The working equations for the calculation of NMR shielding tensors in the framework of auxiliary density functional theory are derived. It is shown that in this approach the numerical integration over gauge-including atomic orbitals can be avoided without the loss of accuracy. New integral recurrence relations for the required analytic electric-field-type integrals are derived. The computational performance of the resulting formalism permits shielding tensor calculations of systems with more than 1000 atoms and 15,000 basis functions.