Chapter 2
detailed summary tables of results
2.1 List of integrals sorted by grade for each CAS
2.1.1 Rubi
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84
}
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 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, 74, 76, 79, 82, 83
}
B grade: { 10, 77 }
C grade: { 18, 80 }
F grade: { 75, 78, 81, 84}
2.1.3 Maple
A grade: { 2, 4, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 24, 26, 28, 29, 30, 31, 35, 36, 40, 41, 45, 46,
50, 51, 54, 55, 56, 59, 60, 61, 64, 65, 66, 70, 71, 72 }
B grade: { 27, 69 }
C grade: { 73, 74, 76, 79, 80, 82 }
F grade: { 1, 3, 8, 10, 16, 18, 23, 25, 32, 33, 34, 37, 38, 39, 42, 43, 44, 47, 48, 49, 52, 53, 57, 58, 62, 63, 67,
68, 75, 77, 78, 81, 83, 84 }
2.1.4 Maxima
A grade: { 2, 4, 5, 6, 7, 9, 11, 12, 13, 14, 17, 19, 21, 22, 24, 26, 28, 29, 30, 31, 35, 36, 40, 41, 45, 46, 50, 51,
54, 55, 56, 59, 60, 65, 66, 70, 71, 72
B grade: { 15 }
C grade: { }
F grade: { 1, 3, 8, 10, 16, 18, 20, 23, 25, 27, 32, 33, 34, 37, 38, 39, 42, 43, 44, 47, 48, 49, 52, 53, 57, 58, 61,
62, 63, 64, 67, 68, 69, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84 }
2.1.5 FriCAS
A grade: { 2, 4, 5, 6, 7, 9, 11, 13, 14, 17, 19, 20, 21, 22, 24, 26, 28, 29, 30, 35, 36, 40, 41, 45, 46, 50, 51, 54,
55, 56, 60, 61, 65, 66, 70, 71, 72 }
B grade: { 3, 10, 12, 15, 18, 25, 27, 31, 59, 64, 69, 73, 74, 76, 77, 79, 80, 82, 83 }
C grade: { 1, 8, 16, 23, 75, 78, 81 }
F grade: { 32, 33, 34, 37, 38, 39, 42, 43, 44, 47, 48, 49, 52, 53, 57, 58, 62, 63, 67, 68, 84 }
2.1.6 Sympy
A grade: { 2, 4, 6, 7, 9, 11, 13, 14, 17, 19, 21, 22, 24, 26, 28, 29, 30, 35, 36, 40, 41, 45, 46, 50, 51, 55, 56,
60, 61, 65, 66, 70, 71 }
B grade: { }
C grade: { }
F grade: { 1, 3, 5, 8, 10, 12, 15, 16, 18, 20, 23, 25, 27, 31, 32, 33, 34, 37, 38, 39, 42, 43, 44, 47, 48,
49, 52, 53, 54, 57, 58, 59, 62, 63, 64, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84
}
2.1.7 Giac
A grade: { 2, 4, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 24, 26, 27, 28, 29, 30, 31, 35, 36, 40, 41, 45,
46, 50, 51, 54, 55, 56, 59, 60, 61, 64, 65, 66, 69, 70, 71, 72 }
B grade: { }
C grade: { }
F grade: { 1, 3, 8, 10, 16, 18, 23, 25, 32, 33, 34, 37, 38, 39, 42, 43, 44, 47, 48, 49, 52, 53, 57, 58, 62, 63, 67,
68, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84 }
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 | 125 | 125 | 143 | 0 | 0 | 706 | 0 | 0 |
normalized size | 1 | 1. | 1.14 | 0. | 0. | 5.65 | 0. | 0. |
time (sec) |
N/A |
0.138 |
1.313 |
0.15 |
0. |
2.335 |
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 |
0.018 |
3.807 |
0.046 |
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 | 77 | 77 | 135 | 0 | 0 | 516 | 0 | 0 |
normalized size | 1 | 1. | 1.75 | 0. | 0. | 6.7 | 0. | 0. |
time (sec) |
N/A |
0.077 |
0.178 |
0.043 |
0. |
2.249 |
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 |
0.017 |
3.17 |
0.043 |
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 |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 26 | 30 | 30 | 88 | 0 | 38 |
normalized size | 1 | 1. | 1. | 1.15 | 1.15 | 3.38 | 0. | 1.46 |
time (sec) |
N/A |
0.024 |
0.014 |
0.013 |
1.179 |
2.405 |
0. |
1.131 |
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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 |
0.017 |
2.514 |
0.056 |
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 |
0.018 |
3.413 |
0.045 |
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 | 217 | 217 | 294 | 0 | 0 | 3021 | 0 | 0 |
normalized size | 1 | 1. | 1.35 | 0. | 0. | 13.92 | 0. | 0. |
time (sec) |
N/A |
0.358 |
4.493 |
0.089 |
0. |
2.609 |
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 |
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 |
0.025 |
8.12 |
0.07 |
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 |
B |
F |
F |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 119 | 119 | 273 | 0 | 0 | 2028 | 0 | 0 |
normalized size | 1 | 1. | 2.29 | 0. | 0. | 17.04 | 0. | 0. |
time (sec) |
N/A |
0.159 |
3.175 |
0.105 |
0. |
2.449 |
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 |
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 |
0.025 |
7.197 |
0.059 |
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 |
A |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 44 | 44 | 44 | 51 | 62 | 482 | 0 | 74 |
normalized size | 1 | 1. | 1. | 1.16 | 1.41 | 10.95 | 0. | 1.68 |
time (sec) |
N/A |
0.055 |
0.091 |
0.086 |
1.167 |
2.18 |
0. |
1.172 |
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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 |
0.024 |
19.126 |
0.08 |
0. |
0. |
0. |
0. |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.024 |
9.131 |
0.088 |
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 |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 90 | 90 | 77 | 83 | 246 | 5025 | 0 | 197 |
normalized size | 1 | 1. | 0.86 | 0.92 | 2.73 | 55.83 | 0. | 2.19 |
time (sec) |
N/A |
0.08 |
0.138 |
0.053 |
1.652 |
2.279 |
0. |
1.182 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F(-2) |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 349 | 349 | 376 | 0 | 0 | 1670 | 0 | 0 |
normalized size |
1 |
1. |
1.08 |
0. |
0. |
4.79 |
0. |
0. |
time (sec) |
N/A |
0.858 |
1.481 |
0.053 |
0. |
2.329 |
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 |
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 |
0.028 |
8.065 |
0.049 |
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 |
C |
F |
F(-2) |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 241 | 241 | 843 | 0 | 0 | 1230 | 0 | 0 |
normalized size |
1 |
1. |
3.5 |
0. |
0. |
5.1 |
0. |
0. |
time (sec) |
N/A |
0.508 |
1.673 |
0.047 |
0. |
2.243 |
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 |
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 |
0.028 |
7.186 |
0.046 |
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(-2) |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 67 | 95 | 0 | 702 | 0 | 82 |
normalized size |
1 |
1. |
1.02 |
1.44 |
0. |
10.64 |
0. |
1.24 |
time (sec) |
N/A |
0.103 |
0.119 |
0.043 |
0. |
2.19 |
0. |
1.161 |
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|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.028 |
4.682 |
0.052 |
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 |
0.017 |
0.1 |
0.001 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 994 | 994 | 1565 | 0 | 0 | 8047 | 0 | 0 |
normalized size |
1 |
1. |
1.57 |
0. |
0. |
8.1 |
0. |
0. |
time (sec) |
N/A |
2.115 |
12.648 |
0.116 |
0. |
2.8 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
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 | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.029 |
53.669 |
0.102 |
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 | 555 | 555 | 755 | 0 | 0 | 5169 | 0 | 0 |
normalized size |
1 |
1. |
1.36 |
0. |
0. |
9.31 |
0. |
0. |
time (sec) |
N/A |
1.102 |
6.365 |
0.128 |
0. |
2.532 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
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 | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.028 |
48.42 |
0.085 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 123 | 123 | 220 | 234 | 0 | 2867 | 0 | 200 |
normalized size |
1 |
1. |
1.79 |
1.9 |
0. |
23.31 |
0. |
1.63 |
time (sec) |
N/A |
0.25 |
0.438 |
0.053 |
0. |
2.397 |
0. |
1.216 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
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 | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.026 |
98.241 |
0.115 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
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 | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.027 |
62.244 |
0.124 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
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 | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.027 |
63.176 |
0.133 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 6 | 6 | 6 | 7 | 16 | 80 | 0 | 16 |
normalized size | 1 | 1. | 1. | 1.17 | 2.67 | 13.33 | 0. | 2.67 |
time (sec) |
N/A |
0.022 |
0.018 |
0.012 |
1.139 |
1.992 |
0. |
1.184 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 426 | 426 | 415 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.97 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.403 |
1.861 |
0.097 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 310 | 310 | 311 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1. | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.279 |
1.748 |
0.069 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 194 | 194 | 207 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.07 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.171 |
1.752 |
0.069 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
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 |
0.019 |
6.009 |
0.073 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
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 | 25 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.019 |
7.253 |
0.075 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 677 | 677 | 739 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.09 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.816 |
8.675 |
0.069 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 497 | 497 | 573 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.15 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.602 |
7.815 |
0.063 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 319 | 319 | 459 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.44 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.433 |
8.315 |
0.064 |
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 |
0.025 |
60.247 |
0.066 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
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 |
0.025 |
26.649 |
0.069 |
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 | 961 | 961 | 939 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.98 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.331 |
3.025 |
0.088 |
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 | 721 | 721 | 744 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.03 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.055 |
2.032 |
0.085 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
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 | 481 | 481 | 508 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.06 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.861 |
21.112 |
0.083 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
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 |
0.028 |
5.078 |
0.076 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
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 | 25 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.018 |
1.25 |
0. |
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 | 2851 | 2851 | 3033 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.06 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
3.782 |
18.493 |
0.105 |
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 | 2123 | 2123 | 2245 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.06 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.982 |
16.935 |
0.105 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
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 | 1395 | 1395 | 1393 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.313 |
16.514 |
0.096 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.025 |
116.839 |
0.092 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
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 |
0.026 |
68.316 |
0.093 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 254 | 254 | 288 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.13 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.221 |
1.98 |
0.069 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 140 | 140 | 172 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.23 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.119 |
6.928 |
0.073 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 30 | 23 | 30 | 101 | 0 | 39 |
normalized size | 1 | 1. | 1.15 | 0.88 | 1.15 | 3.88 | 0. | 1.5 |
time (sec) |
N/A |
0.023 |
0.03 |
0.017 |
1.188 |
2.256 |
0. |
1.147 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
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 | 29 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.016 |
8.439 |
0.072 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
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 | 31 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.015 |
9.21 |
0.072 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 407 | 407 | 487 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.2 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.532 |
6.056 |
0.063 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 229 | 229 | 309 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.35 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.321 |
5.647 |
0.065 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 48 | 51 | 65 | 576 | 0 | 74 |
normalized size | 1 | 1. | 1.02 | 1.09 | 1.38 | 12.26 | 0. | 1.57 |
time (sec) |
N/A |
0.057 |
0.113 |
0.032 |
1.136 |
2.201 |
0. |
1.19 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
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 |
0.025 |
26.403 |
0.07 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
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 |
26.574 |
0.068 |
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 | 601 | 601 | 626 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.04 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.969 |
2.172 |
0.083 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
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 | 361 | 361 | 390 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.08 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.776 |
7.686 |
0.082 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 64 |
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 | 68 | 68 | 69 | 95 | 0 | 570 | 0 | 82 |
normalized size |
1 |
1. |
1.01 |
1.4 |
0. |
8.38 |
0. |
1.21 |
time (sec) |
N/A |
0.092 |
0.124 |
0.051 |
0. |
2.697 |
0. |
1.148 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 65 |
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 |
0.028 |
8.337 |
0.076 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 66 |
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 |
0.027 |
8.482 |
0.078 |
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 |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1755 | 1755 | 1769 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.01 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.667 |
15.373 |
0.102 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 68 |
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 | 1027 | 1027 | 986 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.96 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.898 |
13.034 |
0.099 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 127 | 127 | 232 | 236 | 0 | 3187 | 0 | 200 |
normalized size |
1 |
1. |
1.83 |
1.86 |
0. |
25.09 |
0. |
1.57 |
time (sec) |
N/A |
0.2 |
0.483 |
0.066 |
0. |
2.644 |
0. |
1.129 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 70 |
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 |
0.026 |
64.005 |
0.095 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 71 |
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 |
0.027 |
64.029 |
0.096 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 72 |
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 |
0.055 |
17.768 |
0.244 |
0. |
0. |
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 | 44 | 44 | 41 | 155 | 0 | 429 | 0 | 0 |
normalized size |
1 |
1. |
0.93 |
3.52 |
0. |
9.75 |
0. |
0. |
time (sec) |
N/A |
0.056 |
0.05 |
0.226 |
0. |
2.237 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 74 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F(-2) |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 135 | 135 | 260 | 368 | 0 | 2233 | 0 | 0 |
normalized size |
1 |
1. |
1.93 |
2.73 |
0. |
16.54 |
0. |
0. |
time (sec) |
N/A |
0.113 |
0.175 |
0.218 |
0. |
2.578 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 75 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F(-2) |
C |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 217 | 217 | 0 | 0 | 0 | 3557 | 0 | 0 |
normalized size |
1 |
1. |
0. |
0. |
0. |
16.39 |
0. |
0. |
time (sec) |
N/A |
0.183 |
10.08 |
0.254 |
0. |
2.675 |
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 |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 57 | 271 | 0 | 2187 | 0 | 0 |
normalized size |
1 |
1. |
0.72 |
3.43 |
0. |
27.68 |
0. |
0. |
time (sec) |
N/A |
0.102 |
0.201 |
0.112 |
0. |
2.515 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 77 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F(-2) |
B |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 208 | 208 | 501 | 0 | 0 | 9585 | 0 | 0 |
normalized size |
1 |
1. |
2.41 |
0. |
0. |
46.08 |
0. |
0. |
time (sec) |
N/A |
0.208 |
2.849 |
0.31 |
0. |
3.203 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 78 |
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 | 363 | 363 | 0 | 0 | 0 | 17207 | 0 | 0 |
normalized size |
1 |
1. |
0. |
0. |
0. |
47.4 |
0. |
0. |
time (sec) |
N/A |
0.404 |
68.287 |
0.306 |
0. |
3.711 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 79 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 87 | 80 | 317 | 0 | 1381 | 0 | 0 |
normalized size | 1 | 1. | 0.92 | 3.64 | 0. | 15.87 | 0. | 0. |
time (sec) |
N/A |
0.146 |
0.146 |
0.121 |
0. |
2.375 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 80 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 307 | 307 | 859 | 585 | 0 | 3371 | 0 | 0 |
normalized size | 1 | 1. | 2.8 | 1.91 | 0. | 10.98 | 0. | 0. |
time (sec) |
N/A |
0.564 |
2.039 |
0.117 |
0. |
2.422 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 81 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 452 | 452 | 0 | 0 | 0 | 5237 | 0 | 0 |
normalized size | 1 | 1. | 0. | 0. | 0. | 11.59 | 0. | 0. |
time (sec) |
N/A |
0.863 |
6.793 |
0.23 |
0. |
2.646 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 82 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
B |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 233 | 491 | 0 | 9543 | 0 | 0 |
normalized size |
1 |
1. |
1.48 |
3.13 |
0. |
60.78 |
0. |
0. |
time (sec) |
N/A |
0.298 |
0.574 |
0.125 |
0. |
2.915 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 83 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
B |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 717 | 717 | 542 | 0 | 0 | 21863 | 0 | 0 |
normalized size |
1 |
1. |
0.76 |
0. |
0. |
30.49 |
0. |
0. |
time (sec) |
N/A |
1.22 |
30.274 |
0.588 |
0. |
3.716 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 84 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1284 | 1284 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.258 |
116.23 |
0.417 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
2.3 Detailed conclusion table specific for Rubi results
The following table is specific to Rubi. It gives additional statistics for each integral. the column
steps is the number of steps used by Rubi to obtain the antiderivative. The rules
column is the number of unique rules used. The integrand size column is the leaf size
of the integrand. Finally the ratio \(\frac{\text{number of rules}}{\text{integrand size}}\) is given. The larger this ratio is, the harder the
integral was to solve. In this test, problem number [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 |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
32 |
A |
20 |
7 |
1. |
18 |
0.389 |
|
|
|
|
|
|
|
33 |
A |
16 |
7 |
1. |
18 |
0.389 |
|
|
|
|
|
|
|
34 |
A |
12 |
7 |
1. |
16 |
0.438 |
|
|
|
|
|
|
|
35 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
36 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
37 |
A |
30 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
38 |
A |
24 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
39 |
A |
18 |
10 |
1. |
18 |
0.556 |
|
|
|
|
|
|
|
40 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
41 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
42 |
A |
23 |
9 |
1. |
20 |
0.45 |
|
|
|
|
|
|
|
43 |
A |
19 |
9 |
1. |
20 |
0.45 |
|
|
|
|
|
|
|
44 |
A |
15 |
9 |
1. |
18 |
0.5 |
|
|
|
|
|
|
|
45 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
46 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
47 |
A |
61 |
11 |
1. |
20 |
0.55 |
|
|
|
|
|
|
|
48 |
A |
49 |
11 |
1. |
20 |
0.55 |
|
|
|
|
|
|
|
49 |
A |
37 |
11 |
1. |
18 |
0.611 |
|
|
|
|
|
|
|
50 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
51 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
52 |
A |
14 |
7 |
1. |
20 |
0.35 |
|
|
|
|
|
|
|
53 |
A |
10 |
6 |
1. |
20 |
0.3 |
|
|
|
|
|
|
|
54 |
A |
4 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
55 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
56 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
57 |
A |
21 |
10 |
1. |
22 |
0.454 |
|
|
|
|
|
|
|
58 |
A |
15 |
11 |
1. |
22 |
0.5 |
|
|
|
|
|
|
|
59 |
A |
5 |
5 |
1. |
22 |
0.227 |
|
|
|
|
|
|
|
60 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
61 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
62 |
A |
17 |
9 |
1. |
22 |
0.409 |
|
|
|
|
|
|
|
63 |
A |
13 |
8 |
1. |
22 |
0.364 |
|
|
|
|
|
|
|
64 |
A |
4 |
4 |
1. |
22 |
0.182 |
|
|
|
|
|
|
|
65 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
66 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
67 |
A |
43 |
11 |
1. |
22 |
0.5 |
|
|
|
|
|
|
|
68 |
A |
31 |
12 |
1. |
22 |
0.546 |
|
|
|
|
|
|
|
69 |
A |
6 |
6 |
1. |
22 |
0.273 |
|
|
|
|
|
|
|
70 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
71 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
72 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
73 |
A |
5 |
4 |
1. |
20 |
0.2 |
|
|
|
|
|
|
|
74 |
A |
9 |
6 |
1. |
22 |
0.273 |
|
|
|
|
|
|
|
75 |
A |
11 |
7 |
1. |
22 |
0.318 |
|
|
|
|
|
|
|
76 |
A |
6 |
6 |
1. |
22 |
0.273 |
|
|
|
|
|
|
|
77 |
A |
11 |
8 |
1. |
24 |
0.333 |
|
|
|
|
|
|
|
78 |
A |
16 |
12 |
1. |
24 |
0.5 |
|
|
|
|
|
|
|
79 |
A |
5 |
5 |
1. |
22 |
0.227 |
|
|
|
|
|
|
|
80 |
A |
12 |
8 |
1. |
24 |
0.333 |
|
|
|
|
|
|
|
|
81 |
A |
14 |
9 |
1. |
24 |
0.375 |
|
|
|
|
|
|
|
82 |
A |
7 |
7 |
1. |
22 |
0.318 |
|
|
|
|
|
|
|
83 |
A |
23 |
11 |
1. |
24 |
0.458 |
|
|
|
|
|
|
|
84 |
A |
32 |
13 |
1. |
24 |
0.542 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|