On Monday, March 20, 2017 at 3:38:01 AM UTC+1, saad khalid wrote:
> ... Also, Sage often gives solutions that are not as simple as possible, 
> in the sense that they look ugly often. I think this would help with that.

Note that an alternative for this could be to implement special 
smplification algorithms to be applied after integration. Additional 
benefit of this would be that these are then available for any 
simplification task.

> If the main concern is that students don't need integration algorithms so 
> advanced, well, I'm certain there are researchers that do. Clearly 
> Mathematica thinks there are researches that need these algorithms, as they 
> support much more integration than Sage does, which is good enough reason 
> for me to support this for Sage. I don't understand why we would want to be 
> mediocre, it makes sense to be cutting edge.


> And, as I said at the beginning of this message, there is ample reason for 
> this integration method to be useful for students too, as it shows the 
> steps to the solution (which is excellent for studying). 
> ... However, I think that this integration method is good as it provides 
> cleaner output and it gives steps to the solution and it is easy to 
> advertise. The fact that it also performs far more integrals than we 
> currently can is also a great bonus. 
> As far as I can tell, real progress in converting Rubi to python will be 
> possible once it is converted to using binary search instead of pattern 
> matching, right? 

I have looked a bit into this (with GiNaC C++) the last weeks. On first 
glance the main motivation for converting the Rubi rules into a decision 
tree is to avoid the big chunk of work on a sophisticated pattern matcher 
plus a catch-all rule converter in favor of sequential work on single parts 
of Rubi. In principle there can be fast progress if the first version only 
implements general fallback rules like the mentioned 2F1 solutions. Many 
Rubi rules only specialize 2F1 solutions, a sort of 
simplify_hypergeometric() if you want. But then, with only the 
hypergeometric (H) rules the output is ugly as well. You'll get more 
integrals solved than usual algorithms, however, so this low-hanging fruit 
would have a place *after eg Maxima returns an unsolved integral.

Full conversion to decisions of eg a single Rubi chapter like 1.2.1 
(a+bx+cx^2)^p needs other chapters like 1.1.1, 1.1.2, 1.2.2, 1.3.1, 1.3.2 
to get optimal results in the sense of "most simplified". Contrary to my 
own fears I think it would be straightforward after that to add 
step-by-step output in the sense of Rubi steps. They are well documented, 
even if I find that the accompanying code is more accurate, so beware.


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