Symbolic algebra: Sage, Maxima, Ginac, Sympy?

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Concerning relations between Sage, maxima and ginac, you can have a look to this question.

Concerning sympy, some computations can be done by sympy but you need to specify it, see for example:

sage: integral?

sympy is not imported in Sage by default, i guess the main reason is startup speed. If you want to use sympy within Sage, just do:

sage: import sympy

Now, if you have an element of the Symbolic Ring, you can transform it into a sympy object as follows:

sage: a = cos(x) + pi
sage: b = a._sympy_()
sage: b
cos(x) + pi

You can check:

sage: type(b)
<class 'sympy.core.add.Add'>

You can come back to the symbolic ring as follows:

sage: b._sage_()
pi + cos(x)