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 9 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 | 0 | 2 |

6 | 2 | 2 | 2 | 0 | 0 |

7a | 2 | 2 | 2 | 2 | 2 |

7b | 2 | 2 | 0 | 2 | 2 |

8 | 2 | 2 | 0 | 2 | 2 |

9 | 2 | 2 | 2 | 2 | 2 |

10 | 2 | 2 | 2 | 2 | 2 |

11 | 2 | 2 | 0 | 2 | 2 |

12 | 2 | 1 | 0 | x | x |

13 | 2 | 2 | 2 | 2 | 2 |

14 | 2 | 2 | 2 | 2 | 2 |

15 | 2 | 2 | 2 | 2 | 2 |

16a | 2 | 2 | 2 | 2 | 2 |

16b | 2 | 2 | 2 | 2 | 2 |

17 | 2 | 2 | 2 | 2 | 2 |

18 | 2 | 2 | 2 | 2 | 2 |

19 | 2 | 1 | 2 | x | x |

20 | 2 | 2 | 2 | 2 | 2 |

21 | 2 | 2 | 0 | 2 | 2 |

22 | 2 | 2 | 2 | 0 | 2 |

23 | 2 | 2 | 2 | 2 | 2 |

24 | 2 | 2 | 2 | 2 | 2 |

25 | 2 | 2 | 2 | 2 | 2 |

26 | 2 | 2 | 2 | x | x |

27 | 2 | 2 | 2 | 2 | 2 |

28 | 2 | 1 | 2 | x | x |

29 | 2 | 1 | 2 | x | x |

30 | 2 | 2 | 2 | 2 | 2 |

31 | 2 | 2 | 2 | 2 | 2 |

32 | 2 | 2 | 2 | 2 | x |

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 | 0 | 0 | x |

38 | 2 | 2 | 0 | 0 | 2 |

39 | 2 | 2 | 0 | 0 | 2 |

40 | 2 | 2 | 0 | 0 | 2 |

41 | 2 | 2 | 0 | 0 | 2 |

42 | 2 | 2 | 0 | 0 | 2 |

43 | 2 | 1 | 0 | 0 | x |

44 | 2 | 2 | 0 | 0 | 2 |

45 | 2 | 2 | 0 | 0 | 2 |

46 | 2 | 2 | 0 | 0 | 0 |

47 | 2 | 2 | 0 | 0 | 0 |

48 | 2 | 2 | 0 | 0 | 0 |

49 | 1 | 1 | 1 | 1 | 0 |

50 | 2 | 1 | 2 | 2 | 0 |

51 | 2 | 2 | 2 | 2 | 2 |

52 | 2 | 1 | 2 | 2 | x |

53 | 0 | 2 | 0 | 0 | 2 |

54 | 2 | 2 | 2 | 2 | 2 |

55 | 2 | 2 | 2 | 2 | 2 |

56 | 2 | 2 | 2 | 2 | 2 |

57 | 2 | 2 | 0 | 0 | 2 |

58 | 2 | 1 | 1 | 1 | 2 |

59 | 1 | 1 | 0 | 2 | 2 |

Totals: | 118 | 112 | 80 | 76 | 92 |

Percent: | 96.7% | 91.8% | 65.6% | 62.3% | 75.4% |

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**,