Chapter 2
detailed summary tables of results
2.1 List of integrals sorted by grade for each CAS
2.1.1 Rubi
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57,
58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83
}
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 78, 81
}
B grade: { 18, 79, 82 }
C grade: { }
F grade: { 74, 77, 80, 83}
2.1.3 Maple
A grade: { 2, 4, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 24, 26, 27, 28, 29, 30, 34, 35, 39, 40, 44, 45,
49, 50, 53, 54, 55, 58, 59, 60, 63, 64, 65, 68, 69, 70, 71 }
B grade: { }
C grade: { 72, 73, 75, 76, 78, 79, 81, 82 }
F grade: { 1, 3, 8, 10, 16, 18, 23, 25, 31, 32, 33, 36, 37, 38, 41, 42, 43, 46, 47, 48, 51, 52, 56, 57, 61, 62, 66,
67, 74, 77, 80, 83 }
2.1.4 Maxima
A grade: { 2, 4, 5, 6, 7, 13, 22, 34, 39, 40, 53, 58, 59, 71
B grade: { 12, 15, 27, 31, 32, 33, 36, 37, 38, 51, 52, 56, 57 }
C grade: { }
F grade: { 1, 3, 8, 9, 10, 11, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 28, 29, 30, 35, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 54, 55, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83
}
2.1.5 FriCAS
A grade: { 2, 4, 5, 6, 7, 9, 11, 13, 14, 15, 17, 19, 20, 21, 22, 24, 26, 28, 29, 30, 34, 35, 39, 40, 44, 45, 49, 50,
53, 54, 55, 59, 60, 63, 64, 65, 69, 70, 71, 72, 75, 78, 81 }
B grade: { 3, 10, 12, 18, 25, 27, 58, 68, 73, 76, 79, 82 }
C grade: { 1, 8, 16, 23, 74, 77, 80, 83 }
F grade: { 31, 32, 33, 36, 37, 38, 41, 42, 43, 46, 47, 48, 51, 52, 56, 57, 61, 62, 66, 67 }
2.1.6 Sympy
A grade: { 2, 4, 5, 6, 7, 9, 11, 13, 14, 17, 19, 21, 22, 24, 26, 28, 29, 30, 34, 35, 39, 40, 44, 45, 49, 50, 53, 54,
55, 58, 59, 60, 64, 65, 69 }
B grade: { }
C grade: { }
F grade: { 1, 3, 8, 10, 12, 15, 16, 18, 20, 23, 25, 27, 31, 32, 33, 36, 37, 38, 41, 42, 43, 46, 47, 48, 51, 52, 56,
57, 61, 62, 63, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83 }
2.1.7 Giac
A grade: { 2, 4, 6, 7, 9, 11, 13, 14, 15, 17, 19, 20, 21, 22, 24, 26, 27, 28, 29, 30, 34, 35, 39, 40, 44, 45, 49,
50, 54, 55, 59, 60, 63, 64, 65, 68, 69, 70, 71 }
B grade: { 5, 12, 53, 58 }
C grade: { }
F grade: { 1, 3, 8, 10, 16, 18, 23, 25, 31, 32, 33, 36, 37, 38, 41, 42, 43, 46, 47, 48, 51, 52, 56, 57, 61, 62, 66,
67, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83 }
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 |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 143 | 143 | 146 | 0 | 0 | 1269 | 0 | 0 |
normalized size | 1 | 1. | 1.02 | 0. | 0. | 8.87 | 0. | 0. |
time (sec) |
N/A |
0.162 |
0.044 |
0.113 |
0. |
2.015 |
0. |
0. |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.015 |
0.908 |
0.092 |
0. |
0. |
0. |
0. |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 92 | 92 | 95 | 0 | 0 | 884 | 0 | 0 |
normalized size | 1 | 1. | 1.03 | 0. | 0. | 9.61 | 0. | 0. |
time (sec) |
N/A |
0.088 |
0.021 |
0.092 |
0. |
1.994 |
0. |
0. |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.015 |
0.728 |
0.078 |
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 |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 26 | 39 | 42 | 103 | 42 | 68 |
normalized size | 1 | 1. | 1. | 1.5 | 1.62 | 3.96 | 1.62 | 2.62 |
time (sec) |
N/A |
0.023 |
0.014 |
0.014 |
1.173 |
2.058 |
3.257 |
1.347 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.016 |
0.72 |
0.066 |
0. |
0. |
0. |
0. |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.016 |
0.694 |
0.08 |
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 |
A |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 242 | 242 | 229 | 0 | 0 | 1960 | 0 | 0 |
normalized size | 1 | 1. | 0.95 | 0. | 0. | 8.1 | 0. | 0. |
time (sec) |
N/A |
0.37 |
0.59 |
0.206 |
0. |
2.297 |
0. |
0. |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.023 |
9.059 |
0.163 |
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(-1) |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 133 | 133 | 123 | 0 | 0 | 1331 | 0 | 0 |
normalized size |
1 |
1. |
0.92 |
0. |
0. |
10.01 |
0. |
0. |
time (sec) |
N/A |
0.164 |
0.426 |
0.291 |
0. |
2.125 |
0. |
0. |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.024 |
7.433 |
0.147 |
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 |
A |
A |
B |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 44 | 44 | 41 | 59 | 130 | 220 | 0 | 119 |
normalized size | 1 | 1. | 0.93 | 1.34 | 2.95 | 5. | 0. | 2.7 |
time (sec) |
N/A |
0.052 |
0.205 |
0.023 |
1.148 |
1.714 |
0. |
1.344 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.022 |
17.023 |
0.174 |
0. |
0. |
0. |
0. |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.022 |
8.889 |
0.197 |
0. |
0. |
0. |
0. |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 90 | 90 | 62 | 92 | 3831 | 254 | 0 | 115 |
normalized size | 1 | 1. | 0.69 | 1.02 | 42.57 | 2.82 | 0. | 1.28 |
time (sec) |
N/A |
0.073 |
0.201 |
0.02 |
2.165 |
1.708 |
0. |
1.285 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F(-1) |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 382 | 382 | 472 | 0 | 0 | 3491 | 0 | 0 |
normalized size |
1 |
1. |
1.24 |
0. |
0. |
9.14 |
0. |
0. |
time (sec) |
N/A |
0.878 |
1.091 |
0.151 |
0. |
2.615 |
0. |
0. |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.026 |
1.33 |
0.122 |
0. |
0. |
0. |
0. |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
F |
F(-2) |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 261 | 261 | 845 | 0 | 0 | 2566 | 0 | 0 |
normalized size |
1 |
1. |
3.24 |
0. |
0. |
9.83 |
0. |
0. |
time (sec) |
N/A |
0.54 |
1.338 |
0.11 |
0. |
2.517 |
0. |
0. |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.025 |
1.104 |
0.098 |
0. |
0. |
0. |
0. |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-1) |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 67 | 70 | 0 | 535 | 0 | 140 |
normalized size |
1 |
1. |
1.02 |
1.06 |
0. |
8.11 |
0. |
2.12 |
time (sec) |
N/A |
0.109 |
0.154 |
0.046 |
0. |
1.762 |
0. |
1.251 |
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|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.025 |
0.922 |
0.085 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.014 |
0.137 |
0.001 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1092 | 1092 | 895 | 0 | 0 | 6750 | 0 | 0 |
normalized size | 1 | 1. | 0.82 | 0. | 0. | 6.18 | 0. | 0. |
time (sec) |
N/A |
2.288 |
6.169 |
0.322 |
0. |
3.777 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.024 |
6.375 |
0.272 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 596 | 596 | 1069 | 0 | 0 | 4228 | 0 | 0 |
normalized size |
1 |
1. |
1.79 |
0. |
0. |
7.09 |
0. |
0. |
time (sec) |
N/A |
1.204 |
9.139 |
0.335 |
0. |
3.132 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.024 |
5.36 |
0.188 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 123 | 123 | 153 | 214 | 11976 | 1112 | 0 | 263 |
normalized size | 1 | 1. | 1.24 | 1.74 | 97.37 | 9.04 | 0. | 2.14 |
time (sec) |
N/A |
0.254 |
0.6 |
0.06 |
175.746 |
1.99 |
0. |
1.244 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.024 |
10.453 |
0.271 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.024 |
6.829 |
0.317 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.024 |
7.303 |
0.369 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 476 | 476 | 479 | 0 | 2041 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.01 | 0. | 4.29 | 0. | 0. | 0. |
time (sec) |
N/A |
0.457 |
0.136 |
0.121 |
2.556 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 348 | 348 | 351 | 0 | 1304 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.01 | 0. | 3.75 | 0. | 0. | 0. |
time (sec) |
N/A |
0.311 |
0.081 |
0.112 |
2.184 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 220 | 220 | 223 | 0 | 729 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.01 | 0. | 3.31 | 0. | 0. | 0. |
time (sec) |
N/A |
0.186 |
0.06 |
0.103 |
2.191 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.017 |
2.014 |
0.108 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.018 |
2.111 |
0.127 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
B |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 749 | 749 | 739 | 0 | 8608 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.99 |
0. |
11.49 |
0. |
0. |
0. |
time (sec) |
N/A |
0.86 |
2.06 |
0.087 |
4.261 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 551 | 551 | 543 | 0 | 5266 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.99 | 0. | 9.56 | 0. | 0. | 0. |
time (sec) |
N/A |
0.634 |
1.203 |
0.087 |
3.034 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 355 | 355 | 347 | 0 | 2712 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.98 | 0. | 7.64 | 0. | 0. | 0. |
time (sec) |
N/A |
0.451 |
0.663 |
0.087 |
2.543 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.023 |
38.083 |
0.088 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.023 |
19.96 |
0.092 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1041 | 1041 | 1122 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.08 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.476 |
2.565 |
0.087 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 781 | 781 | 858 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.1 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.18 |
2.042 |
0.091 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 521 | 521 | 632 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.21 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.966 |
20.196 |
0.083 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.025 |
1.919 |
0.081 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.015 |
0.355 |
0.001 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 3123 | 3123 | 3737 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.2 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
4.118 |
15.076 |
0.106 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 2323 | 2323 | 2777 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.2 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
3.228 |
13.162 |
0.104 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1523 | 1523 | 1767 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.16 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.504 |
15.711 |
0.112 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.026 |
36.689 |
0.099 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.025 |
26.245 |
0.099 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 284 | 284 | 281 | 0 | 996 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.99 | 0. | 3.51 | 0. | 0. | 0. |
time (sec) |
N/A |
0.248 |
0.195 |
0.11 |
2.312 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 158 | 158 | 155 | 0 | 505 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.98 | 0. | 3.2 | 0. | 0. | 0. |
time (sec) |
N/A |
0.132 |
0.099 |
0.106 |
2.085 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
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 | 26 | 26 | 26 | 32 | 42 | 113 | 58 | 68 |
normalized size | 1 | 1. | 1. | 1.23 | 1.62 | 4.35 | 2.23 | 2.62 |
time (sec) |
N/A |
0.021 |
0.057 |
0.018 |
1.038 |
2.122 |
4.81 |
1.449 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 29 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.014 |
2.329 |
0.115 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.014 |
2.345 |
0.133 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 451 | 451 | 443 | 0 | 3885 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.98 | 0. | 8.61 | 0. | 0. | 0. |
time (sec) |
N/A |
0.537 |
1.326 |
0.086 |
2.486 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 255 | 255 | 247 | 0 | 1727 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.97 | 0. | 6.77 | 0. | 0. | 0. |
time (sec) |
N/A |
0.325 |
0.694 |
0.081 |
2.103 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 45 | 60 | 68 | 261 | 90 | 119 |
normalized size | 1 | 1. | 0.96 | 1.28 | 1.45 | 5.55 | 1.91 | 2.53 |
time (sec) |
N/A |
0.052 |
0.213 |
0.036 |
1.142 |
1.859 |
8.604 |
1.481 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.022 |
19.331 |
0.091 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.023 |
20.171 |
0.091 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 653 | 653 | 725 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.11 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.049 |
2.072 |
0.083 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 393 | 393 | 486 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.24 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.826 |
5.531 |
0.084 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 68 | 68 | 69 | 71 | 0 | 625 | 0 | 140 |
normalized size |
1 |
1. |
1.01 |
1.04 |
0. |
9.19 |
0. |
2.06 |
time (sec) |
N/A |
0.082 |
0.214 |
0.063 |
0. |
1.861 |
0. |
1.287 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.025 |
3.369 |
0.082 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.024 |
3.378 |
0.079 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 66 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1925 | 1925 | 2231 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.16 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.858 |
13.257 |
0.103 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1125 | 1125 | 1210 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.08 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.106 |
8.049 |
0.102 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 127 | 127 | 163 | 216 | 0 | 1307 | 0 | 265 |
normalized size |
1 |
1. |
1.28 |
1.7 |
0. |
10.29 |
0. |
2.09 |
time (sec) |
N/A |
0.199 |
0.754 |
0.075 |
0. |
1.944 |
0. |
1.275 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.023 |
24.899 |
0.1 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.023 |
25.283 |
0.098 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 71 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.048 |
2.723 |
0.331 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 72 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F(-2) |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 44 | 44 | 38 | 159 | 0 | 149 | 0 | 0 |
normalized size |
1 |
1. |
0.86 |
3.61 |
0. |
3.39 |
0. |
0. |
time (sec) |
N/A |
0.047 |
0.092 |
0.22 |
0. |
1.763 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 73 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F(-2) |
B |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 149 | 149 | 188 | 873 | 0 | 1170 | 0 | 0 |
normalized size |
1 |
1. |
1.26 |
5.86 |
0. |
7.85 |
0. |
0. |
time (sec) |
N/A |
0.114 |
0.539 |
0.346 |
0. |
2.078 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 74 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F(-2) |
C |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 235 | 235 | 0 | 0 | 0 | 1640 | 0 | 0 |
normalized size |
1 |
1. |
0. |
0. |
0. |
6.98 |
0. |
0. |
time (sec) |
N/A |
0.19 |
1.18 |
0.378 |
0. |
2.193 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 75 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F(-2) |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 54 | 276 | 0 | 271 | 0 | 0 |
normalized size |
1 |
1. |
0.68 |
3.49 |
0. |
3.43 |
0. |
0. |
time (sec) |
N/A |
0.09 |
0.366 |
0.18 |
0. |
1.775 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 76 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F(-2) |
B |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 221 | 221 | 347 | 1096 | 0 | 1631 | 0 | 0 |
normalized size |
1 |
1. |
1.57 |
4.96 |
0. |
7.38 |
0. |
0. |
time (sec) |
N/A |
0.2 |
4.627 |
0.267 |
0. |
2.401 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 77 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F(-2) |
C |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 390 | 390 | 0 | 0 | 0 | 2492 | 0 | 0 |
normalized size |
1 |
1. |
0. |
0. |
0. |
6.39 |
0. |
0. |
time (sec) |
N/A |
0.396 |
10.419 |
0.774 |
0. |
2.568 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 78 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F(-1) |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 87 | 80 | 314 | 0 | 633 | 0 | 0 |
normalized size |
1 |
1. |
0.92 |
3.61 |
0. |
7.28 |
0. |
0. |
time (sec) |
N/A |
0.147 |
0.26 |
0.192 |
0. |
1.886 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 79 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
C |
F(-1) |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 328 | 328 | 861 | 1308 | 0 | 2938 | 0 | 0 |
normalized size |
1 |
1. |
2.62 |
3.99 |
0. |
8.96 |
0. |
0. |
time (sec) |
N/A |
0.595 |
1.731 |
0.345 |
0. |
2.72 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 80 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F(-1) |
C |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 485 | 485 | 0 | 0 | 0 | 3954 | 0 | 0 |
normalized size |
1 |
1. |
0. |
0. |
0. |
8.15 |
0. |
0. |
time (sec) |
N/A |
0.923 |
1.68 |
0.361 |
0. |
2.856 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 81 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F(-1) |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 191 | 706 | 0 | 1347 | 0 | 0 |
normalized size |
1 |
1. |
1.22 |
4.5 |
0. |
8.58 |
0. |
0. |
time (sec) |
N/A |
0.287 |
0.775 |
0.257 |
0. |
2.012 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 82 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
C |
F(-1) |
B |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 757 | 757 | 2450 | 2819 | 0 | 5435 | 0 | 0 |
normalized size |
1 |
1. |
3.24 |
3.72 |
0. |
7.18 |
0. |
0. |
time (sec) |
N/A |
1.273 |
10.034 |
0.332 |
0. |
3.336 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 83 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F(-1) |
C |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1384 | 1384 | 0 | 0 | 0 | 8319 | 0 | 0 |
normalized size |
1 |
1. |
0. |
0. |
0. |
6.01 |
0. |
0. |
time (sec) |
N/A |
2.367 |
10.051 |
0.961 |
0. |
4.064 |
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 [23] had the largest ratio of [ 0.6667
]
|
|
|
|
|
|
|
| Table 2.1Rubi specific breakdown of results for each
integral
|
|
|
|
|
|
|
# |
grade |
|
|
normalized | antiderivative |
leaf size | |
|
\(\frac{\text{number of rules}}{\text{integrand leaf size}}\) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
A |
10 |
6 |
1. |
16 |
0.375 |
|
|
|
|
|
|
|
2 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
3 |
A |
8 |
5 |
1. |
16 |
0.312 |
|
|
|
|
|
|
|
4 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
5 |
A |
4 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
6 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
7 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
8 |
A |
15 |
11 |
1. |
18 |
0.611 |
|
|
|
|
|
|
|
9 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
10 |
A |
10 |
7 |
1. |
18 |
0.389 |
|
|
|
|
|
|
|
11 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
12 |
A |
5 |
5 |
1. |
16 |
0.312 |
|
|
|
|
|
|
|
13 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
14 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
15 |
A |
5 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
16 |
A |
13 |
8 |
1. |
18 |
0.444 |
|
|
|
|
|
|
|
17 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
18 |
A |
11 |
7 |
1. |
18 |
0.389 |
|
|
|
|
|
|
|
19 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
20 |
A |
4 |
4 |
1. |
16 |
0.25 |
|
|
|
|
|
|
|
|
21 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
22 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
23 |
A |
31 |
12 |
1. |
18 |
0.667 |
|
|
|
|
|
|
|
24 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
25 |
A |
22 |
10 |
1. |
18 |
0.556 |
|
|
|
|
|
|
|
26 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
27 |
A |
6 |
6 |
1. |
16 |
0.375 |
|
|
|
|
|
|
|
28 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
29 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
30 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
31 |
A |
20 |
7 |
1. |
18 |
0.389 |
|
|
|
|
|
|
|
32 |
A |
16 |
7 |
1. |
18 |
0.389 |
|
|
|
|
|
|
|
33 |
A |
12 |
7 |
1. |
16 |
0.438 |
|
|
|
|
|
|
|
34 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
35 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
36 |
A |
30 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
37 |
A |
24 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
38 |
A |
18 |
10 |
1. |
18 |
0.556 |
|
|
|
|
|
|
|
39 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
40 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
41 |
A |
23 |
9 |
1. |
20 |
0.45 |
|
|
|
|
|
|
|
42 |
A |
19 |
9 |
1. |
20 |
0.45 |
|
|
|
|
|
|
|
43 |
A |
15 |
9 |
1. |
18 |
0.5 |
|
|
|
|
|
|
|
44 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
45 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
46 |
A |
61 |
11 |
1. |
20 |
0.55 |
|
|
|
|
|
|
|
47 |
A |
49 |
11 |
1. |
20 |
0.55 |
|
|
|
|
|
|
|
48 |
A |
37 |
11 |
1. |
18 |
0.611 |
|
|
|
|
|
|
|
49 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
50 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
51 |
A |
14 |
7 |
1. |
20 |
0.35 |
|
|
|
|
|
|
|
52 |
A |
10 |
6 |
1. |
20 |
0.3 |
|
|
|
|
|
|
|
53 |
A |
4 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
54 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
55 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
56 |
A |
21 |
10 |
1. |
22 |
0.454 |
|
|
|
|
|
|
|
57 |
A |
15 |
11 |
1. |
22 |
0.5 |
|
|
|
|
|
|
|
58 |
A |
5 |
5 |
1. |
22 |
0.227 |
|
|
|
|
|
|
|
59 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
60 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
61 |
A |
17 |
9 |
1. |
22 |
0.409 |
|
|
|
|
|
|
|
62 |
A |
13 |
8 |
1. |
22 |
0.364 |
|
|
|
|
|
|
|
63 |
A |
4 |
4 |
1. |
22 |
0.182 |
|
|
|
|
|
|
|
64 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
65 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
66 |
A |
43 |
11 |
1. |
22 |
0.5 |
|
|
|
|
|
|
|
67 |
A |
31 |
12 |
1. |
22 |
0.546 |
|
|
|
|
|
|
|
68 |
A |
6 |
6 |
1. |
22 |
0.273 |
|
|
|
|
|
|
|
69 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
70 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
71 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
72 |
A |
5 |
4 |
1. |
20 |
0.2 |
|
|
|
|
|
|
|
73 |
A |
9 |
6 |
1. |
22 |
0.273 |
|
|
|
|
|
|
|
74 |
A |
11 |
7 |
1. |
22 |
0.318 |
|
|
|
|
|
|
|
75 |
A |
6 |
6 |
1. |
22 |
0.273 |
|
|
|
|
|
|
|
76 |
A |
11 |
8 |
1. |
24 |
0.333 |
|
|
|
|
|
|
|
77 |
A |
16 |
12 |
1. |
24 |
0.5 |
|
|
|
|
|
|
|
78 |
A |
5 |
5 |
1. |
22 |
0.227 |
|
|
|
|
|
|
|
79 |
A |
12 |
8 |
1. |
24 |
0.333 |
|
|
|
|
|
|
|
80 |
A |
14 |
9 |
1. |
24 |
0.375 |
|
|
|
|
|
|
|
|
81 |
A |
7 |
7 |
1. |
22 |
0.318 |
|
|
|
|
|
|
|
82 |
A |
23 |
11 |
1. |
24 |
0.458 |
|
|
|
|
|
|
|
83 |
A |
32 |
13 |
1. |
24 |
0.542 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|