {-4 a c +b^{2}}}\, \sqrt {b +2 c -\sqrt {-4 a c +b^{2}}}}+\frac {2 b c \arctan \left (\frac {\sqrt {b -2 c +\sqrt {-4 a c +b^{2}}}\, \tan \left (\frac {x}{2}\right )}{\sqrt {b +2 c +\sqrt {-4 a c +b^{2}}}}\right ) \left (1+\frac {2 a c -b^{2}}{b \sqrt {-4 a c +b^{2}}}\right )}{a^{2} \sqrt {b -2 c +\sqrt {-4 a c +b^{2}}}\, \sqrt {b +2 c +\sqrt {-4 a c +b^{2}}}}+\frac {\tan \left (x \right )}{a} \]
command
integrate(sec(x)^2/(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 \[ \text {Timed out} \]_______________________________________________________