Best Answer:
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.).

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