\[ \int \frac {\tan ^5(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx \]
Optimal antiderivative \[ \frac {a^{2}}{5 f \left (a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}-\frac {2 a}{3 f \left (a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}+\frac {1}{f \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}} \]
command
integrate(tan(f*x+e)^5/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {16 \, {\left (10 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 5 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}}{15 \, {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{5} \sqrt {a} f \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________