\[ \int \frac {\tan ^2(x)}{\sqrt {a+b \cot ^2(x)}} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {\cot \left (x \right ) \sqrt {a -b}}{\sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}\right )}{\sqrt {a -b}}+\frac {\sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}\, \tan \left (x \right )}{a} \]
command
integrate(tan(x)^2/(a+b*cot(x)^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (a \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + \sqrt {-a + b} \sqrt {b} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - b \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + 2 \, a - 2 \, b\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (a \sqrt {-a + b} - a \sqrt {b} - \sqrt {-a + b} b + b^{\frac {3}{2}}\right )}} - \frac {\frac {\log \left ({\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2}\right )}{\sqrt {-a + b}} + \frac {4 \, \sqrt {-a + b}}{{\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2} - a}}{2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________