\[ \int \frac {\sec (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\sin \left (x \right )\right )}{a}-\frac {2 c \arctan \left (\frac {\sqrt {b -2 c -\sqrt {-4 a c +b^{2}}}\, \tan \left (\frac {x}{2}\right )}{\sqrt {b +2 c -\sqrt {-4 a c +b^{2}}}}\right ) \left (1+\frac {b}{\sqrt {-4 a c +b^{2}}}\right )}{a \sqrt {b -2 c -\sqrt {-4 a c +b^{2}}}\, \sqrt {b +2 c -\sqrt {-4 a c +b^{2}}}}-\frac {2 c \arctan \left (\frac {\sqrt {b -2 c +\sqrt {-4 a c +b^{2}}}\, \tan \left (\frac {x}{2}\right )}{\sqrt {b +2 c +\sqrt {-4 a c +b^{2}}}}\right ) \left (1-\frac {b}{\sqrt {-4 a c +b^{2}}}\right )}{a \sqrt {b -2 c +\sqrt {-4 a c +b^{2}}}\, \sqrt {b +2 c +\sqrt {-4 a c +b^{2}}}} \]
command
integrate(sec(x)/(a+b*cos(x)+c*cos(x)^2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]