I am trying to solve this integration numericallly:
$$I=\int_0^{\infty}\frac{1}{t}\frac{e^{\frac{x^2t^2}{1+2t^2\sigma^2}}}{\sqrt{1+2t^2\sigma^2}}\frac{e^{\frac{y^2t^2}{1+2t^2\sigma^2}}}{\sqrt{1+2t^2\sigma^2}}\, dt$$
but the lower value of the interval makes the Integration unstable in its neighbour.
I tried using the quad integerator from mpmath using tanh-sinh quadrature, For $\sigma =10^{-4},\ x=y=0,\ I=76.109010\pm0.01$ but i am not sure of the answer at all.
I am wondering is there any numerical integration method or integrator devoted to such cases?
When $x = y = 0$ the integral becomes $$ I = \int_0^{+\infty} \dfrac{dt}{t(1+2t^2 \sigma^2)} = \int_0^1 \dfrac{dt}{t(1+2t^2 \sigma^2)} + \int_1^{+\infty} \dfrac{dt}{t(1+2t^2 \sigma^2)}, $$
where the first term is divergent... The same goes for other values of $x,y$: when $t$ is close to zero, the integrand behaves as $\frac 1t$.