# Integration Test Results

The results of running Rubi, Mathematica and Maple’s built-in integrators on the integration test-suite are available for viewing by clicking on the links below. For each problem on which a system produces a deficient result, the following items are shown:

- The problem number followed by a brief explanation why the result is deficient
- The problem shown as an indefinite integral
- The optimal antiderivative shown in blue and preceded by its type number, leaf count size, and the number of steps Rubi uses to integrate it
- The deficient result shown in black and preceded by its type number, leaf count size, and if Rubi is the system being tested the number of steps Rubi uses to integrate it

The type number of an expression indicates the highest function type level it involves. A test result is considered deficient if its type number is greater than the type number of the optimal antiderivative. The following summarizes the levels of the various function types:

- 1 = Rational functions
- 2 = Algebraic functions
- 3 = Elementary functions
- 4 = Special functions
- 5 = Hypergeometric functions
- 6 = Appell functions
- 7 = Nonclosed-form functions (e.g. RootSum)
- 8 = Integrate functions
- 9 = Unrecognized functions

## Rubi Test Results

Test results for both the current and previous versions of Rubi on the over 72,000 problems in the current integration test-suite are available for viewing as pdf files. The results are divided into separate pdf files based on the type of integrand being tested.

At the end of each file is a pie chart summarizing the results. The chart is divided into colored slices as follows:

- green = the number of optimal results
- light green = the number of valid but suboptimal results
- yellow = the number of results unnecessarily involving higher level functions or complex numbers
- orange = the number of integrable problems that were not integrated
- magenta = the number of problems that timed out after 120 seconds
- red = the number of results that were
*not*valid antiderivatives

Rubi 4.16.1 results Rubi 4.16.0 results on independent test-suites

Rubi 4.16.1 results Rubi 4.16.0 results on algebraic function integration problems

Rubi 4.16.1 results Rubi 4.16.0 results on exponential function integration problems

Rubi 4.16.1 results Rubi 4.16.0 results on logarithm function integration problems

Rubi 4.16.1 results Rubi 4.16.0 results on trig function integration problems

Rubi 4.16.1 results Rubi 4.16.0 results on inverse trig function integration problems

Rubi 4.16.1 results Rubi 4.16.0 results on hyperbolic function integration problems

Rubi 4.16.1 results Rubi 4.16.0 results on inverse hyperbolic function integration problems

Rubi 4.16.1 results Rubi 4.16.0 results on special function integration problems

When new capabilities are incorporated into Rubi, problems are also added to the integration test-suite to thoroughly test them. Thus the test results showing the deficiencies of the previous version of Rubi highlight the new capabilities of the current version. But note that when “?” is shown as the number of steps required to find the optimal antiderivative, neither the current nor previous version of Rubi is able to produce the optimal antiderivative.

## Mathematica and Maple Test Results

Test results for Mathematica 11.3 and Maple 2018.2 on the current integration test-suite are also available for viewing as pdf files. The results are divided into separate pdf files based on the type of integrand being tested. Because of the large size and number of deficient results, Maple was tested on a randomly chosen quarter of the problems in the test-suite to avoid crashing the system.

At the end of each file is a pie chart summarizing the results. The chart is divided into colored slices as follows:

- green = the number of optimal results
- light green = the number of results more than twice the size of the optimal antiderivative
- yellow = the number of results unnecessarily involving higher level functions or complex numbers
- orange = the number of integrable problems that were not integrated
- red = the number of problems that timed out after 120 seconds

Mathematica results Maple results on independent test-suites

Mathematica results Maple results on linear binomial products integration problems

Mathematica results Maple results on quadratic binomial products integration problems

Mathematica results Maple results on general binomial products integration problems

Mathematica results Maple results on improper binomial products integration problems

Mathematica results Maple results on quadratic trinomial products integration problems

Mathematica results Maple results on quartic trinomial products integration problems

Mathematica results Maple results on general trinomial products integration problems

Mathematica results Maple results on improper trinomial products integration problems

Mathematica results Maple results on miscellaneous algebraic functions integration problems

Mathematica results Maple results on exponential function integration problems

Mathematica results Maple results on logarithm function integration problems

Mathematica results Maple results on sine integration problems

Mathematica results Maple results on cosine integration problems

Mathematica results Maple results on tangent integration problems

Mathematica results Maple results on cotangent integration problems

Mathematica results Maple results on secant integration problems

Mathematica results Maple results on cosecant integration problems

Mathematica results Maple results on miscellaneous trig function integration problems

Mathematica results Maple results on inverse sine integration problems

Mathematica results Maple results on inverse cosine integration problems

Mathematica results Maple results on inverse tangent integration problems

Mathematica results Maple results on inverse cotangent integration problems

Mathematica results Maple results on inverse secant integration problems

Mathematica results Maple results on inverse cosecant integration problems

Mathematica results Maple results on hyperbolic sine integration problems

Mathematica results Maple results on hyperbolic cosine integration problems

Mathematica results Maple results on hyperbolic tangent integration problems

Mathematica results Maple results on hyperbolic cotangent integration problems

Mathematica results Maple results on hyperbolic secant integration problems

Mathematica results Maple results on hyperbolic cosecant integration problems

Mathematica results Maple results on miscellaneous hyperbolic function integration problems

Mathematica results Maple results on inverse hyperbolic sine integration problems

Mathematica results Maple results on inverse hyperbolic cosine integration problems

Mathematica results Maple results on inverse hyperbolic tangent integration problems

Mathematica results Maple results on inverse hyperbolic cotangent integration problems

Mathematica results Maple results on inverse hyperbolic secant integration problems

Mathematica results Maple results on inverse hyperbolic cosecant integration problems

Mathematica results Maple results on special function integration problems

## Integration Test Result Summary

The following table summarizes the results produced by Rubi, Mathematica and Maple on the integration test-suite. The first column gives the total number of problems for the integrand type specified in the last column. The column for each integrator gives the following separated by colons:

- the number of optimal results
- the number of results more than twice the size of the optimal antiderivative
- the number of results unnecessarily involving higher level functions or complex numbers
- the number of integrable problems that were not integrated or timed out after 120 seconds

Total | Rubi 4.16.1 | Mathematica 11.3 | Maple 2018.2 | Integrand type |
---|---|---|---|---|

1892 | 1838 : 28 : 7 : 19 | 1579 : 123 : 149 : 41 | 1481 : 181 : 57 : 173 | Independent test-suites |

26125 | 26092 : 9 : 13 : 11 | 19886 : 1252 : 4342 : 645 | 15613 : 5909 : 221 : 4382 | Algebraic functions |

965 | 963 : 0 : 0 : 2 | 844 : 50 : 9 : 62 | 618 : 154 : 0 : 193 | Exponentials |

3085 | 3078 : 1 : 0 : 6 | 2387 : 377 : 199 : 122 | 710 : 890 : 0 : 1485 | Logarithms |

22551 | 22515 : 12 : 5 : 19 | 13443 : 3445 : 4436 : 1227 | 10634 : 7773 : 431 : 3713 | Trig functions |

4585 | 4582 : 1 : 0 : 2 | 3965 : 225 : 193 : 202 | 2785 : 950 : 0 : 850 | Inverse trig functions |

5166 | 5164 : 0 : 2 : 0 | 3989 : 503 : 386 : 288 | 2571 : 1462 : 42 : 1091 | Hyperbolic functions |

6626 | 6624 : 0 : 0 : 2 | 5544 : 158 : 559 : 365 | 3809 : 1466 : 6 : 1345 | Inverse hyperbolic functions |

1949 | 1947 : 2 : 0 : 0 | 1541 : 71 : 21 : 316 | 1237 : 145 : 31 : 536 | Special functions |

72944 | 72803 : 53 : 27 : 61 | 53178 : 6204 : 10294 : 3268 | 39458 : 18930 : 788 : 13768 | Entire test-suite |