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I use Octave daily (in the last hour, will return to it when I finish my soup), but have never seen anything that matched Mathematica's symbolic/pure math functionality.

Will check out Sage, haven't used Maxima since 2004. Thanks!

Edit: Sage looks nifty, Maxima looks unchanged, at least at screenshot-level.

SymPy (sympy.org) is a well-developed Python package that works with symbolic/analytical mathematics on a level comparable to Mathematica's.

Octave is more of an open source Matlab equivalent. That is, mostly linear algebra.

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