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, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87,
88, 89, 90, 91, 92, 93 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
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, 55, 56, 57, 61, 62, 63, 64,
65, 66, 67, 68, 69, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93
}
B grade: { }
C grade: { }
F grade: { 52, 53, 54, 58, 59, 60, 70, 71, 72}
2.1.3 Maple
A grade: { 1, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 26, 27, 28, 34, 35, 42, 49, 50, 51,
55, 56, 57, 61, 62, 63, 64, 65, 66, 67, 68, 69, 73, 74, 75, 76, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92,
93 }
B grade: { 2, 3, 10, 11, 19, 46, 47, 48, 52, 53, 54, 58, 59, 60, 77, 78, 83 }
C grade: { }
F grade: { 25, 29, 30, 31, 32, 33, 36, 37, 38, 39, 40, 41, 43, 44, 45, 70, 71, 72 }
2.1.4 Maxima
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, 35, 42, 49, 50, 51,
55, 56, 57, 61, 62, 63, 64, 65, 66, 67, 68, 69, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
90, 91, 92, 93
B grade: { }
C grade: { }
F grade: { 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 52, 53, 54, 58,
59, 60, 70, 71, 72 }
2.1.5 FriCAS
A grade: { 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 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, 48, 49, 50, 51, 55, 56, 57, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72,
73, 74, 75, 78, 79, 80, 81, 83, 84, 85, 86, 87, 90, 91, 92, 93 }
B grade: { 19, 54, 60, 89 }
C grade: { 1, 2, 9, 10, 17, 18, 46, 47, 52, 53, 58, 59, 76, 77, 82, 88 }
F grade: { }
2.1.6 Sympy
A grade: { 4, 5, 6, 7, 8, 12, 13, 14, 15, 16, 20, 21, 22, 23, 24, 25, 26, 27, 28, 32, 33, 34, 35, 39, 40, 41, 42,
49, 50, 55, 61, 64, 65, 66, 67, 68, 69, 79, 80, 81, 84, 85, 86, 87, 90, 92, 93 }
B grade: { 91 }
C grade: { }
F grade: { 1, 2, 3, 9, 10, 11, 17, 18, 19, 29, 30, 31, 36, 37, 38, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 56, 57,
58, 59, 60, 62, 63, 70, 71, 72, 73, 74, 75, 76, 77, 78, 82, 83, 88, 89 }
2.1.7 Giac
A grade: { 4, 5, 6, 7, 8, 12, 13, 14, 15, 16, 20, 21, 22, 23, 24, 28, 50, 51, 56, 57, 62, 63, 64, 65, 66, 67, 68,
69, 73, 74, 75, 79, 80, 81, 85, 86, 87, 91, 92, 93 }
B grade: { }
C grade: { 25, 26, 27, 33, 34, 35, 40, 41, 42 }
F grade: { 1, 2, 3, 9, 10, 11, 17, 18, 19, 29, 30, 31, 32, 36, 37, 38, 39, 43, 44, 45, 46, 47, 48, 49, 52, 53, 54,
55, 58, 59, 60, 61, 70, 71, 72, 76, 77, 78, 82, 83, 84, 88, 89, 90 }
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 |
A |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 112 | 191 | 127 | 297 | 0 | 0 |
normalized size | 1 | 1. | 1.02 | 1.74 | 1.15 | 2.7 | 0. | 0. |
time (sec) |
N/A |
0.189 |
0.009 |
0.012 |
1.087 |
1.564 |
0. |
0. |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 83 | 166 | 97 | 238 | 0 | 0 |
normalized size | 1 | 1. | 0.99 | 1.98 | 1.15 | 2.83 | 0. | 0. |
time (sec) |
N/A |
0.168 |
0.006 |
0.003 |
1.079 |
1.527 |
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 |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 53 | 133 | 65 | 177 | 0 | 0 |
normalized size | 1 | 1. | 0.91 | 2.29 | 1.12 | 3.05 | 0. | 0. |
time (sec) |
N/A |
0.099 |
0.003 |
0.005 |
1.086 |
1.494 |
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 |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 26 | 35 | 43 | 51 | 17 | 45 |
normalized size | 1 | 1. | 1. | 1.35 | 1.65 | 1.96 | 0.65 | 1.73 |
time (sec) |
N/A |
0.018 |
0.004 |
0.003 |
1.047 |
1.49 |
0.189 |
1.243 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.048 |
0.043 |
0.033 |
0. |
0. |
0. |
0. |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.045 |
0.114 |
0.023 |
0. |
0. |
0. |
0. |
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Problem 7 |
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 | 26 | 26 | 21 | 37 | 46 | 53 | 17 | 47 |
normalized size | 1 | 1. | 0.81 | 1.42 | 1.77 | 2.04 | 0.65 | 1.81 |
time (sec) |
N/A |
0.021 |
0.015 |
0.003 |
1.084 |
1.505 |
0.138 |
1.334 |
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Problem 8 |
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 | 28 | 28 | 19 | 41 | 46 | 53 | 19 | 47 |
normalized size | 1 | 1. | 0.68 | 1.46 | 1.64 | 1.89 | 0.68 | 1.68 |
time (sec) |
N/A |
0.023 |
0.015 |
0.003 |
1.108 |
1.464 |
0.134 |
1.28 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 217 | 217 | 158 | 382 | 263 | 756 | 0 | 0 |
normalized size | 1 | 1. | 0.73 | 1.76 | 1.21 | 3.48 | 0. | 0. |
time (sec) |
N/A |
0.507 |
0.201 |
0.101 |
1.197 |
1.524 |
0. |
0. |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 165 | 165 | 113 | 324 | 201 | 608 | 0 | 0 |
normalized size | 1 | 1. | 0.68 | 1.96 | 1.22 | 3.68 | 0. | 0. |
time (sec) |
N/A |
0.399 |
0.233 |
0.063 |
1.217 |
1.486 |
0. |
0. |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
A |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 107 | 107 | 85 | 231 | 128 | 420 | 0 | 0 |
normalized size | 1 | 1. | 0.79 | 2.16 | 1.2 | 3.93 | 0. | 0. |
time (sec) |
N/A |
0.201 |
0.118 |
0.023 |
1.112 |
1.532 |
0. |
0. |
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Problem 12 |
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 | 46 | 46 | 40 | 54 | 69 | 146 | 39 | 70 |
normalized size | 1 | 1. | 0.87 | 1.17 | 1.5 | 3.17 | 0.85 | 1.52 |
time (sec) |
N/A |
0.033 |
0.039 |
0.001 |
1.057 |
1.485 |
0.201 |
1.237 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.047 |
0.777 |
0.1 |
0. |
0. |
0. |
0. |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.044 |
0.614 |
0.103 |
0. |
0. |
0. |
0. |
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Problem 15 |
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 | 48 | 48 | 42 | 58 | 74 | 151 | 39 | 76 |
normalized size | 1 | 1. | 0.88 | 1.21 | 1.54 | 3.15 | 0.81 | 1.58 |
time (sec) |
N/A |
0.034 |
0.053 |
0.002 |
1.06 |
1.543 |
0.219 |
1.252 |
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Problem 16 |
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 | 52 | 52 | 35 | 64 | 77 | 151 | 42 | 78 |
normalized size | 1 | 1. | 0.67 | 1.23 | 1.48 | 2.9 | 0.81 | 1.5 |
time (sec) |
N/A |
0.035 |
0.057 |
0.001 |
1.051 |
1.489 |
0.274 |
1.265 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 333 | 333 | 241 | 548 | 409 | 1558 | 0 | 0 |
normalized size | 1 | 1. | 0.72 | 1.65 | 1.23 | 4.68 | 0. | 0. |
time (sec) |
N/A |
1.078 |
0.226 |
0.092 |
1.119 |
1.571 |
0. |
0. |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 243 | 243 | 203 | 385 | 316 | 1170 | 0 | 0 |
normalized size | 1 | 1. | 0.84 | 1.58 | 1.3 | 4.81 | 0. | 0. |
time (sec) |
N/A |
0.764 |
0.168 |
0.077 |
1.055 |
1.532 |
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 |
A |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 159 | 159 | 120 | 393 | 201 | 760 | 0 | 0 |
normalized size | 1 | 1. | 0.75 | 2.47 | 1.26 | 4.78 | 0. | 0. |
time (sec) |
N/A |
0.332 |
0.134 |
0.026 |
1.106 |
1.477 |
0. |
0. |
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Problem 20 |
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 | 69 | 69 | 62 | 74 | 113 | 300 | 76 | 93 |
normalized size | 1 | 1. | 0.9 | 1.07 | 1.64 | 4.35 | 1.1 | 1.35 |
time (sec) |
N/A |
0.042 |
0.059 |
0.003 |
1.05 |
1.524 |
0.336 |
1.22 |
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|
|
|
|
|
|
|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.045 |
0.966 |
0.112 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.044 |
0.745 |
0.123 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
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 | 72 | 72 | 62 | 79 | 119 | 309 | 78 | 99 |
normalized size | 1 | 1. | 0.86 | 1.1 | 1.65 | 4.29 | 1.08 | 1.38 |
time (sec) |
N/A |
0.044 |
0.097 |
0.001 |
1.07 |
1.509 |
0.25 |
1.304 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 78 | 78 | 54 | 87 | 124 | 309 | 85 | 104 |
normalized size | 1 | 1. | 0.69 | 1.12 | 1.59 | 3.96 | 1.09 | 1.33 |
time (sec) |
N/A |
0.047 |
0.091 |
0.001 |
1.065 |
1.531 |
0.364 |
1.36 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 153 | 153 | 130 | 0 | 0 | 559 | 332 | 7738 |
normalized size |
1 |
1. |
0.85 |
0. |
0. |
3.65 |
2.17 |
50.58 |
time (sec) |
N/A |
0.238 |
0.267 |
0.029 |
0. |
1.803 |
0.312 |
1.498 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 115 | 115 | 91 | 199 | 0 | 360 | 196 | 3680 |
normalized size |
1 |
1. |
0.79 |
1.73 |
0. |
3.13 |
1.7 |
32. |
time (sec) |
N/A |
0.154 |
0.167 |
0.029 |
0. |
1.794 |
0.212 |
1.332 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 73 | 105 | 0 | 200 | 94 | 1508 |
normalized size |
1 |
1. |
0.95 |
1.36 |
0. |
2.6 |
1.22 |
19.58 |
time (sec) |
N/A |
0.076 |
0.162 |
0.02 |
0. |
1.805 |
0.462 |
1.541 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 30 | 30 | 30 | 31 | 0 | 80 | 32 | 43 |
normalized size |
1 |
1. |
1. |
1.03 |
0. |
2.67 |
1.07 |
1.43 |
time (sec) |
N/A |
0.016 |
0.025 |
0.001 |
0. |
1.641 |
0.267 |
1.25 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 68 | 68 | 56 | 0 | 0 | 109 | 0 | 0 |
normalized size | 1 | 1. | 0.82 | 0. | 0. | 1.6 | 0. | 0. |
time (sec) |
N/A |
0.134 |
0.143 |
0.046 |
0. |
1.574 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 100 | 100 | 78 | 0 | 0 | 190 | 0 | 0 |
normalized size | 1 | 1. | 0.78 | 0. | 0. | 1.9 | 0. | 0. |
time (sec) |
N/A |
0.174 |
0.274 |
0.05 |
0. |
1.588 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 147 | 147 | 111 | 0 | 0 | 338 | 0 | 0 |
normalized size |
1 |
1. |
0.76 |
0. |
0. |
2.3 |
0. |
0. |
time (sec) |
N/A |
0.24 |
0.327 |
0.026 |
0. |
1.585 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
A |
A |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 322 | 322 | 239 | 0 | 0 | 999 | 709 | 0 |
normalized size |
1 |
1. |
0.74 |
0. |
0. |
3.1 |
2.2 |
0. |
time (sec) |
N/A |
0.486 |
0.545 |
0.022 |
0. |
1.569 |
0.563 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 239 | 239 | 171 | 0 | 0 | 620 | 439 | 7694 |
normalized size |
1 |
1. |
0.72 |
0. |
0. |
2.59 |
1.84 |
32.19 |
time (sec) |
N/A |
0.32 |
0.365 |
0.023 |
0. |
1.612 |
0.453 |
1.828 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 156 | 156 | 117 | 220 | 0 | 317 | 233 | 3121 |
normalized size |
1 |
1. |
0.75 |
1.41 |
0. |
2.03 |
1.49 |
20.01 |
time (sec) |
N/A |
0.157 |
0.256 |
0.034 |
0. |
1.543 |
0.32 |
1.575 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 52 | 90 | 101 | 136 | 94 | 914 |
normalized size | 1 | 1. | 0.78 | 1.34 | 1.51 | 2.03 | 1.4 | 13.64 |
time (sec) |
N/A |
0.037 |
0.037 |
0.015 |
1.299 |
1.515 |
0.211 |
1.428 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 134 | 134 | 108 | 0 | 0 | 208 | 0 | 0 |
normalized size | 1 | 1. | 0.81 | 0. | 0. | 1.55 | 0. | 0. |
time (sec) |
N/A |
0.254 |
0.299 |
0.056 |
0. |
1.53 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 202 | 202 | 136 | 0 | 0 | 389 | 0 | 0 |
normalized size | 1 | 1. | 0.67 | 0. | 0. | 1.93 | 0. | 0. |
time (sec) |
N/A |
0.341 |
0.684 |
0.026 |
0. |
1.532 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 286 | 286 | 217 | 0 | 0 | 671 | 0 | 0 |
normalized size |
1 |
1. |
0.76 |
0. |
0. |
2.35 |
0. |
0. |
time (sec) |
N/A |
0.475 |
0.629 |
0.026 |
0. |
1.595 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
A |
A |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 496 | 496 | 341 | 0 | 0 | 1463 | 1074 | 0 |
normalized size |
1 |
1. |
0.69 |
0. |
0. |
2.95 |
2.17 |
0. |
time (sec) |
N/A |
0.705 |
0.721 |
0.023 |
0. |
1.657 |
0.985 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 366 | 366 | 248 | 0 | 0 | 890 | 653 | 11953 |
normalized size |
1 |
1. |
0.68 |
0. |
0. |
2.43 |
1.78 |
32.66 |
time (sec) |
N/A |
0.464 |
0.513 |
0.023 |
0. |
1.6 |
0.646 |
1.902 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 236 | 236 | 161 | 0 | 0 | 447 | 350 | 4822 |
normalized size |
1 |
1. |
0.68 |
0. |
0. |
1.89 |
1.48 |
20.43 |
time (sec) |
N/A |
0.239 |
0.351 |
0.014 |
0. |
1.558 |
0.421 |
1.799 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 103 | 103 | 74 | 124 | 155 | 189 | 153 | 1393 |
normalized size | 1 | 1. | 0.72 | 1.2 | 1.5 | 1.83 | 1.49 | 13.52 |
time (sec) |
N/A |
0.049 |
0.049 |
0.002 |
1.23 |
1.519 |
0.292 |
1.378 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 200 | 200 | 160 | 0 | 0 | 306 | 0 | 0 |
normalized size |
1 |
1. |
0.8 |
0. |
0. |
1.53 |
0. |
0. |
time (sec) |
N/A |
0.354 |
0.477 |
0.026 |
0. |
1.544 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 305 | 305 | 250 | 0 | 0 | 589 | 0 | 0 |
normalized size |
1 |
1. |
0.82 |
0. |
0. |
1.93 |
0. |
0. |
time (sec) |
N/A |
0.491 |
1.331 |
0.026 |
0. |
1.57 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 447 | 447 | 325 | 0 | 0 | 1018 | 0 | 0 |
normalized size |
1 |
1. |
0.73 |
0. |
0. |
2.28 |
0. |
0. |
time (sec) |
N/A |
0.695 |
1.023 |
0.027 |
0. |
1.582 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
C |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 192 | 192 | 166 | 3227 | 0 | 888 | 0 | 0 |
normalized size |
1 |
1. |
0.86 |
16.81 |
0. |
4.62 |
0. |
0. |
time (sec) |
N/A |
0.326 |
0.107 |
0.094 |
0. |
1.654 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 145 | 145 | 121 | 1765 | 0 | 597 | 0 | 0 |
normalized size | 1 | 1. | 0.83 | 12.17 | 0. | 4.12 | 0. | 0. |
time (sec) |
N/A |
0.276 |
0.033 |
0.09 |
0. |
1.749 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 98 | 98 | 74 | 719 | 0 | 344 | 0 | 0 |
normalized size | 1 | 1. | 0.76 | 7.34 | 0. | 3.51 | 0. | 0. |
time (sec) |
N/A |
0.157 |
0.011 |
0.06 |
0. |
1.69 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 40 | 40 | 40 | 65 | 89 | 92 | 27 | 0 |
normalized size | 1 | 1. | 1. | 1.62 | 2.22 | 2.3 | 0.68 | 0. |
time (sec) |
N/A |
0.03 |
0.006 |
0.005 |
1.068 |
1.792 |
0.201 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.137 |
0.117 |
0.094 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.128 |
0.331 |
0.111 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
B |
F |
C |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 388 | 388 | 0 | 3519 | 0 | 2936 | 0 | 0 |
normalized size | 1 | 1. | 0. | 9.07 | 0. | 7.57 | 0. | 0. |
time (sec) |
N/A |
0.867 |
1.723 |
0.138 |
0. |
1.815 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
B |
F |
C |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 294 | 294 | 0 | 1754 | 0 | 1823 | 0 | 0 |
normalized size | 1 | 1. | 0. | 5.97 | 0. | 6.2 | 0. | 0. |
time (sec) |
N/A |
0.671 |
1.124 |
0.085 |
0. |
1.643 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 191 | 191 | 0 | 631 | 0 | 918 | 0 | 0 |
normalized size | 1 | 1. | 0. | 3.3 | 0. | 4.81 | 0. | 0. |
time (sec) |
N/A |
0.34 |
0.998 |
0.089 |
0. |
1.595 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 62 | 99 | 135 | 248 | 66 | 0 |
normalized size | 1 | 1. | 0.84 | 1.34 | 1.82 | 3.35 | 0.89 | 0. |
time (sec) |
N/A |
0.049 |
0.059 |
0.003 |
1.027 |
1.563 |
0.202 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.127 |
0.989 |
0.469 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.12 |
1.017 |
0.336 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
B |
F |
C |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 594 | 594 | 0 | 4237 | 0 | 5646 | 0 | 0 |
normalized size | 1 | 1. | 0. | 7.13 | 0. | 9.51 | 0. | 0. |
time (sec) |
N/A |
1.901 |
1.728 |
0.147 |
0. |
2.414 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
B |
F |
C |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 439 | 439 | 0 | 1999 | 0 | 3272 | 0 | 0 |
normalized size | 1 | 1. | 0. | 4.55 | 0. | 7.45 | 0. | 0. |
time (sec) |
N/A |
1.319 |
1.302 |
0.089 |
0. |
2.099 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 276 | 276 | 0 | 708 | 0 | 1565 | 0 | 0 |
normalized size | 1 | 1. | 0. | 2.57 | 0. | 5.67 | 0. | 0. |
time (sec) |
N/A |
0.564 |
0.938 |
0.056 |
0. |
1.709 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 111 | 111 | 84 | 134 | 196 | 464 | 116 | 0 |
normalized size | 1 | 1. | 0.76 | 1.21 | 1.77 | 4.18 | 1.05 | 0. |
time (sec) |
N/A |
0.063 |
0.106 |
0.003 |
1.168 |
1.591 |
0.227 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.128 |
1.801 |
0.297 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.121 |
1.434 |
0.574 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 64 |
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 | 71 | 71 | 71 | 77 | 111 | 167 | 85 | 93 |
normalized size | 1 | 1. | 1. | 1.08 | 1.56 | 2.35 | 1.2 | 1.31 |
time (sec) |
N/A |
0.088 |
0.094 |
0.004 |
1.192 |
1.636 |
1.9 |
1.312 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 65 |
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 | 145 | 145 | 134 | 144 | 216 | 327 | 184 | 182 |
normalized size | 1 | 1. | 0.92 | 0.99 | 1.49 | 2.26 | 1.27 | 1.26 |
time (sec) |
N/A |
0.185 |
0.311 |
0.006 |
1.839 |
1.639 |
2.871 |
1.256 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 66 |
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 | 224 | 224 | 196 | 211 | 321 | 486 | 291 | 271 |
normalized size | 1 | 1. | 0.88 | 0.94 | 1.43 | 2.17 | 1.3 | 1.21 |
time (sec) |
N/A |
0.264 |
0.669 |
0.005 |
1.972 |
1.563 |
3.916 |
1.231 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.04 |
0.57 |
0.024 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.04 |
1.211 |
0.052 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.039 |
2.265 |
0.066 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F(-2) |
A |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 340 | 340 | 0 | 0 | 0 | 656 | 0 | 0 |
normalized size |
1 |
1. |
0. |
0. |
0. |
1.93 |
0. |
0. |
time (sec) |
N/A |
0.467 |
0.329 |
0.04 |
0. |
1.663 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 71 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F(-2) |
A |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 228 | 228 | 0 | 0 | 0 | 463 | 0 | 0 |
normalized size |
1 |
1. |
0. |
0. |
0. |
2.03 |
0. |
0. |
time (sec) |
N/A |
0.278 |
0.208 |
0.039 |
0. |
1.591 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 72 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F(-2) |
A |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 116 | 116 | 0 | 0 | 0 | 266 | 0 | 0 |
normalized size |
1 |
1. |
0. |
0. |
0. |
2.29 |
0. |
0. |
time (sec) |
N/A |
0.135 |
0.119 |
0.039 |
0. |
1.556 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 73 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.123 |
0.103 |
0.071 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 74 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.12 |
0.129 |
0.454 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 75 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.119 |
0.265 |
0.184 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 76 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 115 | 115 | 115 | 225 | 180 | 327 | 0 | 0 |
normalized size | 1 | 1. | 1. | 1.96 | 1.57 | 2.84 | 0. | 0. |
time (sec) |
N/A |
0.132 |
0.012 |
0.053 |
1.124 |
1.548 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 77 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 85 | 194 | 144 | 259 | 0 | 0 |
normalized size | 1 | 1. | 1. | 2.28 | 1.69 | 3.05 | 0. | 0. |
time (sec) |
N/A |
0.109 |
0.009 |
0.047 |
1.116 |
1.534 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 78 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 54 | 154 | 107 | 182 | 0 | 0 |
normalized size | 1 | 1. | 1. | 2.85 | 1.98 | 3.37 | 0. | 0. |
time (sec) |
N/A |
0.064 |
0.006 |
0.044 |
1.131 |
1.542 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 79 |
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 | 23 | 23 | 23 | 24 | 31 | 50 | 17 | 32 |
normalized size | 1 | 1. | 1. | 1.04 | 1.35 | 2.17 | 0.74 | 1.39 |
time (sec) |
N/A |
0.037 |
0.007 |
0.003 |
1.059 |
1.507 |
0.133 |
1.288 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 80 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.07 |
0.085 |
0.05 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 81 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.069 |
0.065 |
0.05 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 82 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 140 | 140 | 137 | 274 | 181 | 575 | 0 | 0 |
normalized size | 1 | 1. | 0.98 | 1.96 | 1.29 | 4.11 | 0. | 0. |
time (sec) |
N/A |
0.248 |
0.157 |
0.03 |
1.152 |
1.553 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 83 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 107 | 107 | 103 | 231 | 143 | 443 | 0 | 0 |
normalized size | 1 | 1. | 0.96 | 2.16 | 1.34 | 4.14 | 0. | 0. |
time (sec) |
N/A |
0.181 |
0.091 |
0.027 |
1.098 |
1.558 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 84 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 54 | 67 | 97 | 171 | 58 | 0 |
normalized size | 1 | 1. | 0.78 | 0.97 | 1.41 | 2.48 | 0.84 | 0. |
time (sec) |
N/A |
0.074 |
0.07 |
0.01 |
1.07 |
1.519 |
0.17 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 85 |
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 | 25 | 25 | 25 | 26 | 34 | 62 | 26 | 34 |
normalized size | 1 | 1. | 1. | 1.04 | 1.36 | 2.48 | 1.04 | 1.36 |
time (sec) |
N/A |
0.035 |
0.02 |
0.003 |
1.181 |
1.477 |
0.115 |
1.273 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 86 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.118 |
0.125 |
0.052 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 87 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.118 |
0.154 |
0.058 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 88 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 261 | 261 | 220 | 501 | 355 | 1310 | 0 | 0 |
normalized size | 1 | 1. | 0.84 | 1.92 | 1.36 | 5.02 | 0. | 0. |
time (sec) |
N/A |
0.504 |
0.323 |
0.066 |
1.141 |
1.637 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 89 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 182 | 182 | 177 | 304 | 289 | 882 | 0 | 0 |
normalized size | 1 | 1. | 0.97 | 1.67 | 1.59 | 4.85 | 0. | 0. |
time (sec) |
N/A |
0.291 |
0.162 |
0.056 |
1.127 |
1.572 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 90 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 98 | 127 | 203 | 342 | 122 | 0 |
normalized size | 1 | 1. | 0.92 | 1.2 | 1.92 | 3.23 | 1.15 | 0. |
time (sec) |
N/A |
0.089 |
0.085 |
0.019 |
1.166 |
1.55 |
0.204 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 91 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
B |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 27 | 26 | 34 | 115 | 53 | 34 |
normalized size | 1 | 1. | 1. | 0.96 | 1.26 | 4.26 | 1.96 | 1.26 |
time (sec) |
N/A |
0.035 |
0.018 |
0.001 |
1.025 |
1.507 |
0.137 |
1.287 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 92 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 64 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.116 |
0.687 |
0.158 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 93 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 62 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.113 |
0.729 |
0.185 |
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 [19] had the largest ratio of [ 0.7333
]
|
|
|
|
|
|
|
| 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 |
6 |
6 |
1. |
17 |
0.353 |
|
|
|
|
|
|
|
2 |
A |
5 |
5 |
1. |
17 |
0.294 |
|
|
|
|
|
|
|
3 |
A |
4 |
4 |
1. |
15 |
0.267 |
|
|
|
|
|
|
|
4 |
A |
4 |
4 |
1. |
13 |
0.308 |
|
|
|
|
|
|
|
5 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
6 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
7 |
A |
4 |
4 |
1. |
14 |
0.286 |
|
|
|
|
|
|
|
8 |
A |
4 |
4 |
1. |
16 |
0.25 |
|
|
|
|
|
|
|
9 |
A |
13 |
8 |
1. |
17 |
0.471 |
|
|
|
|
|
|
|
10 |
A |
11 |
9 |
1. |
17 |
0.529 |
|
|
|
|
|
|
|
11 |
A |
10 |
10 |
1. |
15 |
0.667 |
|
|
|
|
|
|
|
12 |
A |
3 |
2 |
1. |
13 |
0.154 |
|
|
|
|
|
|
|
13 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
14 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
15 |
A |
3 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
16 |
A |
3 |
2 |
1. |
16 |
0.125 |
|
|
|
|
|
|
|
17 |
A |
26 |
10 |
1. |
17 |
0.588 |
|
|
|
|
|
|
|
18 |
A |
23 |
12 |
1. |
17 |
0.706 |
|
|
|
|
|
|
|
19 |
A |
15 |
11 |
1. |
15 |
0.733 |
|
|
|
|
|
|
|
20 |
A |
3 |
2 |
1. |
13 |
0.154 |
|
|
|
|
|
|
|
|
21 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
22 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
23 |
A |
3 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
24 |
A |
3 |
2 |
1. |
16 |
0.125 |
|
|
|
|
|
|
|
25 |
A |
6 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
26 |
A |
5 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
27 |
A |
4 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
28 |
A |
2 |
1 |
1. |
15 |
0.067 |
|
|
|
|
|
|
|
29 |
A |
4 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
30 |
A |
5 |
4 |
1. |
23 |
0.174 |
|
|
|
|
|
|
|
31 |
A |
6 |
4 |
1. |
23 |
0.174 |
|
|
|
|
|
|
|
32 |
A |
10 |
3 |
1. |
25 |
0.12 |
|
|
|
|
|
|
|
33 |
A |
8 |
3 |
1. |
25 |
0.12 |
|
|
|
|
|
|
|
34 |
A |
6 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
35 |
A |
4 |
3 |
1. |
17 |
0.176 |
|
|
|
|
|
|
|
36 |
A |
6 |
3 |
1. |
25 |
0.12 |
|
|
|
|
|
|
|
37 |
A |
8 |
4 |
1. |
25 |
0.16 |
|
|
|
|
|
|
|
38 |
A |
10 |
4 |
1. |
25 |
0.16 |
|
|
|
|
|
|
|
39 |
A |
14 |
3 |
1. |
25 |
0.12 |
|
|
|
|
|
|
|
40 |
A |
11 |
3 |
1. |
25 |
0.12 |
|
|
|
|
|
|
|
|
41 |
A |
8 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
42 |
A |
4 |
3 |
1. |
17 |
0.176 |
|
|
|
|
|
|
|
43 |
A |
8 |
3 |
1. |
25 |
0.12 |
|
|
|
|
|
|
|
44 |
A |
11 |
4 |
1. |
25 |
0.16 |
|
|
|
|
|
|
|
45 |
A |
14 |
4 |
1. |
25 |
0.16 |
|
|
|
|
|
|
|
46 |
A |
6 |
6 |
1. |
25 |
0.24 |
|
|
|
|
|
|
|
47 |
A |
5 |
5 |
1. |
25 |
0.2 |
|
|
|
|
|
|
|
48 |
A |
4 |
4 |
1. |
23 |
0.174 |
|
|
|
|
|
|
|
49 |
A |
5 |
5 |
1. |
17 |
0.294 |
|
|
|
|
|
|
|
50 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
51 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
52 |
A |
13 |
8 |
1. |
25 |
0.32 |
|
|
|
|
|
|
|
53 |
A |
11 |
9 |
1. |
25 |
0.36 |
|
|
|
|
|
|
|
54 |
A |
11 |
11 |
1. |
23 |
0.478 |
|
|
|
|
|
|
|
55 |
A |
4 |
3 |
1. |
17 |
0.176 |
|
|
|
|
|
|
|
56 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
57 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
58 |
A |
26 |
10 |
1. |
25 |
0.4 |
|
|
|
|
|
|
|
59 |
A |
24 |
13 |
1. |
25 |
0.52 |
|
|
|
|
|
|
|
60 |
A |
17 |
12 |
1. |
23 |
0.522 |
|
|
|
|
|
|
|
|
61 |
A |
4 |
3 |
1. |
17 |
0.176 |
|
|
|
|
|
|
|
62 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
63 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
64 |
A |
5 |
4 |
1. |
17 |
0.235 |
|
|
|
|
|
|
|
65 |
A |
8 |
4 |
1. |
19 |
0.21 |
|
|
|
|
|
|
|
66 |
A |
11 |
4 |
1. |
19 |
0.21 |
|
|
|
|
|
|
|
67 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
68 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
69 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
70 |
A |
8 |
3 |
1. |
25 |
0.12 |
|
|
|
|
|
|
|
71 |
A |
6 |
3 |
1. |
25 |
0.12 |
|
|
|
|
|
|
|
72 |
A |
4 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
73 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
74 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
75 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
76 |
A |
5 |
5 |
1. |
24 |
0.208 |
|
|
|
|
|
|
|
77 |
A |
4 |
4 |
1. |
24 |
0.167 |
|
|
|
|
|
|
|
78 |
A |
3 |
3 |
1. |
22 |
0.136 |
|
|
|
|
|
|
|
79 |
A |
2 |
2 |
1. |
21 |
0.095 |
|
|
|
|
|
|
|
80 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
81 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
82 |
A |
6 |
6 |
1. |
24 |
0.25 |
|
|
|
|
|
|
|
83 |
A |
5 |
5 |
1. |
24 |
0.208 |
|
|
|
|
|
|
|
84 |
A |
5 |
5 |
1. |
22 |
0.227 |
|
|
|
|
|
|
|
85 |
A |
2 |
2 |
1. |
21 |
0.095 |
|
|
|
|
|
|
|
86 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
87 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
88 |
A |
12 |
9 |
1. |
24 |
0.375 |
|
|
|
|
|
|
|
89 |
A |
11 |
10 |
1. |
24 |
0.417 |
|
|
|
|
|
|
|
90 |
A |
4 |
3 |
1. |
22 |
0.136 |
|
|
|
|
|
|
|
91 |
A |
2 |
2 |
1. |
21 |
0.095 |
|
|
|
|
|
|
|
92 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
93 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|