I have faced difficulties to compute the following integral, which comes when I studied some diffraction/scattering problem :
$$\int_0^l\int_0 ^{2\pi} \frac {\exp -i\big[kr\sin\phi\big] \times \exp i \big[k\sqrt{r^2+2r d \sin \phi+d^2}\big]}{\sqrt{r^2 + 2rd \sin \phi+d^2}} \,r\,dr\,d\phi~.$$
Where, $r$ and $d$ have the same dimension of length and $r$ varies from $0$ to $l$ and $\phi$ varies from $0$ to $2\pi$. Further, $d$ is comparable to $r$.
I went through the Bessel function for this type of problem but I could not do this due to the presence of the factor under square root.
However, I don't know whether it will be helpful if the integral will be written in cartesian form such as:
$$\int_{z=-l}^l\int_{y=-\sqrt{l^2-z^2}} ^{\sqrt{l^2-z^2}} \frac {\exp -i\big[kz\big] \times \exp i \big[k\sqrt{y^2+z^2+2z d+d^2}\big]}{\sqrt{y^2+z^2+ 2zd+d^2}} \,dy\,dz~.$$
While going through the Hankel function I have got something like this: $$\int_{-\infty}^{\infty}\frac{\exp ik[|\vec{r}-\vec{r}'|]}{|\vec{r}-\vec{r}'|}\,dy'=i\pi H_0 (k|\vec{r}-\vec{r}'|)$$
Will this be helpful for this case?
Would you kindly suggest me how to solve this kind of integration analytically (even by approximation) or any relevant sources or books from where I can get some help.