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