[wenc]

I used a bit of Mathematica but settled on Waterloo Maple because it had a cheaper academic license. Mathematica has stronger algorithms than Maple (like cylindrical algebraic decomposition), but for what I was using it for, they were both equally capable.

I was working on mathematical models (large scale optimization). These are usually solved numerically, and in numerical mathematics, how you write an equation matters tremendously (for instance, the equality x/y = z is much worse than x = y * z for solvers especially if y is a variable that can take on 0 as a value because during iteration this might create a lot of NaNs in your Jacobian or Hessian matrices).

I was using symbolic math to find better (but mathematically equivalent) ways to pose equations that would be numerically expedient. One example is using Groebner bases to do the equivalent of Gaussian elimination on a system of polynomial, which produces a row-echelon form and has many nice properties.