$\displaystyle \int \frac{dx}{(x^2+2x+2)^2}$

$\displaystyle u=x+1$ & $\displaystyle du=dx$

$\displaystyle \int \frac{du}{(u^2+1)^2}$

What should I do at this point? I considered expanding the square, but that seems to make things more messy and it seems like something else should be going on. It also didn't seem to lead me to an answer either. Thanks!