Chapter 2
detailed summary tables of 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, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59, 60, 61, 62, 63, 64 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 3, 4, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 26, 27, 28, 29, 39, 40, 41, 42, 43, 44, 46,
47, 49, 60, 61, 62, 63, 64 }
B grade: { 8, 9, 12, 22, 30, 34, 38, 45, 50 }
C grade: { 2, 23, 24, 25, 31, 32, 33, 35, 36, 37, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59 }
F grade: { }
2.1.3 Maple
A grade: { 2, 3, 4, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44,
45, 46, 49, 50, 51, 54, 56, 59, 60, 62 }
B grade: { 6, 7, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 48, 53, 55, 58, 61, 63, 64 }
C grade: { 1, 21, 29, 47, 52 }
F grade: { 57 }
2.1.4 Maxima
A grade: { 2, 3, 4, 10, 12, 13, 14, 15, 16, 18, 59
B grade: { 8, 9, 11, 40, 43 }
C grade: { }
F grade: { 1, 5, 6, 7, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41,
42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64 }
2.1.5 FriCAS
A grade: { 5, 10, 14, 18, 21, 24, 25, 29, 30, 47, 48, 59 }
B grade: { 1, 2, 3, 4, 6, 7, 8, 9, 15, 16, 17, 19, 20, 22, 23, 26, 27, 28, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64 }
C grade: { 11, 12, 13 }
F grade: { }
2.1.6 Sympy
A grade: { 2, 3, 4, 5, 6, 10, 13, 16, 51, 56, 59 }
B grade: { }
C grade: { }
F grade: { 1, 7, 8, 9, 11, 12, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 57, 58, 60, 61, 62, 63, 64
}
2.1.7 Giac
A grade: { 1, 2, 6, 7, 8, 9, 14, 17, 18, 59, 62 }
B grade: { 3, 4, 5, 10, 15, 16, 19, 20, 21, 23, 24, 25, 28, 38, 41, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57,
58, 63, 64 }
C grade: { 11, 12, 13, 39, 40 }
F grade: { 22, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 42, 45, 50, 55, 60, 61 }
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 \(\frac{\text{antiderivative leaf size}}{\text{optimal antiderivative leaf size}}\)
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Problem 1 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 233 | 233 | 148 | 2454 | 0 | 2959 | 0 | 335 |
normalized size |
1 |
1. |
0.64 |
10.53 |
0. |
12.7 |
0. |
1.44 |
time (sec) |
N/A |
0.299 |
0.802 |
0.338 |
0. |
2.571 |
0. |
1.389 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
A |
B |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 20 | 34 | 31 | 31 | 105 | 22 | 54 |
normalized size | 1 | 1. | 1.7 | 1.55 | 1.55 | 5.25 | 1.1 | 2.7 |
time (sec) |
N/A |
0.013 |
0.022 |
0.004 |
1.465 |
2.124 |
0.144 |
1.216 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 71 | 68 | 85 | 296 | 68 | 154 |
normalized size | 1 | 1. | 1.51 | 1.45 | 1.81 | 6.3 | 1.45 | 3.28 |
time (sec) |
N/A |
0.033 |
1.132 |
0.004 |
1.476 |
1.649 |
0.313 |
1.182 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 78 | 78 | 111 | 116 | 151 | 583 | 126 | 309 |
normalized size | 1 | 1. | 1.42 | 1.49 | 1.94 | 7.47 | 1.62 | 3.96 |
time (sec) |
N/A |
0.047 |
2.752 |
0.004 |
1.47 |
1.668 |
0.698 |
1.248 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-2) |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 49 | 49 | 49 | 64 | 0 | 574 | 279 | 178 |
normalized size |
1 |
1. |
1. |
1.31 |
0. |
11.71 |
5.69 |
3.63 |
time (sec) |
N/A |
0.073 |
0.053 |
0.021 |
0. |
1.78 |
2.166 |
1.215 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-2) |
B |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 97 | 97 | 90 | 173 | 0 | 1184 | 2086 | 166 |
normalized size |
1 |
1. |
0.93 |
1.78 |
0. |
12.21 |
21.51 |
1.71 |
time (sec) |
N/A |
0.1 |
0.877 |
0.028 |
0. |
1.893 |
151.394 |
1.237 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-2) |
B |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 150 | 150 | 138 | 363 | 0 | 2361 | 0 | 278 |
normalized size |
1 |
1. |
0.92 |
2.42 |
0. |
15.74 |
0. |
1.85 |
time (sec) |
N/A |
0.157 |
0.286 |
0.026 |
0. |
2.255 |
0. |
1.241 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
B |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 22 | 51 | 19 | 405 | 266 | 0 | 43 |
normalized size | 1 | 1. | 2.32 | 0.86 | 18.41 | 12.09 | 0. | 1.95 |
time (sec) |
N/A |
0.018 |
0.093 |
0.031 |
1.663 |
1.621 |
0. |
1.262 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
B |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 5 | 5 | 28 | 6 | 47 | 163 | 0 | 14 |
normalized size | 1 | 1. | 5.6 | 1.2 | 9.4 | 32.6 | 0. | 2.8 |
time (sec) |
N/A |
0.014 |
0.013 |
0.021 |
1.649 |
1.852 |
0. |
1.247 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 12 | 12 | 12 | 13 | 14 | 63 | 14 | 38 |
normalized size | 1 | 1. | 1. | 1.08 | 1.17 | 5.25 | 1.17 | 3.17 |
time (sec) |
N/A |
0.021 |
0.009 |
0.017 |
1.493 |
1.744 |
0.724 |
1.182 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
C |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 35 | 35 | 48 | 32 | 383 | 232 | 0 | 46 |
normalized size | 1 | 1. | 1.37 | 0.91 | 10.94 | 6.63 | 0. | 1.31 |
time (sec) |
N/A |
0.026 |
0.08 |
0.016 |
1.647 |
1.74 |
0. |
1.158 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
A |
C |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 14 | 14 | 30 | 15 | 23 | 55 | 0 | 15 |
normalized size | 1 | 1. | 2.14 | 1.07 | 1.64 | 3.93 | 0. | 1.07 |
time (sec) |
N/A |
0.02 |
0.018 |
0.014 |
1.622 |
1.812 |
0. |
1.184 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
C |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 14 | 14 | 14 | 15 | 16 | 45 | 15 | 38 |
normalized size | 1 | 1. | 1. | 1.07 | 1.14 | 3.21 | 1.07 | 2.71 |
time (sec) |
N/A |
0.023 |
0.007 |
0.008 |
1.467 |
1.502 |
0.547 |
1.174 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 28 | 19 | 29 | 32 | 73 | 0 | 41 |
normalized size | 1 | 1. | 0.68 | 1.04 | 1.14 | 2.61 | 0. | 1.46 |
time (sec) |
N/A |
0.096 |
0.024 |
0.036 |
0.966 |
1.528 |
0. |
1.198 |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 32 | 38 | 36 | 205 | 0 | 73 |
normalized size | 1 | 1. | 1.03 | 1.23 | 1.16 | 6.61 | 0. | 2.35 |
time (sec) |
N/A |
0.1 |
0.032 |
0.028 |
1.701 |
1.653 |
0. |
1.27 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 10 | 10 | 10 | 11 | 11 | 74 | 12 | 39 |
normalized size | 1 | 1. | 1. | 1.1 | 1.1 | 7.4 | 1.2 | 3.9 |
time (sec) |
N/A |
0.048 |
0.011 |
0.019 |
0.962 |
1.58 |
2.774 |
1.267 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-2) |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 36 | 36 | 49 | 56 | 0 | 225 | 0 | 66 |
normalized size |
1 |
1. |
1.36 |
1.56 |
0. |
6.25 |
0. |
1.83 |
time (sec) |
N/A |
0.087 |
0.035 |
0.122 |
0. |
1.62 |
0. |
1.233 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 29 | 29 | 19 | 33 | 24 | 97 | 0 | 55 |
normalized size | 1 | 1. | 0.66 | 1.14 | 0.83 | 3.34 | 0. | 1.9 |
time (sec) |
N/A |
0.101 |
0.03 |
0.069 |
1.494 |
1.548 |
0. |
1.328 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-2) |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 65 | 84 | 0 | 782 | 0 | 267 |
normalized size |
1 |
1. |
0.98 |
1.27 |
0. |
11.85 |
0. |
4.05 |
time (sec) |
N/A |
0.116 |
0.178 |
0.055 |
0. |
2.273 |
0. |
2.38 |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-2) |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 48 | 71 | 0 | 587 | 0 | 128 |
normalized size |
1 |
1. |
1. |
1.48 |
0. |
12.23 |
0. |
2.67 |
time (sec) |
N/A |
0.067 |
0.026 |
0.022 |
0. |
2.227 |
0. |
1.312 |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 60 | 591 | 0 | 949 | 0 | 252 |
normalized size | 1 | 1. | 1. | 9.85 | 0. | 15.82 | 0. | 4.2 |
time (sec) |
N/A |
0.097 |
0.027 |
0.289 |
0. |
1.749 |
0. |
1.321 |
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|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
B |
F |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 2105 | 174 | 0 | 1901 | 0 | 0 |
normalized size |
1 |
1. |
23.65 |
1.96 |
0. |
21.36 |
0. |
0. |
time (sec) |
N/A |
0.125 |
22.246 |
0.035 |
0. |
1.842 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-2) |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 65 | 65 | 167 | 137 | 0 | 1285 | 0 | 284 |
normalized size |
1 |
1. |
2.57 |
2.11 |
0. |
19.77 |
0. |
4.37 |
time (sec) |
N/A |
0.045 |
0.391 |
0.026 |
0. |
1.815 |
0. |
21.938 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 44 | 750 | 0 | 521 | 0 | 323 |
normalized size | 1 | 1. | 0.86 | 14.71 | 0. | 10.22 | 0. | 6.33 |
time (sec) |
N/A |
0.089 |
0.087 |
0.219 |
0. |
2.1 |
0. |
1.549 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
A |
F |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 174 | 951 | 0 | 618 | 0 | 643 |
normalized size | 1 | 1. | 2.05 | 11.19 | 0. | 7.27 | 0. | 7.56 |
time (sec) |
N/A |
0.141 |
1.592 |
0.151 |
0. |
2.113 |
0. |
1.401 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
B |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 88 | 88 | 91 | 150 | 0 | 1152 | 0 | 0 |
normalized size |
1 |
1. |
1.03 |
1.7 |
0. |
13.09 |
0. |
0. |
time (sec) |
N/A |
0.137 |
0.497 |
0.023 |
0. |
2.323 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 127 | 127 | 253 | 286 | 0 | 2689 | 0 | 0 |
normalized size |
1 |
1. |
1.99 |
2.25 |
0. |
21.17 |
0. |
0. |
time (sec) |
N/A |
0.23 |
1.15 |
0.026 |
0. |
2.014 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 63 | 136 | 0 | 803 | 0 | 285 |
normalized size |
1 |
1. |
0.91 |
1.97 |
0. |
11.64 |
0. |
4.13 |
time (sec) |
N/A |
0.091 |
0.165 |
0.018 |
0. |
2.283 |
0. |
5.725 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
A |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 75 | 2628 | 0 | 1531 | 0 | 0 |
normalized size |
1 |
1. |
1. |
35.04 |
0. |
20.41 |
0. |
0. |
time (sec) |
N/A |
0.128 |
0.07 |
0.226 |
0. |
7.137 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
A |
F(-1) |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 80 | 80 | 222 | 1276 | 0 | 1520 | 0 | 0 |
normalized size |
1 |
1. |
2.78 |
15.95 |
0. |
19. |
0. |
0. |
time (sec) |
N/A |
0.125 |
0.705 |
0.147 |
0. |
6.815 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 171 | 171 | 259 | 462 | 0 | 3595 | 0 | 0 |
normalized size | 1 | 1. | 1.51 | 2.7 | 0. | 21.02 | 0. | 0. |
time (sec) |
N/A |
0.186 |
1.726 |
0.057 |
0. |
2.221 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 126 | 126 | 234 | 298 | 0 | 2558 | 0 | 0 |
normalized size | 1 | 1. | 1.86 | 2.37 | 0. | 20.3 | 0. | 0. |
time (sec) |
N/A |
0.098 |
1.387 |
0.02 |
0. |
1.991 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-2) |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 87 | 202 | 170 | 0 | 1674 | 0 | 0 |
normalized size |
1 |
1. |
2.32 |
1.95 |
0. |
19.24 |
0. |
0. |
time (sec) |
N/A |
0.055 |
0.595 |
0.027 |
0. |
1.867 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
F(-2) |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 111 | 68 | 0 | 558 | 0 | 0 |
normalized size |
1 |
1. |
2.36 |
1.45 |
0. |
11.87 |
0. |
0. |
time (sec) |
N/A |
0.033 |
0.418 |
0.027 |
0. |
2.146 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-2) |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 231 | 104 | 0 | 1169 | 0 | 0 |
normalized size |
1 |
1. |
2.72 |
1.22 |
0. |
13.75 |
0. |
0. |
time (sec) |
N/A |
0.061 |
3.66 |
0.028 |
0. |
2.303 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-2) |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 135 | 135 | 367 | 176 | 0 | 1963 | 0 | 0 |
normalized size |
1 |
1. |
2.72 |
1.3 |
0. |
14.54 |
0. |
0. |
time (sec) |
N/A |
0.11 |
7.904 |
0.027 |
0. |
3.189 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-2) |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 190 | 190 | 2553 | 284 | 0 | 3156 | 0 | 0 |
normalized size |
1 |
1. |
13.44 |
1.49 |
0. |
16.61 |
0. |
0. |
time (sec) |
N/A |
0.194 |
14.746 |
0.027 |
0. |
3.487 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
F |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 123 | 51 | 0 | 317 | 0 | 347 |
normalized size | 1 | 1. | 2.28 | 0.94 | 0. | 5.87 | 0. | 6.43 |
time (sec) |
N/A |
0.046 |
0.391 |
0.041 |
0. |
1.981 |
0. |
1.395 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 32 | 32 | 62 | 34 | 0 | 193 | 0 | 230 |
normalized size |
1 |
1. |
1.94 |
1.06 |
0. |
6.03 |
0. |
7.19 |
time (sec) |
N/A |
0.027 |
0.069 |
0.028 |
0. |
1.839 |
0. |
1.387 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
B |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 28 | 42 | 31 | 122 | 171 | 0 | 46 |
normalized size | 1 | 1. | 1.5 | 1.11 | 4.36 | 6.11 | 0. | 1.64 |
time (sec) |
N/A |
0.019 |
0.063 |
0.026 |
1.679 |
2.166 |
0. |
1.43 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 121 | 48 | 0 | 477 | 0 | 242 |
normalized size | 1 | 1. | 1.98 | 0.79 | 0. | 7.82 | 0. | 3.97 |
time (sec) |
N/A |
0.043 |
0.124 |
0.03 |
0. |
1.927 |
0. |
2.49 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 42 | 42 | 60 | 35 | 0 | 351 | 0 | 0 |
normalized size |
1 |
1. |
1.43 |
0.83 |
0. |
8.36 |
0. |
0. |
time (sec) |
N/A |
0.025 |
0.044 |
0.027 |
0. |
1.894 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 45 | 21 | 193 | 177 | 0 | 61 |
normalized size | 1 | 1. | 1.73 | 0.81 | 7.42 | 6.81 | 0. | 2.35 |
time (sec) |
N/A |
0.017 |
0.033 |
0.025 |
1.694 |
2.106 |
0. |
1.788 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 52 | 52 | 52 | 44 | 0 | 659 | 0 | 135 |
normalized size |
1 |
1. |
1. |
0.85 |
0. |
12.67 |
0. |
2.6 |
time (sec) |
N/A |
0.098 |
0.145 |
0.025 |
0. |
2.603 |
0. |
1.722 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 64 | 64 | 158 | 80 | 0 | 1413 | 0 | 0 |
normalized size | 1 | 1. | 2.47 | 1.25 | 0. | 22.08 | 0. | 0. |
time (sec) |
N/A |
0.09 |
0.268 |
0.028 |
0. |
2.097 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 33 | 33 | 33 | 29 | 0 | 304 | 0 | 82 |
normalized size |
1 |
1. |
1. |
0.88 |
0. |
9.21 |
0. |
2.48 |
time (sec) |
N/A |
0.064 |
0.013 |
0.024 |
0. |
2.02 |
0. |
1.464 |
|
|
|
|
|
|
|
|
|
|
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|
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|
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|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 60 | 376 | 0 | 1079 | 0 | 277 |
normalized size | 1 | 1. | 1. | 6.27 | 0. | 17.98 | 0. | 4.62 |
time (sec) |
N/A |
0.094 |
0.045 |
0.14 |
0. |
2.251 |
0. |
1.654 |
|
|
|
|
|
|
|
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|
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|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
A |
F |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 134 | 330 | 0 | 593 | 0 | 132 |
normalized size | 1 | 1. | 2.48 | 6.11 | 0. | 10.98 | 0. | 2.44 |
time (sec) |
N/A |
0.091 |
1.552 |
0.147 |
0. |
2.41 |
0. |
1.516 |
|
|
|
|
|
|
|
|
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|
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|
|
|
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|
|
|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 59 | 68 | 0 | 841 | 0 | 163 |
normalized size |
1 |
1. |
1. |
1.15 |
0. |
14.25 |
0. |
2.76 |
time (sec) |
N/A |
0.111 |
0.218 |
0.025 |
0. |
2.117 |
0. |
1.403 |
|
|
|
|
|
|
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|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
F(-2) |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 137 | 99 | 0 | 884 | 0 | 0 |
normalized size |
1 |
1. |
2.32 |
1.68 |
0. |
14.98 |
0. |
0. |
time (sec) |
N/A |
0.096 |
0.721 |
0.021 |
0. |
3.008 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
|
|
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|
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|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-2) |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 55 | 55 | 44 | 56 | 0 | 774 | 48 | 165 |
normalized size |
1 |
1. |
0.8 |
1.02 |
0. |
14.07 |
0.87 |
3. |
time (sec) |
N/A |
0.075 |
0.041 |
0.016 |
0. |
2.113 |
15.251 |
1.282 |
|
|
|
|
|
|
|
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|
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|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 75 | 962 | 0 | 2087 | 0 | 404 |
normalized size | 1 | 1. | 0.89 | 11.45 | 0. | 24.85 | 0. | 4.81 |
time (sec) |
N/A |
0.131 |
0.056 |
0.194 |
0. |
3.025 |
0. |
1.485 |
|
|
|
|
|
|
|
|
|
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|
|
|
|
|
|
|
|
Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-1) |
B |
F |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 92 | 92 | 674 | 421 | 0 | 945 | 0 | 228 |
normalized size |
1 |
1. |
7.33 |
4.58 |
0. |
10.27 |
0. |
2.48 |
time (sec) |
N/A |
0.154 |
6.903 |
0.204 |
0. |
2.694 |
0. |
1.474 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-2) |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 82 | 82 | 69 | 88 | 0 | 1538 | 0 | 370 |
normalized size |
1 |
1. |
0.84 |
1.07 |
0. |
18.76 |
0. |
4.51 |
time (sec) |
N/A |
0.127 |
0.109 |
0.025 |
0. |
2.219 |
0. |
1.424 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-2) |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 94 | 94 | 200 | 166 | 0 | 1589 | 0 | 0 |
normalized size |
1 |
1. |
2.13 |
1.77 |
0. |
16.9 |
0. |
0. |
time (sec) |
N/A |
0.124 |
6.638 |
0.022 |
0. |
3.079 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-2) |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 78 | 78 | 47 | 75 | 0 | 1416 | 70 | 270 |
normalized size |
1 |
1. |
0.6 |
0.96 |
0. |
18.15 |
0.9 |
3.46 |
time (sec) |
N/A |
0.092 |
0.039 |
0.017 |
0. |
2.131 |
51.106 |
1.355 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 118 | 118 | 78 | 0 | 0 | 3457 | 0 | 686 |
normalized size | 1 | 1. | 0.66 | 0. | 0. | 29.3 | 0. | 5.81 |
time (sec) |
N/A |
0.188 |
0.058 |
0.186 |
0. |
4.233 |
0. |
1.65 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-1) |
B |
F |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 141 | 141 | 1450 | 1040 | 0 | 1477 | 0 | 351 |
normalized size |
1 |
1. |
10.28 |
7.38 |
0. |
10.48 |
0. |
2.49 |
time (sec) |
N/A |
0.241 |
8.058 |
0.48 |
0. |
2.994 |
0. |
1.409 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 37 | 37 | 57 | 37 | 45 | 82 | 34 | 46 |
normalized size | 1 | 1. | 1.54 | 1. | 1.22 | 2.22 | 0.92 | 1.24 |
time (sec) |
N/A |
0.062 |
0.033 |
0.022 |
1.63 |
1.824 |
0.454 |
1.212 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 90 | 90 | 86 | 139 | 0 | 2849 | 0 | 0 |
normalized size | 1 | 1. | 0.96 | 1.54 | 0. | 31.66 | 0. | 0. |
time (sec) |
N/A |
0.135 |
0.151 |
0.057 |
0. |
3.621 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 126 | 126 | 167 | 312 | 0 | 3946 | 0 | 0 |
normalized size | 1 | 1. | 1.33 | 2.48 | 0. | 31.32 | 0. | 0. |
time (sec) |
N/A |
0.202 |
4.444 |
0.033 |
0. |
4.316 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 41 | 41 | 65 | 0 | 686 | 0 | 78 |
normalized size | 1 | 1. | 1. | 1.59 | 0. | 16.73 | 0. | 1.9 |
time (sec) |
N/A |
0.069 |
0.019 |
0.035 |
0. |
3.295 |
0. |
1.518 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 73 | 248 | 0 | 1628 | 0 | 211 |
normalized size | 1 | 1. | 0.99 | 3.35 | 0. | 22. | 0. | 2.85 |
time (sec) |
N/A |
0.111 |
0.294 |
0.055 |
0. |
3.807 |
0. |
5.869 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 117 | 117 | 114 | 602 | 0 | 3077 | 0 | 564 |
normalized size | 1 | 1. | 0.97 | 5.15 | 0. | 26.3 | 0. | 4.82 |
time (sec) |
N/A |
0.19 |
0.7 |
0.063 |
0. |
4.736 |
0. |
14.283 |
|
|
|
|
|
|
|
|
|
|
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 [59] had the largest ratio of [ 0.75
]
|
|
|
|
|
|
|
| Table 2.1Rubi specific breakdown of results for each
integral
|
|
|
|
|
|
|
# |
grade |
|
|
normalized | antiderivative |
leaf size | |
|
\(\frac{\text{number of rules}}{\text{integrand leaf size}}\) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
A |
15 |
12 |
1. |
25 |
0.48 |
|
|
|
|
|
|
|
2 |
A |
3 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
3 |
A |
4 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
4 |
A |
4 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
5 |
A |
3 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
6 |
A |
5 |
5 |
1. |
14 |
0.357 |
|
|
|
|
|
|
|
7 |
A |
6 |
6 |
1. |
14 |
0.429 |
|
|
|
|
|
|
|
8 |
A |
4 |
4 |
1. |
10 |
0.4 |
|
|
|
|
|
|
|
9 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
10 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
11 |
A |
5 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
12 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
13 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
14 |
A |
4 |
3 |
1. |
17 |
0.176 |
|
|
|
|
|
|
|
15 |
A |
5 |
5 |
1. |
17 |
0.294 |
|
|
|
|
|
|
|
16 |
A |
3 |
3 |
1. |
15 |
0.2 |
|
|
|
|
|
|
|
17 |
A |
5 |
5 |
1. |
15 |
0.333 |
|
|
|
|
|
|
|
18 |
A |
5 |
4 |
1. |
17 |
0.235 |
|
|
|
|
|
|
|
19 |
A |
6 |
6 |
1. |
17 |
0.353 |
|
|
|
|
|
|
|
20 |
A |
5 |
5 |
1. |
15 |
0.333 |
|
|
|
|
|
|
|
|
21 |
A |
7 |
5 |
1. |
15 |
0.333 |
|
|
|
|
|
|
|
22 |
A |
7 |
7 |
1. |
17 |
0.412 |
|
|
|
|
|
|
|
23 |
A |
6 |
6 |
1. |
12 |
0.5 |
|
|
|
|
|
|
|
24 |
A |
5 |
5 |
1. |
17 |
0.294 |
|
|
|
|
|
|
|
25 |
A |
6 |
6 |
1. |
17 |
0.353 |
|
|
|
|
|
|
|
26 |
A |
7 |
6 |
1. |
17 |
0.353 |
|
|
|
|
|
|
|
27 |
A |
8 |
8 |
1. |
17 |
0.471 |
|
|
|
|
|
|
|
28 |
A |
6 |
5 |
1. |
15 |
0.333 |
|
|
|
|
|
|
|
29 |
A |
8 |
6 |
1. |
15 |
0.4 |
|
|
|
|
|
|
|
30 |
A |
7 |
7 |
1. |
17 |
0.412 |
|
|
|
|
|
|
|
31 |
A |
8 |
8 |
1. |
16 |
0.5 |
|
|
|
|
|
|
|
32 |
A |
7 |
7 |
1. |
16 |
0.438 |
|
|
|
|
|
|
|
33 |
A |
6 |
6 |
1. |
16 |
0.375 |
|
|
|
|
|
|
|
34 |
A |
3 |
3 |
1. |
16 |
0.188 |
|
|
|
|
|
|
|
35 |
A |
4 |
4 |
1. |
16 |
0.25 |
|
|
|
|
|
|
|
36 |
A |
6 |
6 |
1. |
16 |
0.375 |
|
|
|
|
|
|
|
37 |
A |
7 |
6 |
1. |
16 |
0.375 |
|
|
|
|
|
|
|
38 |
A |
6 |
6 |
1. |
12 |
0.5 |
|
|
|
|
|
|
|
39 |
A |
5 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
40 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
|
41 |
A |
7 |
6 |
1. |
10 |
0.6 |
|
|
|
|
|
|
|
42 |
A |
6 |
5 |
1. |
10 |
0.5 |
|
|
|
|
|
|
|
43 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
44 |
A |
5 |
5 |
1. |
17 |
0.294 |
|
|
|
|
|
|
|
45 |
A |
6 |
6 |
1. |
17 |
0.353 |
|
|
|
|
|
|
|
46 |
A |
4 |
4 |
1. |
15 |
0.267 |
|
|
|
|
|
|
|
47 |
A |
7 |
5 |
1. |
15 |
0.333 |
|
|
|
|
|
|
|
48 |
A |
5 |
5 |
1. |
17 |
0.294 |
|
|
|
|
|
|
|
49 |
A |
5 |
5 |
1. |
17 |
0.294 |
|
|
|
|
|
|
|
50 |
A |
4 |
4 |
1. |
17 |
0.235 |
|
|
|
|
|
|
|
51 |
A |
5 |
5 |
1. |
15 |
0.333 |
|
|
|
|
|
|
|
52 |
A |
8 |
6 |
1. |
15 |
0.4 |
|
|
|
|
|
|
|
53 |
A |
6 |
6 |
1. |
17 |
0.353 |
|
|
|
|
|
|
|
54 |
A |
6 |
6 |
1. |
17 |
0.353 |
|
|
|
|
|
|
|
55 |
A |
6 |
6 |
1. |
17 |
0.353 |
|
|
|
|
|
|
|
56 |
A |
6 |
5 |
1. |
15 |
0.333 |
|
|
|
|
|
|
|
57 |
A |
9 |
7 |
1. |
15 |
0.467 |
|
|
|
|
|
|
|
58 |
A |
7 |
7 |
1. |
17 |
0.412 |
|
|
|
|
|
|
|
59 |
A |
7 |
6 |
1. |
8 |
0.75 |
|
|
|
|
|
|
|
60 |
A |
8 |
7 |
1. |
15 |
0.467 |
|
|
|
|
|
|
|
|
61 |
A |
9 |
8 |
1. |
15 |
0.533 |
|
|
|
|
|
|
|
62 |
A |
4 |
4 |
1. |
15 |
0.267 |
|
|
|
|
|
|
|
63 |
A |
6 |
6 |
1. |
15 |
0.4 |
|
|
|
|
|
|
|
64 |
A |
7 |
7 |
1. |
15 |
0.467 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|