  We present new IRAM 30m spectroscopic observations of the $\sim88$ GHz band,
including emission from the CCH (n=1-0) multiplet, HCN (1-0), HCO+ (1-0), and
HNC (1-0), for a sample of 58 local luminous and ultraluminous infrared
galaxies from the Great Observatories All-sky LIRG Survey (GOALS). By combining
our new IRAM data with literature data and Spitzer/IRS spectroscopy, we study
the correspondence between these putative tracers of dense gas and the relative
contribution of active galactic nuclei (AGN) and star formation to the
mid-infrared luminosity of each system. We find the HCN (1-0) emission to be
enhanced in AGN-dominated systems ($\langle$L'$_{HCN (1-0)}$/L'$_{HCO^+
(1-0)}\rangle=1.84$), compared to composite and starburst-dominated systems
($\langle$L'$_{HCN (1-0)}$/L'$_{HCO^+ (1-0)}\rangle=1.14$, and 0.88,
respectively). However, some composite and starburst systems have L'$_{HCN
(1-0)}$/L'$_{HCO^+ (1-0)}$ ratios comparable to those of AGN, indicating that
enhanced HCN emission is not uniquely associated with energetically dominant
AGN. After removing AGN-dominated systems from the sample, we find a linear
relationship (within the uncertainties) between $\log_{10}$(L'$_{HCN (1-0)}$)
and $\log_{10}$(L$_{IR}$), consistent with most previous findings. L'$_{HCN
(1-0)}$/L$_{IR}$, typically interpreted as the dense gas depletion time,
appears to have no systematic trend with L$_{IR}$ for our sample of luminous
and ultraluminous infrared galaxies, and has significant scatter. The
galaxy-integrated HCN (1-0) and HCO+ (1-0) emission do not appear to have a
simple interpretation, in terms of the AGN dominance or the star formation
rate, and are likely determined by multiple processes, including density and
radiative effects.
