I'm currently reading a paper where they construct a $C^\infty$ function $\varphi$ on $\mathbb{R}$ that is symmetric and decreasing on $\mathbb{R}^+$ supported on $[-1,1]$ such that $0\leq \varphi(x)\leq 1$ and $\varphi(x)=1$ if $x\in [-\frac12,\frac12]$.
They create an example of such a function and say the following:
"We realized the function $\varphi$ of §2 by connecting the levels $0$ and $1$ with a function that is the primitive, suitably rescaled of the function $x → e^{−(1−x^2)^{−1}}$, set to be equal to $0$ outside $[−1, 1]$."
They go on to use this function heavily in simulations but never give a closed form for it.
I am fine with the idea of this function's construction in almost every aspect except for one. There is no closed form for the primitive of $e^{\frac{-1}{1-x^2}}$ right? Is there a clever trick here that I'm missing? Or perhaps, am I misunderstanding the language used here and we aren't working with an antiderivative of this function.
Also, I believe I am right in saying one could just use the function itself, rather than its primitive, in much the same way, in order to construct another example of such a $\varphi$. Is this correct?
Any help would be much appreciated.