\[ \int \frac {\sin ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (15 a^{2}+40 a b +24 b^{2}\right ) \cos \left (f x +e \right )}{15 a^{3} f \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}+\frac {2 \left (5 a +3 b \right ) \left (\cos ^{3}\left (f x +e \right )\right )}{15 a^{2} f \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}-\frac {\cos ^{5}\left (f x +e \right )}{5 a f \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}-\frac {2 b \left (15 a^{2}+40 a b +24 b^{2}\right ) \sec \left (f x +e \right )}{15 a^{4} f \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}} \]
command
integrate(sin(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________