unexpected logarithmic identities and other surprises

This is a journey through integrals of involutions and surprising consequences of the Lagrange inversion theorem. On the way, we meet unexpected logarithmic identities, hypergeometric functions with a linear regime and other mysterious objects. This study was inspired by some results from the fascinating article A.E. Holroyd, T.M. Liggett and D. Romik, Integrals, Partitions, and Cellular Automata, Transactions of the American Mathematical Society, 356-8, 3349-3368, 2004