You can simplify your algebra a little bit by working in terms of angles. If $\theta$ is the half of the central angle of the arc overlapping with the other circle, then $d=2r\cos\theta$ and your equation is further simplified into
Of course, this is still implicit; one cannot really solve this equation for $\theta$ without a numerical method.
In addition, you may be interested to know that there are various "tricks", ways to write an explicit solutions to transcendental equation like yours in terms of custom special functions, basically, integrals that would still need to be computed numerically. An early paper in this area by E.E. Burniston & C.E. Siewert is called "Exact analytical solution of the transcendental equation $\alpha\sin\zeta=\zeta$" and was published in SIAM J. Appl. Math., Vol.24, No.4 (1973). It can be downloaded from C.E. Siewert's web page.