Abstract We present a new exploration of the cosmic star formation history and dust obscuration in massive galaxies at redshifts 0.5 < z < 6. We utilize the deepest 450- and 850-μm imaging from SCUBA-2 CLS, covering 230 arcmin2 in the AEGIS, COSMOS and UDS fields, together with 100–250 μm imaging from Herschel. We demonstrate the capability of the t-phot deconfusion code to reach below the confusion limit, using multiwavelength prior catalogues from CANDELS/3D-HST. By combining IR and UV data, we measure the relationship between total star formation rate (SFR) and stellar mass up to z ∼ 5, indicating that UV-derived dust corrections underestimate the SFR in massive galaxies. We investigate the relationship between obscuration and the UV slope (the IRX–β relation) in our sample, which is similar to that of low-redshift starburst galaxies, although it deviates at high stellar masses. Our data provide new measurements of the total SFR density (SFRD) in $M_{\ast }>10^{10}\,\textrm{M}_{\odot }$ galaxies at 0.5 < z < 6. This is dominated by obscured star formation by a factor of >10. One third of this is accounted for by 450-μm-detected sources, while one-fifth is attributed to UV-luminous sources (brighter than $L_{\rm UV}^\ast$), although even these are largely obscured. By extrapolating our results to include all stellar masses, we estimate a total SFRD that is in good agreement with previous results from IR and UV data at z ≲ 3, and from UV-only data at z ∼ 5. The cosmic star formation history undergoes a transition at z ∼ 3–4, as predominantly unobscured growth in the early Universe is overtaken by obscured star formation, driven by the build-up of the most massive galaxies during the peak of cosmic assembly.