Integration invlolving gaussian kernel

Question

I have been trying to solve this integration . I have tried via changing the coordinate system to cartesian coordinates system but it gets complicated. The integration is $$\int\limits_{0}^{2\pi}\int\limits_{-\infty}^{\infty} r^2e^{\frac{-(r-r_0)^2}{w^2}}(cos(k_xrcos\theta)+cos(k_yrsin\theta))drd\theta$$ where $r_0,w,k_x,k_y$ are constants