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Library-based methods are known to be very effective for fast motion planning by adapting an experience retrieved from a precomputed library. This article presents CoverLib, a principled approach for constructing and utilizing such a library. CoverLib iteratively adds an experience-classifier-pair to the library, where each classifier corresponds to an adaptable region of the experience within the problem space. This iterative process is an active procedure, as it selects the next experience based on its ability to effectively cover the uncovered region. During the query phase, these classifiers are utilized to select an experience that is expected to be adaptable for a given problem. Experimental results demonstrate that CoverLib effectively mitigates the trade-off between plannability and speed observed in global (e.g. sampling-based) and local (e.g. optimization-based) methods. As a result, it achieves both fast planning and high success rates over the problem domain. Moreover, due to its adaptation-algorithm-agnostic nature, CoverLib seamlessly integrates with various adaptation methods, including nonlinear programming-based and sampling-based algorithms.