Non-Elementary Integration · Superchallenge problem

The standard normal density is $\varphi(x)=\dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}$. Using the known second moment $\displaystyle\int_{-\infty}^{\infty}x^2\varphi(x)\,dx=1$ and the reduction $\displaystyle\int_{-\infty}^{\infty}x^n\varphi(x)\,dx=(n-1)\displaystyle\int_{-\infty}^{\infty}x^{n-2}\varphi(x)\,dx$ (for even $n\ge2$, from integration by parts), evaluate $\displaystyle\int_{-\infty}^{\infty}x^4\varphi(x)\,dx$.