If systematically applied, the integration rules provided on this website can determine the
antiderivative of a wide variety of mathematical expressions.
As proof, a **ru**le-**b**ased **i**ntegrator nicknamed ** Rubi** was
implemented using these rules.
Organized as a decision tree based on the type of the integrand, the 6000+ rules

** Rubi** dramatically out-performs

The following chart shows the percentage of optimal antiderivatives found by the three integrators for various types of integrands:

For example, it shows ** Mathematica** has the most difficulty with integrands involving trig functions; whereas

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:

**Problems**: the number of integration problems for each type of integrand.**Optimal**: for, 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.*Rubi***Nonoptimal**: for, the number of results that are correct*Rubi**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.**Fail**: the number of problems the system returns unintegrated but that are actually integrable in terms of functions known by the system, plus the number of problems the system fails to integrate within a 120 second time limit.

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