\[ \int \frac {\cot ^3(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}{\sqrt {a}}\right )}{2 f \sqrt {a}}-\frac {\left (\csc ^{2}\left (f x +e \right )\right ) \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}{2 a f} \]
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
\[ -\frac {\frac {\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )} - \frac {2 \, \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )} + \frac {2 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}}{8 \, \sqrt {a} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________