This paper numerically investigates the self-propelled swimming of a flexible filament driven by coupled pitching and plunging motions at the leading edge. The influences of bending rigidity and some actuation parameters (including the phase offset between pitching and plunging, and the amplitudes of pitching and plunging motions) on the swimming performance are explored. It is found that with increasing rigidity, the swimming style gradually transits from the undulatory mode to the oscillatory mode. The plunging-pitching actuation is found to be superior to the plunging-only actuation, in the sense that it prevents the decrease of speed at high rigidity and achieves a higher efficiency across a wide range of rigidity. The comparison of the body kinematics with those of animal swimmers, and the classification of the wake structures are discussed. The results of this study provide some novel insights for the bio-inspired design of autonomous underwater vehicles.