I have to evaluate the expression $$f(|\vec{c}|) = \int_0^\infty \int_0^{2\pi} (z(\vec{a})+z(\vec{a}+\vec{c})) \frac{(1-\cos(\theta_{\vec{a}+\vec{c}} - \theta_{\vec{a}}))}{|\vec{a}|+|\vec{a}+\vec{c}|} |a| \,\mathrm{d}\phi \,\mathrm{d}|a| $$ over a function $z(\vec{r}) = z(r)$ (meaning the value of $z$ only depends on the length of the vector $\vec{r}$). The integration is over the modulus of $ \vec a$ and the direction of $\vec{c}$. $\vec{a}$, and $\vec{c}$ are vectors in a 2-dimensional cartesian space, $\phi$ is the angle between $\vec{c}$ and $\vec{a}$ and therefore $\cos(\theta_{\vec{a}+\vec{c}} - \theta_{\vec{a}}) = ({a + c·\cos \phi})/|\vec{a}+\vec{c}|$.
In the practice, I have to do this on a discretized space, and the values of $z(r)$ vary on an enclosed domain ($r \leq r_{max}$), and $z(r > r_{max}) = 1$. Conveniently, for each $\vec{c}$, $r_{max} \rightarrow |\vec{a}|+|\vec{c}|$ and the calculation in $|\vec{a}|\in [0, r_{max}]$ is computed numericaly. So, in order to accelerate the calculations I am searching the analytical result of $$ f^+(c) \int_{r_{max}}^\infty \int_0^{2\pi} 2 \frac{(1-\cos(\theta_{\vec{a}+\vec{c}} - \theta_{\vec{a}}))}{|\vec{a}|+|\vec{a}+\vec{c}|} |a| \,\mathrm{d}\phi \,\mathrm{d}|a| .$$ I have been struggling with the problem for a lot of time. Please!! Any hint to tackle this is welcome! Thanks in advance.