[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.
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