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 μJy beam-1. 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 L1.4 GHz ∝ (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 (100 M⊙ yr-1 < SFR < 1000 M⊙ yr-1) contribute up to ~25% to the total SFRD in the same redshift range. Hyperluminous infrared galaxies (SFR > 1000 M⊙ yr-1) 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 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.