符号积分用的Rubi软件包(Mathematica下的Package)

http://www.apmaths.uwo.ca/~arich/

Rule-based Mathematics

Symbolic Integration Rules

Crafted by Albert D. Rich, Applied Logician

If systematically applied, the integration rules provided on this website can determine the antiderivative for a wide variety of mathematical expressions. As proof, a rule-based integrator nicknamed Rubi was implemented using these rules. Rubi dramatically out-performs Maple and Mathematica (the two major commercial computer algebra systems) on a test suite of over 43 thousand integration problems.

The following table summarizes the result of running the integration test suite on these systems as of 8 May 2013:

     Integration Test Suite Results           Rubi 4 Maple 17 Mathematica 9  Integrand TypesProblems OptimalMessyUnableTimeout OptimalMessyUnableTimeout OptimalMessyUnableTimeout  Algebraic binomial functions8832 8832000 731710944210 8275529280  Algebraic trinomial functions7013 7013000 5320155911222 6569425181  Algebraic functions1635 1624650 12742311300 1416167520  Exponential functions838 834040 669551140 696114271  Logarithm functions1049 1047020 4031764673 92299271  Trig functions13183 131404390 76573936155436 87333670167613  Inverse trig functions1612 1609030 11221863004 14061115738  Hyperbolic functions3761 37471130 27044885627 30635730125  Inverse hyperbolic functions4099 4093060 2343295145110 37261999876  Special functions1241 1241000 865563200 9862721810  Contributed problems200 1852130 15215330 15526172  Totals43463 4336513850 298268091546482 359475940709867  Percentages  99.8%0.0%0.2%0.0% 68.6%18.6%12.6%0.2% 82.7%13.7%1.6%2.0% �                 �

The following summarizes the meaning of the numbers under the column headings in the above table:

• Problems: the number of integration problems for each integrand type.
• Optimal: for Rubi 4, the number of results identical to the optimal antiderivative; for the other systems, the number of results no more than twice the size of the optimal antiderivative, based on leaf counts.
• Messy: for Rubi 4, the number of results that are correct but not identical to the optimal antiderivative; for the other systems, the number of results that are more than twice the size of the optimal antideriative.
• Unable: the number of problems the system returns unintegrated but that are actually integrable in terms of functions known by the system.
• Timeout: the number of problems the system fails to integrate within a 25 second timelimit.

Click on Highlights of Integration Test Results to see numerous examples comparing the Rubi, Maple and Mathematica integrators.

To see a table showing how recent versions of Rubi and four other symbolic integrators perform on a test-suite of symbolic integration problems written independently of Rubi, click on one of the following:

• Charlwood Integration Problems & Test Results
• Timofeev Integration Problems & Test Results

Click on A Knowledge Repository for Indefinite Integration Based on Transformation Rules to see an article describing the principles used to the build system of over 5000 rules Rubi uses to efficiently integrate a wide variety of mathematical expressions.

To view or download the rules Rubi uses to integrate expressions, click on one of the following file types:

• Rules in portable document format (pdf) files
• Rules in Mathematica notebook files

To view or download the indefinite integration problems in the test suite, click on one of the following formats:

• Problems in Axiom syntax text files
• Problems in Maple syntax text files
• Problems in Mathematica syntax package files
• Problems in Maxima syntax text files

To view or download the raw indefinite integration test results as generated by these systems, click on one of the following:

• Rubi 4 test results
• Maple 17 test results
• Mathematica 9 test results

If you have access to Mathematica 7 or better, you can download and run Rubi 4.7 for yourself. In addition to being able to integrate problems like those in the test suite, Rubi provides the option of showing the rules required to integrate expressions, along with the intermediate results. Extract the contents of the(下载->) Rubi 4.7 zip file into a directory of your choice. Then from Mathematica, open the notebook file Rubi4.7.nb. Click on the sample integration problem at the end of the notebook and press Shift-Enter to evaluate it. After a minute or so depending on the speed of your computer, the first step of the integration should be displayed. To see successive steps, click on the intermediate results and press Shift-Enter.

I encourage the submission of new rules and test problems, preferably in the same format as the files on this website. Please send your comments and suggestions toAlbert Rich.

The mathematical knowledge on this website is freely available for any educational, academic or commercial use. Please include the website address and appropriately acknowledge its author in any product incorporating its contents.

Maple is a registered trademark of Maplesoft.
Mathematica is a registered trademark of Wolfram Research, Inc. who generously provided a copy of Mathematica to support this research.