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 }
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, 15, 16, 18, 19, 20, 21, 23, 27, 28, 29, 30, 33, 34, 35, 43, 44,
45, 46, 47, 48, 49, 50 }
B grade: { 14, 22, 31, 32, 36, 38, 39, 40, 41, 42 }
C grade: { 17, 24, 25, 26 }
F grade: { 37}
2.1.3 Maple
A grade: { 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 21, 22, 23, 27, 28, 29, 30, 32, 33, 34, 38, 39, 40, 42, 50
}
B grade: { 11, 12, 18, 19, 20, 24, 25, 26 }
C grade: { }
F grade: { 1, 31, 35, 36, 37, 41, 43, 44, 45, 46, 47, 48, 49 }
2.1.4 Maxima
A grade: { 2, 3, 4, 5, 7, 12, 14, 22, 38, 39, 40, 41
B grade: { 8, 9 }
C grade: { }
F grade: { 1, 6, 10, 11, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
42, 43, 44, 45, 46, 47, 48, 49, 50 }
2.1.5 FriCAS
A grade: { 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 15, 18, 19, 20, 21, 25, 26, 38, 39, 40, 41, 47, 48, 49 }
B grade: { 14, 16, 22, 24 }
C grade: { }
F grade: { 1, 6, 13, 17, 23, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 42, 43, 44, 45, 46, 50 }
2.1.6 Sympy
A grade: { 10, 11, 12 }
B grade: { }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 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 }
2.1.7 Giac
A grade: { 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 15, 16, 47 }
B grade: { 14, 24, 25, 26 }
C grade: { }
F grade: { 1, 6, 13, 17, 18, 19, 20, 21, 22, 23, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 48, 49, 50 }
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 |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 62 | 62 | 56 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.9 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.087 |
0.034 |
0.26 |
0. |
0. |
0. |
0. |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 40 | 43 | 89 | 97 | 0 | 55 |
normalized size |
1 |
1. |
0.69 |
0.74 |
1.53 |
1.67 |
0. |
0.95 |
time (sec) |
N/A |
0.02 |
0.026 |
0.114 |
1.001 |
2.281 |
0. |
1.101 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 35 | 38 | 70 | 84 | 0 | 46 |
normalized size |
1 |
1. |
0.74 |
0.81 |
1.49 |
1.79 |
0. |
0.98 |
time (sec) |
N/A |
0.018 |
0.023 |
0.108 |
1.001 |
2.198 |
0. |
1.132 |
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Problem 4 |
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 | 36 | 36 | 28 | 31 | 51 | 69 | 0 | 36 |
normalized size | 1 | 1. | 0.78 | 0.86 | 1.42 | 1.92 | 0. | 1. |
time (sec) |
N/A |
0.011 |
0.018 |
0.108 |
0.984 |
2.156 |
0. |
1.116 |
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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 | 18 | 18 | 18 | 25 | 28 | 45 | 0 | 20 |
normalized size | 1 | 1. | 1. | 1.39 | 1.56 | 2.5 | 0. | 1.11 |
time (sec) |
N/A |
0.004 |
0.005 |
0.107 |
0.971 |
2.178 |
0. |
1.117 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 54 | 63 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.96 | 1.12 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.079 |
0.026 |
0.2 |
0. |
0. |
0. |
0. |
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Problem 7 |
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 | 38 | 38 | 32 | 45 | 69 | 63 | 0 | 41 |
normalized size | 1 | 1. | 0.84 | 1.18 | 1.82 | 1.66 | 0. | 1.08 |
time (sec) |
N/A |
0.016 |
0.02 |
0.11 |
1.463 |
2.19 |
0. |
1.108 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 55 | 57 | 108 | 86 | 0 | 51 |
normalized size |
1 |
1. |
1.02 |
1.06 |
2. |
1.59 |
0. |
0.94 |
time (sec) |
N/A |
0.019 |
0.032 |
0.111 |
1.517 |
2.241 |
0. |
1.116 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 68 | 68 | 45 | 67 | 143 | 105 | 0 | 61 |
normalized size |
1 |
1. |
0.66 |
0.99 |
2.1 |
1.54 |
0. |
0.9 |
time (sec) |
N/A |
0.024 |
0.049 |
0.108 |
1.504 |
2.398 |
0. |
1.122 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 42 | 66 | 0 | 88 | 51 | 63 |
normalized size |
1 |
1. |
0.75 |
1.18 |
0. |
1.57 |
0.91 |
1.12 |
time (sec) |
N/A |
0.042 |
0.039 |
0.175 |
0. |
2.469 |
0.599 |
1.088 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 44 | 93 | 0 | 86 | 41 | 53 |
normalized size |
1 |
1. |
0.94 |
1.98 |
0. |
1.83 |
0.87 |
1.13 |
time (sec) |
N/A |
0.02 |
0.027 |
0.152 |
0. |
2.55 |
0.266 |
1.097 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 26 | 51 | 32 | 55 | 22 | 38 |
normalized size | 1 | 1. | 1. | 1.96 | 1.23 | 2.12 | 0.85 | 1.46 |
time (sec) |
N/A |
0.009 |
0.013 |
0.151 |
0.964 |
2.606 |
0.165 |
1.081 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 59 | 76 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1. |
1.29 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.06 |
0.02 |
0.236 |
0. |
0. |
0. |
0. |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
A |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 93 | 41 | 70 | 225 | 0 | 82 |
normalized size | 1 | 1. | 3. | 1.32 | 2.26 | 7.26 | 0. | 2.65 |
time (sec) |
N/A |
0.03 |
0.136 |
0.145 |
0.962 |
2.411 |
0. |
1.108 |
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Problem 15 |
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 | 38 | 38 | 36 | 54 | 0 | 84 | 0 | 82 |
normalized size |
1 |
1. |
0.95 |
1.42 |
0. |
2.21 |
0. |
2.16 |
time (sec) |
N/A |
0.021 |
0.021 |
0.172 |
0. |
2.285 |
0. |
1.11 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-2) |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 69 | 98 | 0 | 294 | 0 | 108 |
normalized size |
1 |
1. |
1.15 |
1.63 |
0. |
4.9 |
0. |
1.8 |
time (sec) |
N/A |
0.042 |
0.043 |
0.178 |
0. |
2.401 |
0. |
1.104 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
F |
F(-2) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 60 | 98 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.87 |
1.42 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.093 |
0.084 |
0.336 |
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 |
A |
B |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 197 | 197 | 173 | 509 | 0 | 374 | 0 | 0 |
normalized size | 1 | 1. | 0.88 | 2.58 | 0. | 1.9 | 0. | 0. |
time (sec) |
N/A |
0.232 |
0.183 |
0.237 |
0. |
2.498 |
0. |
0. |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 155 | 155 | 150 | 359 | 0 | 316 | 0 | 0 |
normalized size | 1 | 1. | 0.97 | 2.32 | 0. | 2.04 | 0. | 0. |
time (sec) |
N/A |
0.139 |
0.28 |
0.217 |
0. |
2.61 |
0. |
0. |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 116 | 116 | 131 | 273 | 0 | 284 | 0 | 0 |
normalized size | 1 | 1. | 1.13 | 2.35 | 0. | 2.45 | 0. | 0. |
time (sec) |
N/A |
0.09 |
0.174 |
0.212 |
0. |
2.535 |
0. |
0. |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 78 | 78 | 110 | 126 | 0 | 254 | 0 | 0 |
normalized size | 1 | 1. | 1.41 | 1.62 | 0. | 3.26 | 0. | 0. |
time (sec) |
N/A |
0.053 |
0.118 |
0.208 |
0. |
2.438 |
0. |
0. |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
A |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 37 | 37 | 115 | 51 | 74 | 184 | 0 | 0 |
normalized size | 1 | 1. | 3.11 | 1.38 | 2. | 4.97 | 0. | 0. |
time (sec) |
N/A |
0.023 |
0.139 |
0.203 |
0.945 |
2.541 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 200 | 200 | 284 | 374 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.42 | 1.87 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.309 |
0.319 |
0.404 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
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|
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|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-1) |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 70 | 70 | 112 | 154 | 0 | 666 | 0 | 174 |
normalized size |
1 |
1. |
1.6 |
2.2 |
0. |
9.51 |
0. |
2.49 |
time (sec) |
N/A |
0.097 |
0.301 |
0.219 |
0. |
2.583 |
0. |
1.635 |
|
|
|
|
|
|
|
|
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|
|
|
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|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 125 | 125 | 198 | 452 | 0 | 999 | 0 | 375 |
normalized size | 1 | 1. | 1.58 | 3.62 | 0. | 7.99 | 0. | 3. |
time (sec) |
N/A |
0.192 |
1.04 |
0.243 |
0. |
2.682 |
0. |
1.665 |
|
|
|
|
|
|
|
|
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|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 181 | 181 | 241 | 760 | 0 | 1241 | 0 | 792 |
normalized size | 1 | 1. | 1.33 | 4.2 | 0. | 6.86 | 0. | 4.38 |
time (sec) |
N/A |
0.288 |
0.406 |
0.226 |
0. |
3.046 |
0. |
1.841 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 381 | 381 | 667 | 734 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.75 | 1.93 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.302 |
9.374 |
0.97 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 288 | 288 | 473 | 540 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.64 | 1.88 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.232 |
6.625 |
0.572 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 154 | 154 | 142 | 327 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.92 | 2.12 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.131 |
0.145 |
0.524 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 94 | 94 | 111 | 179 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.18 | 1.9 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.063 |
0.108 |
0.325 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 310 | 310 | 813 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 2.62 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.485 |
2.23 |
0.627 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 244 | 244 | 686 | 341 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
2.81 |
1.4 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.395 |
2.222 |
0.404 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 494 | 494 | 442 | 770 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.89 | 1.56 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.397 |
0.491 |
0.59 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 278 | 278 | 248 | 429 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.89 | 1.54 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.259 |
0.423 |
0.476 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
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 | 154 | 154 | 160 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.04 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.104 |
0.093 |
0.454 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 430 | 430 | 1058 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 2.46 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.533 |
3.481 |
0.697 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 362 | 362 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0. | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.595 |
180.003 |
0.884 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 148 | 81 | 96 | 227 | 0 | 0 |
normalized size |
1 |
1. |
2.55 |
1.4 |
1.66 |
3.91 |
0. |
0. |
time (sec) |
N/A |
0.073 |
0.223 |
0.254 |
0.971 |
2.841 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 148 | 81 | 96 | 227 | 0 | 0 |
normalized size |
1 |
1. |
2.55 |
1.4 |
1.66 |
3.91 |
0. |
0. |
time (sec) |
N/A |
0.076 |
0.147 |
0.262 |
0.997 |
2.628 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 137 | 81 | 96 | 227 | 0 | 0 |
normalized size |
1 |
1. |
2.36 |
1.4 |
1.66 |
3.91 |
0. |
0. |
time (sec) |
N/A |
0.078 |
0.367 |
0.265 |
0.998 |
2.49 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
A |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 49 | 49 | 130 | 0 | 89 | 216 | 0 | 0 |
normalized size |
1 |
1. |
2.65 |
0. |
1.82 |
4.41 |
0. |
0. |
time (sec) |
N/A |
0.072 |
0.335 |
0.35 |
0.95 |
2.596 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
F(-1) |
F(-2) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 280 | 116 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
3.29 |
1.36 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.081 |
0.85 |
0.392 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
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 | 99 | 99 | 95 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.96 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.117 |
0.301 |
0.181 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
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 | 91 | 91 | 107 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.18 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.107 |
0.191 |
0.176 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
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 | 91 | 91 | 54 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.59 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.092 |
0.06 |
0.175 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
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 | 45 | 45 | 79 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.76 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.058 |
0.054 |
0.175 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 39 | 39 | 34 | 0 | 0 | 63 | 0 | 69 |
normalized size | 1 | 1. | 0.87 | 0. | 0. | 1.62 | 0. | 1.77 |
time (sec) |
N/A |
0.03 |
0.041 |
0.176 |
0. |
2.434 |
0. |
1.159 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 41 | 30 | 0 | 0 | 80 | 0 | 0 |
normalized size | 1 | 1. | 0.73 | 0. | 0. | 1.95 | 0. | 0. |
time (sec) |
N/A |
0.042 |
0.05 |
0.177 |
0. |
2.484 |
0. |
0. |
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Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 54 | 0 | 0 | 100 | 0 | 0 |
normalized size | 1 | 1. | 0.64 | 0. | 0. | 1.19 | 0. | 0. |
time (sec) |
N/A |
0.065 |
0.141 |
0.179 |
0. |
2.711 |
0. |
0. |
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Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 59 | 92 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.86 |
1.33 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.091 |
0.051 |
0.347 |
0. |
0. |
0. |
0. |
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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 [34] had the largest ratio of [ 1.2
]
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| Table 2.1Rubi specific breakdown of results for each
integral
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# |
grade |
|
|
normalized | antiderivative |
leaf size | |
|
\(\frac{\text{number of rules}}{\text{integrand leaf size}}\) |
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1 |
A |
7 |
6 |
1. |
10 |
0.6 |
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2 |
A |
4 |
3 |
1. |
10 |
0.3 |
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3 |
A |
4 |
3 |
1. |
10 |
0.3 |
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4 |
A |
4 |
3 |
1. |
8 |
0.375 |
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5 |
A |
3 |
3 |
1. |
6 |
0.5 |
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6 |
A |
7 |
6 |
1. |
10 |
0.6 |
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7 |
A |
5 |
5 |
1. |
10 |
0.5 |
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8 |
A |
6 |
5 |
1. |
10 |
0.5 |
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9 |
A |
7 |
5 |
1. |
10 |
0.5 |
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10 |
A |
5 |
4 |
1. |
10 |
0.4 |
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11 |
A |
4 |
4 |
1. |
8 |
0.5 |
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12 |
A |
3 |
3 |
1. |
6 |
0.5 |
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13 |
A |
6 |
6 |
1. |
10 |
0.6 |
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14 |
A |
5 |
5 |
1. |
10 |
0.5 |
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15 |
A |
3 |
3 |
1. |
10 |
0.3 |
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16 |
A |
6 |
6 |
1. |
10 |
0.6 |
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17 |
A |
7 |
6 |
1. |
10 |
0.6 |
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18 |
A |
9 |
8 |
1. |
10 |
0.8 |
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19 |
A |
8 |
7 |
1. |
10 |
0.7 |
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20 |
A |
7 |
6 |
1. |
10 |
0.6 |
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21 |
A |
6 |
6 |
1. |
8 |
0.75 |
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22 |
A |
5 |
5 |
1. |
6 |
0.833 |
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23 |
A |
14 |
8 |
1. |
10 |
0.8 |
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24 |
A |
5 |
5 |
1. |
10 |
0.5 |
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25 |
A |
7 |
7 |
1. |
10 |
0.7 |
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26 |
A |
8 |
8 |
1. |
10 |
0.8 |
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27 |
A |
20 |
9 |
1. |
12 |
0.75 |
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28 |
A |
17 |
9 |
1. |
12 |
0.75 |
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29 |
A |
11 |
8 |
1. |
10 |
0.8 |
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30 |
A |
8 |
6 |
1. |
8 |
0.75 |
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31 |
A |
17 |
9 |
1. |
12 |
0.75 |
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32 |
A |
12 |
8 |
1. |
12 |
0.667 |
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33 |
A |
25 |
14 |
1. |
12 |
1.167 |
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34 |
A |
16 |
12 |
1. |
10 |
1.2 |
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35 |
A |
10 |
7 |
1. |
8 |
0.875 |
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36 |
A |
20 |
10 |
1. |
12 |
0.833 |
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37 |
A |
14 |
9 |
1. |
12 |
0.75 |
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38 |
A |
7 |
6 |
1. |
14 |
0.429 |
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39 |
A |
7 |
6 |
1. |
16 |
0.375 |
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40 |
A |
7 |
6 |
1. |
16 |
0.375 |
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41 |
A |
6 |
6 |
1. |
14 |
0.429 |
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42 |
A |
7 |
7 |
1. |
10 |
0.7 |
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43 |
A |
6 |
4 |
1. |
10 |
0.4 |
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44 |
A |
6 |
4 |
1. |
8 |
0.5 |
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45 |
A |
5 |
3 |
1. |
6 |
0.5 |
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46 |
A |
6 |
5 |
1. |
10 |
0.5 |
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47 |
A |
3 |
3 |
1. |
10 |
0.3 |
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|
48 |
A |
5 |
4 |
1. |
10 |
0.4 |
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|
49 |
A |
6 |
4 |
1. |
10 |
0.4 |
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|
50 |
A |
8 |
8 |
1. |
19 |
0.421 |
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