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