$$\int_{-\infty}^{\infty}e^{-y\left(x\right)}\left(\log\int_{-\infty}^{\infty}e^{-y\left(z\right)}N\left(x-z,v\right)dz\right)dx $$
where $y(x)\equiv\sum_{i=1}^{n}c_{i}x^{i}$, $n$ is even and finite, and $c_{n}>0$ so the integral converges. $N(x,v)$ is a Normal distribution in $x$ with variance $v$. Note that $y(\cdot)$ appears twice with different arguments.