= Exponential spherical mass factor
{title2=$f(q)=1-q+qe^{-1/q}$}
For radial <mass density> $\rho(r)=Ar^{-2}(1-e^{-r/r_0})$ within radius $R>0$, integration against the spherical <volume element> gives $M=4\pi ARf(r_0/R)$. The <dimensionless quantity> $q=r_0/R>0$ has
$$
f(q)=\int_0^1(1-e^{-s/q})\,ds,\quad f'(q)=-1+(1+1/q)e^{-1/q}<0,\quad f''(q)=q^{-3}e^{-1/q}>0.
$$
Thus $f$ is positive, strictly decreasing and convex, with $f(0^+)=1$ and $f(\infty)=0$. Its asymptotic forms are $f(q)=1-q+qe^{-1/q}$ near zero and $f(q)=1/(2q)-1/(6q^2)+O(q^{-3})$ at infinity. The density's central $1/r$ divergence is integrable against the factor $r^2dr$.
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