The identity under question can be found in the paper S. Boettner, V.H. Moll *The integrals in Gradshteyn and Ryzhik. Part 16: Complete elliptic integrals*, pages 11-12 https://arxiv.org/abs/1005.2941

[![enter image description here][1]][1]

The proof given by OP is miraculously close to the one in this paper, not only in the method used, but also the wording and formatting of equations. By the way, one of the authors V. Moll, is a frequent coauthor of the OP. Coincidences happen, and with some people they happen more often than with others, this is just law of probabilities. We suspect no foul play. But it is really intriguing to know, how is the identity under question related to Digamma function, as the OP claims? Unfortunately his proof does not contain any digamma functions. But in the paper it naturally came out of some work on the elliptic integral. Please, @T.Amdeberhan, would you share your insight? It is really intriguing to know, how is digamma function related here.


  [1]: https://i.sstatic.net/p08F7.png