One thing you could try is to set \(x=\sqrt y\). This makes \(y=x^2\), so that \(\mathrm dy=2x\,\mathrm dx\), and \(y^2=x^4\). So the integral is
\(\displaystyle\int\frac x{x^4+1}\,\mathrm dx=\frac12\int\frac{\mathrm dy}{y^2+1}=\frac12\arctan y+C=\frac12\arctan(x^2)+C\)
A "trickier" way to do it is to write
\(x^4+1=(x^2+\sqrt2x+1)(x^2-\sqrt2x+1)\)
so you could decompose the integrand into partial fractions. But that's more work than needed.
