\[ \int \frac {1}{(a+b \cos (e+f x)) (c+d \sec (e+f x))^2} \, dx \]
Optimal antiderivative \[ -\frac {2 d \left (2 a \,c^{2}-a \,d^{2}-b c d \right ) \arctanh \left (\frac {\sqrt {c -d}\, \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {c +d}}\right )}{\left (c -d \right )^{\frac {3}{2}} \left (c +d \right )^{\frac {3}{2}} \left (a c -b d \right )^{2} f}+\frac {d^{2} \sin \left (f x +e \right )}{\left (a c -b d \right ) \left (c^{2}-d^{2}\right ) f \left (d +c \cos \left (f x +e \right )\right )}+\frac {2 a^{2} \arctan \left (\frac {\sqrt {a -b}\, \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {a +b}}\right )}{\left (a c -b d \right )^{2} f \sqrt {a -b}\, \sqrt {a +b}} \]
command
integrate(1/(a+b*cos(f*x+e))/(c+d*sec(f*x+e))^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 \[ \text {Timed out} \]