Maxima is a surprisingly good alternative to SymPy. We had to learn that the hard way at my job. Both systems offer similar functionality, but it turns out Maxima is typically 50-200x as fast as SymPy for many tasks (I guess because it was written at a time when computers were slow and expensive). For our application, switching from SymPy to Maxima shortened the time-to-solution from hours to minutes.

Also, it shows that Maxima has been in use for a very long time, and maintained by very bright people. I have yet to encounter a bug in Maxima. And I have already encountered plenty of bugs and inconsistent behavior in SymPy.

Maxima sees serious 'more free' versions of computational algorithms that migrate out from less free acedemic systems.

On the computational real actual algebra side (ie. a bit past computing a quadractic at 'x') a number of things that appear in Magma follow into Maxima (but not all the things, although with time they will come).

https://en.wikipedia.org/wiki/Hidden_Field_Equations

https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis

    The first HFE cryptosystem challenge of J. Patarin is solved by Magma in 82.5 seconds, using 1.3GB memory; with 4 threads the time taken is just 29.0 seconds. This involves solving a system of 80 quadratic equations in 80 variables over GF(2) (each input polynomial has about 1600 terms). The challenge was first solved by J.-C. Faugère in 52 hours on an 1GHz Alpha in 2002 using his unpublished F5/2 algorithm. At the time of writing, there is apparently still no publically-available software besides Magma which can solve the challenge.

http://magma.maths.usyd.edu.au/magma/overview/2/27/13/#subsu...

https://maxima.sourceforge.io/docs/manual/maxima_257.html

    Notes on the grobner package: The code is a little bit out of date now. Modern implementation use the fast F4 algorithm described in "A new efficient algorithm for computing Gröbner bases (F4)" Jean-Charles Faugère

( *NB a somewhat old example, I'm not keeping up on the comp.algebra lists and I'm assuming here the online manuals are up to date )

Can check: https://github.com/sympy/sympy/wiki/SymPy-vs.-Maxima

If you need programmability or interoperability, Sympy is way nicer. If you just want an interactive symbolic calculator, Maxima is fine but sometimes quirky (has odd conventions due mainly to its age). As heisig points out, Maxima can be quite a bit faster (but I run into slow things with it too). Using Maxima via Sage is in some ways the best of both worlds.

You may also be interested in SymEngine: https://github.com/symengine/symengine

It is better at symbolic integration:

https://www.12000.org/my_notes/CAS_integration_tests/reports...

From FOSS CAS Fricas seems to fair best. But damn, Mathematica is on another level. And somehow Rubi looks even more impressive and puts a foundation for shareable high-quality integration to other systems.