\[ \int \left (-1-\cot ^2(x)\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {\cot \left (x \right )}{\sqrt {-\left (\csc ^{2}\left (x \right )\right )}}\right )}{2}+\frac {\cot \left (x \right ) \sqrt {-\left (\csc ^{2}\left (x \right )\right )}}{2} \]
command
integrate((-1-cot(x)^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{4} \, {\left (\frac {2 i \, \cos \left (x\right )}{\cos \left (x\right )^{2} - 1} - i \, \log \left (\cos \left (x\right ) + 1\right ) + i \, \log \left (-\cos \left (x\right ) + 1\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________