SymPy is a symbolic manipulation package, written in pure Python. Its aim is to become a full featured CAS in Python, while keeping the code as simple as possible in order to be comprehensible and easily extensible.
http://sympy.org/media/logo.png
Homepage: http://sympy.org/
Source Code: http://github.com/sympy/sympy
Mailing list: https://groups.google.com/forum/?fromgroups#!forum/sympy
Examples
>>> from sympy import *
>>> x, y, z, t = symbols('x y z t')
>>> k, m, n = symbols('k m n', integer=True)>>> integrate(exp(x)*x**2, x) ⎛ 2 ⎞ x ⎝x - 2⋅x + 2⎠⋅e
>>> expand((x + y)**3) 3 2 2 3 x + 3⋅x ⋅y + 3⋅x⋅y + y >>> factor(_) 3 (x + y)
>>> Sum((-1)**n*x**n/factorial(n), (n, 0, oo)) ∞ ____ ╲ ╲ n n ╲ (-1) ⋅x ╱ ──────── ╱ n! ╱ ̅ ̅ ̅ ̅ n = 0 >>> _.doit() -x e