We make use of the deep Karl G. Jansky Very Large Array (VLA) COSMOS radio observations at 3 GHz to infer radio luminosity functions of star-forming galaxies up to redshifts of z~5 based on approximately 6000 detections with reliable optical counterparts. This is currently the largest radio-selected sample available out to z~5 across an area of 2 square degrees with a sensitivity of rms=2.3 ujy/beam. By fixing the faint and bright end shape of the radio luminosity function to the local values, we find a strong redshift trend that can be fitted with a pure luminosity evolution L~(1+z)^{(3.16 +- 0.2)-(0.32 +- 0.07) z}. We estimate star formation rates (SFRs) from our radio luminosities using an infrared (IR)-radio correlation that is redshift dependent. By integrating the parametric fits of the evolved luminosity function we calculate the cosmic SFR density (SFRD) history since z~5. Our data suggest that the SFRD history peaks between 2<z<3 and that the ultraluminous infrared galaxies (ULIRGs; 100 Msol/yr<SFR<1000 Msol/yr) contribute up to ~25% to the total SFRD in the same redshift range. Hyperluminous infrared galaxies (HyLIRGs; SFR>1000 Msol/yr) contribute an additional <2% in the entire observed redshift range. We find evidence of a potential underestimation of SFRD based on ultraviolet (UV) rest-frame observations of Lyman break galaxies (LBGs) at high redshifts (z>4) on the order of 15-20%, owing to appreciable star formation in highly dust-obscured galaxies, which might remain undetected in such UV observations.
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