I am trying to introduce a tortoise coordinate for a modified Schwarzschild metric
$$\mathrm{d}s^2=\left(1-\frac{2M\mathop{}\!\mathrm{erf}(r)}{r}\right) \mathrm{d}t^2 + \left(1-\frac{2M\mathop{}\!\mathrm{erf}(r)}{r}\right)^{-1} \mathrm{d}r^2 +r^2\mathrm{d}\Omega^2,$$
where $\mathop{}\!\mathrm{erf}$ is the error function.
For standard Schwarzschild, one would make the transform
$$r*=r+2M \ln\left(\frac{r}{2M} -1\right),$$
to get the metric in the form
$$\mathrm{d}s^2=-\left(1-\frac{2M }{r}\right) \left(\mathrm{d}t^2 - \mathrm{d}r^{*2} \right) +r^2\mathrm{d}\Omega^2.$$
However, here there is a derivative of the error function which messes everything up if we try
$$r*=\frac{r}{\mathrm{erf}(r)} +2M \ln\left(\frac{r}{2M\mathop{}\!\mathrm{erf}(r)} -1\right).$$
To do the transform properly, we really need to find $r^*$ such that $dr^*=\left(1-\frac{2M\mathop{}\!\mathrm{erf}(r)}{r}\right)^{-1}dr$,
i.e. solve
$$\int \mathrm{d}r \left(1-\frac{2M\mathop{}\!\mathrm{erf}(r)}{r}\right)^{-1}$$
which I cannot calculate analytically.
Does anyone know how I can make a transformation that would get my modified Schwarzschild metric into the form
$$f(r) -(\mathrm{d}t^2 -\mathrm{d}r^{*2}) +r^2 \mathrm{d}\Omega^2?$$
