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