Echoing Bramble: don't ever write an integral without a differential, it doesn't mean anything.
If I assume you meant the integral of
(x-1) dx / [(x^4 - 1)^2],
probably your best approach will be "partial fractions."
1/ [(x+1)(x-1)^2 (x^2+1)^2] =
= A/(x+1) + B/(x-1) + C/(x-1)^2 +
+ (Ex+F)/(x^2 + 1) + (Gx+H)/ (x^2+1)^2
Then multiply all terms by the denominator of the LHS,
and equate coefficients for each separate power of x.
I admit this is UGLY, but I'm sure it's tractable, since the Wolfram Integrator gives a closed-form answer in elementary functions which are the obvious ones you'd expect from the above formulation (i.e., ln(x-1), ln(x+1), arctan(x), etc.).