You can make the substitution $\displaystyle u=x, v=y, w=z-1$ and notice that this does not change the Jacobian. That leaves us with the unit sphere to work with: $\displaystyle u^2+v^2+w^2\leq 1$
To solve the integral you need to start integrating with respect to $\displaystyle \theta \in [0, \pi]$ once you switch to spherical coordinates.