Abstract The star formation rate density (SFRD) history of the universe is well constrained up to redshift z ∼ 2. At earlier cosmic epochs, the picture has been largely inferred from UV-selected galaxies (e.g., Lyman-break galaxies; LBGs). However, the inferred star formation rates of LBGs strongly depend on the assumed dust extinction correction, which is not well constrained at high z, while observations in the radio domain are not affected by this issue. In this work we measure the SFRD from a 1.4 GHz selected sample of ∼600 galaxies in the GOODS-N field up to redshift ∼3.5. We take into account the contribution of active galactic nuclei from the infrared-radio correlation. We measure the radio luminosity function, fitted with a modified Schechter function, and derive the SFRD. The cosmic SFRD shows an increase up to z ∼ 2 and then an almost flat plateau up to z ∼ 3.5. Our SFRD is in agreement with those from other far-IR/radio surveys and a factor 2 higher than those from LBG samples. We also estimate that galaxies lacking a counterpart in the HST/WFC3 H-band (H-dark) make up ∼25% of the ϕ-integrated SFRD relative to the full sample at z ∼ 3.2, and up to 58% relative to LBG samples.