Context. Radio used as a tracer of the star formation rate (SFR) presents enormous advantages because it is not affected by dust and radio sources that are located at the subarcsecond level. The interpretation of the low-frequency 1.4 GHz luminosity is hampered by the difficulty of modeling the paths of cosmic rays in the interstellar medium, however, and by their interactions with the magnetic field. Aims. We compare the SFR derived from radio observations and the SFRs derived from spectral energy distribution (SED) modeling. We aim at better understanding the behavior of the SFR radio tracer, with a specific emphasis on the link to star formation histories (SFHs). Methods. The analysis is based on a subsample of 1584 star-forming galaxies extracted from the Cosmic Evolution Survey (COSMOS) with observations of the Very large array project at 3 GHz. We used the SED modeling code investigating galaxy emission, CIGALE , with a nonparametric model for the SFH and fit the data over the wavelength range from the ultraviolet (UV) to the mid-infrared (mid-IR). We interpret the difference between radio and SED-based SFR tracers in the light of recent gradients in the derived SFH. To validate the robustness of the results, we searched for any remaining contribution of active galaxy nuclei and tested the impact of our SFH modeling approach. Results. Approximately 27% our galaxies present a radio SFR (SFR radio ) that is at least ten times higher than the instantaneous SFR from SED fitting (SFR SED ). This trend primarily affects the galaxies whose SFH activity decreased over the last 300 Myr. Both SFR indicators converge toward a consistent value when the SFHs are averaged over a period longer than 150 Myr to derive SFR SED . Conclusions. Although the radio at a low frequency of 1.4 GHz is a good tracer of the star formation activity of galaxies with a constant or increasing SFH, our results indicate that this is not the case for quenched galaxies. Our analysis suggests that the star formation time sensitivity of the low radio frequency might be longer than 150 Myr. Interestingly, the discrepancy between the SFR radio and SFR SED can be used as diagnostic to select post-starburst galaxies.