definite integrals - Help solving integration: $I=int_{-infty}^{infty}phileft(xright)Phileft(a/sqrt{b+cmathrm{e}^{frac{x-mu}{sigma}}}right)dx$

My work has arrived at needing to solve the integral below for $a,b,c,\sigma>0$

$$I=\int_{-\infty}^{\infty}\phi\left(x\right)\Phi\left(\frac{a}{\sqrt{b+c\mathrm{e}^{(x-\mu)/\sigma}}}\right)dx$$

I have tried substitution: $u=\frac{a}{\sqrt{b+c\mathrm{e}^{\frac{x-\mu}{\sigma}}}}$ and then a couple of rounds of integration by parts. However it does not seem to be getting closer to finding an integration that can be done directly (i.e. without needing a further by parts integration).

Is there another route to solving this, or does it not have a closed-form solution?

Thanks