Two widely cited ICA algorithms are (extended) Infomax and fastICA. Both can be seen as maximizing the same likelihood (with fastICA enforcing an additional decorrelation constraint). They can be slow or even fail to converge when applied on real data.
Together with Pierre Ablin and Alexandre Gramfort, we have developped PICARD: an ICA algorithm which solves the same optimization problems as fastICA and Infomax but which does it fast and reliably even on real data and in large dimension.
Check the fastest ICA solver on Earth (papers and codes) on Picard github.