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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 3, 4, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 40, 41, 46, 47, 50, 51, 53, 54, 55, 59, 60, 61, 62 }
B grade: { 2, 39, 45 }
C grade: { 5, 6, 7, 18, 25, 26, 27, 28, 42, 43, 44, 49, 56, 57, 58 }
F grade: { 48, 52}
2.1.3 Maple
A grade: { 4, 6, 7, 12, 13, 14, 29, 34, 35, 36, 37, 41, 43, 44, 50, 51, 55, 56, 59, 60 }
B grade: { 1, 2, 3, 9, 10, 11, 15, 16, 17, 19, 20, 21, 22, 30, 31, 32, 33, 38, 39, 40, 47, 54, 61, 62
}
C grade: { 5, 8, 18, 23, 24, 25, 26, 27, 28, 42, 45, 46, 48, 49, 53, 57, 58 }
F grade: { 52 }
2.1.4 Maxima
A grade: { 1, 2, 3, 4, 6, 7, 13, 32, 33, 34, 36, 37, 55, 56
B grade: { 9, 10, 11, 14, 15, 16, 17, 20, 22, 24, 29, 30, 31, 38, 39, 40 }
C grade: { }
F grade: { 5, 8, 12, 18, 19, 21, 23, 25, 26, 27, 28, 35, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 57,
58, 59, 60, 61, 62 }
2.1.5 FriCAS
A grade: { 11, 13, 14, 33, 34, 50, 51 }
B grade: { 9, 10, 15, 17, 20, 22, 31, 32, 36, 37 }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 12, 16, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 30, 35, 38, 39, 40, 41, 42, 43, 44,
45, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62 }
2.1.6 Sympy
A grade: { 9, 10, 11, 13, 17, 31, 32, 33, 34 }
B grade: { }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 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
}
2.1.7 Giac
A grade: { 13, 34, 36, 50, 51 }
B grade: { 9, 10, 11, 14, 15, 17, 22, 31, 32, 33, 37 }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 12, 16, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 35, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62 }
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 |
B |
A |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 263 | 263 | 187 | 967 | 432 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.71 | 3.68 | 1.64 | 0. | 0. | 0. |
time (sec) |
N/A |
0.339 |
1.175 |
0.059 |
0.982 |
0. |
0. |
0. |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
B |
A |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 204 | 204 | 463 | 729 | 350 | 0 | 0 | 0 |
normalized size | 1 | 1. | 2.27 | 3.57 | 1.72 | 0. | 0. | 0. |
time (sec) |
N/A |
0.269 |
2.195 |
0.052 |
1.008 |
0. |
0. |
0. |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
A |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 136 | 136 | 98 | 365 | 273 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.72 | 2.68 | 2.01 | 0. | 0. | 0. |
time (sec) |
N/A |
0.2 |
0.265 |
0.052 |
0.987 |
0. |
0. |
0. |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 81 | 81 | 55 | 103 | 188 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.68 | 1.27 | 2.32 | 0. | 0. | 0. |
time (sec) |
N/A |
0.086 |
0.022 |
0.075 |
0.986 |
0. |
0. |
0. |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 148 | 148 | 634 | 1022 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 4.28 | 6.91 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.09 |
2.967 |
0.537 |
0. |
0. |
0. |
0. |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 251 | 251 | 208 | 342 | 329 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.83 | 1.36 | 1.31 | 0. | 0. | 0. |
time (sec) |
N/A |
0.699 |
1.321 |
0.135 |
1.016 |
0. |
0. |
0. |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 370 | 370 | 271 | 467 | 486 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.73 | 1.26 | 1.31 | 0. | 0. | 0. |
time (sec) |
N/A |
0.801 |
3.532 |
0.096 |
1.021 |
0. |
0. |
0. |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 75 | 160 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.34 | 2.86 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.128 |
0.091 |
0.271 |
0. |
0. |
0. |
0. |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 72 | 72 | 78 | 242 | 482 | 342 | 231 | 362 |
normalized size | 1 | 1. | 1.08 | 3.36 | 6.69 | 4.75 | 3.21 | 5.03 |
time (sec) |
N/A |
0.067 |
0.104 |
0.035 |
0.996 |
2.235 |
26.819 |
1.294 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 59 | 174 | 304 | 323 | 180 | 258 |
normalized size | 1 | 1. | 0.86 | 2.52 | 4.41 | 4.68 | 2.61 | 3.74 |
time (sec) |
N/A |
0.065 |
0.073 |
0.038 |
0.962 |
2.431 |
6.206 |
1.275 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 77 | 107 | 153 | 169 | 95 | 192 |
normalized size | 1 | 1. | 1.6 | 2.23 | 3.19 | 3.52 | 1.98 | 4. |
time (sec) |
N/A |
0.034 |
0.046 |
0.038 |
0.966 |
2.311 |
2.297 |
1.234 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 54 | 89 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1. | 1.65 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.045 |
0.025 |
0.041 |
0. |
0. |
0. |
0. |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 69 | 86 | 128 | 205 | 2749 | 115 |
normalized size | 1 | 1. | 1.1 | 1.37 | 2.03 | 3.25 | 43.63 | 1.83 |
time (sec) |
N/A |
0.059 |
0.068 |
0.036 |
0.971 |
2.339 |
152.211 |
1.142 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 100 | 88 | 177 | 194 | 0 | 196 |
normalized size |
1 |
1. |
1.59 |
1.4 |
2.81 |
3.08 |
0. |
3.11 |
time (sec) |
N/A |
0.049 |
0.056 |
0.04 |
0.982 |
2.28 |
0. |
1.226 |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 159 | 159 | 148 | 732 | 1116 | 830 | 0 | 944 |
normalized size |
1 |
1. |
0.93 |
4.6 |
7.02 |
5.22 |
0. |
5.94 |
time (sec) |
N/A |
0.248 |
0.181 |
0.054 |
1.976 |
2.129 |
0. |
1.511 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 179 | 179 | 150 | 583 | 836 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.84 | 3.26 | 4.67 | 0. | 0. | 0. |
time (sec) |
N/A |
0.237 |
0.45 |
0.058 |
1.947 |
0. |
0. |
0. |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 95 | 95 | 134 | 390 | 427 | 436 | 238 | 470 |
normalized size | 1 | 1. | 1.41 | 4.11 | 4.49 | 4.59 | 2.51 | 4.95 |
time (sec) |
N/A |
0.135 |
0.102 |
0.052 |
1.903 |
1.988 |
6.195 |
1.344 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 168 | 168 | 424 | 893 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 2.52 | 5.32 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.303 |
0.369 |
0.316 |
0. |
0. |
0. |
0. |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 104 | 104 | 126 | 396 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.21 | 3.81 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.181 |
0.257 |
0.063 |
0. |
0. |
0. |
0. |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
B |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 119 | 119 | 136 | 371 | 444 | 598 | 0 | 0 |
normalized size |
1 |
1. |
1.14 |
3.12 |
3.73 |
5.03 |
0. |
0. |
time (sec) |
N/A |
0.174 |
0.178 |
0.064 |
1.023 |
2.239 |
0. |
0. |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 180 | 180 | 218 | 492 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.21 | 2.73 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.27 |
0.535 |
0.066 |
0. |
0. |
0. |
0. |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 172 | 172 | 218 | 431 | 828 | 1181 | 0 | 608 |
normalized size |
1 |
1. |
1.27 |
2.51 |
4.81 |
6.87 |
0. |
3.53 |
time (sec) |
N/A |
0.262 |
0.267 |
0.076 |
1.132 |
2.418 |
0. |
1.647 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 263 | 263 | 336 | 1768 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.28 |
6.72 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.467 |
0.721 |
0.468 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 160 | 160 | 213 | 6834 | 849 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.33 | 42.71 | 5.31 | 0. | 0. | 0. |
time (sec) |
N/A |
0.263 |
1.161 |
0.43 |
1.949 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 257 | 257 | 599 | 1842 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 2.33 | 7.17 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.492 |
0.508 |
0.256 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 143 | 143 | 248 | 2001 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.73 | 13.99 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.304 |
0.598 |
0.338 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 166 | 166 | 335 | 5796 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
2.02 |
34.92 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.335 |
1.121 |
0.437 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 269 | 269 | 393 | 2247 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.46 | 8.35 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.514 |
1.245 |
0.519 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 21 | 31 | 34 | 78 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.48 | 1.62 | 3.71 | 0. | 0. | 0. |
time (sec) |
N/A |
0.022 |
0.003 |
0.04 |
0.965 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 32 | 32 | 52 | 59 | 178 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.62 | 1.84 | 5.56 | 0. | 0. | 0. |
time (sec) |
N/A |
0.031 |
0.006 |
0.043 |
0.978 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 168 | 168 | 270 | 786 | 450 | 857 | 644 | 1003 |
normalized size | 1 | 1. | 1.61 | 4.68 | 2.68 | 5.1 | 3.83 | 5.97 |
time (sec) |
N/A |
0.331 |
0.268 |
0.041 |
0.986 |
2.175 |
27.428 |
2.653 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 120 | 120 | 174 | 477 | 279 | 549 | 369 | 597 |
normalized size | 1 | 1. | 1.45 | 3.98 | 2.32 | 4.58 | 3.08 | 4.97 |
time (sec) |
N/A |
0.204 |
0.157 |
0.037 |
0.969 |
2.08 |
8.348 |
1.677 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 97 | 97 | 138 | 184 | 147 | 320 | 173 | 294 |
normalized size | 1 | 1. | 1.42 | 1.9 | 1.52 | 3.3 | 1.78 | 3.03 |
time (sec) |
N/A |
0.169 |
0.046 |
0.033 |
0.959 |
2.115 |
2.34 |
1.311 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
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 | 40 | 40 | 48 | 44 | 49 | 158 | 46 | 82 |
normalized size | 1 | 1. | 1.2 | 1.1 | 1.22 | 3.95 | 1.15 | 2.05 |
time (sec) |
N/A |
0.024 |
0.016 |
0.028 |
0.945 |
1.943 |
0.775 |
1.176 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 130 | 130 | 126 | 202 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.97 | 1.55 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.136 |
0.073 |
0.211 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 115 | 115 | 125 | 141 | 163 | 585 | 0 | 242 |
normalized size |
1 |
1. |
1.09 |
1.23 |
1.42 |
5.09 |
0. |
2.1 |
time (sec) |
N/A |
0.169 |
0.195 |
0.037 |
0.961 |
3.196 |
0. |
1.227 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 167 | 167 | 174 | 236 | 393 | 1767 | 0 | 2056 |
normalized size |
1 |
1. |
1.04 |
1.41 |
2.35 |
10.58 |
0. |
12.31 |
time (sec) |
N/A |
0.254 |
0.343 |
0.073 |
1.015 |
9.639 |
0. |
29.816 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 562 | 562 | 1082 | 4401 | 1840 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.93 |
7.83 |
3.27 |
0. |
0. |
0. |
time (sec) |
N/A |
1.041 |
7.194 |
0.082 |
1.991 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 374 | 374 | 795 | 2694 | 1088 | 0 | 0 | 0 |
normalized size |
1 |
1. |
2.13 |
7.2 |
2.91 |
0. |
0. |
0. |
time (sec) |
N/A |
0.64 |
2.981 |
0.069 |
1.955 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 221 | 220 | 271 | 857 | 560 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.23 | 3.88 | 2.53 | 0. | 0. | 0. |
time (sec) |
N/A |
0.462 |
0.54 |
0.053 |
1.929 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 97 | 97 | 98 | 174 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.01 | 1.79 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.121 |
0.107 |
0.073 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 214 | 214 | 2404 | 1984 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
11.23 |
9.27 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.154 |
23.53 |
0.878 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 480 | 485 | 425 | 783 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1.01 |
0.89 |
1.63 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.713 |
7.452 |
0.168 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 750 | 750 | 1970 | 1428 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
2.63 |
1.9 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.126 |
14.663 |
0.179 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 546 | 546 | 1868 | 12111 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
3.42 |
22.18 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.046 |
9.386 |
4.24 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 326 | 325 | 566 | 20255 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.74 |
62.13 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.726 |
1.295 |
1.31 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 132 | 132 | 194 | 346 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.47 | 2.62 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.23 |
0.149 |
0.106 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 308 | 308 | 0 | 4064 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0. |
13.19 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.193 |
30.178 |
0.607 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1089 | 1094 | 2683 | 5728 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
2.46 |
5.26 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.827 |
23.168 |
0.84 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-2) |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.068 |
4.239 |
1.266 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-2) |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.069 |
0.289 |
1.103 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F(-2) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 162 | 162 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.257 |
0.083 |
1.027 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F(-2) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 780 | 780 | 623 | 587 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.8 |
0.75 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.395 |
0.728 |
0.493 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 481 | 481 | 365 | 1300 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.76 |
2.7 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.606 |
0.291 |
0.352 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 120 | 120 | 138 | 176 | 259 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.15 | 1.47 | 2.16 | 0. | 0. | 0. |
time (sec) |
N/A |
0.13 |
0.012 |
0.213 |
0.989 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 186 | 186 | 759 | 297 | 259 | 0 | 0 | 0 |
normalized size | 1 | 1. | 4.08 | 1.6 | 1.39 | 0. | 0. | 0. |
time (sec) |
N/A |
0.237 |
3.927 |
0.197 |
0.961 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F(-2) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 545 | 545 | 1456 | 20505 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
2.67 |
37.62 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.911 |
24.507 |
3.524 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F(-2) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 832 | 832 | 917 | 650 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.1 |
0.78 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.43 |
7.935 |
0.579 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 585 | 585 | 549 | 738 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.94 |
1.26 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.013 |
0.519 |
0.239 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 661 | 661 | 598 | 970 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.9 |
1.47 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.072 |
0.607 |
0.181 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 335 | 335 | 403 | 2140 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.2 |
6.39 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.721 |
0.54 |
0.622 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 83 | 83 | 114 | 194 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.37 | 2.34 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.263 |
0.116 |
0.058 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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 [7] had the largest ratio of [ 1.333
]
|
|
|
|
|
|
|
| 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 |
19 |
15 |
1. |
12 |
1.25 |
|
|
|
|
|
|
|
2 |
A |
15 |
13 |
1. |
12 |
1.083 |
|
|
|
|
|
|
|
3 |
A |
12 |
10 |
1. |
10 |
1. |
|
|
|
|
|
|
|
4 |
A |
6 |
6 |
1. |
8 |
0.75 |
|
|
|
|
|
|
|
5 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
6 |
A |
17 |
15 |
1. |
12 |
1.25 |
|
|
|
|
|
|
|
7 |
A |
21 |
16 |
1. |
12 |
1.333 |
|
|
|
|
|
|
|
8 |
A |
4 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
9 |
A |
6 |
5 |
1. |
21 |
0.238 |
|
|
|
|
|
|
|
10 |
A |
6 |
5 |
1. |
21 |
0.238 |
|
|
|
|
|
|
|
11 |
A |
5 |
5 |
1. |
19 |
0.263 |
|
|
|
|
|
|
|
12 |
A |
3 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
13 |
A |
7 |
7 |
1. |
21 |
0.333 |
|
|
|
|
|
|
|
14 |
A |
5 |
5 |
1. |
21 |
0.238 |
|
|
|
|
|
|
|
15 |
A |
13 |
9 |
1. |
23 |
0.391 |
|
|
|
|
|
|
|
16 |
A |
11 |
10 |
1. |
23 |
0.435 |
|
|
|
|
|
|
|
17 |
A |
8 |
7 |
1. |
21 |
0.333 |
|
|
|
|
|
|
|
18 |
A |
8 |
7 |
1. |
23 |
0.304 |
|
|
|
|
|
|
|
19 |
A |
6 |
6 |
1. |
23 |
0.261 |
|
|
|
|
|
|
|
20 |
A |
10 |
9 |
1. |
23 |
0.391 |
|
|
|
|
|
|
|
|
21 |
A |
10 |
9 |
1. |
23 |
0.391 |
|
|
|
|
|
|
|
22 |
A |
15 |
10 |
1. |
23 |
0.435 |
|
|
|
|
|
|
|
23 |
A |
14 |
11 |
1. |
23 |
0.478 |
|
|
|
|
|
|
|
24 |
A |
10 |
10 |
1. |
21 |
0.476 |
|
|
|
|
|
|
|
25 |
A |
10 |
8 |
1. |
23 |
0.348 |
|
|
|
|
|
|
|
26 |
A |
7 |
8 |
1. |
23 |
0.348 |
|
|
|
|
|
|
|
27 |
A |
9 |
8 |
1. |
23 |
0.348 |
|
|
|
|
|
|
|
28 |
A |
16 |
13 |
1. |
23 |
0.565 |
|
|
|
|
|
|
|
29 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
30 |
A |
3 |
3 |
1. |
19 |
0.158 |
|
|
|
|
|
|
|
31 |
A |
7 |
5 |
1. |
18 |
0.278 |
|
|
|
|
|
|
|
32 |
A |
7 |
5 |
1. |
18 |
0.278 |
|
|
|
|
|
|
|
33 |
A |
7 |
5 |
1. |
16 |
0.312 |
|
|
|
|
|
|
|
34 |
A |
4 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
35 |
A |
5 |
5 |
1. |
18 |
0.278 |
|
|
|
|
|
|
|
36 |
A |
7 |
5 |
1. |
18 |
0.278 |
|
|
|
|
|
|
|
37 |
A |
5 |
4 |
1. |
18 |
0.222 |
|
|
|
|
|
|
|
38 |
A |
20 |
15 |
1. |
20 |
0.75 |
|
|
|
|
|
|
|
39 |
A |
16 |
13 |
1. |
20 |
0.65 |
|
|
|
|
|
|
|
40 |
A |
13 |
10 |
1. |
18 |
0.556 |
|
|
|
|
|
|
|
|
41 |
A |
6 |
6 |
1. |
12 |
0.5 |
|
|
|
|
|
|
|
42 |
A |
2 |
2 |
1. |
20 |
0.1 |
|
|
|
|
|
|
|
43 |
A |
21 |
19 |
1.01 |
20 |
0.95 |
|
|
|
|
|
|
|
44 |
A |
26 |
18 |
1. |
20 |
0.9 |
|
|
|
|
|
|
|
45 |
A |
21 |
14 |
1. |
20 |
0.7 |
|
|
|
|
|
|
|
46 |
A |
15 |
11 |
1. |
18 |
0.611 |
|
|
|
|
|
|
|
47 |
A |
6 |
7 |
1. |
12 |
0.583 |
|
|
|
|
|
|
|
48 |
A |
2 |
2 |
1. |
20 |
0.1 |
|
|
|
|
|
|
|
49 |
A |
30 |
18 |
1. |
20 |
0.9 |
|
|
|
|
|
|
|
50 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
51 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
52 |
A |
6 |
4 |
1. |
18 |
0.222 |
|
|
|
|
|
|
|
53 |
A |
23 |
5 |
1. |
16 |
0.312 |
|
|
|
|
|
|
|
54 |
A |
17 |
5 |
1. |
16 |
0.312 |
|
|
|
|
|
|
|
55 |
A |
5 |
5 |
1. |
14 |
0.357 |
|
|
|
|
|
|
|
56 |
A |
15 |
7 |
1. |
16 |
0.438 |
|
|
|
|
|
|
|
57 |
A |
25 |
7 |
1. |
16 |
0.438 |
|
|
|
|
|
|
|
58 |
A |
31 |
7 |
1. |
16 |
0.438 |
|
|
|
|
|
|
|
59 |
A |
31 |
13 |
1. |
18 |
0.722 |
|
|
|
|
|
|
|
60 |
A |
37 |
16 |
1. |
18 |
0.889 |
|
|
|
|
|
|
|
|
61 |
A |
12 |
8 |
1. |
19 |
0.421 |
|
|
|
|
|
|
|
62 |
A |
6 |
5 |
1. |
32 |
0.156 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|