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