\[ \int \cot ^4(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx \]
Optimal antiderivative \[ \frac {2 \csc \left (f x +e \right ) \sec \left (f x +e \right ) \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}{f}-\frac {\left (\csc ^{3}\left (f x +e \right )\right ) \sec \left (f x +e \right ) \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}{3 f}+\frac {\sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )}{f} \]
command
integrate(cot(f*x+e)^4*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left ({\left (\frac {1}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right ) - 24 \, {\left (\frac {1}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right ) - \frac {48 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )}{\frac {1}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}\right )} \sqrt {a}}{24 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________