\[ \int \frac {\cot ^6(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\cot \left (f x +e \right ) \left (\csc ^{2}\left (f x +e \right )\right )}{3 a f \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}-\frac {\cot \left (f x +e \right ) \left (\csc ^{4}\left (f x +e \right )\right )}{5 a f \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}} \]
command
integrate(cot(f*x+e)^6/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {30 \, \sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 5 \, \sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, \sqrt {a}}{a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}} - \frac {3 \, a^{\frac {17}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 5 \, a^{\frac {17}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 30 \, a^{\frac {17}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{10} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )}}{480 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________