I know that the following result holds
Given $a,b,\lambda\in \mathbb{R}$, $\Phi$ the cumulative distribution function of a standard normal random variable, and $f_X$ the probability density function of a normal random variable $X\sim {N}(\mu_X,\sigma_X^2)$ \begin{equation} \int_{-\infty}^\infty \exp\left(\lambda x\right)\Phi\left(ax + b\right) f_X(x) d x = \exp\left( \mu_X \lambda+ \frac{\sigma_X^2\lambda^2}{2}\right) \Phi\left( \frac{a\left(\mu_X+\sigma_X^2\lambda\right)+b}{\sqrt{1+a^2\sigma_X^2}}\right) \end{equation}
However, I was wondering if, by any chance, something can be said about a more general integral, i.e. \begin{equation} \int_{-\infty}^\infty \exp\left(\lambda x\right)\Phi\left(g(x)\right) f_X(x) d x \end{equation} with $g$ a mononote function.
Any suggestions would be very much appreciated. Thank you