\[ \int \frac {\sin ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (5 a^{2}+10 a b +b^{2}\right ) \cos \left (f x +e \right )}{5 \left (a -b \right )^{3} f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}+\frac {2 \left (5 a -b \right ) \left (\cos ^{3}\left (f x +e \right )\right )}{15 \left (a -b \right )^{2} f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}-\frac {\cos ^{5}\left (f x +e \right )}{5 \left (a -b \right ) f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}-\frac {4 b \left (5 a^{2}+10 a b +b^{2}\right ) \sec \left (f x +e \right )}{15 \left (a -b \right )^{4} f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}-\frac {8 b \left (5 a^{2}+10 a b +b^{2}\right ) \sec \left (f x +e \right )}{15 \left (a -b \right )^{5} f \sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}} \]
command
integrate(sin(f*x+e)^5/(a+b*tan(f*x+e)^2)^(5/2),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 \[ \text {Timed out} \]_______________________________________________________