Cosmic rays (CRs) have strong influences on the chemistry of dense molecular clouds (MCs). To study the detailed chemistry induced by CRs, we conducted a Yebes W-band line survey towards an unshocked MC (which we named 3C391:NML) associated with supernova remnant 3C 391. We detected emission lines of 18 molecular species in total and estimated their column densities with local thermodynamic equilibrium (LTE) and non-LTE analysis. Using the abundance ratio N(HCO+)/N(CO) and an upper limit of N(DCO+)/N(HCO+), we estimated that the CR ionization rate of 3C391:NML is ζ ≳ 2.7 × 10−14 s−1 with an analytic method. However, we caution against adopting this value because chemical equilibrium, which is a prerequisite of using the equations, is not necessarily reached in 3C391:NML. We observed lower N(HCO+)/N(HOC+), higher N(HCS+)/N(CS), and higher X(l-C3H+) by an order of magnitude in 3C391:NML than the typical values in quiescent dense MCs. We found that an enhanced CR ionization rate (of order ∼10−15 or ∼10−14 s−1) is needed to reproduce the observation with a chemical model. This is higher than the values found in typical MCs by 2–3 orders of magnitude.