\[ \int \frac {\cot ^6(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {\cot \left (f x +e \right )}{f \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 \cot \left (f x +e \right ) \left (\csc ^{2}\left (f x +e \right )\right )}{3 f \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}-\frac {\cot \left (f x +e \right ) \left (\csc ^{4}\left (f x +e \right )\right )}{5 f \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}} \]
command
integrate(cot(f*x+e)^6/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {3 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 25 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 150 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )} + \frac {150 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 25 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}}}{480 \, \sqrt {a} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________