Dear Stephen,
thanks a lot for your long reply. Unfortunately, we are talking about different integrals. Your are dealing with the zero-counting integral \int_\alpha f ' (s)/f(s) ds, I work on the integral \int_\alpha s*f ' (s)/f(s) ds, where * is the common multiplication. From complex function theory we know that it equals a, if a is the only root of f in the rectangle. We want to compute this root, thus method 2 doesn't work for us. Remains method 1: what is the best (most efficient and stable) built in integration command?