Context. Most of our knowledge on the stellar component of galaxies is based on the analysis of distant systems and comes from integrated light data. It is important to test whether the results of the star formation histories (SFH) obtained with standard full-spectrum fitting methods are in agreement with those obtained through colour-magnitude diagram (CMD) fitting (usually considered the most reliable approach). Aims. We compare SFHs recovered from both techniques in Leo~A, a Local Group dwarf galaxy whose majority of stars formed during the last 8 Gyrs. This complements our previous findings in a field in the Large Magellanic Cloud bar, where star formation has been on-going since early epochs though at varying rates. Methods. We have used GTC/OSIRIS in long-slit mode to obtain a high-quality integrated light spectrum by scanning a selected region within Leo~A, for which a CMD reaching the old main mequence turn-off (oMSTO) is available from HST. We compared the SFH obtained from the two datasets, using state-of-art methods of integrated light ({tt STECKMAP}) and resolved stellar population analysis. In the case of the CMD, we computed the SFH both from a deep CMD (observed with HST/ACS), and from a shallower one (archival data from HST/WFPC2). Results. The agreement between the SFHs recovered from the oMSTO CMD and from full spectrum fitting is remarkable, particularly regarding the time evolution of the star formation rate. The overall extremely low metallicity of Leo~A is recovered up to the last 2 Gyrs, when some discrepancies appear. A relatively high metallicity found for the youngest stars from the integrated data is a recurring feature that might indicate that the current models or synthesis codes should be revised, but that can be significantly mitigated using a more restrictive metallicity range... [Abridged]
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