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Gradient-based planners are widely used for quadrotor local planning, in which a Euclidean Signed Distance Field (ESDF) is crucial for evaluating gradient magnitude and direction. Nevertheless, computing such a field has much redundancy since the trajectory optimization procedure only covers a very limited subspace of the ESDF updating range. In this paper, an ESDF-free gradient-based planning framework is proposed, which significantly reduces computation time. The main improvement is that the collision term in the penalty function is formulated by comparing the colliding trajectory with a collision-free guiding path. The resulting obstacle information will be stored only if the trajectory hits new obstacles, making the planner only extract necessary obstacle information. Then, we lengthen the time allocation if dynamical feasibility is violated. An anisotropic curve fitting algorithm is introduced to adjust higher-order derivatives of the trajectory while maintaining the original shape. Benchmark comparisons and real-world experiments verify its robustness and high-performance. The source code is released as ROS packages.