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Chapter 2
detailed summary tables of results

 2.1 List of integrals sorted by grade for each CAS
  2.1.1 Rubi
  2.1.2 Mathematica
  2.1.3 Maple
  2.1.4 Maxima
  2.1.5 FriCAS
  2.1.6 Sympy
  2.1.7 Giac
 2.2 Detailed conclusion table per each integral for all CAS systems
 2.3 Detailed conclusion table specific for Rubi results

2.1 List of integrals sorted by grade for each CAS

2.1.1 Rubi

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 }

B grade: { }

C grade: { }

F grade: { }

2.1.2 Mathematica

A grade: { 1, 2, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 23, 24 }

B grade: { 3, 4, 9, 15, 21, 22 }

C grade: { }

F grade: { }

2.1.3 Maple

A grade: { 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 15, 16, 17, 19, 20, 21, 22, 23, 24 }

B grade: { 8, 12, 14, 18 }

C grade: { }

F grade: { }

2.1.4 Maxima

A grade: { 1, 2, 3, 4, 5, 6, 9, 10, 11, 15, 16, 17, 19, 20, 21, 22, 23, 24

B grade: { 7, 8, 12, 13, 14, 18 }

C grade: { }

F grade: { }

2.1.5 FriCAS

A grade: { 1, 2, 4, 5, 6, 10, 11, 12, 16, 17, 18, 19, 22, 23, 24 }

B grade: { 3, 7, 8, 9, 13, 14, 15, 20, 21 }

C grade: { }

F grade: { }

2.1.6 Sympy

A grade: { 8, 11, 14, 16, 17, 18, 22, 23, 24 }

B grade: { }

C grade: { }

F grade: { 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 15, 19, 20, 21 }

2.1.7 Giac

A grade: { 1, 2, 3, 4, 6, 7, 8, 11, 13, 14, 17, 19, 20, 21, 22, 24 }

B grade: { 5, 9, 10, 12, 15, 16, 18, 23 }

C grade: { }

F grade: { }

2.2 Detailed conclusion table per each integral for all CAS systems

Detailed conclusion table per each integral is given by table below. The elapsed time is in seconds. For failed result it is given as F(-1) if the failure was due to timeout. It is given as F(-2) if the failure was due to an exception being raised, which could indicate a bug in the system. If the failure was due to integral not being evaluated within the time limit, then it is given just an F.

In this table,the column normalized size is defined as \(\frac{\text{antiderivative leaf size}}{\text{optimal antiderivative leaf size}}\)

Problem 1 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 91 91 163 99 196 439 0 209
normalized size 1 1. 1.79 1.09 2.15 4.82 0. 2.3
time (sec) N/A 0.102 0.057 0.051 0.984 0.502 0. 1.446
Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 72 72 116 77 117 363 0 166
normalized size 1 1. 1.61 1.07 1.62 5.04 0. 2.31
time (sec) N/A 0.096 0.045 0.043 0.988 0.5 0. 1.357
Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B A A B F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 51 51 137 57 108 273 0 126
normalized size 1 1. 2.69 1.12 2.12 5.35 0. 2.47
time (sec) N/A 0.058 0.04 0.045 0.999 0.493 0. 1.368
Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 33 33 72 50 68 147 0 63
normalized size 1 1. 2.18 1.52 2.06 4.45 0. 1.91
time (sec) N/A 0.059 0.023 0.079 0.989 0.508 0. 1.356
Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 42 42 33 49 63 97 0 107
normalized size 1 1. 0.79 1.17 1.5 2.31 0. 2.55
time (sec) N/A 0.073 0.063 0.078 0.99 0.476 0. 1.497
Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 54 54 43 62 81 128 0 128
normalized size 1 1. 0.8 1.15 1.5 2.37 0. 2.37
time (sec) N/A 0.094 0.084 0.089 0.962 0.504 0. 1.391
Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B B F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 61 61 117 65 162 344 0 144
normalized size 1 1. 1.92 1.07 2.66 5.64 0. 2.36
time (sec) N/A 0.092 0.045 0.026 1.014 0.497 0. 1.381
Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B B B A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 17 17 17 38 57 85 54 20
normalized size 1 1. 1. 2.24 3.35 5. 3.18 1.18
time (sec) N/A 0.072 0.016 0.02 1.057 0.456 2.153 1.367
Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B A A B F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 38 38 79 44 92 223 0 136
normalized size 1 1. 2.08 1.16 2.42 5.87 0. 3.58
time (sec) N/A 0.051 0.035 0.023 1.116 0.496 0. 1.333
Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 27 27 46 38 54 132 0 81
normalized size 1 1. 1.7 1.41 2. 4.89 0. 3.
time (sec) N/A 0.053 0.024 0.041 1.112 0.501 0. 1.366
Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 29 29 25 40 50 68 54 47
normalized size 1 1. 0.86 1.38 1.72 2.34 1.86 1.62
time (sec) N/A 0.055 0.022 0.036 1.04 0.476 13.381 1.259
Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B B A F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 17 17 17 35 49 35 0 76
normalized size 1 1. 1. 2.06 2.88 2.06 0. 4.47
time (sec) N/A 0.061 0.007 0.036 1.077 0.471 0. 1.359
Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B B F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 61 61 117 65 162 344 0 144
normalized size 1 1. 1.92 1.07 2.66 5.64 0. 2.36
time (sec) N/A 0.09 0.038 0.027 1. 0.497 0. 1.426
Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B B B A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 17 17 17 38 57 85 54 20
normalized size 1 1. 1. 2.24 3.35 5. 3.18 1.18
time (sec) N/A 0.071 0.012 0.021 1.022 0.456 2.146 1.385
Problem 15 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B A A B F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 38 38 79 44 92 223 0 136
normalized size 1 1. 2.08 1.16 2.42 5.87 0. 3.58
time (sec) N/A 0.049 0.035 0.025 1.001 0.496 0. 1.413
Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 27 27 46 38 45 132 75 81
normalized size 1 1. 1.7 1.41 1.67 4.89 2.78 3.
time (sec) N/A 0.052 0.022 0.023 1.015 0.507 11.496 1.393
Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 29 29 25 40 50 68 85 47
normalized size 1 1. 0.86 1.38 1.72 2.34 2.93 1.62
time (sec) N/A 0.055 0.011 0.021 0.994 0.479 6.966 1.453
Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B B A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 17 17 17 35 49 35 83 76
normalized size 1 1. 1. 2.06 2.88 2.06 4.88 4.47
time (sec) N/A 0.061 0.006 0.028 0.981 0.472 10.56 1.315
Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 91 91 163 99 196 439 0 209
normalized size 1 1. 1.79 1.09 2.15 4.82 0. 2.3
time (sec) N/A 0.096 0.057 0.049 0.988 0.504 0. 1.479
Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A B F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 70 70 116 77 117 363 0 166
normalized size 1 1. 1.66 1.1 1.67 5.19 0. 2.37
time (sec) N/A 0.09 0.043 0.045 1.05 0.505 0. 1.647
Problem 21 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B A A B F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 51 51 137 57 108 274 0 126
normalized size 1 1. 2.69 1.12 2.12 5.37 0. 2.47
time (sec) N/A 0.055 0.035 0.048 1.004 0.496 0. 1.498
Problem 22 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 33 33 72 50 55 149 82 66
normalized size 1 1. 2.18 1.52 1.67 4.52 2.48 2.
time (sec) N/A 0.058 0.027 0.041 0.995 0.503 12.314 1.372
Problem 23 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 42 42 35 49 63 97 105 107
normalized size 1 1. 0.83 1.17 1.5 2.31 2.5 2.55
time (sec) N/A 0.069 0.055 0.041 0.994 0.473 10.545 1.434
Problem 24 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 54 54 44 62 81 128 139 128
normalized size 1 1. 0.81 1.15 1.5 2.37 2.57 2.37
time (sec) N/A 0.09 0.11 0.048 1.007 0.482 13.726 1.527

2.3 Detailed conclusion table specific for Rubi results

The following table is specific to Rubi. It gives additional statistics for each integral. the column steps is the number of steps used by Rubi to obtain the antiderivative. The rules column is the number of unique rules used. The integrand size column is the leaf size of the integrand. Finally the ratio \(\frac{\text{number of rules}}{\text{integrand size}}\) is given. The larger this ratio is, the harder the integral was to solve. In this test, problem number [2] had the largest ratio of [ 0.2414 ]

Table 2.1Rubi specific breakdown of results for each integral
# grade
number of
steps
used
number of
unique
rules
normalized
antiderivative
leaf size
integrand
leaf size
\(\frac{\text{number of rules}}{\text{integrand leaf size}}\)
1 A 7 6 1. 29 0.207
2 A 7 7 1. 29 0.241
3 A 6 6 1. 27 0.222
4 A 5 5 1. 27 0.185
5 A 5 5 1. 29 0.172
6 A 7 6 1. 29 0.207
7 A 4 4 1. 30 0.133
8 A 3 3 1. 30 0.1
9 A 3 3 1. 28 0.107
10 A 4 4 1. 28 0.143
11 A 3 3 1. 30 0.1
12 A 3 3 1. 30 0.1
13 A 4 4 1. 30 0.133
14 A 3 3 1. 30 0.1
15 A 3 3 1. 28 0.107
16 A 4 4 1. 28 0.143
17 A 3 3 1. 30 0.1
18 A 3 3 1. 30 0.1
19 A 7 6 1. 31 0.194
20 A 7 7 1. 31 0.226
21 A 6 6 1. 29 0.207
22 A 5 5 1. 29 0.172
23 A 5 5 1. 31 0.161
24 A 7 6 1. 31 0.194

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