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