Is it possible to integrate this function without using spherical coordinates? I am looking to solve the integral

${\int\int\int{\rm e}^{-\sqrt {{x}^{2}+{y}^{2}+{z}^{2}}}} dxdydz$

My boundary conditions are the boundaries of a cube (not nicely aligned with the origin), so I'd like to avoid muddying the waters with spherical coordinates.

Any advice?

If you're interested, I'm essentially trying to solve the following problem.

I'm holding a flashlight underwater in a cubic swimming pool, what's the total amount of light being absorbed by the pool, assuming that some light escapes the pool? Treat the flashlight as an isotropic point source of light.

Attenuation of a beam in 1D is given by a simple exponential decay.
$\int \exp{(- \sqrt{x^2 + y^2 + z^2} ) } dx$

$ u = x^2 + y^2 + z^2$

$ du = 2x dx$

$ dx = du/2x$

$\int (\exp{(- \sqrt{u} ) }/u)  du =2 Ei(-\sqrt{u}) + C$ with $Ei(x)$ being the Exponential Integral
$\int \exp{(- \sqrt{x^2 + y^2 + z^2} ) } dx$

$ u = x^2 + y^2 + z^2$

$ du = 2x dx$

$ dx = du/2x$

$\int (\exp{(- \sqrt{u} ) }/u)  du =2 Ei(-\sqrt{u}) + C$ with $Ei(x)$ being the Exponential Integral