I'm trying to evaluate the following integral:
$$ \int_{-\infty}^{\infty} e^{-\frac{1}{2}x^2} \left[ \text{erf } \left(a x + b\right) \right]^{k} dx.$$
And I found a formula for a similar integral in this book (formulas 3, 4 in section 2.3.3. (p. 55)), however it's not exactly what I'm looking for.
Is there a closed-form expression for that integral? I would appreciate any suggestions and/or hints.
To give some more background, I started from this integral: $$ \int_{-\infty}^{\infty} \phi(x) \left[ \Phi \left(c x + d\right) \right]^{k} dx$$ where $\phi(\cdot), \Phi(\cdot)$ are the PDF and CDF of the standard normal distribution. I developed it to a sum of integrals, based on the relation between $\Phi(x)$ and $\text{erf}(x)$, where each one of the integrals in the sum is of the form of the integral above.