If systematically applied, the integration rules provided on this website can determine the antiderivative of a wide variety of mathematical expressions. As proof, a rule-based integrator nicknamed Rubi was implemented using these rules. Organized as a decision tree based on the type of the integrand, the 6000+ rules Rubi uses can be viewed in human-readable form or downloaded in machine-readable form. To read about the feasibility and desirability of organizing mathematical knowledge as a rule-based decision tree, click on the About>Vision menu option above.
Rubi dramatically out-performs Maple and Mathematica (the two major commercial computer algebra systems) on a grueling integration test suite. Consisting of over 55 thousand integrands and optimal antiderivatives, the entire test suite is also available for downloading. This chart shows the percentage of test suite problems for which these systems were able to find optimal antiderivatives. What constitutes an optimal antiderivative is defined following the table below.
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The following chart shows the percentage of optimal antiderivatives found by the three integrators for various types of integrands:
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For example, it shows Mathematica has the most difficulty with integrands involving trig functions; whereas Maple has the most difficulty with logarithms.
Both the above bar charts are based on the following test suite results:
| Integration Test Suite Results | |||||||||||||||
| Maple 18 | Mathematica 10 | Rubi 4.9 | |||||||||||||
| Integrand Types | Problems | Optimal | Nonoptimal | Fail | Optimal | Nonoptimal | Fail | Optimal | Nonoptimal | Fail | |||||
| Algebraic linear functions | 4258 | 3533 | 392 | 333 | 3940 | 304 | 14 | 4253 | 5 | 0 | |||||
| Algebraic quadratic functions | 5935 | 4591 | 1130 | 214 | 5626 | 283 | 26 | 5935 | 0 | 0 | |||||
| Algebraic binomial functions | 6481 | 4985 | 874 | 622 | 5976 | 483 | 22 | 6475 | 6 | 0 | |||||
| Algebraic trinomial functions | 2022 | 1447 | 451 | 124 | 1779 | 213 | 30 | 2021 | 1 | 0 | |||||
| Miscellaneous algebraic functions | 1722 | 1185 | 344 | 193 | 1401 | 192 | 129 | 1708 | 7 | 7 | |||||
| Exponential functions | 856 | 650 | 69 | 137 | 746 | 76 | 34 | 852 | 0 | 4 | |||||
| Trig functions | 18050 | 8776 | 6826 | 2448 | 11878 | 5395 | 777 | 18026 | 14 | 10 | |||||
| Hyperbolic functions | 3905 | 2098 | 1274 | 533 | 3240 | 626 | 39 | 3905 | 0 | 0 | |||||
| Logarithm functions | 1058 | 410 | 153 | 495 | 909 | 117 | 32 | 1057 | 0 | 1 | |||||
| Inverse trig functions | 3603 | 2567 | 407 | 629 | 3236 | 224 | 143 | 3599 | 2 | 2 | |||||
| Inverse hyperbolic functions | 4877 | 2401 | 597 | 1879 | 4439 | 253 | 185 | 4872 | 1 | 4 | |||||
| Special functions | 1239 | 644 | 59 | 536 | 991 | 29 | 219 | 1239 | 0 | 0 | |||||
| Independent test problems | 1424 | 1146 | 173 | 105 | 1247 | 158 | 19 | 1373 | 28 | 23 | |||||
| Totals | 55430 | 34433 | 12749 | 8248 | 45408 | 8353 | 1669 | 55315 | 64 | 51 | |||||
| Percentages | 62.1% | 23.0% | 14.9% | 81.9% | 15.1% | 3.0% | 99.8% | 0.1% | 0.1% | ||||||
The following summarizes the meaning of the numbers under the column headings in the above table:
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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.