\[ \int \frac {\tan ^2(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {\cot \left (x \right ) \sqrt {a -b}}{\sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}\right )}{\left (a -b \right )^{\frac {3}{2}}}+\frac {b \tan \left (x \right )}{a \left (a -b \right ) \sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}+\frac {\left (a -2 b \right ) \sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}\, \tan \left (x \right )}{a^{2} \left (a -b \right )} \]
command
integrate(tan(x)^2/(a+b*cot(x)^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (a^{3} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + a^{2} \sqrt {-a + b} \sqrt {b} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - a^{2} b \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + 2 \, a^{3} - 4 \, a^{2} b + 2 \, a \sqrt {-a + b} b^{\frac {3}{2}} - 2 \, \sqrt {-a + b} b^{\frac {5}{2}} + 2 \, b^{3}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (a^{4} \sqrt {-a + b} - a^{4} \sqrt {b} - 2 \, a^{3} \sqrt {-a + b} b + 2 \, a^{3} b^{\frac {3}{2}} + a^{2} \sqrt {-a + b} b^{2} - a^{2} b^{\frac {5}{2}}\right )}} + \frac {\frac {2 \, \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a} b^{2} \cos \left (x\right )}{{\left (a^{3} - a^{2} b\right )} {\left (a \cos \left (x\right )^{2} - b \cos \left (x\right )^{2} - a\right )}} - \frac {\log \left ({\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2}\right )}{{\left (a - b\right )} \sqrt {-a + b}} - \frac {4 \, \sqrt {-a + b}}{{\left ({\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2} - a\right )} a}}{2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________