integrate x/(1-x^2 + squareroot(1-x^2).
I suppose you mean $\displaystyle \int \frac x{1 - x^2 + \sqrt{1 - x^2}}~dx$ ?
A substitution of $\displaystyle u^2 = 1 - x^2$ yields the integral $\displaystyle - \int \frac 1{u + 1}~du$ (i leave it to you to verify this).
from there, you should be able to handle it