As the title, $X$ is an random variable subject to $N(0,1)$, $N$ is an random variable subject to $N(0,\sigma^2)$, and $X$ and $N$ are independent. I want to calculate the mutual information $I(X,Y)$ between $X$ and $Y=X^2+N$. The problem can be reduced to calculate the differential entropy $h(Y)$ of $Y$. However, $h(Y)=-\int_{-\infty}^{+\infty}p_Y(y)log(p_Y(y))\mathrm{d}y$, and the $p_Y(y)$ can only be simplified to a integral form? How can I calculate $h(Y)$ or $I(X,Y)$ can be calculated in other ways?

As the title, $X$ is an random variable subject to $N(0,1)$, $N$ is an random variable subject to $N(0,\sigma^2)$, and $X$ and $N$ are independent. I want to calculate the mutual information $I(X,Y)$ between $X$ and $Y=X^2+N$. The problem can be reduced to calculate the differential entropy $h(Y)$ of $Y$. However, $h(Y)=-\int_{-\infty}^{+\infty}p_Y(y)log(p_Y(y))\mathrm{d}y$, and the $p_Y(y)$ can not get the explicit form. How can I calculate $h(Y)$ or $I(X,Y)$ can be calculated in other ways?