Well, here goes:
$$\int\frac{x^2-1}{(x+1)^3\sqrt{1+3x^2+x^4}}\,dx$$
I wrote down a solution ... I am mentioning the basic steps.
a) Apply the sub $x=1/t$ and get the integral into a much a better form.
b) In the nominator there appears to be the derivative of $t+1/t$ which is substituted and then the integral is reduced down to this $\displaystyle \int \frac{dx}{(1+x)\sqrt{1+x^2}}$.
However, well what I don't like to this problem is that from step a) to step b) by applying a suitable sub I get to add the two integrals since they have the same denominator. Is this acceptable?
Any other ideas on how to attack this integral which seems pretty tough to me.