can't really see an elegant way to do this one. but integrating by parts should work (you will need to do several substitutions afterwards, ok, like 2, "several" makes it sound like a lot), so yeah, this will be messy.
let $\displaystyle \frac x{\sqrt{1 - x^2}}$ be the functoin you integrate, i think it is convention to call this your $\displaystyle dv$ and let $\displaystyle u$ (the function you differentiate) be $\displaystyle \ln x$
i'm sure one of the best integrator nominees (
) can find a better way to do this