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We show that every integrally closed -primary ideal I in a commutative Noetherian local ring has maximal complexity and curvature, i.e., and . As a consequence, we characterize complete intersection local rings in terms of complexity, curvature, and complete intersection dimension of such ideals. The analogous results on projective, injective, and Gorenstein dimensions are known. However, we provide short proofs of these results as well.