\[ \int \frac {1}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\sqrt {a +b}\, \tanh \left (x \right )}{\sqrt {a +b \left (\tanh ^{2}\left (x \right )\right )}}\right )}{\left (a +b \right )^{\frac {5}{2}}}+\frac {b \left (5 a +2 b \right ) \tanh \left (x \right )}{3 a^{2} \left (a +b \right )^{2} \sqrt {a +b \left (\tanh ^{2}\left (x \right )\right )}}+\frac {b \tanh \left (x \right )}{3 a \left (a +b \right ) \left (a +b \left (\tanh ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}} \]
command
integrate(1/(a+b*tanh(x)^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {a + b} \log \left ({\left | -{\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} {\left (a + b\right )} - \sqrt {a + b} {\left (a - b\right )} \right |}\right )}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {\sqrt {a + b} \log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} + \sqrt {a + b} \right |}\right )}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {\sqrt {a + b} \log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} - \sqrt {a + b} \right |}\right )}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {2 \, {\left ({\left ({\left (\frac {{\left (3 \, a^{6} b^{3} + 16 \, a^{5} b^{4} + 35 \, a^{4} b^{5} + 40 \, a^{3} b^{6} + 25 \, a^{2} b^{7} + 8 \, a b^{8} + b^{9}\right )} e^{\left (2 \, x\right )}}{a^{8} b^{2} + 6 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 20 \, a^{5} b^{5} + 15 \, a^{4} b^{6} + 6 \, a^{3} b^{7} + a^{2} b^{8}} + \frac {3 \, {\left (a^{6} b^{3} + 2 \, a^{5} b^{4} - 3 \, a^{4} b^{5} - 12 \, a^{3} b^{6} - 13 \, a^{2} b^{7} - 6 \, a b^{8} - b^{9}\right )}}{a^{8} b^{2} + 6 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 20 \, a^{5} b^{5} + 15 \, a^{4} b^{6} + 6 \, a^{3} b^{7} + a^{2} b^{8}}\right )} e^{\left (2 \, x\right )} - \frac {3 \, {\left (a^{6} b^{3} + 2 \, a^{5} b^{4} - 3 \, a^{4} b^{5} - 12 \, a^{3} b^{6} - 13 \, a^{2} b^{7} - 6 \, a b^{8} - b^{9}\right )}}{a^{8} b^{2} + 6 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 20 \, a^{5} b^{5} + 15 \, a^{4} b^{6} + 6 \, a^{3} b^{7} + a^{2} b^{8}}\right )} e^{\left (2 \, x\right )} - \frac {3 \, a^{6} b^{3} + 16 \, a^{5} b^{4} + 35 \, a^{4} b^{5} + 40 \, a^{3} b^{6} + 25 \, a^{2} b^{7} + 8 \, a b^{8} + b^{9}}{a^{8} b^{2} + 6 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 20 \, a^{5} b^{5} + 15 \, a^{4} b^{6} + 6 \, a^{3} b^{7} + a^{2} b^{8}}\right )}}{3 \, {\left (a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b\right )}^{\frac {3}{2}}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________