\[ \int \sqrt {b \sec (e+f x)} \sin ^5(e+f x) \, dx \]
Optimal antiderivative \[ -\frac {2 b^{5}}{9 f \left (b \sec \left (f x +e \right )\right )^{\frac {9}{2}}}+\frac {4 b^{3}}{5 f \left (b \sec \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {2 b}{f \sqrt {b \sec \left (f x +e \right )}} \]
command
integrate(sin(f*x+e)^5*(b*sec(f*x+e))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, {\left (5 \, \sqrt {b \cos \left (f x + e\right )} b^{4} \cos \left (f x + e\right )^{4} - 18 \, \sqrt {b \cos \left (f x + e\right )} b^{4} \cos \left (f x + e\right )^{2} + 45 \, \sqrt {b \cos \left (f x + e\right )} b^{4}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{45 \, b^{4} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________