  [Abridged] This paper aims at providing new conservative constraints to the
cosmic star-formation history from the empirical modeling of mid- and
far-infrared data. We perform a non-parametric inversion of galaxy counts at
15, 24, 70, 160, and 850 microns simultaneously. It is a "blind" search (no
redshift information is required) of all possible evolutions of the infrared
luminosity function of galaxies, from which the evolution of the star-formation
rate density and its uncertainties are derived. The cosmic infrared background
(CIRB) measurements are used a posteriori to tighten the range of solutions.
The inversion relies only on two hypotheses: (1) the luminosity function
remains smooth both in redshift and luminosity, (2) a set of infrared spectral
energy distributions (SEDs) of galaxies must be assumed. The range of
star-formation histories that we derive is well constrained and consistent with
redshift-based measurements from deep surveys. The redshift decompositions of
the counts are also recovered successfully. Therefore, multi-wavelength counts
and CIRB (both projected observations) alone seem to contain enough information
to recover the cosmic star-formation history with quantifiable errors. A peak
of the SFRD at z~2 is preferred, although higher redshifts are not excluded. We
also find a good consistency between the observed evolution of the stellar mass
density and the prediction from our model. Finally, the inability of the
inversion to model perfectly and simultaneously all the multi-wavelength
infrared counts (especially at 160 microns where an excess is seen around 20
mJ) implies either (i) the existence of a sub-population of colder galaxies,
(ii) a larger dispersion of dust temperatures among local galaxies than
expected, (iii) or a redshift evolution of the infrared SEDs of galaxies.
