  We explore the effect of tunable integrability breaking dipole-dipole
interactions in the equilibrium states of highly magnetic 1D Bose gases of
dysprosium at low temperatures. We experimentally observe that in the strongly
correlated Tonks-Girardeau regime, rapidity and momentum distributions are
nearly unaffected by the dipolar interactions. By contrast, we also observe
that significant changes of these distributions occur when decreasing the
strength of the contact interactions. We show that the main experimental
observations are captured by modeling the system as an array of 1D gases with
only contact interactions, dressed by the contribution of the short-range part
of the dipolar interactions. Improvements to theory-experiment correspondence
will require new tools tailored to near-integrable models possessing both short
and long-range interactions.
