\[ \int \sin ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx \]
Optimal antiderivative \[ -\frac {\left (a^{2}-6 a b +6 b^{2}\right ) \cos \left (f x +e \right )}{f}+\frac {2 \left (a -2 b \right ) \left (a -b \right ) \left (\cos ^{3}\left (f x +e \right )\right )}{3 f}-\frac {\left (a -b \right )^{2} \left (\cos ^{5}\left (f x +e \right )\right )}{5 f}+\frac {2 \left (a -2 b \right ) b \sec \left (f x +e \right )}{f}+\frac {b^{2} \left (\sec ^{3}\left (f x +e \right )\right )}{3 f} \]
command
integrate(sin(f*x+e)^5*(a+b*tan(f*x+e)^2)^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 \[ \text {Timed out} \]_______________________________________________________