A half-shift reflection identity for the digamma function

We prove the identity $$2W_1(x)+\log 4+\psi (\frac{1}{2}+x)+\psi (\frac{3}{2}-x)=0,$$ where $\psi$ is the digamma function and $$W_1(x)=2{\int }_0^{\mathrm{\infty }}\mathrm{\Re }\left(\frac{y}{(y^2+1)(e^{\pi (y+2ix)}-1)}\right)dy\mathrm{.}$$ The identity was first conjectured while studying class number $h(D)$ for $D=m^2$ from two complementary perspectives. Our proof, however, is purely analytic: we compute cosine-series expansions of both sides, expressed in terms of the cosine integral Ci$(z)$. Using the above identity and Möbius inversion we find an elementary formula for $$\sum _{\begin{array}{c}1\le r<m\\ (r,m)=1\end{array}}{W}_1\text{\hspace{0.17em}⁣}\left(\frac{r}{m}\right)\mathrm{.}$$