  The one-loop worldsheet quantum corrections to the energy of spinning strings
on R x S^3 within AdS_5 x S^5 are reexamined. The explicit expansion in the
effective 't Hooft coupling \lambda'= \lambda/J^2 is rigorously derived. The
expansion contains both analytic and non-analytic terms in \lambda', as well as
exponential corrections. Furthermore, we pin down the origin of the terms that
are not captured by the quantum string Bethe ansatz, which only produces
analytic terms in \lambda'. It is shown that the analytic terms arise from
string fluctuations within the S^3, whereas the non-analytic and exponential
terms, which are not captured by the Bethe ansatz, originate from the
fluctuations in all directions within the supersymmetric sigma model on AdS_5 x
S^5. We also comment on the case of spinning string in AdS_3 x S^1.
