Interesting integral related to the Omega Constant/Lambert W Function I ran across an interesting integral and I am wondering if anyone knows where I may find its derivation or proof.  I looked through the site. If it is here and I overlooked it, I am sorry.
$$\displaystyle\frac{1}{\int_{-\infty}^{\infty}\frac{1}{(e^{x}-x)^{2}+{\pi}^{2}}dx}-1=W(1)=\Omega$$
$W(1)=\Omega$ is often referred to as the Omega Constant. Which is the solution to 
$xe^{x}=1$.  Which is $x\approx .567$
Thanks much. 
EDIT:  Sorry, I had the integral written incorrectly. Thanks for the catch.
I had also seen this:
$\displaystyle\int_{-\infty}^{\infty}\frac{dx}{(e^{x}-x)^{2}+{\pi}^{2}}=\frac{1}{1+W(1)}=\frac{1}{1+\Omega}\approx .638$
EDIT:  I do not what is wrong, but I am trying to respond, but can not. All the buttons are unresponsive but this one. I have been trying to leave a greenie and add a comment, but neither will respond. I just wanted you to know this before you thought I was an ingrate.
Thank you. That is an interesting site.

The identity is due to Victor Adamchik, see

http://mathworld.wolfram.com/OmegaConstant.html

You may want to contact Dr Adamchik himself via the e-mail at

http://www.cs.cmu.edu/~adamchik/research.html

because this particular paper doesn't seem to be in the list, as far as I can see.
