The function $$ f(x)\ :=\ e^{1-\frac{1}{\sqrt{1-x^2}}} $$ has the properties that

- $\frac{d^nf}{dx^n}(\pm 1)=0$,
- $\frac{d^n}{dx^n}\left(\sqrt{1-x^2}-f(x)\right) = 0$ at $x=0$,

and its integral $F(x) := \int_{-1}^{x}f(t)\,\mathrm{d}t$ provides $C^\infty$ transition between the two non-identical horizontal lines $y(x)=0$ and $y(x)\approx1.48756$.

Questions:

- can the integral $$ F(x) := \int_{-1}^{x}e^{1-\frac{1}{\sqrt{1-t^2}}}\,\mathrm{d}t $$ be expressed by standard functions or known special functions?
- can at least $F(1)$ be expressed via already known functions?
- do $f(x)$ or $F(x)$ already appear as the solution to a mathematical or physical problem?

Remark: I "discovered" $f(x)$ when trying to find a $C^\infty$ connection between two parallel lines.