I wish to prove that for any $a\in\mathbb R$ $$ \lim_{A\to\infty}\int_{-A}^A \frac{e^{-x^2/2-(a+x)^2/2}(a+2x)}{e^{-x^2/2}+e^{-(a+x)^2/2}}dx = 0. $$
I have verified the above equality for multiple $a\in\mathbb R$ values using Monte Carlo, but don't know how to prove it rigorously. Any help is greatly appreciated.
Edit: Mathematica cannot evaluate the indefinite integral via symbolic computation :(
With a shift of the variable
$$ I=\lim_{A\to\infty}\int_{-A+a/2}^{A+a/2} \frac{e^{-(x-a/2)^2/2-(a/2+x)^2/2}\,x}{e^{-(x-a/2)^2/2}+e^{-(a/2+x)^2/2}}dx = 0. $$
and the integrand is odd.