I am interested in a problem that involves computing the expectation of of the CDF $\Phi$ (or equivalently erfc shifted and scaled) for the standard normal distribution, for $x$ normal distributed with mean $\mu$ and variance $\sigma^2$, but shifted and scaled as $E\left[\Phi \left(\frac{|x-a|}{b}\right)\right]$. The bounds for the expressions in https://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn1p1_A1b.pdf don't seem to allow for the usage of any analytical expression, so I instead looked for a decent approximation.
7.1.25 onward in https://personal.math.ubc.ca/%7Ecbm/aands/page_299.htm give rational approximations that are integrable, but seemingly not over a Gaussian measure. The same goes for an approximation of the error function using $\tanh(x)$.
My question is, is such an approximation to a reasonable degree of accuracy even possible? Or is my best bet to just continue using quadrature integration?