I don't really see sympy and the like surpassing Mathematica. Even maple is kinda having trouble. If when I'm buying Mathematica I'm paying for a proper implementation of the risch (integration) algorithm, it will have been money well spent.
* Even when it succeeds, it doesn't necessarily give you the simplest possible form of the solution.
* It is concerned with elementary functions, but real-world problem often involve non-elementary functions (like Bessel and hypergeometric functions). You can extend the Risch algorithm to work with such functions, but this is complicated.
* It doesn't allow computing parametric answers for common parametric families of integrals (this needs to be done using heuristics and lookup tables).
* It doesn't give you an algorithm for definite integration, except in simple cases. Even with proper definite integrals, you need to be extremely careful with branch cuts when applying the fundamental theorem of calculus. For improper definite integrals, systems like Mathematica generally try to go via convolutions of Meijer G-functions, using huge lookup tables and simplification heuristics.
* Even more fundamentally, the Risch algorithm (and symbolic integration more generally) puts heavy demands on the underlying symbolic computation engine. In particular, it depends on the ability to decide whether an expression is equivalent to zero, which in fact is an undecidable problem, and this strictly speaking makes the Risch "algorithm" a non-algorithm (though it is a proper algorithm when restricted to a ground field where zero testing is effective, such as Q).
Given enough time, it will happen. There are several reasons. One, people who publish mathematics academically are reluctant to use a closed-source tool to create their results, because those results can't be examined and verified at their source. This issue pushes people toward sympy/Sage/Jupyter and other similar open environments, even though at the moment they aren't as powerful as the Wolfram environment.
For example, imagine that the four-color map problem (the first significant computer-aided result in mathematics) had been solved by a close-source environment (instead of an open, freely readable source as it was). If that had been true, people would still be arguing about whether the result was valid.
Two, an open-source environment like sympy attracts technically skilled people intent on improving it, knowing that they're contributing to a tangible kind of progress that's clear to everyone, and then freely copied by like-minded people into different environments, all with the intent to advance human knowledge, rather than make stockholders happy.
Three, this may not be apparent, but the existence of environments like Sage and sympy are putting tremendous pressure on Wolfram to price their products more reasonably. Instead, and so far, Wolfram's strategy has been to design enticing free samplers like Wolfram Alpha as gateways to their expensive offerings.
> If when I'm buying Mathematica I'm paying for a proper implementation of the risch (integration) algorithm, it will have been money well spent.
This is a perfect example -- until Mathematica is open-sourced, no one can know whether the existing algorithm is either complete or optimal. Also, given that research into this algorithm has enormous practical value in both mathematics and computer science, it's a shame that a large segment of the effort is being carried out in secret. I'm sure those responsible for its present form would love to publish their results, if only for the fact that this would get them fired and possibly prosecuted.
There was a time when people who made an original contribution to human knowledge would publish their results in the open (think Einstein) and take pride in their contribution to the common good. It seems those days are past.
Mathematica isn't magic and the source code for your mathematica program is perfectly readable text. Reproducing the result from a mathematica program without mathematica is often time consuming and occasionally painful, but it's rarely a hard problem. So if the four-color map problem had first been solved using Mathematica, it wouldn't have taken many weeks for the solution to get reproduced in a different languages. Hell you could probably reproduce the necessary code just from reading the paper without ever looking at their source code.
Yes, but the user's program only tells Mathematica what to do, not how to do it. How Mathematica does it is proprietary, and this represents a serious transparency problem for academic work and publication.
> Hell you could probably reproduce the necessary code just from reading the paper without ever looking at their source code.
If this were true, Wolfram wouldn't be able to charge thousands of dollars per copy of Mathematica. The reason Mathematica is expensive is because of what it conceals, not what it reveals.