A brief survey of the ones I have tried, with the student version price (as of mid 2008).
- Computer algebra systems (see wikipedia comparison)
- derive. Very simple, very easy. Unfortunately, it seems discontinued.
- Mathematica. Incredibly powerful. But: crazy syntax, annoying graphic interface (it used to be impossible to rotate 3D objects, I don’t know if this has been fixed). Owned by Wolfram Research, founded by visionary Dr. Wolfram (see his humble book A New Kind of Science). 145$
- maple. Really powerful, nicer syntax than Mathematica, much better graphics (rotating 3D figures since the early 90s at least). Owned by a spinoff from a university (Waterloo, Canada, hence the name). 99$
- mupad. Used to be free, not any longer. I never had the time to play with it much. 88€
- sage. I have not tried, but it’s very high on my to-try list. Open source, powerful (it seems), and a project based at the University of Washington, where I spent three years of my life (my wife and very good friends were actually at the Department of Mathematics, where the project is hosted, but I never met anyone working on this).
- Numerical analysis (see wikipedia comparison)
- matlab. Very powerful, very popular. Quite simple syntax. Specially useful for linear algebra. 99$
- octave. A matlab clone: an open source program whose syntax mimics matlab’s (but with different underlying code, hence no copyright infringements). Of course it lacks many advanced matlab features, but, hey, it’s free. 0$