We propose an information-theoretic planning approach that enables mobile robots to autonomously construct dense 3D maps in a computationally efficient manner. Inspired by prior work, we accomplish this task by formulating an information-theoretic objective function based on CauchySchwarz quadratic mutual information (CSQMI) that guides robots to obtain measurements in uncertain regions of the map. We then contribute a two stage approach for active mapping. First, we generate a candidate set of trajectories using a combination of global planning and generation of local motion primitives. From this set, we choose a trajectory that maximizes the information-theoretic objective. Second, we employ a gradientbased trajectory optimization technique to locally refine the chosen trajectory such that the CSQMI objective is maximized while satisfying the robot’s motion constraints. We evaluated our approach through a series of simulations and experiments on a ground robot and an aerial robot mapping unknown 3D environments. Real-world experiments suggest our approach reduces the time to explore an environment by 70% compared to a closest frontier exploration strategy and 57% compared to an information-based strategy that uses global planning, while simulations demonstrate the approach extends to aerial robots with higher-dimensional state.