  We use a factorization scheme analogous to one proposed for single inclusive
forward hadron production to factorize the soft gluon divergence present in the
deep inelastic scattering cross sections in the dipole picture at
next-to-leading order (NLO). We show numerically that in this carefully
constructed scheme it is possible to obtain meaningful results for the DIS
cross sections at NLO, and so we are able to quantitatively study the recently
derived NLO corrections to the DIS cross sections. We find that the NLO
corrections can be significant and sensitive to the details of the
factorization scheme used for the resummation of the large logarithms into the
BK evolution equation. In the case of an approximative factorization scheme we
observe a problematic behavior of the DIS cross sections similar to what has
been seen with analogously factorized single inclusive cross sections.
