Which is better? Vedic mathematics or Trachtenberg

This is what Alon Amit has to say on this question. 

Alon Amit 

Better" in what sense? Both systems consist of various tricks for doing calculations somewhat faster than you would without them. In either case, practice is more important than the particular system you choose to work with.

And in either case, the proficiency they give you is quite irrelevant for anything to do with mathematics or pretty much anything. Doing mental arithmetic is fun for some, so practicing it is (for them) a neat pastime. Other than that, digging into either "system" is a waste of time.

(Some folks, hearing how John von Neumann or Richard Feynman were very quick mental calculators, start believing that learning mental arithmetic could make them better mathematicians, physicists, computer scientists or people. It doesn't. Both von Neumann and Feynman had a deep interest in mathematics from an early age, and they played with it as other kids play with toys. This helped them develop a facility with mental calculation among other things, but more importantly they were eager to learn actual mathematics and became very good at it. The causal relationship doesn't work in the other direction.)