Definite Integral from 0 to 1 of x^4 (1 - x)^4 over (1 + x^2)

Proof
We then have:

which should take us far enough.

Then it is a matter of evaluating the individual integrals.

It remains to gather up the terms.

First the terms in $x^5$ and $x^7$, as there is some important cancelling out:

Thus we continue: