Sympy is AWESOME! In particular, http://live.sympy.org is a great way to learn and teach math... I often send links to my students with an entire sequence of steps to find the solution. Can you factor x^2-5x+6 ? (find numbers a,b such that (x-a)(x-b)=x^2-5x+6) http://live.sympy.org/?evaluate=solve(%20x**2-5*x%2B6%2C%20x... (it's like an entire iPython notebook in a URL)On the topic of sympy, I'm working on this short tutorial---an introduction to sympy based around topics from the standard high school and first-year university curriculum: http://minireference.com/static/sympy_tutorial.pdfPlease don't post the tutorial on HN yet---I'm working out some last typos and I want to time the "official" announcement on HN with the beginning of the school year.

 SymPy Gamma is also very interesting. Yes, it's no real competitor to WolframAlpha, but it still has quite a few useful features.http://www.sympygamma.com/input/?i=integrate+log%28x^2%29For instance, as you can see in the link above, it can provide steps for derivatives and integrals. WA has the same feature, but you need to subscribe to actually use it without limits.
 Just FYI, Sage also has the Sage Cell Server, which is similar in that you can do calculations on a webpage, but also allows embedding live cells in webpages easily, sage interacts, 3d graphics, etc. For example, here is your calculation: http://sagecell.sagemath.org/?q=mthhin. Click the "About" link in the upper right for more information.
 I'm a Google Summer of Code mentor for SymPy. There's a number of good additions to it being done this summer. My student is working on systems of ODEs. There's 10 different projects in total, including work on optics, 3D geometry, and vectors (and hopefully vector calculus).The optics and the 3D geometry should help a lot with teaching and learning.
 Seconded!I used Sympy recently to solve the rocket equation. It involved some integrals that my rusty math skills couldn't solve so I used Sympy to solve them.Then I ran a side by side comparison between the integrals I solved with Sympy against a numerical solution using Scipy and Numpy to verify that my results are correct.The only negative thing about Sympy is that it's rather slow.Here's the code if you're interested: https://gist.github.com/rikusalminen/6d6bb834d48b9664b38d

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