  Both Galactic and extragalactic studies on star formation suggest that stars
form directly from dense molecular gas. To trace such high volume density gas,
HCN and HCO+ J=1-0 have been widely used for their high dipole moments,
relatively high abundances, and often being the strongest lines after CO.
However, HCN and HCO+ J=1-0 emission could be arguably dominated by the gas
components at low volume densities. HCN J=2-1 and HCO+ J=2-1, with more
suitable critical densities and excitation requirements, would trace typical
dense gas closely related to star formation. Here we report new observations of
HCN J=2-1 and HCO+ J=2-1 towards 17 nearby infrared-bright galaxies with the
APEX 12-m telescope. The correlation slopes between luminosities of HCN J=2-1,
and HCO+ J=2-1 and total infrared emission are 1.03 +- 0.05 and 1.00 +- 0.05,
respectively. The correlations of their surface densities, normalised with the
area of radio/sub-millimeter continuum, show even tighter relations (Slopes:
0.99 +- 0.03 and 1.02 +- 0.03). The eight AGN-dominated galaxies show no
significant difference from the eleven star-formation dominated galaxies in
above relations. The average HCN/HCO+ ratios are 1.15 +- 0.26 and 0.98 +- 0.42
for AGN-dominated and star-formation dominated galaxies, respectively, without
obvious dependencies on infrared luminosity, dust temperature, or infrared
pumping. The Magellanic Clouds roughly follow the same correlations, expanding
to eight orders of magnitude. On the other hand, ultra-luminous infrared
galaxies with active galactic nucleus (AGN) systematically lay above the
correlations, indicating potential biases introduced by AGNs.
