# Chapter 5 of Timofeev Integration Test Suite

## Summary of Test Results

A. F. Timofeev's 1948 book Integration of Functions includes numerous indefinite integration problems involving a wide variety of rational, algebraic and elementary function integrands. The following table shows the score given Rubi and a variety of computer algebra systems for each of the problems in Chapter 5 of Timofeev's book:

 Problem # Rubi 4.8 Mathematica 10 Maple 18 Maxima 5.28 Derive 6.1 1 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 4 2 2 2 2 2 5 2 2 2 2 2 6 2 2 2 2 2 7 2 2 2 2 2 8 2 2 2 2 2 9 2 2 2 2 2 10 2 2 2 2 2 11 2 1 2 2 2 12 2 2 2 2 2 13 2 2 2 2 2 14 2 2 2 2 2 15 2 2 2 2 2 16 2 2 2 2 2 17 2 2 2 2 2 18 2 2 2 2 2 19.1 2 2 2 2 2 19.2 2 2 2 2 2 19.3 2 2 2 2 2 19.4 2 2 2 2 2 19.m 2 2 0 x x 20 2 1 2 2 0 21 2 2 2 2 2 22 2 1 2 2 2 23 2 2 2 2 2 24 2 2 2 2 2 25 2 2 2 2 2 26 2 2 2 2 2 27 2 2 2 2 2 28 2 2 2 2 2 29 2 1 2 2 0 30 2 2 2 2 2 31 2 2 2 2 2 32 2 2 2 2 2 33 2 2 2 2 2 34 2 2 2 2 2 35 2 2 2 2 2 36 2 2 2 2 2 37 2 2 2 2 2 38 2 2 2 2 2 39 2 2 2 2 2 40 2 2 2 2 2 41 2 2 2 2 2 42 2 2 2 2 2 43 2 2 2 2 2 44 2 2 2 2 2 45 2 2 2 2 2 46 2 2 2 2 2 47 2 2 2 2 2 48 2 1 2 2 2 49 2 2 2 2 2 50 2 2 2 0 2 51 2 1 2 2 2 52 2 2 2 2 2 53 2 2 2 2 2 54 2 2 2 2 2 55 2 1 2 2 2 56 2 1 2 2 2 57a 2 2 2 0 2 57b 2 2 2 0 2 58a 2 1 1 2 2 58b 2 1 2 1 2 59 2 1 2 1 2 60 2 2 2 0 2 61 2 2 0 2 2 62 2 2 2 2 2 63 2 2 2 2 2 64 2 1 2 1 2 65 2 2 1 2 2 66 1 1 1 1 1 67a 2 1 2 0 0 67b 2 1 2 0 0 68 2 1 1 0 0 69 2 2 2 0 0 70 2 1 1 0 0 71 2 1 1 0 0 72 2 2 2 0 0 73 2 2 2 0 0 74 2 2 2 0 0 75 0 1 1 0 0 76 2 2 2 2 2 77 2 2 1 2 2 78 2 2 0 2 2 79 2 2 0 2 2 80 1 1 2 2 2 81 2 0 1 2 2 82 2 2 0 2 2 83 2 2 2 2 2 84 2 2 2 2 2 85 2 2 2 2 2 86 2 2 2 2 2 87 2 2 2 1 2 88 2 1 2 2 2 89 2 2 2 2 2 90 2 1 2 2 0 91 1 1 0 0 2 92 2 1 2 2 2 93 2 2 2 0 2 94 2 2 2 1 2 95 2 2 2 2 2 96 2 2 2 0 2 97 2 2 2 1 2 98 2 1 1 0 0 99 2 1 1 0 0 100 2 2 1 0 0 101 2 2 2 0 2 102 2 1 1 0 -1 103 2 2 2 2 2 104 2 2 2 0 2 105 2 2 2 0 2 106 2 1 0 0 2 107 2 1 0 0 2 108a 2 1 0 2 2 108b 2 1 0 2 2 109 2 2 0 0 0 110 1 1 1 0 2 111 2 1 2 2 2 112 2 1 0 2 2 113 2 2 0 2 2 114 2 1 2 2 2 115 1 1 0 0 0 116 2 1 0 2 2 117 2 1 0 0 2 118 1 1 0 0 2 119 1 1 1 1 1 120 2 2 2 0 2 Totals: 247 214 207 182 213 Percent: 96.5% 83.6% 80.9% 71.1% 83.2%

The result produced by the various systems on each problem is assigned a numeric grade based on a comparison with the optimal antiderivative as follows:

• If the system gives a result as good as the optimal antiderivative, 2 points are earned.
• If the system gives a mathematically correct result (i.e. its derivative equals the integrand) but is substantially more complex than the optimal antiderivative (e.g. significantly larger or involves the imaginary unit), 1 point is earned.
• If the system is unable to integrate the expression in closed-form or times-out after 120 seconds, 0 points are earned.
• If the system returns a mathematically incorrect result, 1 point is decremented.
• If the system is unable to integrate the expression because it does not support the special function required to express the antiderivative, a grade of "x" is given.

Note that simplification of results using functions and/or domain restrictions on variables was not done. In other words, the comparisons are based on the results a novice user would see.

The raw test results generated by running Rubi, Mathematica and Maple on the Timofeev Test Suite are available for viewing or downloading by clicking on the Test Results menu option on the Rubi homepage.