  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
