I just created the Rubi-5 repository on GitHub. It contains an actual functioning prototype of Rubi 5 that shows the structure of the proposed integrator. Also it provides an example how to compile Rubi 4 pattern matching rules into a Rubi 5 if-then-else decision tree. The repository's README file provides more details.
@rfateman Thank you for your interest in Rubi 5. The utility routines needed for the Rubi 5 prototype are defined at the end of its source file in Rubi-5.zip.
Of course, the number of utility routines will increase as Rubi 5 is fully implemented. However, their implementation should not be that difficult.
Looks like Expand, Together, Simplify, polynomial division with remainder are needed. Implementing a substitute for Mathematica's Simplify[] might be a chore. I haven't run (parts of ) Rubi in Lisp recently, or (ever) using Mathematica. I do have a concern that has caused some discussion in the Maxima world. In Maxima there is an attempt to provide an "optimal" solution which has different forms depending on the answers to some questions. e.g. integrate( (ax^2+bx+c)^(3/2),x) can use log or asinh. The current integration program can either use pre-declared assumptions, or asks for the sign of a, whether b is nonzero, and if the discriminant b^2-4ac is zero or not. Interrupting the user with questions is unpopular, especially when the questions are a good deal more complicated! I suppose one could try to answer all questions so as to guide the solution toward the "most generic" answer. I suppose that might include, somehow, all others, even if a far simpler result is sometimes possible. Is there a mechanism to steer Rubi through a potential maze of symbolic inequalities e.g. b^2-4ac > 0 ?? Thanks for keeping up on this project! --RJF
However to select between antiderivatives which are both valid, Rubi assumes the integrand' s variables are positive. For example, Int[1/Sqrt[a + b x + c x^2], x] returns
ArcTanh[(b+2*c*x)/(2*Sqrt[c]*Sqrt[a+b*x+c*x^2])]/Sqrt[c]
whereas Int[1/Sqrt[a + b x - c x^2], x] returns
-ArcTan[(b-2*c*x)/(2*Sqrt[c]*Sqrt[a+b*x-c*x^2])]/Sqrt[c]
Note that both results are mathematically valid no matter what nonzero value is substituted for c and both results are free of the imaginary unit if a positive value is substituted for c.
(Unfortunately we can not see the new issues "on the right" in this gitter chat room. Can you set the link to the repository in the gitter settings to make them appear?)
This open source development you've taken on is impressive and very encouraging, thank you.
It's true that the user of symbols a,b,c could have written -a, -b, -c, but the antiderivatives must be valid for any values of a,b,c positive, negative or zero. There is an advantage to some forms of antiderivative which may represent functions continuous in some designated region, as you know. It's a harder problem though, and Rubi's doing great stuff anyway.
@rfateman Thanks for your endorsement of Rubi. Note that I said "both results are mathematically valid no matter what nonzero value is substituted for c..." (emphasis added). Neither antiderivative is valid if 0 is substituted for c. Happily the set on which they are invalid (i.e. {0} ) is just a set of measure zero.
@nasser1 has posted extensive integration test results including performance times for all the major CAS at Computer Algebra Independent Integration Tests.
@nasser1 would you happen to have timing breakdowns by Section? eg Total time taken by a CAS in the Independent tests, Total time taken by a CAS in the Algebraic Binomial tests... so on.
Poking around - why are some of the integrals here commented @AlbertRich ? https://github.com/RuleBasedIntegration/Rubi/blob/master/Rubi/IntegrationRules/8%20Special%20functions/8.8%20Polylogarithm%20function.m
Do those not pass tests?
@miguelraz The RuleBasedIntegration/Rubi GitHub repository is not being maintained. Instead to get the most up-to-date Rubi 4 source files, integration test-suite and test results go to the Rule-based Integration website.
To answer your questions download the relevant Rubi 4 source file as a pdf or notebook file from the Integration Rules page of Rubi's website. Comments in the source files should explain why some code has been commented-out.
Ah, manually revising comments in the pdfs is definitely going to hamper part of the translation workflow.
Oh well :sweat_smile:
Hey @AlbertRich, good news - I've made some headway into the RUBI port to Julia. (Please don't post this link publicly, gimme a week.) I munged all the *.m files in the Rubi rules and dumped them into a huge JSON - this should lessen efforts by others to porting this to their own languages.
We have
{Sqrt[2*x + 1], x, 1, (1/3)*(1 + 2*x)^(3/2)}
{Sin[x]^3, x, 2, -Cos[x] + Cos[x]^3/3}
Would I be correct in assuming that the
- The first field entry
Sqrt[2*x + 1]is the expression to integrate - The second field
xcontains the only variables with respect to integration - The third field
1contains the "highest level function" the optimal antiderivative may contain (as expressed in the Integration Test Results section) - The fourth field
(1/3)*(1 + 2*x)^(3/2)is the answer
@miguelraz I say again: To get the most up-to-date Rubi 4 source files, integration test-suite files and test results go to the Rule-based Integration website. For Rubi 5 go to RuleBasedIntegration/Rubi-5.
Forget the other RuleBasedIntegration repositories on GitHub. As their dates indicate, they are not being maintained.
Your question about the 4 fields of the integration test-suite files is answered in the first paragraph of the Test Problems page of Rubi's website.
No worries! Your work is paving way for an entire ecosystem for us, so we appreciate all you've done. It really is amazing. If I'm not out of line, do consider bumping the zip file on the Rubi website to
4.17 or akin so that others know it is not the same version as those in the github repos (as is common in semantic versioning practice.)
@miguelraz Like I said in the Rubi 5 README file, the next release of Rubi 4 (version
4.17.0) will hopefully be the last. To avoid confusion, I'm starting to think the out-of-date RuleBasedIntegration repositories should all be deleted, or at least hidden...
But perhaps Rubi is now mature enough to be transformed into a collaborative software project focused on implementing Rubi 5 and then porting it to popular computer algebra systems. The Plan for Implementing Rubi 5 outlines how this could be achieved as a crowd-source project.
So if there is sufficient interest, I propose a new repository for Rubi be created and operated as a conventional GitHub site encouraging collaboration (unlike the now dormant RuleBasedIntegration repository).
Meanwhile, I'll keep my focus on perfecting Rubi 4 to my satisfaction before turning my attention to Rubi 5. So I invite others having the time and inclination, create a repository for The Rubi Project and oversee its operation.