@vez There is the dream that modern AI techniques may be able to eventually resolve this problem for fairly general classes of algebraic problems. (Of course, we have special-purpose algorithms already for specific classes of problems, e.g., Gaussian elimination for systems of linear equations.) For instance the AlphaGo model, in which a neural network learns to "score" the favorability of Go positions for the AI player, and then selects moves accordingly, could be potentially adaptable to algebraic simplification problems.
Perhaps one major bottleneck, though, is the insufficient amount of curated datasets of algebraic expressions (ideally labeled with some sort of difficulty level) that would allow current-technology AIs to train to figure out which expressions look easier to solve and which ones look complex..