We present heat capacity measurements on a series of superconducting Cu$_x$TiSe$_2$ single crystals with different Cu content down to 600 mK and up to 1 T performed by ac microcalorimetry. The samples cover a large portion of the phase diagram from a
n underdoped to a slightly overdoped region with an increasing superconducting critical temperature and the charge density wave (CDW) order gradually suppressed. The electronic heat capacity as a function of normalized temperature $T/T_c$ shows no difference regardless of the concentration of copper, i.e., regardless of how much the CDW order is developed in the samples. The data analysis reveals consistently a single s-wave gap with an intermediate coupling strength $2Delta/k_BT_c$ = 3.7 for all samples.
