# Integration of modified bessel function $\int_{x_1-\sigma}^{x_1+\sigma}x\exp(-t_1x^2)\cdot I_0(t_2x)\ dx$

In fact I was originally trying to evaluate the following integral \begin{align} \int_{x_1-\sigma}^{x_1+\sigma}\int_0^{2\pi}x\exp\left(-c\left(x\sin\phi-a\right)^2-c\left(x\cos\phi-b\right)^2\right)\ d\phi\ dx\,, \end{align} where $$c>0$$, $$a$$, $$b$$ are some constants; and $$x_1>0,x_2>0$$.

I was able to simplify the above integration into the following form \begin{align} \int_{x_1-\sigma}^{x_1+\sigma}x\exp(-t_1x^2)\cdot I_0(t_2x)\ dx\,, \end{align} where $$t_1>0$$, $$t_2>0$$ are some constants, $$I_0(t_2x)$$ is the modified bessel function of the first kind. I wonder if there is a way to simplify this further.

Since right now I am use the following to approximate the above integration \begin{align} 2\sigma\cdot x_1\exp(-t_1x_1^2)\cdot I_0(t_2x_1) \end{align} But it is still not accurate enough. Any help is greatly appreciated! Thanks so much!