\[ \int \frac {1}{x^8 \left (1+x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {1}{2 x^{7} \sqrt {x^{4}+1}}-\frac {9 \sqrt {x^{4}+1}}{14 x^{7}}+\frac {15 \sqrt {x^{4}+1}}{14 x^{3}}+\frac {15 \left (x^{2}+1\right ) \sqrt {\frac {\cos \left (4 \arctan \left (x \right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (x \right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{4}+1}{\left (x^{2}+1\right )^{2}}}}{28 \cos \left (2 \arctan \left (x \right )\right ) \sqrt {x^{4}+1}} \]
command
integrate(1/x^8/(x^4+1)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {15 \, \sqrt {i} {\left (i \, x^{11} + i \, x^{7}\right )} {\rm ellipticF}\left (\sqrt {i} x, -1\right ) - {\left (15 \, x^{8} + 6 \, x^{4} - 2\right )} \sqrt {x^{4} + 1}}{14 \, {\left (x^{11} + x^{7}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {x^{4} + 1}}{x^{16} + 2 \, x^{12} + x^{8}}, x\right ) \]