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