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.
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:If you have access to Mathematica 7 or better, you can download and run Rubi 4.4 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 4.4 zip file into a directory of your choice. Then from Mathematica, open the notebook file Rubi4.4.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.