Things on this page are fragmentary and immature notes/thoughts of the author. Please read with your own judgement! Things on this page are fragmentary and immature notes/thoughts of the author. Please read with your own judgement!
Mathematica
The best one that I've ever used, but commercial unfortunately.
sympy
A computer algebra system written in pure Python
Maxima
A more symbolic-centric package that is an open source development of the old commercial Macsyma code.
SageMath
A good open souce alternative to Mathematica. sympy, Maxima, etc. are integrated into SageMath.
Which One to Use
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If you need the most advanced symbolic computation support and can afford to pay, use Mathematica.
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If you want a opensource alternative to Mathematica, use SageMath.
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If you need a symbolic computation library, use sympy.