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!