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