\[ \int \cot ^3(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (\csc ^{2}\left (f x +e \right )\right )}{2 a f}+\frac {3 \arctanh \left (\frac {\sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}{\sqrt {a}}\right ) \sqrt {a}}{2 f}-\frac {3 \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}{2 f} \]
command
integrate(cot(f*x+e)^3*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6 \, \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right ) \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right ) + \frac {3 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 14 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}\right )} \sqrt {a}}{8 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________