This question has been asked (and answered) before but there were errors in the original question, so I am reposting the corrected version. The following integration comes up in geophysics modeling. I need simplification of the expression.
$$\iiint \frac{\rho(z_3)}{\sqrt{(x_1-z_1)^2+z_2^2+z_3^2}\sqrt{z_1^2+z_2^2+z_3^2}}\delta \left( t-\frac1c\left(\sqrt{(x_1-z_1)^2+z_2^2+z_3^2}+\sqrt{z_1^2+z_2^2+z_3^2}\right) \right)dz_1dz_2dz_3$$
where $\rho$ is some random function. $c$ is a positive constant that corresponds to the speed of acoustic wave propagation in the subsurface. The constraints are that $x_1 \geq 0$, and $z_1$, $z_2$ and $z_3$ are also non-negative. The original question had $\rho(z_1)$ instead of $\rho(z_3)$. The change of variables used by one user worked very nicely to simplify the expression, but it does not work anymore with the change in the independent variable from first to third coordinate of $\mathbf{z}$.