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
}
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 3, 5, 7, 9, 11, 13, 20, 23, 24, 25, 26, 27, 28, 30, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 49
}
B grade: { 48 }
C grade: { 2, 4, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19, 21, 22, 29, 31, 32, 39, 46, 47, 50, 51, 52 }
F grade: { }
2.1.3 Maple
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 45 }
B grade: { }
C grade: { 43, 44 }
F grade: { 23, 24, 37, 38, 39, 40, 41, 42, 46, 47, 48, 49, 50, 51, 52 }
2.1.4 Maxima
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 19, 20, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36
B grade: { }
C grade: { }
F grade: { 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,
49, 50, 51, 52 }
2.1.5 FriCAS
A grade: { 1, 3, 25, 33, 43, 44, 45 }
B grade: { 2, 4, 5, 6, 7, 8, 19, 20, 26, 27, 28, 34, 35, 36 }
C grade: { }
F grade: { 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24, 29, 30, 31, 32, 37, 38, 39, 40, 41, 42, 46, 47,
48, 49, 50, 51, 52 }
2.1.6 Sympy
A grade: { 1, 2, 3, 4, 5, 6, 7, 8 }
B grade: { }
C grade: { }
F grade: { 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52 }
2.1.7 Giac
A grade: { 1, 25, 26, 27, 33, 34 }
B grade: { 2, 3, 4, 5, 6, 7, 8, 28 }
C grade: { }
F grade: { 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52 }
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 |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 11 | 11 | 19 | 17 | 15 | 54 | 29 | 16 |
normalized size | 1 | 1. | 1.73 | 1.55 | 1.36 | 4.91 | 2.64 | 1.45 |
time (sec) |
N/A |
0.005 |
0.015 |
0.013 |
1.03 |
1.914 |
0.19 |
1.096 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
A |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 15 | 15 | 29 | 26 | 24 | 93 | 17 | 47 |
normalized size | 1 | 1. | 1.93 | 1.73 | 1.6 | 6.2 | 1.13 | 3.13 |
time (sec) |
N/A |
0.009 |
0.015 |
0.015 |
1.488 |
1.734 |
0.138 |
1.114 |
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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 |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 28 | 34 | 31 | 31 | 126 | 53 | 159 |
normalized size | 1 | 1. | 1.21 | 1.11 | 1.11 | 4.5 | 1.89 | 5.68 |
time (sec) |
N/A |
0.013 |
0.092 |
0.016 |
0.984 |
1.814 |
0.523 |
1.129 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
A |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 33 | 32 | 46 | 198 | 27 | 84 |
normalized size | 1 | 1. | 1.22 | 1.19 | 1.7 | 7.33 | 1. | 3.11 |
time (sec) |
N/A |
0.017 |
0.013 |
0.017 |
1.458 |
1.232 |
0.227 |
1.127 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 42 | 42 | 46 | 44 | 51 | 217 | 66 | 221 |
normalized size | 1 | 1. | 1.1 | 1.05 | 1.21 | 5.17 | 1.57 | 5.26 |
time (sec) |
N/A |
0.028 |
0.107 |
0.012 |
1.005 |
1.39 |
0.924 |
1.158 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
A |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 45 | 45 | 33 | 46 | 59 | 305 | 39 | 123 |
normalized size | 1 | 1. | 0.73 | 1.02 | 1.31 | 6.78 | 0.87 | 2.73 |
time (sec) |
N/A |
0.024 |
0.02 |
0.014 |
1.528 |
1.356 |
0.493 |
1.133 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 56 | 57 | 65 | 319 | 85 | 281 |
normalized size | 1 | 1. | 0.97 | 0.98 | 1.12 | 5.5 | 1.47 | 4.84 |
time (sec) |
N/A |
0.03 |
0.296 |
0.014 |
1.055 |
1.347 |
1.596 |
1.199 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
A |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 57 | 57 | 33 | 52 | 73 | 417 | 51 | 157 |
normalized size | 1 | 1. | 0.58 | 0.91 | 1.28 | 7.32 | 0.89 | 2.75 |
time (sec) |
N/A |
0.034 |
0.009 |
0.013 |
1.605 |
1.4 |
0.947 |
1.253 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-2) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 232 | 232 | 175 | 200 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.75 |
0.86 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.192 |
0.508 |
0.057 |
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 |
C |
A |
F(-2) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 212 | 212 | 40 | 182 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.19 |
0.86 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.143 |
0.072 |
0.033 |
0. |
0. |
0. |
0. |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-2) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 210 | 210 | 159 | 176 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.76 |
0.84 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.14 |
0.191 |
0.03 |
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 |
C |
A |
F(-2) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 192 | 192 | 40 | 160 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.21 |
0.83 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.115 |
0.039 |
0.053 |
0. |
0. |
0. |
0. |
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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(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 192 | 192 | 131 | 166 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.68 |
0.86 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.112 |
0.086 |
0.064 |
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 |
C |
A |
F(-2) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 212 | 212 | 38 | 184 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.18 |
0.87 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.14 |
0.063 |
0.034 |
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 |
C |
A |
F(-2) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 214 | 214 | 40 | 184 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.19 |
0.86 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.143 |
0.069 |
0.033 |
0. |
0. |
0. |
0. |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
F(-2) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 234 | 234 | 40 | 202 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.17 |
0.86 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.171 |
0.112 |
0.035 |
0. |
0. |
0. |
0. |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
F(-2) |
F(-2) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 242 | 242 | 38 | 214 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.16 |
0.88 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.47 |
0.03 |
0.092 |
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 |
A |
F(-2) |
F(-2) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 225 | 225 | 40 | 203 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.18 |
0.9 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.38 |
0.047 |
0.065 |
0. |
0. |
0. |
0. |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
A |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 131 | 131 | 40 | 114 | 162 | 560 | 0 | 0 |
normalized size | 1 | 1. | 0.31 | 0.87 | 1.24 | 4.27 | 0. | 0. |
time (sec) |
N/A |
0.102 |
0.04 |
0.028 |
1.635 |
1.67 |
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 |
A |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 131 | 131 | 98 | 114 | 163 | 1639 | 0 | 0 |
normalized size | 1 | 1. | 0.75 | 0.87 | 1.24 | 12.51 | 0. | 0. |
time (sec) |
N/A |
0.096 |
0.151 |
0.024 |
1.573 |
1.771 |
0. |
0. |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
F(-2) |
F(-2) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 225 | 225 | 38 | 209 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.17 |
0.93 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.31 |
0.026 |
0.059 |
0. |
0. |
0. |
0. |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
F(-2) |
F(-2) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 244 | 244 | 38 | 229 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.16 |
0.94 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.425 |
0.057 |
0.062 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
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 | 46 | 46 | 48 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.04 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.028 |
0.044 |
0.39 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
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 | 51 | 51 | 54 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.06 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.032 |
0.068 |
0.378 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
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 | 36 | 36 | 27 | 29 | 41 | 139 | 0 | 42 |
normalized size | 1 | 1. | 0.75 | 0.81 | 1.14 | 3.86 | 0. | 1.17 |
time (sec) |
N/A |
0.018 |
0.019 |
0.055 |
1.545 |
1.621 |
0. |
1.207 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
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 | 16 | 16 | 16 | 22 | 27 | 126 | 0 | 27 |
normalized size | 1 | 1. | 1. | 1.38 | 1.69 | 7.88 | 0. | 1.69 |
time (sec) |
N/A |
0.021 |
0.007 |
0.073 |
1.626 |
1.649 |
0. |
1.244 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
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 | 17 | 17 | 17 | 28 | 16 | 128 | 0 | 26 |
normalized size | 1 | 1. | 1. | 1.65 | 0.94 | 7.53 | 0. | 1.53 |
time (sec) |
N/A |
0.012 |
0.008 |
0.086 |
1.553 |
1.673 |
0. |
1.267 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 39 | 39 | 30 | 36 | 30 | 198 | 0 | 188 |
normalized size | 1 | 1. | 0.77 | 0.92 | 0.77 | 5.08 | 0. | 4.82 |
time (sec) |
N/A |
0.018 |
0.028 |
0.046 |
1.546 |
1.689 |
0. |
1.54 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 200 | 200 | 39 | 189 | 153 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.2 |
0.94 |
0.76 |
0. |
0. |
0. |
time (sec) |
N/A |
0.097 |
0.055 |
0.062 |
1.562 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 176 | 176 | 122 | 165 | 127 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.69 |
0.94 |
0.72 |
0. |
0. |
0. |
time (sec) |
N/A |
0.087 |
0.105 |
0.079 |
1.621 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 176 | 176 | 28 | 164 | 127 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.16 |
0.93 |
0.72 |
0. |
0. |
0. |
time (sec) |
N/A |
0.09 |
0.011 |
0.078 |
1.692 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 212 | 212 | 30 | 185 | 147 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.14 |
0.87 |
0.69 |
0. |
0. |
0. |
time (sec) |
N/A |
0.096 |
0.014 |
0.051 |
1.578 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
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 | 70 | 70 | 39 | 40 | 50 | 289 | 0 | 77 |
normalized size | 1 | 1. | 0.56 | 0.57 | 0.71 | 4.13 | 0. | 1.1 |
time (sec) |
N/A |
0.027 |
0.135 |
0.06 |
1.655 |
2.177 |
0. |
1.232 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
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 | 32 | 32 | 20 | 27 | 22 | 154 | 0 | 28 |
normalized size | 1 | 1. | 0.62 | 0.84 | 0.69 | 4.81 | 0. | 0.88 |
time (sec) |
N/A |
0.015 |
0.015 |
0.078 |
1.644 |
2.159 |
0. |
1.299 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 21 | 26 | 18 | 203 | 0 | 0 |
normalized size |
1 |
1. |
0.68 |
0.84 |
0.58 |
6.55 |
0. |
0. |
time (sec) |
N/A |
0.016 |
0.022 |
0.084 |
1.522 |
2.109 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 42 | 42 | 39 | 367 | 0 | 0 |
normalized size |
1 |
1. |
0.55 |
0.55 |
0.51 |
4.77 |
0. |
0. |
time (sec) |
N/A |
0.026 |
0.105 |
0.047 |
1.661 |
1.623 |
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 | 60 | 60 | 58 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.97 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.04 |
0.052 |
4.602 |
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 | 62 | 62 | 60 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.97 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.045 |
0.047 |
5.854 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 87 | 289 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
3.32 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.097 |
1.722 |
1.175 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 83 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.99 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.101 |
0.511 |
1.02 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
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 | 64 | 64 | 67 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.05 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.037 |
0.088 |
0.507 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
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 | 90 | 90 | 83 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.92 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.153 |
0.445 |
1.068 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F(-2) |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 76 | 76 | 73 | 21900 | 0 | 358 | 0 | 0 |
normalized size |
1 |
1. |
0.96 |
288.16 |
0. |
4.71 |
0. |
0. |
time (sec) |
N/A |
0.071 |
0.234 |
2.195 |
0. |
1.751 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F(-2) |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 45 | 10907 | 0 | 204 | 0 | 0 |
normalized size |
1 |
1. |
0.88 |
213.86 |
0. |
4. |
0. |
0. |
time (sec) |
N/A |
0.053 |
0.127 |
0.757 |
0. |
1.732 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 25 | 26 | 26 | 0 | 97 | 0 | 0 |
normalized size |
1 |
1. |
1.04 |
1.04 |
0. |
3.88 |
0. |
0. |
time (sec) |
N/A |
0.042 |
0.019 |
0.03 |
0. |
1.728 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 509 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 9.98 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.049 |
3.074 |
1.099 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 1099 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
21.55 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.048 |
7.333 |
1.15 |
0. |
0. |
0. |
0. |
|
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|
|
|
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|
|
Problem 48 |
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 | 79 | 79 | 190 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 2.41 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.041 |
6.513 |
0.551 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
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|
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|
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|
|
|
Problem 49 |
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 | 77 | 77 | 69 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.9 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.029 |
0.131 |
0.499 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
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|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 73 | 73 | 264 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 3.62 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.041 |
1.051 |
1. |
0. |
0. |
0. |
0. |
|
|
|
|
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Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 477 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
6.04 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.042 |
2.308 |
1.074 |
0. |
0. |
0. |
0. |
|
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Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 83 | 83 | 306 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 3.69 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.046 |
1.798 |
1.115 |
0. |
0. |
0. |
0. |
|
|
|
|
|
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|
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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 [29] had the largest ratio of [ 1.
]
|
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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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|
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|
|
|
1 |
A |
1 |
1 |
1. |
6 |
0.167 |
|
|
|
|
|
|
|
2 |
A |
2 |
2 |
1. |
8 |
0.25 |
|
|
|
|
|
|
|
3 |
A |
2 |
2 |
1. |
8 |
0.25 |
|
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|
|
|
|
|
4 |
A |
3 |
2 |
1. |
8 |
0.25 |
|
|
|
|
|
|
|
5 |
A |
3 |
2 |
1. |
8 |
0.25 |
|
|
|
|
|
|
|
6 |
A |
4 |
2 |
1. |
8 |
0.25 |
|
|
|
|
|
|
|
7 |
A |
4 |
2 |
1. |
8 |
0.25 |
|
|
|
|
|
|
|
8 |
A |
5 |
2 |
1. |
8 |
0.25 |
|
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|
|
|
9 |
A |
13 |
9 |
1. |
12 |
0.75 |
|
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|
|
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|
|
10 |
A |
12 |
9 |
1. |
12 |
0.75 |
|
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|
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11 |
A |
12 |
9 |
1. |
12 |
0.75 |
|
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|
|
|
|
|
12 |
A |
11 |
8 |
1. |
12 |
0.667 |
|
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|
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|
|
13 |
A |
11 |
8 |
1. |
12 |
0.667 |
|
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|
|
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|
|
14 |
A |
12 |
9 |
1. |
12 |
0.75 |
|
|
|
|
|
|
|
15 |
A |
12 |
9 |
1. |
12 |
0.75 |
|
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|
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|
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16 |
A |
13 |
9 |
1. |
12 |
0.75 |
|
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|
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|
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17 |
A |
13 |
9 |
1. |
12 |
0.75 |
|
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|
|
18 |
A |
12 |
8 |
1. |
12 |
0.667 |
|
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|
|
19 |
A |
9 |
9 |
1. |
12 |
0.75 |
|
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|
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|
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20 |
A |
9 |
9 |
1. |
12 |
0.75 |
|
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|
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|
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21 |
A |
12 |
8 |
1. |
12 |
0.667 |
|
|
|
|
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|
|
22 |
A |
13 |
9 |
1. |
12 |
0.75 |
|
|
|
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23 |
A |
2 |
2 |
1. |
8 |
0.25 |
|
|
|
|
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|
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24 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
25 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
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|
|
26 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
27 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
28 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
29 |
A |
14 |
10 |
1. |
10 |
1. |
|
|
|
|
|
|
|
30 |
A |
13 |
10 |
1. |
10 |
1. |
|
|
|
|
|
|
|
31 |
A |
13 |
10 |
1. |
10 |
1. |
|
|
|
|
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|
|
32 |
A |
14 |
10 |
1. |
10 |
1. |
|
|
|
|
|
|
|
33 |
A |
5 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
34 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
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|
|
35 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
36 |
A |
5 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
37 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
38 |
A |
3 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
39 |
A |
2 |
2 |
1. |
21 |
0.095 |
|
|
|
|
|
|
|
40 |
A |
2 |
2 |
1. |
21 |
0.095 |
|
|
|
|
|
|
|
|
41 |
A |
3 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
42 |
A |
3 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
43 |
A |
3 |
2 |
1. |
19 |
0.105 |
|
|
|
|
|
|
|
44 |
A |
3 |
2 |
1. |
19 |
0.105 |
|
|
|
|
|
|
|
45 |
A |
2 |
2 |
1. |
19 |
0.105 |
|
|
|
|
|
|
|
46 |
A |
2 |
2 |
1. |
19 |
0.105 |
|
|
|
|
|
|
|
47 |
A |
2 |
2 |
1. |
19 |
0.105 |
|
|
|
|
|
|
|
48 |
A |
1 |
1 |
1. |
19 |
0.053 |
|
|
|
|
|
|
|
49 |
A |
1 |
1 |
1. |
17 |
0.059 |
|
|
|
|
|
|
|
50 |
A |
1 |
1 |
1. |
17 |
0.059 |
|
|
|
|
|
|
|
51 |
A |
1 |
1 |
1. |
19 |
0.053 |
|
|
|
|
|
|
|
52 |
A |
1 |
1 |
1. |
21 |
0.048 |
|
|
|
|
|
|
|
|
|
|
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|