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