Expectation: sigmoid times mixture of Gaussians

Question

Let $Y$ be a random variable where \begin{align} Y&=Z\theta^*+W \end{align} Here $Z$ is a Rademacher random variable, $W\sim \mathcal{N}(0,\sigma^2 I_p)$, and $\theta\in \mathbb{R}^p$ is a fixed parameter. Marginalizing over $Z$ gives the density \begin{align} g_{\theta^*}(y)&=\frac{1}{2}\phi(y-\theta^*;\theta;\sigma^2 I_p)+\frac{1}{2}\phi(y+\theta^*;\theta;\sigma^2 I_p) \end{align} where $\phi(x;\mu,\Sigma)$ is the density of multivariate $\mathcal{N}(\mu,\Sigma)$. I want to compute $\mathbb{E}\frac{1}{1+\exp(-\frac{2\theta^* TY}{\sigma^2})}Y$. I believe from reading A.3 of https://arxiv.org/pdf/1611.00519.pdf that this is equal to $\frac{\theta^*}{2}$. I do \begin{align} \mathbb{E}\frac{1}{1+\exp(-\frac{2\theta^* TY}{\sigma^2})}Y &=\frac{1}{2}\int y\frac{1}{1+\exp(-\frac{2\theta^*T y}{\sigma^2})}\frac{1}{(2\pi\sigma)^{p/2}}\exp(-\frac{1}{2\sigma^2}[\Vert y-\theta^*\Vert^2])dy\\ &\qquad +\frac{1}{2}\int y\frac{1}{1+\exp(-\frac{2\theta^*T y}{\sigma^2})}\frac{1}{(2\pi\sigma)^{p/2}}\exp(-\frac{1}{2\sigma^2}[\Vert y+\theta^*\Vert^2])dy \end{align} but then don't know how to proceed.