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 Types Problems Optimal Messy Unable Timeout Optimal Messy Unable Timeout Optimal Messy Unable Timeout
Algebraic binomial functions 8832 8832 0 0 0 7317 1094 421 0 8275 529 28 0
Algebraic trinomial functions 7013 7013 0 0 0 5320 1559 112 22 6569 425 18 1
Algebraic functions 1635 1624 6 5 0 1274 231 130 0 1416 167 52 0
Exponential functions 838 834 0 4 0 669 55 114 0   696 114 27 1
Logarithm functions 1049 1047 0 2 0 403 176 467 3 922 99 27 1
Trig functions 13183 13140 4 39 0 7657 3936 1554 36 8733 3670 167 613
Inverse trig functions 1612 1609 0 3 0 1122 186 300 4 1406 111 57 38
Hyperbolic functions 3761 3747 1 13 0 2704 488 562 7 3063 573 0 125
Inverse hyperbolic functions 4099 4093 0 6 0 2343 295 1451 10 3726 199 98 76
Special functions 1241 1241 0 0 0 865 56 320 0 986 27 218 10
Contributed problems 200 185 2 13 0 152 15 33 0 155 26 17 2
Totals 43463 43365 13 85 0 29826 8091 5464 82 35947 5940 709 867
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:

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:

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:

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

If you have access to Mathematica 7 or better, you can download and run Rubi 4.6 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 Rubi 4.6 zip file into a directory of your choice. Then from Mathematica, open the notebook file Rubi4.6.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 to Albert 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.

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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.