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