"Asymptotic behavior of int_inf^-inf (x^2 exp( (x−a)^2))/(1+Aa^(−1) exp(−x^2/(1+a^2))) dx as a->0 Let a>0 and A>0, I am looking for the decay rate of the integralint_inf^-inf (x^2 exp( (x−a)^2))/(1+Aa^(−1) exp(−x^2/(1+a^2))) as a->0. Do we have some literature discussing this kind of issue?

Let a>0 and A>0, I am looking for the decay rate of the integral

There is no closed form answer for the integral. I have tried on Matlab that it should converge to zero much faster than power growth. I think the growth should be exponential types. Do we have some literature discussing this kind of issue? Thanks!
I have successfully obtained the growth rate of

be expanding the denominator in power series.
But I do not know to deal with the integral in [−M,M].
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Mario Monroe
Let us assume $a\ll 1$. Approximating the function (or simply plotting it), we can see that the maximum of the integrand is at $|x|\ge 1$ (in fact it is $|x|=1$ for $A\ll a$ and becomes larger when increasing A).
Because if this we can approximate $1+{a}^{2}\approx 1$ and $\left(x-a{\right)}^{2}\approx {x}^{2}$ in the exponents of the integrand to obtain the leading order behavior.
Denoting the integral by I(a,A), we obtain
$I\left(a,A\right)\simeq {I}_{0}\left(a,A\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{x}^{2}\mathrm{exp}\left(-{x}^{2}\right)}{1+A{a}^{-1}\mathrm{exp}\left(-{x}^{2}\right)}dx=-\frac{\sqrt{\pi }a}{2A}{\mathrm{Li}}_{3/2}\left(-A/a\right)\phantom{\rule{thinmathspace}{0ex}}.$
Here, Lis is the polylogarithm function. In fact, I0 is an excellent approximation to I for $a\lesssim 0.1$
The polylogarithm function has a known asymptotic expansions in terms of logx. In particular, we have $x\gg 1$
${\mathrm{Li}}_{s}\left(-x\right)\sim -\frac{{\mathrm{log}}^{s}\left(x\right)}{\mathrm{\Gamma }\left(s\right)}\left(1+O\left({\mathrm{log}}^{-2}\left(x\right)\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$
As a result, we obtain
$I\left(a,A\right)\simeq {I}_{0}\left(a,A\right)\sim \frac{2a}{3A}{\mathrm{log}}^{3/2}\left(A/a\right)\phantom{\rule{thinmathspace}{0ex}}.$
Note that this is not a rigorous mathematical answer. What remains is to show that $I-{I}_{0}$ is small for $a\to 0$. Numerically, it seems to hold that $I-{I}_{0}=o\left({a}^{2}\right)$