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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 17, 18, 19, 20, 22, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37,
41, 42, 43, 44, 45, 46, 47, 48, 49 }
B grade: { 13, 21, 32, 38, 39, 40 }
C grade: { 16, 23, 24, 25, 26 }
F grade: { }
2.1.3 Maple
A grade: { 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 20, 21, 27, 28, 29, 30, 32, 33, 34, 35, 38, 40, 49
}
B grade: { 10, 11, 17, 18, 19, 22, 23, 24, 25, 26 }
C grade: { }
F grade: { 1, 31, 36, 37, 39, 41, 42, 43, 44, 45, 46, 47, 48 }
2.1.4 Maxima
A grade: { 2, 3, 4, 5, 7, 11, 13, 21, 38, 39
B grade: { 8 }
C grade: { }
F grade: { 1, 6, 9, 10, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49 }
2.1.5 FriCAS
A grade: { 2, 3, 4, 5, 7, 8, 9, 10, 11, 14, 15, 17, 18, 19, 20, 24, 25, 26, 45, 46, 47, 48 }
B grade: { 13, 21, 23, 38, 39 }
C grade: { }
F grade: { 1, 6, 12, 16, 22, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 41, 42, 43, 44, 49 }
2.1.6 Sympy
A grade: { 9, 10, 11 }
B grade: { }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 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 }
2.1.7 Giac
A grade: { 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 45 }
B grade: { 23, 24, 25, 26 }
C grade: { }
F grade: { 1, 6, 12, 16, 17, 18, 19, 20, 21, 22, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 46, 47, 48, 49 }
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 |
0.09 |
0.039 |
0.214 |
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 |
0.027 |
0.123 |
0.961 |
3.278 |
0. |
1.115 |
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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 |
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 |
0.019 |
0.021 |
0.118 |
0.992 |
3.072 |
0. |
1.131 |
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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 | 27 | 31 | 51 | 69 | 0 | 36 |
normalized size | 1 | 1. | 0.75 | 0.86 | 1.42 | 1.92 | 0. | 1. |
time (sec) |
N/A |
0.012 |
0.021 |
0.114 |
1.011 |
3.208 |
0. |
1.111 |
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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 | 16 | 16 | 16 | 24 | 27 | 45 | 0 | 20 |
normalized size | 1 | 1. | 1. | 1.5 | 1.69 | 2.81 | 0. | 1.25 |
time (sec) |
N/A |
0.004 |
0.004 |
0.116 |
1.02 |
3.083 |
0. |
1.116 |
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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 | 105 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.96 | 1.88 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.083 |
0.031 |
0.168 |
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 | 46 | 69 | 63 | 0 | 43 |
normalized size | 1 | 1. | 0.84 | 1.21 | 1.82 | 1.66 | 0. | 1.13 |
time (sec) |
N/A |
0.015 |
0.024 |
0.114 |
1.508 |
2.768 |
0. |
1.099 |
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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 |
0.02 |
0.047 |
0.118 |
1.515 |
3.328 |
0. |
1.137 |
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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) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 42 | 66 | 0 | 88 | 51 | 92 |
normalized size |
1 |
1. |
0.75 |
1.18 |
0. |
1.57 |
0.91 |
1.64 |
time (sec) |
N/A |
0.045 |
0.035 |
0.182 |
0. |
3.209 |
0.794 |
1.095 |
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Problem 10 |
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 | 85 | 41 | 65 |
normalized size |
1 |
1. |
0.94 |
1.98 |
0. |
1.81 |
0.87 |
1.38 |
time (sec) |
N/A |
0.02 |
0.025 |
0.188 |
0. |
3.127 |
0.301 |
1.111 |
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Problem 11 |
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 | 25 | 25 | 25 | 52 | 31 | 55 | 22 | 35 |
normalized size | 1 | 1. | 1. | 2.08 | 1.24 | 2.2 | 0.88 | 1.4 |
time (sec) |
N/A |
0.01 |
0.012 |
0.182 |
1.01 |
2.974 |
0.222 |
1.082 |
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Problem 12 |
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 | 54 | 125 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.92 |
2.12 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.064 |
0.033 |
0.238 |
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 |
B |
A |
A |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 32 | 32 | 93 | 42 | 70 | 140 | 0 | 82 |
normalized size | 1 | 1. | 2.91 | 1.31 | 2.19 | 4.38 | 0. | 2.56 |
time (sec) |
N/A |
0.031 |
0.143 |
0.182 |
0.976 |
3.37 |
0. |
1.126 |
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Problem 14 |
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 | 35 | 54 | 0 | 84 | 0 | 82 |
normalized size |
1 |
1. |
0.92 |
1.42 |
0. |
2.21 |
0. |
2.16 |
time (sec) |
N/A |
0.022 |
0.027 |
0.186 |
0. |
3.205 |
0. |
1.115 |
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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 | 60 | 60 | 69 | 98 | 0 | 198 | 0 | 108 |
normalized size |
1 |
1. |
1.15 |
1.63 |
0. |
3.3 |
0. |
1.8 |
time (sec) |
N/A |
0.042 |
0.052 |
0.181 |
0. |
3.684 |
0. |
1.113 |
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Problem 16 |
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 | 63 | 166 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.91 |
2.41 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.094 |
0.089 |
0.278 |
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 |
A |
B |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 197 | 197 | 174 | 507 | 0 | 374 | 0 | 0 |
normalized size | 1 | 1. | 0.88 | 2.57 | 0. | 1.9 | 0. | 0. |
time (sec) |
N/A |
0.232 |
0.214 |
0.233 |
0. |
3.556 |
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 | 155 | 155 | 149 | 360 | 0 | 316 | 0 | 0 |
normalized size | 1 | 1. | 0.96 | 2.32 | 0. | 2.04 | 0. | 0. |
time (sec) |
N/A |
0.137 |
0.274 |
0.224 |
0. |
3.57 |
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 | 116 | 116 | 129 | 272 | 0 | 284 | 0 | 0 |
normalized size | 1 | 1. | 1.11 | 2.34 | 0. | 2.45 | 0. | 0. |
time (sec) |
N/A |
0.089 |
0.179 |
0.227 |
0. |
3.707 |
0. |
0. |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 110 | 127 | 0 | 254 | 0 | 0 |
normalized size | 1 | 1. | 1.39 | 1.61 | 0. | 3.22 | 0. | 0. |
time (sec) |
N/A |
0.053 |
0.133 |
0.222 |
0. |
3.128 |
0. |
0. |
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Problem 21 |
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 | 36 | 36 | 114 | 50 | 74 | 184 | 0 | 0 |
normalized size | 1 | 1. | 3.17 | 1.39 | 2.06 | 5.11 | 0. | 0. |
time (sec) |
N/A |
0.024 |
0.113 |
0.236 |
0.997 |
3.063 |
0. |
0. |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 210 | 210 | 375 | 607 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.79 | 2.89 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.3 |
0.412 |
0.439 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
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 | 69 | 69 | 115 | 154 | 0 | 663 | 0 | 174 |
normalized size |
1 |
1. |
1.67 |
2.23 |
0. |
9.61 |
0. |
2.52 |
time (sec) |
N/A |
0.096 |
0.39 |
0.237 |
0. |
3.199 |
0. |
1.539 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
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 | 123 | 123 | 199 | 453 | 0 | 998 | 0 | 374 |
normalized size | 1 | 1. | 1.62 | 3.68 | 0. | 8.11 | 0. | 3.04 |
time (sec) |
N/A |
0.194 |
0.741 |
0.263 |
0. |
3.462 |
0. |
1.542 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 180 | 180 | 241 | 759 | 0 | 1242 | 0 | 792 |
normalized size | 1 | 1. | 1.34 | 4.22 | 0. | 6.9 | 0. | 4.4 |
time (sec) |
N/A |
0.297 |
0.491 |
0.24 |
0. |
3.809 |
0. |
1.728 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 239 | 239 | 307 | 1172 | 0 | 1551 | 0 | 1508 |
normalized size | 1 | 1. | 1.28 | 4.9 | 0. | 6.49 | 0. | 6.31 |
time (sec) |
N/A |
0.457 |
0.516 |
0.24 |
0. |
4.241 |
0. |
2.02 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 366 | 366 | 420 | 769 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.15 | 2.1 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.305 |
5.168 |
0.81 |
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 | 272 | 272 | 314 | 545 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.15 | 2. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.231 |
4.661 |
0.588 |
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 |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 145 | 145 | 213 | 341 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.47 | 2.35 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.134 |
0.79 |
0.496 |
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 | 86 | 86 | 99 | 167 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.15 | 1.94 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.064 |
0.129 |
0.299 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
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 | 324 | 324 | 408 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.26 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.493 |
0.321 |
0.804 |
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 | 254 | 254 | 802 | 307 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
3.16 |
1.21 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.416 |
3.098 |
0.493 |
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 |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 464 | 464 | 656 | 809 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.41 | 1.74 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.402 |
9.302 |
0.578 |
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 |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 264 | 264 | 314 | 481 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.19 | 1.82 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.256 |
0.806 |
0.425 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
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 | 140 | 140 | 162 | 247 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.16 | 1.76 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.102 |
0.119 |
0.306 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
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 | 448 | 448 | 554 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.24 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.542 |
0.344 |
0.875 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 378 | 378 | 289 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.76 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.617 |
0.532 |
1.043 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
B |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 127 | 65 | 85 | 205 | 0 | 0 |
normalized size |
1 |
1. |
2.65 |
1.35 |
1.77 |
4.27 |
0. |
0. |
time (sec) |
N/A |
0.063 |
0.269 |
0.27 |
0.987 |
3.003 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
A |
B |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 130 | 0 | 89 | 216 | 0 | 0 |
normalized size |
1 |
1. |
2.71 |
0. |
1.85 |
4.5 |
0. |
0. |
time (sec) |
N/A |
0.07 |
0.307 |
0.279 |
0.988 |
3.279 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
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 | 199 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
3.29 |
2.34 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.086 |
0.751 |
0.318 |
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 |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 95 | 95 | 79 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.83 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.121 |
0.451 |
0.198 |
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 |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 87 | 101 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.16 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.106 |
0.268 |
0.188 |
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 | 87 | 87 | 54 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.62 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.094 |
0.104 |
0.18 |
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 | 43 | 43 | 75 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.74 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.059 |
0.054 |
0.191 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
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 | 33 | 0 | 0 | 65 | 0 | 53 |
normalized size | 1 | 1. | 0.85 | 0. | 0. | 1.67 | 0. | 1.36 |
time (sec) |
N/A |
0.03 |
0.04 |
0.183 |
0. |
3.283 |
0. |
1.14 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
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 |
0.043 |
0.046 |
0.18 |
0. |
3.588 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
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 |
0.067 |
0.132 |
0.181 |
0. |
3.385 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 81 | 81 | 50 | 0 | 0 | 123 | 0 | 0 |
normalized size | 1 | 1. | 0.62 | 0. | 0. | 1.52 | 0. | 0. |
time (sec) |
N/A |
0.068 |
0.129 |
0.179 |
0. |
3.537 |
0. |
0. |
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Problem 49 |
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 | 158 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.86 |
2.29 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.095 |
0.056 |
0.332 |
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 |
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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 |
5 |
4 |
1. |
10 |
0.4 |
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10 |
A |
4 |
4 |
1. |
8 |
0.5 |
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11 |
A |
3 |
3 |
1. |
6 |
0.5 |
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12 |
A |
6 |
6 |
1. |
10 |
0.6 |
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13 |
A |
5 |
5 |
1. |
10 |
0.5 |
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14 |
A |
3 |
3 |
1. |
10 |
0.3 |
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15 |
A |
6 |
6 |
1. |
10 |
0.6 |
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16 |
A |
7 |
6 |
1. |
10 |
0.6 |
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17 |
A |
9 |
8 |
1. |
10 |
0.8 |
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18 |
A |
8 |
7 |
1. |
10 |
0.7 |
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19 |
A |
7 |
6 |
1. |
10 |
0.6 |
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20 |
A |
6 |
6 |
1. |
8 |
0.75 |
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21 |
A |
5 |
5 |
1. |
6 |
0.833 |
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22 |
A |
14 |
8 |
1. |
10 |
0.8 |
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23 |
A |
6 |
6 |
1. |
10 |
0.6 |
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24 |
A |
8 |
8 |
1. |
10 |
0.8 |
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25 |
A |
9 |
9 |
1. |
10 |
0.9 |
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26 |
A |
10 |
9 |
1. |
10 |
0.9 |
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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 |
6 |
6 |
1. |
12 |
0.5 |
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39 |
A |
6 |
6 |
1. |
14 |
0.429 |
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40 |
A |
7 |
7 |
1. |
10 |
0.7 |
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41 |
A |
6 |
4 |
1. |
10 |
0.4 |
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42 |
A |
6 |
4 |
1. |
8 |
0.5 |
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43 |
A |
5 |
3 |
1. |
6 |
0.5 |
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44 |
A |
6 |
5 |
1. |
10 |
0.5 |
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45 |
A |
3 |
3 |
1. |
10 |
0.3 |
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46 |
A |
5 |
4 |
1. |
10 |
0.4 |
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47 |
A |
6 |
4 |
1. |
10 |
0.4 |
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48 |
A |
6 |
4 |
1. |
10 |
0.4 |
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49 |
A |
8 |
8 |
1. |
19 |
0.421 |
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