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Aims: We leverage the largest available Atacama LargeMillimetre/submillimetre Array (ALMA) survey from the archive (A$^3$COSMOS) tostudy to study infrared luminosity function and dust-obscured star formationrate density of sub-millimeter/millimeter (sub-mm/mm) galaxies from$z=0.5\,-\,6$. Methods: The A$^3$COSMOS survey utilizes all publicly availableALMA data in the COSMOS field, therefore having inhomogeneous coverage in termsof observing wavelength and depth. In order to derive the luminosity functionsand star formation rate densities, we apply a newly developed method thatcorrects the statistics of an inhomogeously sampled survey of individualpointings to those representing an unbiased blind survey. Results: We find oursample to mostly consist of massive ($M_{\star} \sim 10^{10} - 10^{12}$ $\rmM_{\odot}$), IR-bright ($L_* \sim 10^{11}-10^{13.5} \rm L_{\odot}$), highlystar-forming (SFR $\sim 100-1000$ $\rm M_{\odot}$ $\rm yr^{-1}$) galaxies. Wefind an evolutionary trend in the typical density ($\Phi^*$) and luminosity($L^*$) of the galaxy population, which decrease and increase with redshift,respectively. Our IR LF is in agreement with previous literature results and weare able to extend to high redshift ($z > 3$) the constraints on the knee andbright-end of the LF, derived by using the Herschel data. Finally, we obtainthe SFRD up to $z\sim 6$ by integrating the IR LF, finding a broad peak from $z\sim 1$ to $z \sim 3$ and a decline towards higher redshifts, in agreement withrecent IR/mm-based studies, within the uncertainties, thus implying thepresence of larger quantities of dust than what is expected by optical/UVstudies.

A$^3$COSMOS: the infrared luminosity function and dust-obscured star formation rate density at $0.5