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