Symbolic Math Programs: Which has the least number of bugs?

[elect_eng]

I've been a little old fashioned when it comes to doing symbolic computations. For years I've resisted using programs like Mathematica, Maple, Maxima etc. to handle symbolic manipulations. Over the past several months I've started using Maxima, choosing this program because it is free. I always double check the final answers by hand, but I'm finding that using Maxima as a tool is useful and can speed up my work.

I've recently encountered some bugs in Maxima. One day I found that a symbolic definite integral calculation came out with a wrong sign. Note that I'm not talking about the numerical definite integral calculation which seems to work fine. I then started comparing other examples of symbolic definite integrals. I went through a definite integral table and was alarmed at the high percentage of answers that had the wrong sign. I went on-line to investigate and a quick scan made it clear that all of these symbolic calculators have some bugs.

My question is whether anyone has any experience with which package is the most reliable in general. Also, more particularly, is there a package that handles symbolic definite integrals best.

 

[CRGreathouse]

Out of curiosity, can we have some examples of expressions that Maxima did wrong?

 

[elect_eng]

Out of curiosity, can we have some examples of expressions that Maxima did wrong?

Here is one example:

integrate(sqrt(-log(x)), x, 0, 1);

This yields -sqrt(pi)/2, but it should be sqrt(pi)/2, according to my math table.

Numerical Integration yields 0.8862269254526032 as expected.