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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 6, 10, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
39, 41, 43, 45, 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
}
B grade: { 5, 9, 25 }
C grade: { 4, 7, 8, 11, 12, 13, 23, 38, 40, 42, 44, 46 }
F grade: { }
2.1.3 Maple
A grade: { 1, 2, 3, 14, 15, 16, 17, 19, 20, 21, 22, 27, 28, 29, 34, 36, 47, 49, 50, 51, 57, 61, 70
}
B grade: { 5, 6, 30, 31, 32, 33, 35, 39, 41, 43, 45, 52, 53, 54, 55, 56, 58, 60, 62, 63, 64, 65, 66, 67, 68
}
C grade: { 38, 40, 42, 44, 46 }
F grade: { 4, 7, 8, 9, 10, 11, 12, 13, 18, 23, 24, 25, 26, 37, 48, 59, 69, 71 }
2.1.4 Maxima
A grade: { 14, 15, 16, 17, 19, 20, 21, 22, 27, 28, 29, 30, 31, 32, 34, 36, 39, 43, 47, 49, 50, 51, 52, 53, 55, 57,
61, 63, 66, 68, 70, 71
B grade: { 33, 35, 41, 45, 54, 56, 58, 60, 62, 64 }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 23, 24, 25, 26, 37, 38, 40, 42, 44, 46, 48, 59, 65, 67, 69
}
2.1.5 FriCAS
A grade: { 14, 15, 16, 17, 19, 20, 21, 22, 27, 28, 29, 30, 32, 34, 36, 39, 47, 49, 50, 51, 53, 57, 58, 60, 61, 62,
63, 64, 66, 68 }
B grade: { 1, 2, 3, 5, 6, 31, 33, 35, 41, 43, 45, 52, 54, 55, 56, 65, 67, 70, 71 }
C grade: { }
F grade: { 4, 7, 8, 9, 10, 11, 12, 13, 18, 23, 24, 25, 26, 37, 38, 40, 42, 44, 46, 48, 59, 69 }
2.1.6 Sympy
A grade: { 22, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 39, 41, 43, 45, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58,
68 }
B grade: { }
C grade: { 38, 40, 42, 44, 46 }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 34, 47, 55, 59, 60,
61, 62, 63, 64, 65, 66, 67, 69, 70, 71 }
2.1.7 Giac
A grade: { 27, 28, 29, 30, 31, 35, 39, 41, 43, 47, 60, 61, 62, 63, 64, 66, 68 }
B grade: { 33, 34, 36, 45, 65, 67 }
C grade: { }
F 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, 32, 37, 38,
40, 42, 44, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 69, 70, 71 }
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 |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 147 | 147 | 149 | 227 | 0 | 729 | 0 | 0 |
normalized size | 1 | 1. | 1.01 | 1.54 | 0. | 4.96 | 0. | 0. |
time (sec) |
N/A |
0.152 |
0.281 |
0.249 |
0. |
3.166 |
0. |
0. |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 129 | 170 | 0 | 686 | 0 | 0 |
normalized size | 1 | 1. | 1.17 | 1.55 | 0. | 6.24 | 0. | 0. |
time (sec) |
N/A |
0.095 |
0.176 |
0.219 |
0. |
3.319 |
0. |
0. |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 110 | 97 | 0 | 643 | 0 | 0 |
normalized size | 1 | 1. | 1.47 | 1.29 | 0. | 8.57 | 0. | 0. |
time (sec) |
N/A |
0.061 |
0.085 |
0.234 |
0. |
3.16 |
0. |
0. |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 162 | 162 | 427 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 2.64 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.287 |
0.379 |
0.25 |
0. |
0. |
0. |
0. |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 141 | 154 | 0 | 792 | 0 | 0 |
normalized size | 1 | 1. | 2.24 | 2.44 | 0. | 12.57 | 0. | 0. |
time (sec) |
N/A |
0.102 |
0.163 |
0.233 |
0. |
2.798 |
0. |
0. |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 114 | 114 | 220 | 453 | 0 | 1045 | 0 | 0 |
normalized size | 1 | 1. | 1.93 | 3.97 | 0. | 9.17 | 0. | 0. |
time (sec) |
N/A |
0.21 |
0.456 |
0.231 |
0. |
2.972 |
0. |
0. |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 501 | 501 | 1429 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
2.85 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.887 |
12.661 |
0.425 |
0. |
0. |
0. |
0. |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 351 | 351 | 864 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 2.46 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.506 |
9.142 |
0.403 |
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 |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 194 | 194 | 401 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 2.07 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.273 |
1.388 |
0.222 |
0. |
0. |
0. |
0. |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 160 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.88 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.083 |
0.249 |
0.218 |
0. |
0. |
0. |
0. |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 475 | 475 | 1008 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
2.12 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.073 |
2.783 |
0.611 |
0. |
0. |
0. |
0. |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 448 | 448 | 2061 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
4.6 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.11 |
12.864 |
0.543 |
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 |
C |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1024 | 1024 | 8348 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
8.15 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.296 |
14.164 |
0.53 |
0. |
0. |
0. |
0. |
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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(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 114 | 114 | 47 | 43 | 78 | 142 | 0 | 0 |
normalized size |
1 |
1. |
0.41 |
0.38 |
0.68 |
1.25 |
0. |
0. |
time (sec) |
N/A |
0.032 |
0.033 |
0.128 |
1.025 |
2.583 |
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 |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 42 | 38 | 62 | 128 | 0 | 0 |
normalized size |
1 |
1. |
0.47 |
0.43 |
0.7 |
1.44 |
0. |
0. |
time (sec) |
N/A |
0.025 |
0.03 |
0.118 |
0.996 |
2.632 |
0. |
0. |
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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 |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 64 | 64 | 35 | 31 | 46 | 113 | 0 | 0 |
normalized size | 1 | 1. | 0.55 | 0.48 | 0.72 | 1.77 | 0. | 0. |
time (sec) |
N/A |
0.017 |
0.024 |
0.119 |
1.001 |
2.563 |
0. |
0. |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 24 | 24 | 24 | 89 | 0 | 0 |
normalized size | 1 | 1. | 0.77 | 0.77 | 0.77 | 2.87 | 0. | 0. |
time (sec) |
N/A |
0.008 |
0.009 |
0.113 |
1.004 |
2.574 |
0. |
0. |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 45 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.98 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.097 |
0.035 |
0.17 |
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 |
A |
A |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 42 | 46 | 88 | 109 | 0 | 0 |
normalized size | 1 | 1. | 0.67 | 0.73 | 1.4 | 1.73 | 0. | 0. |
time (sec) |
N/A |
0.024 |
0.023 |
0.112 |
0.998 |
2.692 |
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 |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 90 | 90 | 47 | 57 | 124 | 132 | 0 | 0 |
normalized size |
1 |
1. |
0.52 |
0.63 |
1.38 |
1.47 |
0. |
0. |
time (sec) |
N/A |
0.028 |
0.034 |
0.112 |
1.019 |
2.506 |
0. |
0. |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 115 | 115 | 52 | 67 | 157 | 150 | 0 | 0 |
normalized size |
1 |
1. |
0.45 |
0.58 |
1.37 |
1.3 |
0. |
0. |
time (sec) |
N/A |
0.034 |
0.045 |
0.117 |
0.991 |
2.589 |
0. |
0. |
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Problem 22 |
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 | 16 | 16 | 18 | 29 | 22 | 57 | 14 | 0 |
normalized size | 1 | 1. | 1.12 | 1.81 | 1.38 | 3.56 | 0.88 | 0. |
time (sec) |
N/A |
0.006 |
0.003 |
0.099 |
1. |
2.585 |
0.147 |
0. |
|
|
|
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Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F(-2) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 64 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.05 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.108 |
0.086 |
0.378 |
0. |
0. |
0. |
0. |
|
|
|
|
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Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 49 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.91 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.101 |
0.04 |
0.279 |
0. |
0. |
0. |
0. |
|
|
|
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Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
F(-2) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 236 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
3.06 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.088 |
0.569 |
0.434 |
0. |
0. |
0. |
0. |
|
|
|
|
|
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|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 52 | 52 | 54 | 0 | 0 | 0 | 51 | 0 |
normalized size |
1 |
1. |
1.04 |
0. |
0. |
0. |
0.98 |
0. |
time (sec) |
N/A |
0.04 |
0.045 |
0.255 |
0. |
0. |
8.634 |
0. |
|
|
|
|
|
|
|
|
|
|
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|
|
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|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 49 | 53 | 68 | 115 | 63 | 108 |
normalized size | 1 | 1. | 0.91 | 0.98 | 1.26 | 2.13 | 1.17 | 2. |
time (sec) |
N/A |
0.029 |
0.049 |
0.21 |
1.008 |
2.565 |
3.266 |
1.153 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 76 | 109 | 144 | 171 | 73 | 93 |
normalized size | 1 | 1. | 1.01 | 1.45 | 1.92 | 2.28 | 0.97 | 1.24 |
time (sec) |
N/A |
0.044 |
0.055 |
0.177 |
0.973 |
2.642 |
5.507 |
1.164 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
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 | 31 | 31 | 38 | 43 | 34 | 90 | 41 | 61 |
normalized size | 1 | 1. | 1.23 | 1.39 | 1.1 | 2.9 | 1.32 | 1.97 |
time (sec) |
N/A |
0.022 |
0.038 |
0.171 |
1.021 |
2.535 |
2.457 |
1.148 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 47 | 85 | 105 | 140 | 29 | 70 |
normalized size | 1 | 1. | 1. | 1.81 | 2.23 | 2.98 | 0.62 | 1.49 |
time (sec) |
N/A |
0.026 |
0.03 |
0.194 |
0.967 |
2.57 |
3.453 |
1.171 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
A |
A |
verified |
N/A |
NO |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 31 | 35 | 113 | 86 | 203 | 48 | 82 |
normalized size | 1 | 1.29 | 1.46 | 4.71 | 3.58 | 8.46 | 2. | 3.42 |
time (sec) |
N/A |
0.015 |
0.015 |
0.169 |
1.002 |
2.776 |
2.018 |
1.161 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
F(-2) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 42 | 107 | 73 | 142 | 41 | 0 |
normalized size |
1 |
1. |
1.11 |
2.82 |
1.92 |
3.74 |
1.08 |
0. |
time (sec) |
N/A |
0.035 |
0.028 |
0.18 |
1.001 |
3.016 |
4.265 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
A |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 40 | 40 | 43 | 145 | 116 | 232 | 36 | 111 |
normalized size | 1 | 1. | 1.08 | 3.62 | 2.9 | 5.8 | 0.9 | 2.78 |
time (sec) |
N/A |
0.029 |
0.027 |
0.188 |
1.008 |
2.949 |
4.075 |
1.135 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 37 | 42 | 34 | 103 | 0 | 93 |
normalized size |
1 |
1. |
1.19 |
1.35 |
1.1 |
3.32 |
0. |
3. |
time (sec) |
N/A |
0.022 |
0.034 |
0.179 |
1.033 |
2.41 |
0. |
1.193 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
A |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 65 | 65 | 53 | 173 | 174 | 250 | 83 | 139 |
normalized size | 1 | 1. | 0.82 | 2.66 | 2.68 | 3.85 | 1.28 | 2.14 |
time (sec) |
N/A |
0.043 |
0.049 |
0.183 |
1.022 |
2.586 |
5.326 |
1.152 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 46 | 52 | 55 | 124 | 65 | 167 |
normalized size | 1 | 1. | 0.9 | 1.02 | 1.08 | 2.43 | 1.27 | 3.27 |
time (sec) |
N/A |
0.038 |
0.042 |
0.197 |
0.989 |
2.439 |
3.043 |
1.255 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 55 | 0 | 0 | 0 | 66 | 0 |
normalized size |
1 |
1. |
0.93 |
0. |
0. |
0. |
1.12 |
0. |
time (sec) |
N/A |
0.042 |
0.061 |
0.199 |
0. |
0. |
19.537 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
C |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 202 | 202 | 112 | 150 | 0 | 0 | 48 | 0 |
normalized size | 1 | 1. | 0.55 | 0.74 | 0. | 0. | 0.24 | 0. |
time (sec) |
N/A |
0.125 |
0.249 |
0.211 |
0. |
0. |
3.352 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 52 | 52 | 53 | 94 | 109 | 149 | 36 | 74 |
normalized size | 1 | 1. | 1.02 | 1.81 | 2.1 | 2.87 | 0.69 | 1.42 |
time (sec) |
N/A |
0.038 |
0.054 |
0.29 |
1.032 |
2.595 |
4.294 |
1.133 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
C |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 86 | 86 | 113 | 104 | 0 | 0 | 41 | 0 |
normalized size | 1 | 1. | 1.31 | 1.21 | 0. | 0. | 0.48 | 0. |
time (sec) |
N/A |
0.053 |
0.215 |
0.187 |
0. |
0. |
3.279 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
A |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 40 | 40 | 42 | 116 | 96 | 205 | 58 | 82 |
normalized size | 1 | 1. | 1.05 | 2.9 | 2.4 | 5.12 | 1.45 | 2.05 |
time (sec) |
N/A |
0.036 |
0.031 |
0.285 |
0.989 |
2.548 |
5.24 |
1.132 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
C |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 165 | 165 | 96 | 146 | 0 | 0 | 42 | 0 |
normalized size | 1 | 1. | 0.58 | 0.88 | 0. | 0. | 0.25 | 0. |
time (sec) |
N/A |
0.08 |
0.141 |
0.18 |
0. |
0. |
0.908 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
A |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 22 | 86 | 73 | 163 | 54 | 80 |
normalized size | 1 | 1. | 0.48 | 1.87 | 1.59 | 3.54 | 1.17 | 1.74 |
time (sec) |
N/A |
0.033 |
0.045 |
0.272 |
1.008 |
2.56 |
7.134 |
1.229 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
C |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 91 | 91 | 96 | 111 | 0 | 0 | 42 | 0 |
normalized size | 1 | 1. | 1.05 | 1.22 | 0. | 0. | 0.46 | 0. |
time (sec) |
N/A |
0.047 |
0.16 |
0.18 |
0. |
0. |
2.604 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
A |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 42 | 42 | 24 | 114 | 124 | 227 | 39 | 103 |
normalized size | 1 | 1. | 0.57 | 2.71 | 2.95 | 5.4 | 0.93 | 2.45 |
time (sec) |
N/A |
0.039 |
0.044 |
0.287 |
1.072 |
2.52 |
4.863 |
1.125 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
C |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 181 | 181 | 114 | 171 | 0 | 0 | 44 | 0 |
normalized size | 1 | 1. | 0.63 | 0.94 | 0. | 0. | 0.24 | 0. |
time (sec) |
N/A |
0.095 |
0.193 |
0.215 |
0. |
0. |
3.465 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-2) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 39 | 42 | 34 | 105 | 0 | 34 |
normalized size |
1 |
1. |
1.26 |
1.35 |
1.1 |
3.39 |
0. |
1.1 |
time (sec) |
N/A |
0.022 |
0.053 |
0.273 |
1.037 |
2.519 |
0. |
1.209 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
A |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 64 | 64 | 57 | 0 | 0 | 0 | 71 | 0 |
normalized size |
1 |
1. |
0.89 |
0. |
0. |
0. |
1.11 |
0. |
time (sec) |
N/A |
0.322 |
0.065 |
0.185 |
0. |
0. |
14.121 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F(-2) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 84 | 117 | 158 | 193 | 82 | 0 |
normalized size |
1 |
1. |
0.99 |
1.38 |
1.86 |
2.27 |
0.96 |
0. |
time (sec) |
N/A |
0.254 |
0.063 |
0.174 |
1.035 |
2.562 |
6.686 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 44 | 61 | 43 | 109 | 51 | 0 |
normalized size |
1 |
1. |
1.16 |
1.61 |
1.13 |
2.87 |
1.34 |
0. |
time (sec) |
N/A |
0.225 |
0.047 |
0.183 |
0.992 |
2.536 |
3.489 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F(-2) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 52 | 52 | 57 | 90 | 120 | 159 | 36 | 0 |
normalized size |
1 |
1. |
1.1 |
1.73 |
2.31 |
3.06 |
0.69 |
0. |
time (sec) |
N/A |
0.24 |
0.038 |
0.171 |
0.989 |
2.525 |
4.371 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
A |
F(-2) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 44 | 120 | 101 | 235 | 63 | 0 |
normalized size |
1 |
1. |
1.02 |
2.79 |
2.35 |
5.47 |
1.47 |
0. |
time (sec) |
N/A |
0.158 |
0.047 |
0.179 |
0.964 |
2.498 |
4.363 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
F(-2) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 52 | 112 | 80 | 165 | 49 | 0 |
normalized size |
1 |
1. |
1.11 |
2.38 |
1.7 |
3.51 |
1.04 |
0. |
time (sec) |
N/A |
0.072 |
0.035 |
0.128 |
0.965 |
2.532 |
4.22 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
A |
F(-2) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 39 | 150 | 126 | 252 | 34 | 0 |
normalized size |
1 |
1. |
1.03 |
3.95 |
3.32 |
6.63 |
0.89 |
0. |
time (sec) |
N/A |
0.214 |
0.057 |
0.177 |
1.016 |
2.595 |
4.739 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
F(-2) |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 34 | 54 | 46 | 63 | 38 | 127 | 0 | 0 |
normalized size |
1 |
1.59 |
1.35 |
1.85 |
1.12 |
3.74 |
0. |
0. |
time (sec) |
N/A |
0.204 |
0.042 |
0.19 |
1. |
2.479 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
A |
F(-2) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 73 | 73 | 73 | 176 | 188 | 269 | 92 | 0 |
normalized size |
1 |
1. |
1. |
2.41 |
2.58 |
3.68 |
1.26 |
0. |
time (sec) |
N/A |
0.234 |
0.049 |
0.21 |
1.01 |
2.529 |
6.106 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 54 | 73 | 70 | 146 | 76 | 0 |
normalized size |
1 |
1. |
0.93 |
1.26 |
1.21 |
2.52 |
1.31 |
0. |
time (sec) |
N/A |
0.233 |
0.054 |
0.184 |
1.001 |
2.681 |
3.767 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
F(-2) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 96 | 96 | 74 | 209 | 243 | 296 | 114 | 0 |
normalized size |
1 |
1. |
0.77 |
2.18 |
2.53 |
3.08 |
1.19 |
0. |
time (sec) |
N/A |
0.253 |
0.072 |
0.187 |
1.002 |
2.788 |
8.605 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 88 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.04 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.104 |
0.243 |
0.382 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
F |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 92 | 92 | 85 | 165 | 216 | 209 | 0 | 120 |
normalized size | 1 | 1. | 0.92 | 1.79 | 2.35 | 2.27 | 0. | 1.3 |
time (sec) |
N/A |
0.108 |
0.208 |
0.201 |
1.537 |
3.023 |
0. |
1.204 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 72 | 72 | 64 | 120 | 66 | 127 | 0 | 101 |
normalized size | 1 | 1. | 0.89 | 1.67 | 0.92 | 1.76 | 0. | 1.4 |
time (sec) |
N/A |
0.098 |
0.127 |
0.143 |
1.027 |
2.981 |
0. |
1.148 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
F |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 54 | 133 | 142 | 162 | 0 | 82 |
normalized size | 1 | 1. | 0.92 | 2.25 | 2.41 | 2.75 | 0. | 1.39 |
time (sec) |
N/A |
0.09 |
0.131 |
0.15 |
1.498 |
2.612 |
0. |
1.138 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 36 | 36 | 35 | 89 | 42 | 89 | 0 | 61 |
normalized size | 1 | 1. | 0.97 | 2.47 | 1.17 | 2.47 | 0. | 1.69 |
time (sec) |
N/A |
0.07 |
0.067 |
0.151 |
1.031 |
2.607 |
0. |
1.131 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
F |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 38 | 85 | 80 | 89 | 0 | 46 |
normalized size | 1 | 1. | 1.41 | 3.15 | 2.96 | 3.3 | 0. | 1.7 |
time (sec) |
N/A |
0.059 |
0.049 |
0.14 |
1.556 |
2.552 |
0. |
1.158 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 33 | 33 | 37 | 172 | 0 | 194 | 0 | 95 |
normalized size | 1 | 1. | 1.12 | 5.21 | 0. | 5.88 | 0. | 2.88 |
time (sec) |
N/A |
0.043 |
0.033 |
0.15 |
0. |
2.677 |
0. |
1.159 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 66 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F(-2) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 30 | 30 | 30 | 154 | 46 | 97 | 0 | 58 |
normalized size |
1 |
1. |
1. |
5.13 |
1.53 |
3.23 |
0. |
1.93 |
time (sec) |
N/A |
0.071 |
0.101 |
0.24 |
1.528 |
2.587 |
0. |
1.154 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 58 | 210 | 0 | 293 | 0 | 154 |
normalized size | 1 | 1. | 0.97 | 3.5 | 0. | 4.88 | 0. | 2.57 |
time (sec) |
N/A |
0.097 |
0.109 |
0.195 |
0. |
2.613 |
0. |
1.144 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 54 | 193 | 76 | 155 | 75 | 111 |
normalized size | 1 | 1. | 0.89 | 3.16 | 1.25 | 2.54 | 1.23 | 1.82 |
time (sec) |
N/A |
0.095 |
0.147 |
0.196 |
1.589 |
2.528 |
5.461 |
1.166 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 52 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.85 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.098 |
0.058 |
0.509 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 74 | 63 | 77 | 570 | 0 | 0 |
normalized size |
1 |
1. |
1.61 |
1.37 |
1.67 |
12.39 |
0. |
0. |
time (sec) |
N/A |
0.058 |
0.136 |
0.25 |
1.011 |
2.814 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 71 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
A |
B |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 74 | 0 | 81 | 969 | 0 | 0 |
normalized size |
1 |
1. |
1.61 |
0. |
1.76 |
21.07 |
0. |
0. |
time (sec) |
N/A |
0.07 |
0.174 |
0.349 |
1.031 |
3.012 |
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 [42] had the largest ratio of [ 0.875
]
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| Table 2.1Rubi specific breakdown of results for each
integral
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# |
grade |
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normalized | antiderivative |
leaf size | |
|
\(\frac{\text{number of rules}}{\text{integrand leaf size}}\) |
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1 |
A |
8 |
7 |
1. |
10 |
0.7 |
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2 |
A |
7 |
6 |
1. |
10 |
0.6 |
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3 |
A |
6 |
6 |
1. |
8 |
0.75 |
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4 |
A |
14 |
8 |
1. |
10 |
0.8 |
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5 |
A |
6 |
6 |
1. |
10 |
0.6 |
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6 |
A |
8 |
8 |
1. |
10 |
0.8 |
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7 |
A |
20 |
9 |
1. |
20 |
0.45 |
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8 |
A |
17 |
9 |
1. |
20 |
0.45 |
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9 |
A |
11 |
8 |
1. |
18 |
0.444 |
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10 |
A |
8 |
6 |
1. |
12 |
0.5 |
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11 |
A |
17 |
9 |
1. |
20 |
0.45 |
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12 |
A |
12 |
8 |
1. |
20 |
0.4 |
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13 |
A |
23 |
11 |
1. |
20 |
0.55 |
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14 |
A |
4 |
3 |
1. |
10 |
0.3 |
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15 |
A |
4 |
3 |
1. |
10 |
0.3 |
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16 |
A |
4 |
3 |
1. |
8 |
0.375 |
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17 |
A |
3 |
3 |
1. |
6 |
0.5 |
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18 |
A |
7 |
6 |
1. |
10 |
0.6 |
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19 |
A |
5 |
5 |
1. |
10 |
0.5 |
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20 |
A |
6 |
5 |
1. |
10 |
0.5 |
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21 |
A |
7 |
5 |
1. |
10 |
0.5 |
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22 |
A |
3 |
3 |
1. |
4 |
0.75 |
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23 |
A |
7 |
6 |
1. |
10 |
0.6 |
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24 |
A |
7 |
6 |
1. |
10 |
0.6 |
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25 |
A |
7 |
7 |
1. |
10 |
0.7 |
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26 |
A |
4 |
4 |
1. |
10 |
0.4 |
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27 |
A |
4 |
4 |
1. |
10 |
0.4 |
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28 |
A |
7 |
7 |
1. |
10 |
0.7 |
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29 |
A |
3 |
3 |
1. |
10 |
0.3 |
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30 |
A |
6 |
6 |
1. |
8 |
0.75 |
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31 |
A |
5 |
5 |
1.29 |
6 |
0.833 |
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32 |
A |
6 |
6 |
1. |
10 |
0.6 |
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33 |
A |
5 |
5 |
1. |
10 |
0.5 |
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34 |
A |
3 |
3 |
1. |
10 |
0.3 |
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35 |
A |
6 |
6 |
1. |
10 |
0.6 |
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36 |
A |
5 |
4 |
1. |
10 |
0.4 |
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37 |
A |
4 |
4 |
1. |
12 |
0.333 |
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38 |
A |
8 |
8 |
1. |
12 |
0.667 |
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39 |
A |
6 |
6 |
1. |
12 |
0.5 |
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40 |
A |
5 |
5 |
1. |
12 |
0.417 |
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41 |
A |
6 |
6 |
1. |
10 |
0.6 |
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42 |
A |
7 |
7 |
1. |
8 |
0.875 |
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43 |
A |
6 |
6 |
1. |
12 |
0.5 |
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44 |
A |
5 |
5 |
1. |
12 |
0.417 |
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45 |
A |
6 |
6 |
1. |
12 |
0.5 |
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46 |
A |
7 |
7 |
1. |
12 |
0.583 |
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47 |
A |
3 |
3 |
1. |
12 |
0.25 |
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48 |
A |
5 |
4 |
1. |
12 |
0.333 |
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49 |
A |
8 |
7 |
1. |
12 |
0.583 |
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50 |
A |
4 |
3 |
1. |
12 |
0.25 |
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51 |
A |
7 |
6 |
1. |
12 |
0.5 |
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52 |
A |
6 |
5 |
1. |
10 |
0.5 |
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53 |
A |
7 |
6 |
1. |
8 |
0.75 |
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54 |
A |
6 |
5 |
1. |
12 |
0.417 |
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55 |
A |
4 |
3 |
1.59 |
12 |
0.25 |
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56 |
A |
7 |
6 |
1. |
12 |
0.5 |
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57 |
A |
6 |
4 |
1. |
12 |
0.333 |
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58 |
A |
8 |
6 |
1. |
12 |
0.5 |
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59 |
A |
4 |
3 |
1. |
23 |
0.13 |
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60 |
A |
9 |
7 |
1. |
21 |
0.333 |
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61 |
A |
6 |
5 |
1. |
21 |
0.238 |
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62 |
A |
7 |
7 |
1. |
21 |
0.333 |
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63 |
A |
3 |
3 |
1. |
21 |
0.143 |
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64 |
A |
5 |
5 |
1. |
19 |
0.263 |
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65 |
A |
7 |
7 |
1. |
18 |
0.389 |
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66 |
A |
4 |
4 |
1. |
21 |
0.19 |
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67 |
A |
7 |
6 |
1. |
21 |
0.286 |
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68 |
A |
7 |
5 |
1. |
21 |
0.238 |
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69 |
A |
8 |
8 |
1. |
19 |
0.421 |
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70 |
A |
6 |
6 |
1. |
12 |
0.5 |
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71 |
A |
6 |
6 |
1. |
14 |
0.429 |
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