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