Full duplex (FD) communication has the potential to double the throughput of wireless links while occupying the same bandwidth of half duplex communication. To enable FD communication, the self-interference (S1) signal is to be suppressed down to the noise floor. S1 cancellation can be achieved via antenna suppression, active analog cancellation, and digital cancellation. At high transmit power levels, the nonlinear behavior of the transceiver hardware degrades linear S1 channel estimation; hence, S1 cancellation results in reduced total suppression. In this article, we review nonlinear DSIC solutions in the literature, and we present a comprehensive integrated linear and nonlinear digital S1 cancellation framework with nested and residual approaches, where nonlinear algorithms, memory polynomial, support vector regression, and orthonormalized least mean squares algorithms are implemented together with linear cancellation. The performance of the proposed integrated framework is evaluated in comparison with well-known nonlinear digital cancellation techniques by implementing all techniques on the same software defined FD radio setup, consisting only of a single antenna and a WARP v3 board. It is shown that the integrated residual approach can enhance the total S1 suppression performance significantly by up to 7 dB as compared to linear-only digital S1 cancellation and by up to 4 dB with respect to existing nonlinear-only techniques observed at moderate power levels, and the integrated nested approach can provide an improvement of up to 6 dB over linear-only and 3 dB over nonlinear-on-ly digital S1 cancellation for high power settings. All schemes are also evaluated in terms of training overhead and computational complexity, providing insights about the trade-off between cost and performance at different power levels.