SymPy

SymPy is a Python library for symbolic mathematics which simplifies expressions, compute derivatives, integrals, and limits, solve equations and work with matrices.

SymPy includes features such as modules for plotting such as coordinate modes, Plotting Geometric Entities, 2D and 3D, Interactive interface, Colors and Matplotlib support, printing like 2D pretty printed output of math formulas, or LATEX, code generation, physics, statistics, combinatorics, number theory, geometry and logic, Conversion from Python objects to SymPy objects, Optional implicit multiplication and function application parsing, Limited Mathematica and Maxima parsing: example on SymPy Live and Custom parsing transformations and Shift cipher, Affine cipher, Bifid ciphers, Vigenere’s cipher, Substitution ciphers, Hill’s cipher, RSA, Kid RSA, Linear feedback shift registers and ElGamal encryption for cryptography.

SymPy tries to rewrite mathematically equivalent expressions to a canonical form when evaluating them to make equality testing useful in basic cases. SymPy is efficient enough for interactive use as an advanced calculator. SymPy is written entirely in Python, and the speed seems comparable to Maxima. With SymPy, users can just as easily use it in an interactive Python environment or import it in their own Python application.

SymPy also provides APIs to make it easy to extend it with users own custom functions. Symbolic computation systems (which by the way, are also often called computer algebra systems, or just CASs) such as SymPy are capable of computing symbolic expressions with variables. With a symbolic computation system like SymPy, square roots of numbers that are not perfect squares are left unevaluated by default.