# defined integrals – Integration of the 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 looking to evaluate the next 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}
or $$c> 0$$, $$a$$, $$b$$ are constants; and $$x_1> 0, x_2> 0$$.

I could simplify the integration above in 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}
or $$t_1> 0$$, $$t_2> 0$$ there are some constants, $$I_0 (t_2x)$$ is the modified bessel function of the first type. I wonder if it is possible to further simplify the situation.

Since now I am using the following to approximate the integration above
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 precise enough. Any help is greatly appreciated! Thank you so much!