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