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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 2, 4, 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, 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 }
B grade: { 1, 3, 5, 41, 42 }
C grade: { }
F grade: { }
2.1.3 Maple
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 61, 62, 63,
64, 65, 66, 67 }
B grade: { 39, 40, 41, 42, 43, 51 }
C grade: { 17, 18, 19, 20, 21, 22, 23, 24, 55, 56, 57, 58, 59, 60 }
F grade: { 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 68, 69, 70 }
2.1.4 Maxima
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 43, 44, 45, 46, 50, 51, 61, 62, 63, 64, 65, 66, 67 }
B grade: { 39, 40, 41, 42, 47, 48, 49 }
C grade: { }
F grade: { 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, 52, 53, 54, 55, 56, 57, 58, 59, 60, 68, 69, 70 }
2.1.5 FriCAS
A grade: { 2, 4, 6, 8, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 61, 62, 63, 64, 65, 66, 67 }
B grade: { 1, 3, 5, 7, 39, 40, 41, 42 }
C grade: { }
F grade: { 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, 55, 56, 57, 58, 59, 60, 68, 69, 70 }
2.1.6 Sympy
A grade: { 1, 43, 44, 45, 46, 51, 52, 53, 54 }
B grade: { }
C grade: { }
F grade: { 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, 47, 48, 49, 50, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67,
68, 69, 70 }
2.1.7 Giac
A grade: { 1, 2, 4, 6, 8, 43, 45, 46, 50, 52, 53, 54, 61, 62, 63, 64, 65, 66, 67 }
B grade: { 3, 5, 7, 39, 40, 41, 42, 44, 47, 48, 49, 51 }
C grade: { }
F grade: { 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, 55, 56, 57, 58, 59, 60, 68, 69, 70 }
2.1.8 Mupad
A grade: { }
B grade: { 1, 2, 3, 4, 5, 6, 7, 8, 12, 20, 43, 51, 61, 62, 63, 64 }
C grade: { }
F grade: { 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 65, 66, 67, 68, 69, 70
}
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 |
B |
A |
A |
B |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 12 | 12 | 38 | 20 | 19 | 30 | 37 | 15 | 12 |
normalized size | 1 | 1.00 | 3.17 | 1.67 | 1.58 | 2.50 | 3.08 | 1.25 | 1.00 |
time (sec) |
N/A |
0.004 |
0.017 |
0.043 |
0.320 |
0.672 |
1.930 |
0.167 |
0.401 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 11 | 11 | 11 | 12 | 13 | 19 | 0 | 13 | 11 |
normalized size | 1 | 1.00 | 1.00 | 1.09 | 1.18 | 1.73 | 0.00 | 1.18 | 1.00 |
time (sec) |
N/A |
0.008 |
0.009 |
0.821 |
0.328 |
0.444 |
0.000 |
0.213 |
0.104 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
B |
A |
A |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 34 | 34 | 75 | 40 | 46 | 72 | 0 | 92 | 36 |
normalized size | 1 | 1.00 | 2.21 | 1.18 | 1.35 | 2.12 | 0.00 | 2.71 | 1.06 |
time (sec) |
N/A |
0.014 |
0.013 |
0.850 |
0.320 |
0.781 |
0.000 |
0.469 |
0.114 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 35 | 23 | 25 | 45 | 0 | 25 | 21 |
normalized size | 1 | 1.00 | 1.30 | 0.85 | 0.93 | 1.67 | 0.00 | 0.93 | 0.78 |
time (sec) |
N/A |
0.011 |
0.011 |
0.870 |
0.319 |
0.688 |
0.000 |
0.151 |
0.080 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
B |
A |
A |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 55 | 55 | 113 | 59 | 71 | 112 | 0 | 138 | 59 |
normalized size | 1 | 1.00 | 2.05 | 1.07 | 1.29 | 2.04 | 0.00 | 2.51 | 1.07 |
time (sec) |
N/A |
0.026 |
0.016 |
0.832 |
0.328 |
0.494 |
0.000 |
0.186 |
0.128 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 42 | 42 | 56 | 33 | 35 | 64 | 0 | 35 | 38 |
normalized size | 1 | 1.00 | 1.33 | 0.79 | 0.83 | 1.52 | 0.00 | 0.83 | 0.90 |
time (sec) |
N/A |
0.014 |
0.015 |
0.787 |
0.355 |
0.530 |
0.000 |
0.189 |
0.096 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 76 | 76 | 151 | 78 | 91 | 155 | 0 | 182 | 78 |
normalized size | 1 | 1.00 | 1.99 | 1.03 | 1.20 | 2.04 | 0.00 | 2.39 | 1.03 |
time (sec) |
N/A |
0.036 |
0.014 |
0.782 |
0.336 |
0.757 |
0.000 |
0.206 |
0.155 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 55 | 55 | 77 | 43 | 45 | 87 | 0 | 45 | 41 |
normalized size | 1 | 1.00 | 1.40 | 0.78 | 0.82 | 1.58 | 0.00 | 0.82 | 0.75 |
time (sec) |
N/A |
0.017 |
0.010 |
0.795 |
0.403 |
0.629 |
0.000 |
0.350 |
0.127 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 90 | 90 | 63 | 160 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.70 | 1.78 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.037 |
0.182 |
2.890 |
0.000 |
0.610 |
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 |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 50 | 88 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.75 | 1.31 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.026 |
0.075 |
2.427 |
0.000 |
1.157 |
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 |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 49 | 132 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.78 | 2.10 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.024 |
0.051 |
3.184 |
0.000 |
0.564 |
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 |
A |
A |
F |
F |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 41 | 40 | 69 | 0 | 0 | 0 | 0 | 61 |
normalized size | 1 | 1.00 | 0.98 | 1.68 | 0.00 | 0.00 | 0.00 | 0.00 | 1.49 |
time (sec) |
N/A |
0.017 |
0.032 |
1.760 |
0.000 |
0.838 |
0.000 |
0.000 |
0.121 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 41 | 40 | 91 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.98 | 2.22 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.017 |
0.035 |
2.648 |
0.000 |
0.615 |
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 |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 53 | 88 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.79 | 1.31 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.027 |
0.053 |
2.203 |
0.000 |
0.687 |
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 |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 60 | 142 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.90 | 2.12 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.026 |
0.097 |
2.533 |
0.000 |
0.681 |
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 |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 90 | 90 | 65 | 104 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.72 | 1.16 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.038 |
0.150 |
2.204 |
0.000 |
0.518 |
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 |
C |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 103 | 103 | 67 | 1028 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.65 | 9.98 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.053 |
0.181 |
1.230 |
0.000 |
0.665 |
0.000 |
0.000 |
0.000 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 55 | 313 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.73 | 4.17 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.031 |
0.165 |
1.035 |
0.000 |
0.567 |
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 |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 71 | 71 | 54 | 505 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.76 | 7.11 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.032 |
0.107 |
1.175 |
0.000 |
0.462 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 42 | 165 | 0 | 0 | 0 | 0 | 63 |
normalized size | 1 | 1.00 | 0.98 | 3.84 | 0.00 | 0.00 | 0.00 | 0.00 | 1.47 |
time (sec) |
N/A |
0.018 |
0.025 |
1.223 |
0.000 |
0.613 |
0.000 |
0.000 |
0.199 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 42 | 533 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.98 | 12.40 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.018 |
0.028 |
1.070 |
0.000 |
0.809 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 63 | 186 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.82 | 2.42 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.033 |
0.071 |
0.842 |
0.000 |
0.637 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 60 | 534 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.78 | 6.94 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.032 |
0.136 |
0.852 |
0.000 |
0.606 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 105 | 105 | 70 | 213 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.67 | 2.03 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.049 |
0.120 |
0.849 |
0.000 |
0.671 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 54 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.06 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.022 |
0.070 |
0.649 |
0.000 |
0.859 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 51 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.022 |
0.051 |
0.712 |
0.000 |
1.259 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 51 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.96 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.022 |
0.052 |
0.951 |
0.000 |
0.490 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 51 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.96 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.021 |
0.045 |
0.779 |
0.000 |
0.584 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 51 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.96 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.021 |
0.046 |
0.773 |
0.000 |
0.586 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 68 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.28 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.021 |
0.113 |
0.611 |
0.000 |
0.605 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 57 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.06 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.030 |
0.045 |
0.577 |
0.000 |
0.665 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 59 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.09 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.027 |
0.067 |
0.651 |
0.000 |
0.683 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 59 | 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 |
0.026 |
0.066 |
0.753 |
0.000 |
0.673 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 59 | 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 |
0.026 |
0.065 |
0.642 |
0.000 |
0.850 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 59 | 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 |
0.027 |
0.064 |
0.630 |
0.000 |
0.535 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 76 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.36 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.027 |
0.150 |
0.523 |
0.000 |
0.614 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 59 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.86 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.026 |
0.090 |
2.652 |
0.000 |
0.664 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 72 | 72 | 65 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.90 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.030 |
0.073 |
2.620 |
0.000 |
0.621 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
F(-1) |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 50 | 50 | 92 | 101 | 1669 | 93 | 0 | 129 | -1 |
normalized size | 1 | 1.00 | 1.84 | 2.02 | 33.38 | 1.86 | 0.00 | 2.58 | -0.02 |
time (sec) |
N/A |
0.017 |
0.255 |
0.447 |
0.760 |
0.605 |
0.000 |
0.267 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 36 | 36 | 72 | 78 | 869 | 69 | 0 | 100 | -1 |
normalized size | 1 | 1.00 | 2.00 | 2.17 | 24.14 | 1.92 | 0.00 | 2.78 | -0.03 |
time (sec) |
N/A |
0.012 |
0.255 |
0.442 |
0.601 |
0.663 |
0.000 |
0.505 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 22 | 51 | 52 | 300 | 44 | 0 | 69 | -1 |
normalized size | 1 | 1.00 | 2.32 | 2.36 | 13.64 | 2.00 | 0.00 | 3.14 | -0.05 |
time (sec) |
N/A |
0.009 |
0.093 |
0.463 |
0.574 |
0.503 |
0.000 |
0.492 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 5 | 5 | 28 | 31 | 35 | 19 | 0 | 12 | -1 |
normalized size | 1 | 1.00 | 5.60 | 6.20 | 7.00 | 3.80 | 0.00 | 2.40 | -0.20 |
time (sec) |
N/A |
0.007 |
0.013 |
0.608 |
0.579 |
0.536 |
0.000 |
0.297 |
0.000 |
|
|
|
|
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Problem 43 |
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 | 12 | 12 | 12 | 26 | 10 | 4 | 12 | 11 | 12 |
normalized size | 1 | 1.00 | 1.00 | 2.17 | 0.83 | 0.33 | 1.00 | 0.92 | 1.00 |
time (sec) |
N/A |
0.009 |
0.007 |
0.583 |
0.571 |
0.646 |
0.352 |
0.289 |
0.170 |
|
|
|
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Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 29 | 29 | 23 | 30 | 11 | 11 | 29 | 44 | -1 |
normalized size | 1 | 1.00 | 0.79 | 1.03 | 0.38 | 0.38 | 1.00 | 1.52 | -0.03 |
time (sec) |
N/A |
0.011 |
0.016 |
0.436 |
0.515 |
0.566 |
0.937 |
0.424 |
0.000 |
|
|
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Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 31 | 38 | 17 | 17 | 46 | 61 | -1 |
normalized size | 1 | 1.00 | 0.72 | 0.88 | 0.40 | 0.40 | 1.07 | 1.42 | -0.02 |
time (sec) |
N/A |
0.015 |
0.026 |
0.426 |
0.680 |
0.579 |
9.322 |
0.333 |
0.000 |
|
|
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Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 57 | 57 | 37 | 44 | 23 | 21 | 61 | 78 | -1 |
normalized size | 1 | 1.00 | 0.65 | 0.77 | 0.40 | 0.37 | 1.07 | 1.37 | -0.02 |
time (sec) |
N/A |
0.019 |
0.040 |
0.430 |
0.525 |
0.667 |
138.800 |
0.547 |
0.000 |
|
|
|
|
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Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
F(-1) |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 61 | 102 | 2183 | 106 | 0 | 164 | -1 |
normalized size | 1 | 1.00 | 0.73 | 1.21 | 25.99 | 1.26 | 0.00 | 1.95 | -0.01 |
time (sec) |
N/A |
0.040 |
0.322 |
0.474 |
4.137 |
0.615 |
0.000 |
0.362 |
0.000 |
|
|
|
|
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|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 65 | 65 | 51 | 79 | 1113 | 80 | 0 | 124 | -1 |
normalized size | 1 | 1.00 | 0.78 | 1.22 | 17.12 | 1.23 | 0.00 | 1.91 | -0.02 |
time (sec) |
N/A |
0.029 |
0.214 |
0.437 |
0.868 |
0.658 |
0.000 |
0.290 |
0.000 |
|
|
|
|
|
|
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|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 39 | 53 | 318 | 49 | 0 | 72 | -1 |
normalized size | 1 | 1.00 | 0.85 | 1.15 | 6.91 | 1.07 | 0.00 | 1.57 | -0.02 |
time (sec) |
N/A |
0.022 |
0.073 |
0.462 |
0.539 |
0.586 |
0.000 |
0.283 |
0.000 |
|
|
|
|
|
|
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|
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|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 30 | 32 | 24 | 64 | 0 | 13 | -1 |
normalized size | 1 | 1.00 | 1.15 | 1.23 | 0.92 | 2.46 | 0.00 | 0.50 | -0.04 |
time (sec) |
N/A |
0.016 |
0.016 |
0.683 |
0.789 |
0.547 |
0.000 |
0.561 |
0.000 |
|
|
|
|
|
|
|
|
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|
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|
|
|
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|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 14 | 14 | 14 | 27 | 13 | 22 | 17 | 34 | 15 |
normalized size | 1 | 1.00 | 1.00 | 1.93 | 0.93 | 1.57 | 1.21 | 2.43 | 1.07 |
time (sec) |
N/A |
0.013 |
0.006 |
0.619 |
0.578 |
0.835 |
0.450 |
0.356 |
0.220 |
|
|
|
|
|
|
|
|
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|
|
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|
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|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 36 | 36 | 27 | 31 | 0 | 29 | 39 | 43 | -1 |
normalized size | 1 | 1.00 | 0.75 | 0.86 | 0.00 | 0.81 | 1.08 | 1.19 | -0.03 |
time (sec) |
N/A |
0.018 |
0.018 |
0.408 |
0.000 |
0.534 |
1.039 |
0.430 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 55 | 55 | 36 | 39 | 0 | 37 | 61 | 56 | -1 |
normalized size | 1 | 1.00 | 0.65 | 0.71 | 0.00 | 0.67 | 1.11 | 1.02 | -0.02 |
time (sec) |
N/A |
0.027 |
0.032 |
0.415 |
0.000 |
0.598 |
9.415 |
0.496 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 42 | 45 | 0 | 43 | 82 | 69 | -1 |
normalized size | 1 | 1.00 | 0.57 | 0.61 | 0.00 | 0.58 | 1.11 | 0.93 | -0.01 |
time (sec) |
N/A |
0.036 |
0.045 |
0.454 |
0.000 |
0.815 |
136.156 |
0.721 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 123 | 123 | 58 | 1313 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.47 | 10.67 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.060 |
0.184 |
0.767 |
0.000 |
0.637 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 71 | 71 | 46 | 372 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.65 | 5.24 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.038 |
0.113 |
0.904 |
0.000 |
0.814 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 46 | 343 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.96 | 7.15 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.027 |
0.040 |
0.975 |
0.000 |
0.594 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 50 | 50 | 38 | 125 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.76 | 2.50 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.030 |
0.049 |
1.089 |
0.000 |
0.563 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 52 | 361 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.66 | 4.57 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.038 |
0.094 |
0.788 |
0.000 |
0.510 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 123 | 123 | 63 | 158 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.51 | 1.28 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.061 |
0.144 |
0.793 |
0.000 |
0.736 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 164 | 164 | 59 | 53 | 66 | 118 | 0 | 69 | 603 |
normalized size | 1 | 1.00 | 0.36 | 0.32 | 0.40 | 0.72 | 0.00 | 0.42 | 3.68 |
time (sec) |
N/A |
0.035 |
0.069 |
0.764 |
0.439 |
0.579 |
0.000 |
0.651 |
5.052 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
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 | 118 | 118 | 47 | 41 | 48 | 88 | 0 | 51 | 121 |
normalized size | 1 | 1.00 | 0.40 | 0.35 | 0.41 | 0.75 | 0.00 | 0.43 | 1.03 |
time (sec) |
N/A |
0.028 |
0.041 |
0.513 |
0.436 |
0.975 |
0.000 |
0.308 |
2.419 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
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 | 62 | 62 | 33 | 29 | 30 | 52 | 0 | 23 | 44 |
normalized size | 1 | 1.00 | 0.53 | 0.47 | 0.48 | 0.84 | 0.00 | 0.37 | 0.71 |
time (sec) |
N/A |
0.021 |
0.019 |
0.424 |
0.477 |
0.566 |
0.000 |
0.291 |
0.674 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 64 |
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 | 16 | 16 | 16 | 18 | 9 | 24 | 0 | 9 | 7 |
normalized size | 1 | 1.00 | 1.00 | 1.12 | 0.56 | 1.50 | 0.00 | 0.56 | 0.44 |
time (sec) |
N/A |
0.015 |
0.006 |
0.632 |
0.584 |
0.416 |
0.000 |
0.249 |
0.178 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 36 | 36 | 25 | 24 | 25 | 46 | 0 | 22 | -1 |
normalized size | 1 | 1.00 | 0.69 | 0.67 | 0.69 | 1.28 | 0.00 | 0.61 | -0.03 |
time (sec) |
N/A |
0.016 |
0.026 |
0.742 |
0.468 |
0.546 |
0.000 |
0.414 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 66 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 86 | 86 | 38 | 41 | 58 | 61 | 0 | 42 | -1 |
normalized size | 1 | 1.00 | 0.44 | 0.48 | 0.67 | 0.71 | 0.00 | 0.49 | -0.01 |
time (sec) |
N/A |
0.029 |
0.044 |
0.642 |
0.453 |
0.665 |
0.000 |
0.302 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 132 | 132 | 55 | 57 | 88 | 73 | 0 | 49 | -1 |
normalized size | 1 | 1.00 | 0.42 | 0.43 | 0.67 | 0.55 | 0.00 | 0.37 | -0.01 |
time (sec) |
N/A |
0.045 |
0.099 |
0.780 |
0.506 |
0.470 |
0.000 |
0.269 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 80 | 80 | 71 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.89 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.050 |
0.122 |
3.011 |
0.000 |
0.621 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 82 | 82 | 73 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.89 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.049 |
0.088 |
2.232 |
0.000 |
0.671 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 91 | 91 | 77 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.85 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.045 |
0.142 |
3.901 |
0.000 |
0.667 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
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 [47] had the largest ratio of
[.4000]
|
|
|
|
|
|
|
| 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 |
1 |
1 |
1.00 |
6 |
0.167 |
|
|
|
|
|
|
|
2 |
A |
2 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
3 |
A |
2 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
4 |
A |
2 |
1 |
1.00 |
8 |
0.125 |
|
|
|
|
|
|
|
5 |
A |
3 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
6 |
A |
2 |
1 |
1.00 |
8 |
0.125 |
|
|
|
|
|
|
|
7 |
A |
4 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
8 |
A |
2 |
1 |
1.00 |
8 |
0.125 |
|
|
|
|
|
|
|
9 |
A |
4 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
10 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
11 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
12 |
A |
2 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
13 |
A |
2 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
14 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
15 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
16 |
A |
4 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
17 |
A |
4 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
18 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
19 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
20 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
21 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
22 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
23 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
24 |
A |
4 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
25 |
A |
2 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
26 |
A |
2 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
27 |
A |
2 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
28 |
A |
2 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
29 |
A |
2 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
30 |
A |
2 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
31 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
32 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
33 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
34 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
35 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
36 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
37 |
A |
2 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
38 |
A |
2 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
39 |
A |
5 |
3 |
1.00 |
8 |
0.375 |
|
|
|
|
|
|
|
40 |
A |
4 |
3 |
1.00 |
8 |
0.375 |
|
|
|
|
|
|
|
41 |
A |
3 |
3 |
1.00 |
8 |
0.375 |
|
|
|
|
|
|
|
42 |
A |
2 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
43 |
A |
2 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
44 |
A |
3 |
3 |
1.00 |
8 |
0.375 |
|
|
|
|
|
|
|
45 |
A |
4 |
3 |
1.00 |
8 |
0.375 |
|
|
|
|
|
|
|
46 |
A |
5 |
3 |
1.00 |
8 |
0.375 |
|
|
|
|
|
|
|
47 |
A |
6 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
48 |
A |
5 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
49 |
A |
4 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
50 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
51 |
A |
2 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
52 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
53 |
A |
4 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
54 |
A |
5 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
55 |
A |
7 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
56 |
A |
5 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
57 |
A |
4 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
58 |
A |
4 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
59 |
A |
5 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
60 |
A |
7 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
61 |
A |
3 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
62 |
A |
3 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
63 |
A |
3 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
64 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
65 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
66 |
A |
5 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
67 |
A |
7 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
68 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
69 |
A |
3 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
70 |
A |
3 |
3 |
1.00 |
21 |
0.143 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|