Closed-form expressions for integrals of traveling wave reductions of integrable lattice equations

We give a method to calculate closed-form expressions in terms of multi-sums of products for integrals of ordinary difference equations which are obtained as traveling wave reductions of integrable partial difference equations. Important ingredients are the staircase method, a non-commutative Vieta formula and certain splittings of the Lax matrices. The method is applied to all equations of the Adler-Bobenko-Suris classification, with the exception of Q4.