Is there a free program like Mathematica?

There are several options depending on what you need, here are a few:
Axiom: http://savannah.nongnu.org/projects/axiom
GiNaC: http://www.ginac.de/
Yacas: http://www.xs4all.nl/~apinkus/yacas.html
Fermat: http://www.bway.net/~lewis/

and you can probably find more info at SymbolicNet: http://www.symbolicnet.org/

I personally like R (http://www.r-project.org/) but it more a replacement for S or Matlab than for Maple or Mathematica. Good luck!

Thanks, Stephan. Are there any executables for Windows?

Or can you tell me the easiest way to compile on XP?

If you can't find an windows executables for the respective projects (you might be able to), you can always use cygwin (http://cygwin.com/) to get a decent linux environment inside of windows where you can run all of those. Or just dual-boot.

Another option is SAGE. It is pretty mature. The current version is 2.6.x.

SAGE provides features that are an OSS alternative to Mathematica, MATLAB, Maple and Magma.

SAGE: http://modular.math.washington.edu/sage/

SAGE has installation packages for Windows, Linux and Mac. They have an online notebook if you prefer, as well as, a VMWare virtual appliance.

@rajames429
+1, SAGE is a great and versatile tool.