\[ \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx \]
Optimal antiderivative \[ \frac {2 b^{7}}{13 f \left (b \sec \left (f x +e \right )\right )^{\frac {13}{2}}}-\frac {2 b^{5}}{3 f \left (b \sec \left (f x +e \right )\right )^{\frac {9}{2}}}+\frac {6 b^{3}}{5 f \left (b \sec \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {2 b}{f \sqrt {b \sec \left (f x +e \right )}} \]
command
integrate(sin(f*x+e)^7*(b*sec(f*x+e))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (15 \, \sqrt {b \cos \left (f x + e\right )} b^{6} \cos \left (f x + e\right )^{6} - 65 \, \sqrt {b \cos \left (f x + e\right )} b^{6} \cos \left (f x + e\right )^{4} + 117 \, \sqrt {b \cos \left (f x + e\right )} b^{6} \cos \left (f x + e\right )^{2} - 195 \, \sqrt {b \cos \left (f x + e\right )} b^{6}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{195 \, b^{6} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________