\[ \int \frac {\sin ^5(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {\left (15 a^{2}+20 a b +8 b^{2}\right ) \cos \left (f x +e \right ) \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}{15 a^{3} f}+\frac {2 \left (5 a +2 b \right ) \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}{15 a^{2} f}-\frac {\left (\cos ^{5}\left (f x +e \right )\right ) \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}{5 a f} \]
command
integrate(sin(f*x+e)^5/(a+b*sec(f*x+e)^2)^(1/2),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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________