I am looking for a closed-form version of this formula:
\begin{align*}\text{P} &= \int_{-\infty} ^{\infty} \left[1 - \left(\int_{-\infty} ^{t}\right.\right.\ldots\\ &\left.\left.\ldots\int_{-\infty} ^{t} \dfrac{1}{\sqrt{2\pi}\sigma_{2}} e^{-\frac{(x-\mu_{2})^{2}}{2\sigma_{2}^{2}}} \dfrac{1}{\sqrt{2\pi}\sigma_{3}} e^{-\frac{(x-\mu_{3})^{2}}{2\sigma_{3}^{2}}}\right.\right.\ldots\\ &\left.\left.\ldots \dfrac{1}{\sqrt{2\pi}\sigma_{M}} e^{-\frac{(x-\mu_{M})^{2}}{2\sigma_{M}^{2}}}~\text{d}x\ldots \text{d}x\right)\right] \dfrac{1}{\sqrt{2\pi}\sigma_{1}} e^{-\frac{(t-\mu_{1})^{2}}{2\sigma_{1}^{2}}}~\text{d}t \end{align*}
where the $\mu_i$'s and $\sigma_i^2$'s are the means and variances of the Gaussian distributions.