Try this:
Factorize (x^4+1) like this:
Code:
(x^4+1) = (x^2+ax+b)(x^2+cx+d)
So you'll have:
Code:
1/(x^4+1) = 1/(x^2+ax+b)(x^2+cx+d)
Then, try to write
Code:
1/(x^2+ax+b)(x^2+cx+d)
like this:
Code:
P(x)/(x^2+ax+b) + Q(x)/(x^2+cx+d),
where P(x) and Q(x) are polynomials of first degree or lower.
Then you'll be able to integrate each part of that sum separately. You only need a simple substitution for each one to do it.