A time domain simulator for analyzing non-linear coupled electromagnetic and micromagnetic phenomena in complex magnetic structures is introduced. The simulator employs a new integral formulation that accurately accounts for the Eddy currents and captures the dominant physics. An implicit time stepping scheme that uses an analytic Jacobian method and allows for large time steps is proposed. Finally, the simulator has a low (O(N logN)) computational complexity for a system with N mesh elements and it is implemented on massively parallel GPU computing systems, which permits for the rapid analysis of highly complex magnetic devices.