\[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}-\frac {\arctanh \left (\frac {\sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}{\sqrt {a -b}}\right )}{\left (a -b \right )^{\frac {5}{2}}}+\frac {b}{3 a \left (a -b \right ) \left (a +b \left (\cot ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}}+\frac {\left (2 a -b \right ) b}{a^{2} \left (a -b \right )^{2} \sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}} \]
command
integrate(tan(x)/(a+b*cot(x)^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (2 \, \sqrt {a - b} a^{3} \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) - 6 \, \sqrt {a - b} a^{2} b \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) + 6 \, \sqrt {a - b} a b^{2} \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) - 2 \, \sqrt {a - b} b^{3} \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) + \sqrt {-a^{2} + a b} \sqrt {a - b} a^{2} \log \left (b\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (\sqrt {-a^{2} + a b} a^{5} - 3 \, \sqrt {-a^{2} + a b} a^{4} b + 3 \, \sqrt {-a^{2} + a b} a^{3} b^{2} - \sqrt {-a^{2} + a b} a^{2} b^{3}\right )}} + \frac {\frac {3 \, \sqrt {a - b} \log \left ({\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {2 \, {\left (\frac {{\left (7 \, a^{5} b^{2} - 17 \, a^{4} b^{3} + 13 \, a^{3} b^{4} - 3 \, a^{2} b^{5}\right )} \sin \left (x\right )^{2}}{a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}} + \frac {3 \, {\left (2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} + a^{2} b^{5}\right )}}{a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}}\right )} \sin \left (x\right )}{{\left (a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b\right )}^{\frac {3}{2}}} + \frac {6 \, \sqrt {a - b} \arctan \left (\frac {{\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - 2 \, a + b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} a^{2}}}{6 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________