  We report cosmic ray ionization rates towards ten reddened stars studied
within the framework of the EDIBLES (ESO Diffuse Interstellar Bands Large
Exploration Survey) program, using the VLT-UVES. For each sightline, between 2
and 10 individual rotational lines of OH$^+$ have been detected in its (0,0)
and (1,0) $A^3\Pi-X^3\Sigma^-$ electronic band system. This allows constraining
of OH$^+$ column densities towards different objects. Results are also
presented for 28 additional sightlines for which only one or rather weak
signals are found. An analysis of these data makes it possible to derive the
primary cosmic ray ionization rate $\zeta_p$ in the targeted diffuse
interstellar clouds. For the ten selected targets, we obtain a range of values
for $\zeta_p$ equal to $(3.9-16.4) \times 10^{-16}~\mathrm{s}^{-1}$. These
values are higher than the numbers derived in previous detections of
interstellar OH$^+$ in the far-infrared / sub-millimeter-wave regions and in
other near-ultraviolet studies. This difference is a result of using new OH$^+$
oscillator strength values and a more complete picture of all relevant OH$^+$
formation and destruction routes (including the effect of proton recombinations
on PAHs), and the relatively high $N$(OH$^+$) seen toward those ten targets.
