IR spectra of cationic copper clusters Cun+ (n = 4-7) complexed with hydrogen molecules are recorded via IR multiple-photon dissociation (IRMPD) spectroscopy. To this end, the copper clusters are generated via laser ablation and reacted with H2 and D2 in a flow-tube-type reaction channel. The complexes formed are irradiated using IR light provided by the free-electron laser for intracavity experiments (FELICE). The spectra are interpreted by making use of isotope-induced shifts of the vibrational bands and by comparing them to density functional theory calculated spectra for candidate structures. The structural candidates have been obtained from global sampling with the minima hopping method, and spectra are calculated at the semilocal (PBE) and hybrid (PBE0) functional level. The highest-quality spectra have been recorded for [5Cu, 2H/2D]+, and we find that the semilocal functional provides better agreement for the lowest-energy isomers. The interaction of hydrogen with the copper clusters strongly depends on their size. Binding energies are largest for Cu5+, which goes hand in hand with the observed predominantly dissociative adsorption. Due to smaller binding energies for dissociated H2 and D2 for Cu4+, also a significant amount of molecular adsorption is observed as to be expected according to the Evans-Polanyi principle. This is confirmed by transition-state calculations for Cu4+ and Cu5+, which show that hydrogen dissociation is not hindered by an endothermic reaction barrier for Cu5+ and by a slightly endothermic barrier for Cu4+. For Cu6+ and Cu7+, it was difficult to draw clear conclusions because the IR spectra could not be unambiguously assigned to structures.