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, 60, 61, 62, 63, 64, 66, 68 }
B grade: { 33, 34, 36, 45, 47, 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 }
2.1.8 Mupad
A grade: { }
B grade: { 17, 19, 22, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 39, 41, 42, 43, 44, 45, 47, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71 }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 23, 24, 25, 26, 37, 38, 40, 46, 48, 59,
69 }
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 |
Mupad |
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grade |
A |
A |
A |
A |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 147 | 147 | 149 | 226 | 0 | 325 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.01 | 1.54 | 0.00 | 2.21 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.152 |
0.300 |
0.074 |
0.000 |
0.628 |
0.000 |
0.000 |
0.000 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 129 | 170 | 0 | 306 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.17 | 1.55 | 0.00 | 2.78 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.095 |
0.177 |
0.056 |
0.000 |
0.622 |
0.000 |
0.000 |
0.000 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 110 | 97 | 0 | 285 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.47 | 1.29 | 0.00 | 3.80 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.061 |
0.105 |
0.050 |
0.000 |
0.566 |
0.000 |
0.000 |
0.000 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 162 | 162 | 427 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.64 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.287 |
0.403 |
0.069 |
0.000 |
0.636 |
0.000 |
0.000 |
0.000 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
B |
B |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 141 | 154 | 0 | 343 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.24 | 2.44 | 0.00 | 5.44 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.102 |
0.165 |
0.074 |
0.000 |
0.487 |
0.000 |
0.000 |
0.000 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 114 | 114 | 220 | 453 | 0 | 461 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.93 | 3.97 | 0.00 | 4.04 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.210 |
0.477 |
0.061 |
0.000 |
0.512 |
0.000 |
0.000 |
0.000 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 501 | 501 | 1429 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.85 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.887 |
12.802 |
0.413 |
0.000 |
0.547 |
0.000 |
0.000 |
0.000 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 351 | 351 | 864 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.46 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.506 |
9.244 |
0.429 |
0.000 |
0.581 |
0.000 |
0.000 |
0.000 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
B |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 194 | 194 | 401 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.07 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.273 |
1.461 |
0.055 |
0.000 |
0.688 |
0.000 |
0.000 |
0.000 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 160 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.88 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.083 |
0.263 |
0.054 |
0.000 |
0.485 |
0.000 |
0.000 |
0.000 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 475 | 475 | 1008 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.12 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.073 |
2.980 |
0.537 |
0.000 |
0.492 |
0.000 |
0.000 |
0.000 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 448 | 448 | 2061 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 4.60 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.110 |
12.884 |
0.486 |
0.000 |
0.521 |
0.000 |
0.000 |
0.000 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1024 | 1024 | 8350 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 8.15 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
2.296 |
14.292 |
0.480 |
0.000 |
0.561 |
0.000 |
0.000 |
0.000 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 114 | 114 | 47 | 43 | 58 | 55 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.41 | 0.38 | 0.51 | 0.48 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.032 |
0.036 |
0.048 |
0.309 |
0.584 |
0.000 |
0.000 |
0.000 |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 42 | 38 | 46 | 50 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.47 | 0.43 | 0.52 | 0.56 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.025 |
0.029 |
0.048 |
0.319 |
0.588 |
0.000 |
0.000 |
0.000 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 64 | 64 | 35 | 31 | 34 | 43 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.55 | 0.48 | 0.53 | 0.67 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.017 |
0.027 |
0.045 |
0.320 |
0.758 |
0.000 |
0.000 |
0.000 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 24 | 24 | 18 | 36 | 0 | 0 | 18 |
normalized size | 1 | 1.00 | 0.77 | 0.77 | 0.58 | 1.16 | 0.00 | 0.00 | 0.58 |
time (sec) |
N/A |
0.008 |
0.011 |
0.050 |
0.319 |
0.525 |
0.000 |
0.000 |
2.571 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 45 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.98 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.097 |
0.037 |
0.073 |
0.000 |
0.666 |
0.000 |
0.000 |
0.000 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 42 | 45 | 65 | 44 | 0 | 0 | 33 |
normalized size | 1 | 1.00 | 0.67 | 0.71 | 1.03 | 0.70 | 0.00 | 0.00 | 0.52 |
time (sec) |
N/A |
0.024 |
0.025 |
0.055 |
0.322 |
0.628 |
0.000 |
0.000 |
2.221 |
|
|
|
|
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|
Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 90 | 90 | 47 | 57 | 92 | 53 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.52 | 0.63 | 1.02 | 0.59 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.028 |
0.037 |
0.050 |
0.319 |
0.703 |
0.000 |
0.000 |
0.000 |
|
|
|
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|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 115 | 115 | 52 | 67 | 116 | 58 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.45 | 0.58 | 1.01 | 0.50 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.034 |
0.044 |
0.060 |
0.319 |
0.645 |
0.000 |
0.000 |
0.000 |
|
|
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|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 16 | 16 | 18 | 29 | 16 | 22 | 14 | 0 | 14 |
normalized size | 1 | 1.00 | 1.12 | 1.81 | 1.00 | 1.38 | 0.88 | 0.00 | 0.88 |
time (sec) |
N/A |
0.006 |
0.003 |
0.049 |
0.312 |
0.729 |
0.116 |
0.000 |
0.074 |
|
|
|
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|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 64 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.05 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.108 |
0.104 |
0.238 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
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|
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|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 49 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.91 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.101 |
0.045 |
0.089 |
0.000 |
0.630 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
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|
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|
|
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|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 236 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 3.06 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.088 |
0.603 |
0.269 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
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|
|
|
|
|
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|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
A |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 52 | 52 | 54 | 0 | 0 | 0 | 51 | 0 | -1 |
normalized size | 1 | 1.00 | 1.04 | 0.00 | 0.00 | 0.00 | 0.98 | 0.00 | -0.02 |
time (sec) |
N/A |
0.040 |
0.047 |
0.052 |
0.000 |
0.738 |
5.993 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 49 | 53 | 50 | 53 | 63 | 78 | 41 |
normalized size | 1 | 1.00 | 0.91 | 0.98 | 0.93 | 0.98 | 1.17 | 1.44 | 0.76 |
time (sec) |
N/A |
0.029 |
0.051 |
0.086 |
0.327 |
0.395 |
3.014 |
0.142 |
2.177 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 76 | 109 | 107 | 79 | 73 | 69 | 61 |
normalized size | 1 | 1.00 | 1.01 | 1.45 | 1.43 | 1.05 | 0.97 | 0.92 | 0.81 |
time (sec) |
N/A |
0.044 |
0.059 |
0.049 |
0.317 |
0.397 |
4.601 |
0.155 |
2.407 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 38 | 43 | 25 | 41 | 41 | 44 | 33 |
normalized size | 1 | 1.00 | 1.23 | 1.39 | 0.81 | 1.32 | 1.32 | 1.42 | 1.06 |
time (sec) |
N/A |
0.022 |
0.039 |
0.044 |
0.311 |
0.560 |
2.712 |
0.142 |
2.166 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 47 | 85 | 78 | 64 | 29 | 52 | 39 |
normalized size | 1 | 1.00 | 1.00 | 1.81 | 1.66 | 1.36 | 0.62 | 1.11 | 0.83 |
time (sec) |
N/A |
0.026 |
0.031 |
0.046 |
0.310 |
0.775 |
3.249 |
0.153 |
2.214 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
A |
A |
B |
verified |
N/A |
NO |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 31 | 35 | 113 | 64 | 86 | 48 | 66 | 36 |
normalized size | 1 | 1.29 | 1.46 | 4.71 | 2.67 | 3.58 | 2.00 | 2.75 | 1.50 |
time (sec) |
N/A |
0.015 |
0.017 |
0.046 |
0.314 |
0.448 |
1.143 |
0.127 |
2.253 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
F(-2) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 42 | 107 | 54 | 64 | 41 | 0 | 34 |
normalized size | 1 | 1.00 | 1.11 | 2.82 | 1.42 | 1.68 | 1.08 | 0.00 | 0.89 |
time (sec) |
N/A |
0.035 |
0.030 |
0.045 |
0.312 |
0.595 |
4.813 |
0.000 |
2.456 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
A |
B |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 40 | 40 | 43 | 145 | 86 | 102 | 36 | 82 | 42 |
normalized size | 1 | 1.00 | 1.08 | 3.62 | 2.15 | 2.55 | 0.90 | 2.05 | 1.05 |
time (sec) |
N/A |
0.029 |
0.030 |
0.046 |
0.323 |
0.462 |
3.233 |
0.144 |
2.556 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-2) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 37 | 42 | 25 | 47 | 0 | 69 | 42 |
normalized size | 1 | 1.00 | 1.19 | 1.35 | 0.81 | 1.52 | 0.00 | 2.23 | 1.35 |
time (sec) |
N/A |
0.022 |
0.037 |
0.064 |
0.305 |
0.704 |
0.000 |
0.166 |
2.155 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
A |
A |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 65 | 65 | 53 | 173 | 129 | 113 | 83 | 103 | 61 |
normalized size | 1 | 1.00 | 0.82 | 2.66 | 1.98 | 1.74 | 1.28 | 1.58 | 0.94 |
time (sec) |
N/A |
0.043 |
0.051 |
0.056 |
0.323 |
0.678 |
4.254 |
0.157 |
2.543 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 46 | 52 | 41 | 58 | 65 | 124 | 61 |
normalized size | 1 | 1.00 | 0.90 | 1.02 | 0.80 | 1.14 | 1.27 | 2.43 | 1.20 |
time (sec) |
N/A |
0.038 |
0.043 |
0.050 |
0.315 |
1.017 |
2.700 |
0.204 |
2.201 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
A |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 55 | 0 | 0 | 0 | 66 | 0 | -1 |
normalized size | 1 | 1.00 | 0.93 | 0.00 | 0.00 | 0.00 | 1.12 | 0.00 | -0.02 |
time (sec) |
N/A |
0.042 |
0.063 |
0.072 |
0.000 |
0.666 |
6.029 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 202 | 202 | 112 | 150 | 0 | 0 | 48 | 0 | -1 |
normalized size | 1 | 1.00 | 0.55 | 0.74 | 0.00 | 0.00 | 0.24 | 0.00 | -0.00 |
time (sec) |
N/A |
0.125 |
0.252 |
0.074 |
0.000 |
0.574 |
2.459 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 52 | 52 | 53 | 94 | 81 | 70 | 36 | 57 | 42 |
normalized size | 1 | 1.00 | 1.02 | 1.81 | 1.56 | 1.35 | 0.69 | 1.10 | 0.81 |
time (sec) |
N/A |
0.038 |
0.052 |
0.168 |
0.328 |
0.649 |
3.684 |
0.153 |
2.610 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 86 | 86 | 113 | 104 | 0 | 0 | 41 | 0 | -1 |
normalized size | 1 | 1.00 | 1.31 | 1.21 | 0.00 | 0.00 | 0.48 | 0.00 | -0.01 |
time (sec) |
N/A |
0.053 |
0.232 |
0.048 |
0.000 |
2.605 |
2.155 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
A |
A |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 40 | 40 | 42 | 116 | 71 | 88 | 58 | 61 | 43 |
normalized size | 1 | 1.00 | 1.05 | 2.90 | 1.78 | 2.20 | 1.45 | 1.52 | 1.08 |
time (sec) |
N/A |
0.036 |
0.031 |
0.153 |
0.366 |
0.614 |
6.793 |
0.145 |
2.971 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
C |
F |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 165 | 165 | 96 | 146 | 0 | 0 | 42 | 0 | 24 |
normalized size | 1 | 1.00 | 0.58 | 0.88 | 0.00 | 0.00 | 0.25 | 0.00 | 0.15 |
time (sec) |
N/A |
0.080 |
0.149 |
0.056 |
0.000 |
0.648 |
0.775 |
0.000 |
2.326 |
|
|
|
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Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
A |
A |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 22 | 86 | 54 | 74 | 54 | 66 | 36 |
normalized size | 1 | 1.00 | 0.48 | 1.87 | 1.17 | 1.61 | 1.17 | 1.43 | 0.78 |
time (sec) |
N/A |
0.033 |
0.047 |
0.160 |
0.306 |
0.871 |
10.694 |
0.163 |
2.380 |
|
|
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Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
C |
F |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 91 | 91 | 96 | 111 | 0 | 0 | 42 | 0 | 27 |
normalized size | 1 | 1.00 | 1.05 | 1.22 | 0.00 | 0.00 | 0.46 | 0.00 | 0.30 |
time (sec) |
N/A |
0.047 |
0.162 |
0.053 |
0.000 |
0.648 |
2.276 |
0.000 |
2.369 |
|
|
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Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
A |
B |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 42 | 42 | 24 | 114 | 92 | 101 | 39 | 76 | 42 |
normalized size | 1 | 1.00 | 0.57 | 2.71 | 2.19 | 2.40 | 0.93 | 1.81 | 1.00 |
time (sec) |
N/A |
0.039 |
0.048 |
0.148 |
0.311 |
0.721 |
3.971 |
0.145 |
2.829 |
|
|
|
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Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 181 | 181 | 114 | 171 | 0 | 0 | 44 | 0 | -1 |
normalized size | 1 | 1.00 | 0.63 | 0.94 | 0.00 | 0.00 | 0.24 | 0.00 | -0.01 |
time (sec) |
N/A |
0.095 |
0.206 |
0.056 |
0.000 |
0.521 |
2.563 |
0.000 |
0.000 |
|
|
|
|
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|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-2) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 39 | 42 | 25 | 49 | 0 | 71 | 44 |
normalized size | 1 | 1.00 | 1.26 | 1.35 | 0.81 | 1.58 | 0.00 | 2.29 | 1.42 |
time (sec) |
N/A |
0.022 |
0.054 |
0.135 |
0.321 |
0.563 |
0.000 |
0.144 |
2.163 |
|
|
|
|
|
|
|
|
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|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
A |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 64 | 64 | 57 | 0 | 0 | 0 | 71 | 0 | -1 |
normalized size | 1 | 1.00 | 0.89 | 0.00 | 0.00 | 0.00 | 1.11 | 0.00 | -0.02 |
time (sec) |
N/A |
0.322 |
0.068 |
0.046 |
0.000 |
0.651 |
7.152 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
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|
|
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|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F(-2) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 84 | 114 | 117 | 87 | 82 | 0 | 73 |
normalized size | 1 | 1.00 | 0.99 | 1.34 | 1.38 | 1.02 | 0.96 | 0.00 | 0.86 |
time (sec) |
N/A |
0.254 |
0.070 |
0.041 |
0.312 |
0.665 |
4.721 |
0.000 |
2.171 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 44 | 61 | 32 | 49 | 51 | 0 | 40 |
normalized size | 1 | 1.00 | 1.16 | 1.61 | 0.84 | 1.29 | 1.34 | 0.00 | 1.05 |
time (sec) |
N/A |
0.225 |
0.051 |
0.051 |
0.305 |
0.528 |
2.774 |
0.000 |
2.142 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F(-2) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 52 | 52 | 57 | 90 | 89 | 72 | 36 | 0 | 51 |
normalized size | 1 | 1.00 | 1.10 | 1.73 | 1.71 | 1.38 | 0.69 | 0.00 | 0.98 |
time (sec) |
N/A |
0.240 |
0.037 |
0.051 |
0.322 |
0.768 |
3.333 |
0.000 |
2.128 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
A |
F(-2) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 44 | 120 | 75 | 99 | 63 | 0 | 52 |
normalized size | 1 | 1.00 | 1.02 | 2.79 | 1.74 | 2.30 | 1.47 | 0.00 | 1.21 |
time (sec) |
N/A |
0.158 |
0.051 |
0.055 |
0.309 |
0.613 |
3.794 |
0.000 |
2.198 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
F(-2) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 52 | 112 | 59 | 73 | 49 | 0 | 47 |
normalized size | 1 | 1.00 | 1.11 | 2.38 | 1.26 | 1.55 | 1.04 | 0.00 | 1.00 |
time (sec) |
N/A |
0.072 |
0.038 |
0.053 |
0.312 |
0.937 |
3.631 |
0.000 |
2.260 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
A |
F(-2) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 39 | 150 | 93 | 112 | 34 | 0 | 44 |
normalized size | 1 | 1.00 | 1.03 | 3.95 | 2.45 | 2.95 | 0.89 | 0.00 | 1.16 |
time (sec) |
N/A |
0.214 |
0.061 |
0.050 |
0.318 |
0.597 |
4.077 |
0.000 |
2.250 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
F(-2) |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 34 | 54 | 46 | 63 | 28 | 57 | 0 | 0 | 51 |
normalized size | 1 | 1.59 | 1.35 | 1.85 | 0.82 | 1.68 | 0.00 | 0.00 | 1.50 |
time (sec) |
N/A |
0.204 |
0.044 |
0.050 |
0.307 |
0.587 |
0.000 |
0.000 |
2.208 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
A |
F(-2) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 73 | 73 | 73 | 176 | 139 | 121 | 92 | 0 | 68 |
normalized size | 1 | 1.00 | 1.00 | 2.41 | 1.90 | 1.66 | 1.26 | 0.00 | 0.93 |
time (sec) |
N/A |
0.234 |
0.052 |
0.053 |
0.312 |
0.509 |
4.607 |
0.000 |
2.306 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 54 | 73 | 52 | 67 | 76 | 0 | 67 |
normalized size | 1 | 1.00 | 0.93 | 1.26 | 0.90 | 1.16 | 1.31 | 0.00 | 1.16 |
time (sec) |
N/A |
0.233 |
0.056 |
0.050 |
0.304 |
0.734 |
3.073 |
0.000 |
2.263 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
F(-2) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 96 | 96 | 74 | 209 | 180 | 131 | 114 | 0 | 89 |
normalized size | 1 | 1.00 | 0.77 | 2.18 | 1.88 | 1.36 | 1.19 | 0.00 | 0.93 |
time (sec) |
N/A |
0.253 |
0.077 |
0.053 |
0.306 |
2.315 |
6.325 |
0.000 |
2.371 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 88 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.04 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.104 |
0.245 |
0.464 |
0.000 |
2.164 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
F |
A |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 92 | 92 | 85 | 165 | 162 | 90 | 0 | 89 | 79 |
normalized size | 1 | 1.00 | 0.92 | 1.79 | 1.76 | 0.98 | 0.00 | 0.97 | 0.86 |
time (sec) |
N/A |
0.108 |
0.201 |
0.119 |
0.402 |
1.642 |
0.000 |
0.147 |
2.434 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 72 | 72 | 64 | 120 | 49 | 58 | 0 | 85 | 61 |
normalized size | 1 | 1.00 | 0.89 | 1.67 | 0.68 | 0.81 | 0.00 | 1.18 | 0.85 |
time (sec) |
N/A |
0.098 |
0.129 |
0.059 |
0.322 |
0.557 |
0.000 |
0.146 |
2.347 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
F |
A |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 54 | 133 | 107 | 68 | 0 | 61 | 51 |
normalized size | 1 | 1.00 | 0.92 | 2.25 | 1.81 | 1.15 | 0.00 | 1.03 | 0.86 |
time (sec) |
N/A |
0.090 |
0.131 |
0.059 |
0.492 |
0.888 |
0.000 |
0.138 |
2.339 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 36 | 36 | 35 | 89 | 31 | 38 | 0 | 44 | 31 |
normalized size | 1 | 1.00 | 0.97 | 2.47 | 0.86 | 1.06 | 0.00 | 1.22 | 0.86 |
time (sec) |
N/A |
0.070 |
0.064 |
0.055 |
0.404 |
0.642 |
0.000 |
0.144 |
2.336 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
F |
A |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 38 | 85 | 61 | 38 | 0 | 34 | 21 |
normalized size | 1 | 1.00 | 1.41 | 3.15 | 2.26 | 1.41 | 0.00 | 1.26 | 0.78 |
time (sec) |
N/A |
0.059 |
0.042 |
0.056 |
0.413 |
0.658 |
0.000 |
0.137 |
2.359 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
B |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 33 | 33 | 37 | 172 | 0 | 80 | 0 | 70 | 38 |
normalized size | 1 | 1.00 | 1.12 | 5.21 | 0.00 | 2.42 | 0.00 | 2.12 | 1.15 |
time (sec) |
N/A |
0.043 |
0.032 |
0.063 |
0.000 |
0.668 |
0.000 |
0.161 |
2.356 |
|
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Problem 66 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F(-2) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 30 | 30 | 30 | 154 | 34 | 41 | 0 | 43 | 29 |
normalized size | 1 | 1.00 | 1.00 | 5.13 | 1.13 | 1.37 | 0.00 | 1.43 | 0.97 |
time (sec) |
N/A |
0.071 |
0.095 |
0.066 |
0.427 |
0.543 |
0.000 |
0.136 |
2.146 |
|
|
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Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
B |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 58 | 210 | 0 | 130 | 0 | 114 | 61 |
normalized size | 1 | 1.00 | 0.97 | 3.50 | 0.00 | 2.17 | 0.00 | 1.90 | 1.02 |
time (sec) |
N/A |
0.097 |
0.111 |
0.067 |
0.000 |
1.337 |
0.000 |
0.155 |
2.418 |
|
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|
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|
Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 54 | 193 | 56 | 70 | 75 | 82 | 57 |
normalized size | 1 | 1.00 | 0.89 | 3.16 | 0.92 | 1.15 | 1.23 | 1.34 | 0.93 |
time (sec) |
N/A |
0.095 |
0.141 |
0.071 |
0.426 |
0.551 |
5.231 |
0.165 |
2.218 |
|
|
|
|
|
|
|
|
|
|
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|
|
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|
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|
Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 52 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.85 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.098 |
0.060 |
0.546 |
0.000 |
0.523 |
0.000 |
0.000 |
0.000 |
|
|
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|
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|
Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F(-1) |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 74 | 63 | 57 | 266 | 0 | 0 | 42 |
normalized size | 1 | 1.00 | 1.61 | 1.37 | 1.24 | 5.78 | 0.00 | 0.00 | 0.91 |
time (sec) |
N/A |
0.058 |
0.146 |
0.085 |
0.307 |
0.609 |
0.000 |
0.000 |
2.714 |
|
|
|
|
|
|
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|
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|
|
Problem 71 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
A |
B |
F(-1) |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 74 | 0 | 60 | 334 | 0 | 0 | 40 |
normalized size | 1 | 1.00 | 1.61 | 0.00 | 1.30 | 7.26 | 0.00 | 0.00 | 0.87 |
time (sec) |
N/A |
0.070 |
0.181 |
0.171 |
0.315 |
0.655 |
0.000 |
0.000 |
2.214 |
|
|
|
|
|
|
|
|
|
|
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
[.8750]
|
|
|
|
|
|
|
| Table 2.1:Rubi specific breakdown of results for each
integral
|
|
|
|
|
|
|
# |
grade |
|
|
normalized | antiderivative |
leaf size | |
|
\(\frac {\text {number of rules}}{\text {integrand leaf size}}\) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
A |
8 |
7 |
1.00 |
10 |
0.700 |
|
|
|
|
|
|
|
2 |
A |
7 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
3 |
A |
6 |
6 |
1.00 |
8 |
0.750 |
|
|
|
|
|
|
|
4 |
A |
14 |
8 |
1.00 |
10 |
0.800 |
|
|
|
|
|
|
|
5 |
A |
6 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
6 |
A |
8 |
8 |
1.00 |
10 |
0.800 |
|
|
|
|
|
|
|
7 |
A |
20 |
9 |
1.00 |
20 |
0.450 |
|
|
|
|
|
|
|
8 |
A |
17 |
9 |
1.00 |
20 |
0.450 |
|
|
|
|
|
|
|
9 |
A |
11 |
8 |
1.00 |
18 |
0.444 |
|
|
|
|
|
|
|
10 |
A |
8 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
11 |
A |
17 |
9 |
1.00 |
20 |
0.450 |
|
|
|
|
|
|
|
12 |
A |
12 |
8 |
1.00 |
20 |
0.400 |
|
|
|
|
|
|
|
13 |
A |
23 |
11 |
1.00 |
20 |
0.550 |
|
|
|
|
|
|
|
14 |
A |
4 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
15 |
A |
4 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
16 |
A |
4 |
3 |
1.00 |
8 |
0.375 |
|
|
|
|
|
|
|
17 |
A |
3 |
3 |
1.00 |
6 |
0.500 |
|
|
|
|
|
|
|
18 |
A |
7 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
19 |
A |
5 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
20 |
A |
6 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
21 |
A |
7 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
22 |
A |
3 |
3 |
1.00 |
4 |
0.750 |
|
|
|
|
|
|
|
23 |
A |
7 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
24 |
A |
7 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
25 |
A |
7 |
7 |
1.00 |
10 |
0.700 |
|
|
|
|
|
|
|
26 |
A |
4 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
27 |
A |
4 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
28 |
A |
7 |
7 |
1.00 |
10 |
0.700 |
|
|
|
|
|
|
|
29 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
30 |
A |
6 |
6 |
1.00 |
8 |
0.750 |
|
|
|
|
|
|
|
31 |
A |
5 |
5 |
1.29 |
6 |
0.833 |
|
|
|
|
|
|
|
32 |
A |
6 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
33 |
A |
5 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
34 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
35 |
A |
6 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
36 |
A |
5 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
37 |
A |
4 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
38 |
A |
8 |
8 |
1.00 |
12 |
0.667 |
|
|
|
|
|
|
|
39 |
A |
6 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
40 |
A |
5 |
5 |
1.00 |
12 |
0.417 |
|
|
|
|
|
|
|
41 |
A |
6 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
42 |
A |
7 |
7 |
1.00 |
8 |
0.875 |
|
|
|
|
|
|
|
43 |
A |
6 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
44 |
A |
5 |
5 |
1.00 |
12 |
0.417 |
|
|
|
|
|
|
|
45 |
A |
6 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
46 |
A |
7 |
7 |
1.00 |
12 |
0.583 |
|
|
|
|
|
|
|
47 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
48 |
A |
5 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
49 |
A |
8 |
7 |
1.00 |
12 |
0.583 |
|
|
|
|
|
|
|
50 |
A |
4 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
51 |
A |
7 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
52 |
A |
6 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
53 |
A |
7 |
6 |
1.00 |
8 |
0.750 |
|
|
|
|
|
|
|
54 |
A |
6 |
5 |
1.00 |
12 |
0.417 |
|
|
|
|
|
|
|
55 |
A |
4 |
3 |
1.59 |
12 |
0.250 |
|
|
|
|
|
|
|
56 |
A |
7 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
57 |
A |
6 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
58 |
A |
8 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
59 |
A |
4 |
3 |
1.00 |
23 |
0.130 |
|
|
|
|
|
|
|
60 |
A |
9 |
7 |
1.00 |
21 |
0.333 |
|
|
|
|
|
|
|
61 |
A |
6 |
5 |
1.00 |
21 |
0.238 |
|
|
|
|
|
|
|
62 |
A |
7 |
7 |
1.00 |
21 |
0.333 |
|
|
|
|
|
|
|
63 |
A |
3 |
3 |
1.00 |
21 |
0.143 |
|
|
|
|
|
|
|
64 |
A |
5 |
5 |
1.00 |
19 |
0.263 |
|
|
|
|
|
|
|
65 |
A |
7 |
7 |
1.00 |
18 |
0.389 |
|
|
|
|
|
|
|
66 |
A |
4 |
4 |
1.00 |
21 |
0.190 |
|
|
|
|
|
|
|
67 |
A |
7 |
6 |
1.00 |
21 |
0.286 |
|
|
|
|
|
|
|
68 |
A |
7 |
5 |
1.00 |
21 |
0.238 |
|
|
|
|
|
|
|
69 |
A |
8 |
8 |
1.00 |
19 |
0.421 |
|
|
|
|
|
|
|
70 |
A |
6 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
71 |
A |
6 |
6 |
1.00 |
14 |
0.429 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|