An online method for rapidly learning the inverse kinematics of a redundant robotic arm is presented addressing the special requirements of active vision for visual inspection tasks. The system is initialized with a model covering a small area around the starting position, which is then incrementally extended by exploration. The number of motions during this process is minimized by only exploring configurations required for successful completion of the task at hand. The explored area is automatically extended online and on demand.To achieve this, state of the art methods for learning and numerical optimization are combined in a tight implementation where parts of the learned model, the Jacobians, are used during optimization, resulting in significant synergy effects. In a series of standard experiments, we show that the integrated method performs better than using both methods sequentially.