\[ \int \frac {\cos ^4(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx \]
Optimal antiderivative \[ \frac {x}{2 c}+\frac {\left (-a c +b^{2}\right ) x}{c^{3}}-\frac {b \sin \left (x \right )}{c^{2}}+\frac {\cos \left (x \right ) \sin \left (x \right )}{2 c}-\frac {2 \arctan \left (\frac {\sqrt {b -2 c -\sqrt {-4 a c +b^{2}}}\, \tan \left (\frac {x}{2}\right )}{\sqrt {b +2 c -\sqrt {-4 a c +b^{2}}}}\right ) \left (b^{3}-2 a b c +\frac {-2 a^{2} c^{2}+4 a \,b^{2} c -b^{4}}{\sqrt {-4 a c +b^{2}}}\right )}{c^{3} \sqrt {b -2 c -\sqrt {-4 a c +b^{2}}}\, \sqrt {b +2 c -\sqrt {-4 a c +b^{2}}}}-\frac {2 \arctan \left (\frac {\sqrt {b -2 c +\sqrt {-4 a c +b^{2}}}\, \tan \left (\frac {x}{2}\right )}{\sqrt {b +2 c +\sqrt {-4 a c +b^{2}}}}\right ) \left (b^{3}-2 a b c +\frac {2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}}{\sqrt {-4 a c +b^{2}}}\right )}{c^{3} \sqrt {b -2 c +\sqrt {-4 a c +b^{2}}}\, \sqrt {b +2 c +\sqrt {-4 a c +b^{2}}}} \]
command
integrate(cos(x)^4/(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 \[ \text {Timed out} \]_______________________________________________________