The Woods-Saxon potential is

$f(r) = 1/(1+Exp(\frac{r-R}{a}))$

where $R=r_0 A^{1/3}$ is the half-maximum radius that $r_0$ is the reduced radius and $A$ is the nuclear mass number, and $a$ is the diffusiveness parameter.

$\sqrt{\left} = \sqrt{\int{f(r) r^2 r^2 dr} / \int{f(r) r^2 dr}}$

where the “extract” $r^2$ is because of spherical coordinate.

The integration is a polynomial

$\int{f(r) r^n dr} = -a^{n+1} \Gamma(n+1) \sum_{k=1}^{\infty} \frac{(-Exp(R/a))^k}{k^{n+1}}$

In mathematica, the sum is notated by,

$PolyLog(n, x) = \sum_{k=1}^{\infty} \frac{x^k}{k^n}$

Thus, the rms radius for $a > 0$ is

$\sqrt{\left} = a \sqrt{12 \frac{PolyLog(5, -Exp(R/a))}{PolyLog(3, -Exp(R/a))} }$

For $a = 0$, $f(r) = 1, r < R$,

$\sqrt{\left} = \sqrt{\frac{3}{5}}R$