I am stuck with the following integral: $$\int_{x \ge 1} \frac{18 \exp \left(-\frac12 \left(\phi^{-1}[1 - \frac{1}{x^3}]\phi^{-1}[1 - \frac{1}{(z-x)^3}] \right) \right)}{\sqrt{3}x^3 (z-x)^4} dx $$ where $z$ is a constant and $\phi^{-1}$ is the inverse of the standard normal cdf which can also be expressed in terms of the error function: $$\phi^{-1}[p] = \sqrt{2} \operatorname{erf}^{-1}(2p-1)\qquad\text{for}\quad p \in (0,1)$$
Can this be solved analytically or numerically?
