# 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 resultsRubi 4.16.0 results   on independent test-suites
Rubi 4.16.1 resultsRubi 4.16.0 results   on algebraic function integration problems
Rubi 4.16.1 resultsRubi 4.16.0 results   on exponential function integration problems
Rubi 4.16.1 resultsRubi 4.16.0 results   on logarithm function integration problems
Rubi 4.16.1 resultsRubi 4.16.0 results   on trig function integration problems
Rubi 4.16.1 resultsRubi 4.16.0 results   on inverse trig function integration problems
Rubi 4.16.1 resultsRubi 4.16.0 results   on hyperbolic function integration problems
Rubi 4.16.1 resultsRubi 4.16.0 results   on inverse hyperbolic function integration problems
Rubi 4.16.1 resultsRubi 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 resultsMaple results   on independent test-suites

Mathematica resultsMaple results   on linear binomial products integration problems
Mathematica resultsMaple results   on quadratic binomial products integration problems
Mathematica resultsMaple results   on general binomial products integration problems
Mathematica resultsMaple results   on improper binomial products integration problems
Mathematica resultsMaple results   on quadratic trinomial products integration problems
Mathematica resultsMaple results   on quartic trinomial products integration problems
Mathematica resultsMaple results   on general trinomial products integration problems
Mathematica resultsMaple results   on improper trinomial products integration problems
Mathematica resultsMaple results   on miscellaneous algebraic functions integration problems

Mathematica resultsMaple results   on exponential function integration problems
Mathematica resultsMaple results   on logarithm function integration problems

Mathematica resultsMaple results   on sine integration problems
Mathematica resultsMaple results   on cosine integration problems
Mathematica resultsMaple results   on tangent integration problems
Mathematica resultsMaple results   on cotangent integration problems
Mathematica resultsMaple results   on secant integration problems
Mathematica resultsMaple results   on cosecant integration problems
Mathematica resultsMaple results   on miscellaneous trig function integration problems

Mathematica resultsMaple results   on inverse sine integration problems
Mathematica resultsMaple results   on inverse cosine integration problems
Mathematica resultsMaple results   on inverse tangent integration problems
Mathematica resultsMaple results   on inverse cotangent integration problems
Mathematica resultsMaple results   on inverse secant integration problems
Mathematica resultsMaple results   on inverse cosecant integration problems

Mathematica resultsMaple results   on hyperbolic sine integration problems
Mathematica resultsMaple results   on hyperbolic cosine integration problems
Mathematica resultsMaple results   on hyperbolic tangent integration problems
Mathematica resultsMaple results   on hyperbolic cotangent integration problems
Mathematica resultsMaple results   on hyperbolic secant integration problems
Mathematica resultsMaple results   on hyperbolic cosecant integration problems
Mathematica resultsMaple results   on miscellaneous hyperbolic function integration problems

Mathematica resultsMaple results   on inverse hyperbolic sine integration problems
Mathematica resultsMaple results   on inverse hyperbolic cosine integration problems
Mathematica resultsMaple results   on inverse hyperbolic tangent integration problems
Mathematica resultsMaple results   on inverse hyperbolic cotangent integration problems
Mathematica resultsMaple results   on inverse hyperbolic secant integration problems
Mathematica resultsMaple results   on inverse hyperbolic cosecant integration problems

Mathematica resultsMaple 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