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