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

$${\int}_{M}^{\mathrm{\infty}}+{\int}_{-\mathrm{\infty}}^{-M}\frac{{x}^{2}\mathrm{exp}(-(x-a{)}^{2})}{1+A{a}^{-1}\mathrm{exp}(-{x}^{2}/(1+{a}^{2}))}dx\phantom{\rule{1em}{0ex}}\text{as}a\to 0$$

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

$${\int}_{M}^{\mathrm{\infty}}+{\int}_{-\mathrm{\infty}}^{-M}\frac{{x}^{2}\mathrm{exp}(-(x-a{)}^{2})}{1+A{a}^{-1}\mathrm{exp}(-{x}^{2}/(1+{a}^{2}))}dx\phantom{\rule{1em}{0ex}}\text{as}a\to 0$$

be expanding the denominator in power series.

But I do not know to deal with the integral in [−M,M].

$${\int}_{M}^{\mathrm{\infty}}+{\int}_{-\mathrm{\infty}}^{-M}\frac{{x}^{2}\mathrm{exp}(-(x-a{)}^{2})}{1+A{a}^{-1}\mathrm{exp}(-{x}^{2}/(1+{a}^{2}))}dx\phantom{\rule{1em}{0ex}}\text{as}a\to 0$$

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

$${\int}_{M}^{\mathrm{\infty}}+{\int}_{-\mathrm{\infty}}^{-M}\frac{{x}^{2}\mathrm{exp}(-(x-a{)}^{2})}{1+A{a}^{-1}\mathrm{exp}(-{x}^{2}/(1+{a}^{2}))}dx\phantom{\rule{1em}{0ex}}\text{as}a\to 0$$

be expanding the denominator in power series.

But I do not know to deal with the integral in [−M,M].