Integration by parts for $I_n=\int \frac 1 {(x^2-1) ^n} d x$ and relation with $I_1$
I suspect that you are looking for a solution at an elementary level; such a thing does not exist and your integral does not have a simple solution in closed form. The result involves the hypergeometric function that you most probably have not studied.
For your curiosity, Mathematica 7 produces the following answer: $x \Big( \frac {1-x^2} {-1 + x^2} \Big) ^n {}_2 F _1 (\frac 1 2, n, \frac 3 2, x^2)$, where ${}_2 F _1$ is the hypergeometric function.