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Nonstabilizerness, also known as magic, quantifies the number of non-Clifford operations needed in order to prepare a quantum state. As typical measures either involve minimization procedures or a computational cost exponential in the number of qubits $N$, it is notoriously hard to characterize for many-body states. In this work, we show that nonstabilizerness, as quantified by the recently introduced Stabilizer Rényi Entropies (SREs), can be computed efficiently for matrix product states (MPSs). Specifically, given an MPS of bond dimension $χ$ and integer Rényi index $n>1$, we show that the SRE can be expressed in terms of the norm of an MPS with bond dimension $χ^{2n}$. For translation-invariant states, this allows us to extract it from a single tensor, the transfer matrix, while for generic MPSs this construction yields a computational cost linear in $N$ and polynomial in $χ$. We exploit this observation to revisit the study of ground-state nonstabilizerness in the quantum Ising chain, providing accurate numerical results up to large system sizes. We analyze the SRE near criticality and investigate its dependence on the local computational basis, showing that it is in general not maximal at the critical point.