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, 6, 7, 8, 9, 10, 11, 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, 43, 44, 46, 47, 48, 49, 50 }
B grade: { 4, 42 }
C grade: { 5, 12, 13 }
F grade: { 3, 45}
2.1.2 Mathematica
A grade: { 1, 2, 4, 6, 7, 10, 11, 14, 15, 16, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 40, 42, 43, 44, 45, 46, 48, 49, 50 }
B grade: { 3, 17, 21, 41 }
C grade: { 5, 8, 9, 12, 13, 37, 38, 39, 47 }
F grade: { }
2.1.3 Maple
A grade: { 2, 6, 7, 9, 19, 24, 27, 32, 34, 36, 37, 43, 48 }
B grade: { 1, 3, 4, 8, 40, 42, 44, 47, 49, 50 }
C grade: { 5, 12, 18, 28, 31, 33, 35, 38, 39, 46 }
F grade: { 10, 11, 13, 14, 15, 16, 17, 20, 21, 22, 23, 25, 26, 29, 30, 41, 45 }
2.1.4 Maxima
A grade: { 1, 2, 3, 9, 16, 18, 19, 20, 24, 25, 27, 28, 31, 32, 33, 34, 35, 36, 37, 46, 48
B grade: { 7, 40, 43 }
C grade: { }
F grade: { 4, 5, 6, 8, 10, 11, 12, 13, 14, 15, 17, 21, 22, 23, 26, 29, 30, 38, 39, 41, 42, 44, 45, 47, 49, 50
}
2.1.5 FriCAS
A grade: { 1, 2, 5, 9, 10, 11, 12, 13, 14, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 39,
46, 48 }
B grade: { 4, 6, 7, 15, 16, 17, 21, 22, 37, 38, 40, 41, 42, 43, 44, 47, 49, 50 }
C grade: { }
F grade: { 3, 8, 45 }
2.1.6 Sympy
A grade: { 1, 2, 10, 19, 20, 24, 25, 31, 33, 34, 36, 46, 48 }
B grade: { 28, 37, 40 }
C grade: { 16, 18 }
F grade: { 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 17, 21, 22, 23, 26, 27, 29, 30, 32, 35, 38, 39, 41, 42, 43, 44,
45, 47, 49, 50 }
2.1.7 Giac
A grade: { 2, 6, 10, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 33, 34, 35, 36, 37, 40, 43, 48, 49, 50
}
B grade: { 1, 4, 7, 12, 13, 27, 29, 30, 31, 32, 41 }
C grade: { 8 }
F grade: { 3, 5, 9, 11, 23, 38, 39, 42, 44, 45, 46, 47 }
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 |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 52 | 93 | 78 | 167 | 102 | 367 |
normalized size | 1 | 1. | 1.02 | 1.82 | 1.53 | 3.27 | 2. | 7.2 |
time (sec) |
N/A |
0.029 |
0.021 |
0.039 |
1.422 |
2.518 |
6.864 |
1.176 |
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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 |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 17 | 17 | 17 | 16 | 20 | 42 | 12 | 20 |
normalized size | 1 | 1. | 1. | 0.94 | 1.18 | 2.47 | 0.71 | 1.18 |
time (sec) |
N/A |
0.028 |
0.005 |
0.044 |
1.43 |
2.48 |
0.187 |
1.123 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
F |
B |
B |
A |
F(-1) |
F |
F(-1) |
verified |
N/A |
N/A |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 0 | 205 | 251 | 5 | 0 | 0 | 0 |
normalized size |
1 |
0. |
2.97 |
3.64 |
0.07 |
0. |
0. |
0. |
time (sec) |
N/A |
0.153 |
0.604 |
0.688 |
4.058 |
0. |
0. |
0. |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
B |
A |
B |
F |
B |
F |
B |
verified |
N/A |
NO |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 97 | 332 | 194 | 426 | 0 | 1222 | 0 | 317 |
normalized size | 1 | 3.42 | 2. | 4.39 | 0. | 12.6 | 0. | 3.27 |
time (sec) |
N/A |
0.669 |
0.358 |
0.143 |
0. |
2.903 |
0. |
1.275 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
C |
C |
C |
F |
A |
F(-1) |
F |
verified |
N/A |
NO |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 45 | 289 | 159 | 312 | 0 | 105 | 0 | 0 |
normalized size |
1 |
6.42 |
3.53 |
6.93 |
0. |
2.33 |
0. |
0. |
time (sec) |
N/A |
0.491 |
2.022 |
0.421 |
0. |
3.841 |
0. |
0. |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 74 | 64 | 0 | 263 | 0 | 107 |
normalized size | 1 | 1. | 1.32 | 1.14 | 0. | 4.7 | 0. | 1.91 |
time (sec) |
N/A |
0.069 |
0.11 |
0.055 |
0. |
4.033 |
0. |
1.119 |
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Problem 7 |
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 | 15 | 15 | 15 | 12 | 36 | 92 | 0 | 38 |
normalized size | 1 | 1. | 1. | 0.8 | 2.4 | 6.13 | 0. | 2.53 |
time (sec) |
N/A |
0.031 |
0.014 |
0.043 |
0.961 |
3.134 |
0. |
1.234 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
F(-2) |
F(-1) |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 137 | 137 | 99 | 1605 | 0 | 0 | 0 | 370 |
normalized size |
1 |
1. |
0.72 |
11.72 |
0. |
0. |
0. |
2.7 |
time (sec) |
N/A |
0.18 |
10.03 |
0.154 |
0. |
0. |
0. |
1.298 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
A |
A |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 41 | 285 | 42 | 81 | 116 | 0 | 0 |
normalized size | 1 | 1. | 6.95 | 1.02 | 1.98 | 2.83 | 0. | 0. |
time (sec) |
N/A |
0.022 |
3.812 |
0.041 |
1.44 |
2.825 |
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 |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 44 | 62 | 38 | 0 | 0 | 80 | 56 | 62 |
normalized size | 1 | 1.41 | 0.86 | 0. | 0. | 1.82 | 1.27 | 1.41 |
time (sec) |
N/A |
0.676 |
0.158 |
0.029 |
0. |
2.726 |
2.141 |
1.155 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 68 | 68 | 64 | 0 | 0 | 127 | 0 | 0 |
normalized size |
1 |
1. |
0.94 |
0. |
0. |
1.87 |
0. |
0. |
time (sec) |
N/A |
0.146 |
0.049 |
0.014 |
0. |
2.436 |
0. |
0. |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
C |
C |
C |
F |
A |
F(-1) |
B |
verified |
N/A |
NO |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 141 | 269 | 1822 | 439 | 0 | 529 | 0 | 491 |
normalized size |
1 |
1.91 |
12.92 |
3.11 |
0. |
3.75 |
0. |
3.48 |
time (sec) |
N/A |
0.746 |
3.397 |
0.061 |
0. |
2.63 |
0. |
1.139 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
C |
C |
F |
F |
A |
F(-1) |
B |
verified |
N/A |
NO |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 152 | 286 | 2180 | 0 | 0 | 548 | 0 | 504 |
normalized size |
1 |
1.88 |
14.34 |
0. |
0. |
3.61 |
0. |
3.32 |
time (sec) |
N/A |
0.457 |
4.531 |
0.059 |
0. |
2.638 |
0. |
1.246 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 45 | 51 | 44 | 0 | 0 | 182 | 0 | 77 |
normalized size | 1 | 1.13 | 0.98 | 0. | 0. | 4.04 | 0. | 1.71 |
time (sec) |
N/A |
0.121 |
0.042 |
180. |
0. |
2.803 |
0. |
1.142 |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
B |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 34 | 34 | 64 | 0 | 0 | 153 | 0 | 49 |
normalized size |
1 |
1. |
1.88 |
0. |
0. |
4.5 |
0. |
1.44 |
time (sec) |
N/A |
0.035 |
0.095 |
0.019 |
0. |
2.554 |
0. |
1.14 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
A |
B |
C |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 22 | 22 | 0 | 24 | 146 | 78 | 24 |
normalized size | 1 | 1. | 1. | 0. | 1.09 | 6.64 | 3.55 | 1.09 |
time (sec) |
N/A |
0.02 |
0.027 |
0.106 |
1.448 |
2.324 |
13.066 |
1.128 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
F |
F |
B |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 32 | 32 | 89 | 0 | 0 | 149 | 0 | 49 |
normalized size |
1 |
1. |
2.78 |
0. |
0. |
4.66 |
0. |
1.53 |
time (sec) |
N/A |
0.041 |
0.083 |
0.012 |
0. |
1.982 |
0. |
1.116 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
A |
A |
C |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 43 | 89 | 55 | 86 | 37 | 84 |
normalized size | 1 | 1. | 1. | 2.07 | 1.28 | 2. | 0.86 | 1.95 |
time (sec) |
N/A |
0.045 |
0.023 |
0.053 |
1.435 |
2.147 |
177.985 |
1.117 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 28 | 28 | 21 | 46 | 104 | 48 | 47 |
normalized size | 1 | 1. | 1. | 0.75 | 1.64 | 3.71 | 1.71 | 1.68 |
time (sec) |
N/A |
0.009 |
0.005 |
0.017 |
0.953 |
1.886 |
1.125 |
1.092 |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 26 | 0 | 30 | 57 | 20 | 30 |
normalized size | 1 | 1. | 1. | 0. | 1.15 | 2.19 | 0.77 | 1.15 |
time (sec) |
N/A |
0.036 |
0.021 |
0.024 |
1.057 |
1.998 |
23.102 |
1.1 |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
F |
F |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 85 | 0 | 0 | 311 | 0 | 51 |
normalized size | 1 | 1. | 2.24 | 0. | 0. | 8.18 | 0. | 1.34 |
time (sec) |
N/A |
0.052 |
0.081 |
0.263 |
0. |
2.3 |
0. |
1.112 |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 70 | 94 | 88 | 0 | 0 | 278 | 0 | 70 |
normalized size | 1 | 1.34 | 1.26 | 0. | 0. | 3.97 | 0. | 1. |
time (sec) |
N/A |
0.132 |
0.094 |
0.58 |
0. |
2.32 |
0. |
1.133 |
|
|
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|
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Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 58 | 0 | 0 | 151 | 0 | 0 |
normalized size |
1 |
1. |
1. |
0. |
0. |
2.6 |
0. |
0. |
time (sec) |
N/A |
0.142 |
0.036 |
0.09 |
0. |
2.149 |
0. |
0. |
|
|
|
|
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|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 55 | 55 | 41 | 37 | 49 | 86 | 31 | 49 |
normalized size | 1 | 1. | 0.75 | 0.67 | 0.89 | 1.56 | 0.56 | 0.89 |
time (sec) |
N/A |
0.053 |
0.021 |
0.005 |
0.942 |
2.078 |
7.99 |
1.09 |
|
|
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Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 26 | 0 | 30 | 57 | 20 | 30 |
normalized size | 1 | 1. | 1. | 0. | 1.15 | 2.19 | 0.77 | 1.15 |
time (sec) |
N/A |
0.033 |
0.019 |
0.012 |
1.543 |
2.173 |
23.107 |
1.104 |
|
|
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Problem 26 |
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 | 78 | 78 | 119 | 0 | 0 | 293 | 0 | 165 |
normalized size | 1 | 1. | 1.53 | 0. | 0. | 3.76 | 0. | 2.12 |
time (sec) |
N/A |
0.271 |
0.06 |
0.019 |
0. |
2.157 |
0. |
1.148 |
|
|
|
|
|
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|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 39 | 39 | 25 | 35 | 47 | 95 | 0 | 99 |
normalized size | 1 | 1. | 0.64 | 0.9 | 1.21 | 2.44 | 0. | 2.54 |
time (sec) |
N/A |
0.048 |
0.031 |
0.033 |
1.436 |
2.081 |
0. |
1.092 |
|
|
|
|
|
|
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|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
B |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 17 | 17 | 17 | 54 | 20 | 69 | 29 | 31 |
normalized size | 1 | 1. | 1. | 3.18 | 1.18 | 4.06 | 1.71 | 1.82 |
time (sec) |
N/A |
0.028 |
0.015 |
0.104 |
1.422 |
2.109 |
2.72 |
1.105 |
|
|
|
|
|
|
|
|
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|
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|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 57 | 57 | 77 | 0 | 0 | 211 | 0 | 140 |
normalized size | 1 | 1. | 1.35 | 0. | 0. | 3.7 | 0. | 2.46 |
time (sec) |
N/A |
0.08 |
0.068 |
0.214 |
0. |
2.207 |
0. |
1.107 |
|
|
|
|
|
|
|
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|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 45 | 45 | 45 | 0 | 0 | 182 | 0 | 146 |
normalized size |
1 |
1. |
1. |
0. |
0. |
4.04 |
0. |
3.24 |
time (sec) |
N/A |
0.042 |
0.034 |
0.144 |
0. |
2.152 |
0. |
1.135 |
|
|
|
|
|
|
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|
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|
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|
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|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 29 | 29 | 33 | 56 | 30 | 127 | 19 | 73 |
normalized size | 1 | 1. | 1.14 | 1.93 | 1.03 | 4.38 | 0.66 | 2.52 |
time (sec) |
N/A |
0.048 |
0.025 |
0.099 |
1.432 |
2.077 |
21.48 |
1.088 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 21 | 21 | 20 | 26 | 55 | 0 | 54 |
normalized size | 1 | 1. | 1. | 0.95 | 1.24 | 2.62 | 0. | 2.57 |
time (sec) |
N/A |
0.044 |
0.014 |
0.038 |
1.427 |
2.156 |
0. |
1.09 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 34 | 34 | 27 | 119 | 36 | 80 | 29 | 38 |
normalized size | 1 | 1. | 0.79 | 3.5 | 1.06 | 2.35 | 0.85 | 1.12 |
time (sec) |
N/A |
0.034 |
0.018 |
0.042 |
1.423 |
2.109 |
2.907 |
1.109 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 33 | 33 | 21 | 29 | 39 | 88 | 26 | 78 |
normalized size | 1 | 1. | 0.64 | 0.88 | 1.18 | 2.67 | 0.79 | 2.36 |
time (sec) |
N/A |
0.042 |
0.025 |
0.029 |
1.424 |
2.071 |
6.69 |
1.084 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 25 | 35 | 97 | 20 | 46 | 0 | 30 |
normalized size | 1 | 1. | 1.4 | 3.88 | 0.8 | 1.84 | 0. | 1.2 |
time (sec) |
N/A |
0.029 |
0.042 |
0.355 |
0.974 |
2.173 |
0. |
1.117 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 34 | 34 | 40 | 39 | 34 | 119 | 41 | 59 |
normalized size | 1 | 1. | 1.18 | 1.15 | 1. | 3.5 | 1.21 | 1.74 |
time (sec) |
N/A |
0.035 |
0.017 |
0.025 |
1.427 |
2.062 |
4.548 |
1.09 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
B |
B |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 16 | 16 | 46 | 14 | 32 | 92 | 46 | 36 |
normalized size | 1 | 1. | 2.88 | 0.88 | 2. | 5.75 | 2.88 | 2.25 |
time (sec) |
N/A |
0.018 |
0.051 |
0.012 |
1.426 |
1.993 |
25.318 |
1.091 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
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 | 23 | 23 | 36 | 112 | 0 | 111 | 0 | 0 |
normalized size | 1 | 1. | 1.57 | 4.87 | 0. | 4.83 | 0. | 0. |
time (sec) |
N/A |
0.032 |
0.096 |
0.059 |
0. |
2.332 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 23 | 40 | 112 | 0 | 61 | 0 | 0 |
normalized size | 1 | 1. | 1.74 | 4.87 | 0. | 2.65 | 0. | 0. |
time (sec) |
N/A |
0.032 |
0.088 |
0.015 |
0. |
2.276 |
0. |
0. |
|
|
|
|
|
|
|
|
|
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|
|
|
|
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|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
B |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 22 | 39 | 54 | 111 | 365 | 97 | 49 |
normalized size | 1 | 1. | 1.77 | 2.45 | 5.05 | 16.59 | 4.41 | 2.23 |
time (sec) |
N/A |
0.071 |
0.042 |
0.093 |
1.433 |
2.034 |
1.676 |
1.143 |
|
|
|
|
|
|
|
|
|
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|
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|
|
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|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 42 | 42 | 87 | 0 | 0 | 539 | 0 | 2256 |
normalized size | 1 | 1. | 2.07 | 0. | 0. | 12.83 | 0. | 53.71 |
time (sec) |
N/A |
0.153 |
0.084 |
0.13 |
0. |
2.196 |
0. |
1.666 |
|
|
|
|
|
|
|
|
|
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|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
B |
A |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 59 | 45 | 91 | 0 | 163 | 0 | 0 |
normalized size | 1 | 2.11 | 1.61 | 3.25 | 0. | 5.82 | 0. | 0. |
time (sec) |
N/A |
0.184 |
0.035 |
0.137 |
0. |
2.376 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
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|
|
|
Problem 43 |
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 | 34 | 34 | 55 | 37 | 763 | 635 | 0 | 68 |
normalized size | 1 | 1. | 1.62 | 1.09 | 22.44 | 18.68 | 0. | 2. |
time (sec) |
N/A |
0.045 |
0.068 |
0.03 |
1.781 |
2.219 |
0. |
1.1 |
|
|
|
|
|
|
|
|
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|
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|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 39 | 50 | 65 | 67 | 0 | 193 | 0 | 0 |
normalized size |
1 |
1.28 |
1.67 |
1.72 |
0. |
4.95 |
0. |
0. |
time (sec) |
N/A |
0.046 |
0.085 |
0.242 |
0. |
2.784 |
0. |
0. |
|
|
|
|
|
|
|
|
|
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|
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|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
F |
A |
F |
F |
F(-1) |
F |
F(-1) |
verified |
N/A |
N/A |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 337 | 0 | 552 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
1.64 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.794 |
1.995 |
0.16 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
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|
|
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|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 58 | 3134 | 90 | 173 | 87 | 0 |
normalized size | 1 | 1. | 0.75 | 40.7 | 1.17 | 2.25 | 1.13 | 0. |
time (sec) |
N/A |
0.22 |
0.019 |
0.913 |
1.459 |
2.44 |
2.853 |
0. |
|
|
|
|
|
|
|
|
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|
|
|
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|
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|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 120 | 120 | 136 | 510 | 0 | 782 | 0 | 0 |
normalized size | 1 | 1. | 1.13 | 4.25 | 0. | 6.52 | 0. | 0. |
time (sec) |
N/A |
0.131 |
0.307 |
0.035 |
0. |
2.685 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
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|
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Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
NO |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 37 | 39 | 28 | 35 | 72 | 29 | 36 |
normalized size | 1 | 1.19 | 1.26 | 0.9 | 1.13 | 2.32 | 0.94 | 1.16 |
time (sec) |
N/A |
0.009 |
0.376 |
0.016 |
1.569 |
2.42 |
78.514 |
1.105 |
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Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F |
B |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 29 | 29 | 29 | 138 | 0 | 146 | 0 | 46 |
normalized size |
1 |
1. |
1. |
4.76 |
0. |
5.03 |
0. |
1.59 |
time (sec) |
N/A |
0.024 |
0.011 |
0.072 |
0. |
2.457 |
0. |
1.155 |
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Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F |
B |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 106 | 198 | 0 | 509 | 0 | 150 |
normalized size |
1 |
1. |
1. |
1.87 |
0. |
4.8 |
0. |
1.42 |
time (sec) |
N/A |
0.11 |
0.179 |
0.052 |
0. |
2.192 |
0. |
1.131 |
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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 [1] had the largest ratio of [ 1.4
]
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| Table 2.1Rubi specific breakdown of results for each
integral
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# |
grade |
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|
normalized | antiderivative |
leaf size | |
|
\(\frac{\text{number of rules}}{\text{integrand leaf size}}\) |
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1 |
A |
8 |
7 |
1. |
5 |
1.4 |
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2 |
A |
2 |
2 |
1. |
15 |
0.133 |
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3 |
F |
0 |
0 |
N/A |
0 |
N/A |
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4 |
B |
32 |
13 |
3.42 |
14 |
0.929 |
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5 |
C |
5 |
4 |
6.42 |
19 |
0.21 |
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6 |
A |
7 |
7 |
1. |
13 |
0.538 |
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7 |
A |
4 |
4 |
1. |
13 |
0.308 |
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8 |
A |
9 |
7 |
1. |
14 |
0.5 |
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9 |
A |
7 |
7 |
1. |
12 |
0.583 |
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10 |
A |
5 |
5 |
1.41 |
19 |
0.263 |
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11 |
A |
7 |
8 |
1. |
29 |
0.276 |
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12 |
C |
40 |
15 |
1.91 |
14 |
1.071 |
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13 |
C |
32 |
16 |
1.88 |
27 |
0.593 |
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14 |
A |
9 |
11 |
1.13 |
18 |
0.611 |
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15 |
A |
3 |
4 |
1. |
24 |
0.167 |
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16 |
A |
3 |
4 |
1. |
12 |
0.333 |
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17 |
A |
3 |
4 |
1. |
22 |
0.182 |
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18 |
A |
4 |
4 |
1. |
15 |
0.267 |
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19 |
A |
4 |
4 |
1. |
13 |
0.308 |
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20 |
A |
2 |
3 |
1. |
23 |
0.13 |
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21 |
A |
5 |
6 |
1. |
17 |
0.353 |
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22 |
A |
7 |
8 |
1.34 |
15 |
0.533 |
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23 |
A |
4 |
9 |
1. |
25 |
0.36 |
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24 |
A |
5 |
5 |
1. |
27 |
0.185 |
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25 |
A |
2 |
3 |
1. |
23 |
0.13 |
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26 |
A |
18 |
11 |
1. |
27 |
0.407 |
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27 |
A |
3 |
3 |
1. |
17 |
0.176 |
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28 |
A |
2 |
2 |
1. |
13 |
0.154 |
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29 |
A |
7 |
6 |
1. |
17 |
0.353 |
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30 |
A |
5 |
5 |
1. |
15 |
0.333 |
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31 |
A |
4 |
4 |
1. |
15 |
0.267 |
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32 |
A |
2 |
2 |
1. |
17 |
0.118 |
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33 |
A |
5 |
5 |
1. |
13 |
0.385 |
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34 |
A |
3 |
3 |
1. |
15 |
0.2 |
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35 |
A |
2 |
2 |
1. |
13 |
0.154 |
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36 |
A |
5 |
5 |
1. |
13 |
0.385 |
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37 |
A |
2 |
2 |
1. |
11 |
0.182 |
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38 |
A |
2 |
2 |
1. |
24 |
0.083 |
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39 |
A |
2 |
2 |
1. |
24 |
0.083 |
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40 |
A |
4 |
5 |
1. |
10 |
0.5 |
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41 |
A |
6 |
7 |
1. |
12 |
0.583 |
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42 |
B |
5 |
5 |
2.11 |
13 |
0.385 |
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43 |
A |
4 |
4 |
1. |
13 |
0.308 |
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44 |
A |
4 |
4 |
1.28 |
15 |
0.267 |
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45 |
F |
0 |
0 |
N/A |
0 |
N/A |
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46 |
A |
13 |
10 |
1. |
12 |
0.833 |
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47 |
A |
12 |
8 |
1. |
12 |
0.667 |
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48 |
A |
6 |
6 |
1.19 |
18 |
0.333 |
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49 |
A |
4 |
4 |
1. |
14 |
0.286 |
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50 |
A |
6 |
6 |
1. |
14 |
0.429 |
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