I am working on a mathematical problem involving the evaluation of a double integral, and I am seeking an analytical solution or techniques to solve it. The integral I'm dealing with is as follows:

​$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} [L(C1+0.5W+0.5Z) - L(C2+0.5W+0.5Z) ] \cdot p(W=w | X=x) p(Z=z | X=x) dw dz$$

Here, $C1$ and $C2$ are constants, $L(\cdot)$ represents the logistic function $L(u)=\frac{1}{1+\exp(-u)}$, and $W$ and $Z$ follow Gaussian distributions with mean $0.5x$ and variances $\sigma_W^2 \ll 1$ and $\sigma_Z^2 \ll 1$ respectively.

I am looking for an approach to analytically solve this integral or simplify it further, if possible. Any insights, techniques, or relevant references would be greatly appreciated.

Thank you for your time and assistance!