$\displaystyle \int \frac{2x^3}{1-x^4} = 2 \int \frac{x^3}{1-x^4}$
There is a crafty shortcut in this example.
Set $\displaystyle u = 1-x^4$. When we differentiate with respect to x we get $\displaystyle du = 4x^3 \, dx$
$\displaystyle 2\int \frac{x^3}{u} \cdot \frac{du}{4x^3}$
This simplifies to $\displaystyle \frac{1}{2}\int \frac{du}{u}
$ which is a standard integral