Integration of expression with root of second order polynomial in denominator

Hi all,

I'm trying to help a friend out and a bit rusty on my calc. Trying to find indefinite integral:

$\displaystyle \int \frac{1}{(x-1)\sqrt{4x^2-8x+3}}dx$

I've run it through Wolfram Alpha, but the solution seems overly complex to me...it seems like I should be able to get there a little easier than that, but I could be wrong.

My out of practice intuition says that there might be some way to solve this thing by doing a u substitution for $\displaystyle 4x^2-8x+3$, such that $\displaystyle \frac{du}{dx}=8x-8=8(x-1)$, and relating that back somehow to the x-1 term...but for the life of me I'm just not getting there.

Anyway, if I really need to go the route of completing the square and making a second substitution as Wolfram suggests, that's fine...but I can't shake the feeling that a more elegant solution exists.

Thanks in advance for any help!

Andre