  The low energy behavior of a recently proposed model for the massive analytic
running coupling of QCD is studied. This running coupling has no unphysical
singularities, and in the absence of masses displays infrared enhancement. The
inclusion of the effects due to the mass of the lightest hadron is accomplished
by employing the dispersion relation for the Adler D function. The presence of
the nonvanishing pion mass tames the aforementioned enhancement, giving rise to
a finite value for the running coupling at the origin. In addition, the
effective charge acquires a "plateau-like" behavior in the low energy region of
the timelike domain. This plateau is found to be in agreement with a number of
phenomenological models for the strong running coupling. The developed
invariant charge is applied in the processing of experimental data on the
inclusive $\tau$ lepton decay. The effects due to the pion mass play an
essential role here as well, affecting the value of the QCD scale parameter
$\Lambda$ extracted from these data. Finally, the massive analytic running
coupling is compared with the effective coupling arising from the study of
Schwinger-Dyson equations, whose infrared finiteness is due to a dynamically
generated gluon mass. A qualitative picture of the possible impact of the
former coupling on the chiral symmetry breaking is presented.
