\[ \int \frac {\sin ^7(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 b^{7}}{19 f \left (b \sec \left (f x +e \right )\right )^{\frac {19}{2}}}-\frac {2 b^{5}}{5 f \left (b \sec \left (f x +e \right )\right )^{\frac {15}{2}}}+\frac {6 b^{3}}{11 f \left (b \sec \left (f x +e \right )\right )^{\frac {11}{2}}}-\frac {2 b}{7 f \left (b \sec \left (f x +e \right )\right )^{\frac {7}{2}}} \]
command
integrate(sin(f*x+e)^7/(b*sec(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (385 \, \sqrt {b \cos \left (f x + e\right )} b^{9} \cos \left (f x + e\right )^{9} - 1463 \, \sqrt {b \cos \left (f x + e\right )} b^{9} \cos \left (f x + e\right )^{7} + 1995 \, \sqrt {b \cos \left (f x + e\right )} b^{9} \cos \left (f x + e\right )^{5} - 1045 \, \sqrt {b \cos \left (f x + e\right )} b^{9} \cos \left (f x + e\right )^{3}\right )}}{7315 \, b^{12} f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________