I am looking for a quick-to-compute approximation of the CDF of $X+Y$, where $X \sim N(0,\sigma_1^2)$ and $Y$ is a truncated gaussian, more specifically, a gaussian with mean $0$, standard deviation of $\sigma_2$, truncated on one side, at $a$ ($a < 0$). I have found a closed-form density function for the distribution of $X+Y$ here, but not a CDF.
Unfortunately, using the density function means that I have to numerically integrate it repeatedly, which is computationally very intensive, if done many times. Is there an approximation to the CDF of $X+Y$, which I could use instead?
I'd appreciate any suggestions or pointers on how to solve this problem.