\[ \int \frac {\sin ^7(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx \]
Optimal antiderivative \[ \frac {2 b^{7}}{15 f \left (b \sec \left (f x +e \right )\right )^{\frac {15}{2}}}-\frac {6 b^{5}}{11 f \left (b \sec \left (f x +e \right )\right )^{\frac {11}{2}}}+\frac {6 b^{3}}{7 f \left (b \sec \left (f x +e \right )\right )^{\frac {7}{2}}}-\frac {2 b}{3 f \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}} \]
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
\[ \frac {2 \, {\left (77 \, \sqrt {b \cos \left (f x + e\right )} b^{7} \cos \left (f x + e\right )^{7} - 315 \, \sqrt {b \cos \left (f x + e\right )} b^{7} \cos \left (f x + e\right )^{5} + 495 \, \sqrt {b \cos \left (f x + e\right )} b^{7} \cos \left (f x + e\right )^{3} - 385 \, \sqrt {b \cos \left (f x + e\right )} b^{7} \cos \left (f x + e\right )\right )}}{1155 \, b^{8} f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________