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
}
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 16, 17, 18, 19, 22, 39, 48, 49, 50, 51 }
B grade: { 8, 9, 13, 32, 33, 43, 44, 45, 47 }
C grade: { 14, 15, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 40, 41, 42, 46
}
F grade: { }
2.1.3 Maple
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 29, 30, 40, 41
}
B grade: { 15, 22, 25, 26, 27, 28, 32, 33, 34, 35, 36, 37, 38, 39, 43, 44, 45, 47, 49, 50 }
C grade: { 31, 42, 46, 48 }
F grade: { 51 }
2.1.4 Maxima
A grade: { 1, 2, 3, 4, 5, 6, 7, 10, 11, 20, 21
B grade: { 8, 9, 16, 17 }
C grade: { 22 }
F grade: { 12, 13, 14, 15, 18, 19, 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 }
2.1.5 FriCAS
A grade: { 1, 2, 3, 4, 5, 6, 12, 15, 16, 17, 18, 19, 22, 27 }
B grade: { 7, 8, 9, 10, 11, 13, 14, 23, 24, 25, 26, 28, 35, 36, 37, 38 }
C grade: { }
F grade: { 20, 21, 29, 30, 31, 32, 33, 34, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51 }
2.1.6 Sympy
A grade: { 1, 2, 3, 4 }
B grade: { }
C grade: { }
F grade: { 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 }
2.1.7 Giac
A grade: { 1, 2, 3, 4, 5, 9, 10, 39 }
B grade: { 6, 7, 8, 11, 13 }
C grade: { }
F grade: { 12, 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,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51 }
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 | 121 | 121 | 77 | 147 | 198 | 225 | 415 | 161 |
normalized size | 1 | 1. | 0.64 | 1.21 | 1.64 | 1.86 | 3.43 | 1.33 |
time (sec) |
N/A |
0.172 |
0.124 |
0.026 |
0.976 |
1.948 |
4.782 |
1.275 |
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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 | 96 | 96 | 57 | 126 | 166 | 189 | 301 | 109 |
normalized size | 1 | 1. | 0.59 | 1.31 | 1.73 | 1.97 | 3.14 | 1.14 |
time (sec) |
N/A |
0.143 |
0.086 |
0.026 |
0.978 |
1.905 |
2.498 |
1.275 |
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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 |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 47 | 106 | 135 | 150 | 245 | 84 |
normalized size | 1 | 1. | 0.61 | 1.38 | 1.75 | 1.95 | 3.18 | 1.09 |
time (sec) |
N/A |
0.102 |
0.111 |
0.021 |
0.989 |
1.992 |
1.278 |
1.168 |
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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 |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 52 | 52 | 43 | 78 | 104 | 112 | 133 | 84 |
normalized size | 1 | 1. | 0.83 | 1.5 | 2. | 2.15 | 2.56 | 1.62 |
time (sec) |
N/A |
0.06 |
0.364 |
0.022 |
0.985 |
1.942 |
0.618 |
1.321 |
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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 | 63 | 63 | 61 | 78 | 99 | 201 | 0 | 150 |
normalized size | 1 | 1. | 0.97 | 1.24 | 1.57 | 3.19 | 0. | 2.38 |
time (sec) |
N/A |
0.09 |
0.084 |
0.039 |
0.974 |
2.136 |
0. |
1.164 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 97 | 72 | 93 | 263 | 0 | 185 |
normalized size |
1 |
1. |
1.83 |
1.36 |
1.75 |
4.96 |
0. |
3.49 |
time (sec) |
N/A |
0.12 |
0.04 |
0.038 |
0.969 |
2.06 |
0. |
1.296 |
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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 |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 64 | 64 | 95 | 80 | 142 | 343 | 0 | 166 |
normalized size |
1 |
1. |
1.48 |
1.25 |
2.22 |
5.36 |
0. |
2.59 |
time (sec) |
N/A |
0.111 |
0.711 |
0.049 |
0.969 |
2.063 |
0. |
1.324 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
B |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 172 | 86 | 162 | 348 | 0 | 198 |
normalized size |
1 |
1. |
2.82 |
1.41 |
2.66 |
5.7 |
0. |
3.25 |
time (sec) |
N/A |
0.162 |
0.074 |
0.048 |
0.973 |
1.943 |
0. |
1.315 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
B |
B |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 86 | 86 | 179 | 109 | 223 | 425 | 0 | 200 |
normalized size |
1 |
1. |
2.08 |
1.27 |
2.59 |
4.94 |
0. |
2.33 |
time (sec) |
N/A |
0.15 |
0.062 |
0.046 |
0.969 |
1.985 |
0. |
1.307 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
B |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 105 | 105 | 204 | 132 | 252 | 520 | 0 | 248 |
normalized size |
1 |
1. |
1.94 |
1.26 |
2.4 |
4.95 |
0. |
2.36 |
time (sec) |
N/A |
0.162 |
0.056 |
0.149 |
0.966 |
2.085 |
0. |
1.254 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 130 | 130 | 204 | 155 | 313 | 602 | 0 | 346 |
normalized size |
1 |
1. |
1.57 |
1.19 |
2.41 |
4.63 |
0. |
2.66 |
time (sec) |
N/A |
0.192 |
0.056 |
0.151 |
0.978 |
2.036 |
0. |
1.321 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 128 | 165 | 101 | 78 | 0 | 405 | 0 | 0 |
normalized size |
1 |
1.29 |
0.79 |
0.61 |
0. |
3.16 |
0. |
0. |
time (sec) |
N/A |
0.347 |
0.845 |
0.753 |
0. |
1.986 |
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 |
F |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 157 | 79 | 0 | 539 | 0 | 575 |
normalized size |
1 |
1. |
2.28 |
1.14 |
0. |
7.81 |
0. |
8.33 |
time (sec) |
N/A |
0.306 |
0.358 |
0.82 |
0. |
2.101 |
0. |
1.434 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
F |
B |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 120 | 120 | 234 | 124 | 0 | 892 | 0 | 0 |
normalized size |
1 |
1. |
1.95 |
1.03 |
0. |
7.43 |
0. |
0. |
time (sec) |
N/A |
0.443 |
0.466 |
1.125 |
0. |
2.243 |
0. |
0. |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 103 | 103 | 194 | 914 | 0 | 1211 | 0 | 0 |
normalized size | 1 | 1. | 1.88 | 8.87 | 0. | 11.76 | 0. | 0. |
time (sec) |
N/A |
0.468 |
0.945 |
0.444 |
0. |
4.17 |
0. |
0. |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 40 | 45 | 417 | 95 | 0 | 0 |
normalized size | 1 | 1. | 0.93 | 1.05 | 9.7 | 2.21 | 0. | 0. |
time (sec) |
N/A |
0.203 |
0.22 |
0.301 |
1.574 |
2. |
0. |
0. |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 114 | 114 | 133 | 187 | 365 | 1006 | 0 | 0 |
normalized size | 1 | 1. | 1.17 | 1.64 | 3.2 | 8.82 | 0. | 0. |
time (sec) |
N/A |
0.49 |
0.326 |
0.336 |
1.58 |
3.319 |
0. |
0. |
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Problem 18 |
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 | 118 | 118 | 132 | 117 | 0 | 1017 | 0 | 0 |
normalized size | 1 | 1. | 1.12 | 0.99 | 0. | 8.62 | 0. | 0. |
time (sec) |
N/A |
0.496 |
0.288 |
0.331 |
0. |
3.732 |
0. |
0. |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
A |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 62 | 73 | 0 | 539 | 0 | 0 |
normalized size |
1 |
1. |
1.35 |
1.59 |
0. |
11.72 |
0. |
0. |
time (sec) |
N/A |
0.171 |
0.109 |
0.339 |
0. |
3.692 |
0. |
0. |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 102 | 102 | 144 | 111 | 80 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.41 | 1.09 | 0.78 | 0. | 0. | 0. |
time (sec) |
N/A |
0.46 |
1.312 |
0.276 |
1.531 |
0. |
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 |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 100 | 100 | 145 | 113 | 80 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.45 | 1.13 | 0.8 | 0. | 0. | 0. |
time (sec) |
N/A |
0.447 |
1.32 |
0.277 |
1.532 |
0. |
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 |
C |
A |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 63 | 111 | 257 | 475 | 0 | 0 |
normalized size |
1 |
1. |
1.37 |
2.41 |
5.59 |
10.33 |
0. |
0. |
time (sec) |
N/A |
0.194 |
0.193 |
0.328 |
1.978 |
2.773 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
B |
F(-1) |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 105 | 105 | 746 | 120 | 0 | 1953 | 0 | 0 |
normalized size |
1 |
1. |
7.1 |
1.14 |
0. |
18.6 |
0. |
0. |
time (sec) |
N/A |
0.289 |
5.299 |
1.32 |
0. |
5.943 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
B |
F(-1) |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 165 | 165 | 331 | 208 | 0 | 2661 | 0 | 0 |
normalized size |
1 |
1. |
2.01 |
1.26 |
0. |
16.13 |
0. |
0. |
time (sec) |
N/A |
0.465 |
2.154 |
1.433 |
0. |
19.975 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 149 | 149 | 661 | 1025 | 0 | 8049 | 0 | 0 |
normalized size | 1 | 1. | 4.44 | 6.88 | 0. | 54.02 | 0. | 0. |
time (sec) |
N/A |
0.508 |
55.384 |
0.522 |
0. |
15.711 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 83 | 83 | 436 | 526 | 0 | 3043 | 0 | 0 |
normalized size | 1 | 1. | 5.25 | 6.34 | 0. | 36.66 | 0. | 0. |
time (sec) |
N/A |
0.228 |
54.394 |
0.316 |
0. |
8.676 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
A |
F |
F(-2) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 166 | 166 | 61316 | 614 | 0 | 7360 | 0 | 0 |
normalized size |
1 |
1. |
369.37 |
3.7 |
0. |
44.34 |
0. |
0. |
time (sec) |
N/A |
0.515 |
39.443 |
0.329 |
0. |
14.272 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 168 | 168 | 99997 | 621 | 0 | 7646 | 0 | 0 |
normalized size | 1 | 1. | 595.22 | 3.7 | 0. | 45.51 | 0. | 0. |
time (sec) |
N/A |
0.536 |
40.236 |
0.379 |
0. |
18.083 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F(-2) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 238 | 238 | 611 | 593 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
2.57 |
2.49 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.503 |
6.546 |
5.068 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 246 | 246 | 625 | 587 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 2.54 | 2.39 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.484 |
6.567 |
4.468 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 267 | 267 | 13199 | 22961 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
49.43 |
86. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.5 |
34.367 |
0.803 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 116 | 116 | 4679 | 6804 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 40.34 | 58.66 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.204 |
40.042 |
0.395 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 252 | 252 | 5708 | 6817 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 22.65 | 27.05 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.511 |
33.585 |
0.442 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 256 | 256 | 1659 | 9043 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 6.48 | 35.32 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.52 |
9.939 |
0.431 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 123 | 123 | 567 | 4568 | 0 | 8421 | 0 | 0 |
normalized size | 1 | 1. | 4.61 | 37.14 | 0. | 68.46 | 0. | 0. |
time (sec) |
N/A |
0.459 |
3.108 |
0.46 |
0. |
8.72 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
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 | 61 | 61 | 367 | 238 | 0 | 2439 | 0 | 0 |
normalized size | 1 | 1. | 6.02 | 3.9 | 0. | 39.98 | 0. | 0. |
time (sec) |
N/A |
0.188 |
1.824 |
0.371 |
0. |
3.866 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 140 | 140 | 472502 | 347 | 0 | 6734 | 0 | 0 |
normalized size | 1 | 1. | 3375.01 | 2.48 | 0. | 48.1 | 0. | 0. |
time (sec) |
N/A |
0.502 |
35.077 |
0.315 |
0. |
5.667 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 140 | 140 | 309693 | 364 | 0 | 7247 | 0 | 0 |
normalized size | 1 | 1. | 2212.09 | 2.6 | 0. | 51.76 | 0. | 0. |
time (sec) |
N/A |
0.471 |
34.404 |
0.296 |
0. |
6.584 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F(-1) |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 181 | 181 | 178 | 606 | 0 | 0 | 0 | 404 |
normalized size |
1 |
1. |
0.98 |
3.35 |
0. |
0. |
0. |
2.23 |
time (sec) |
N/A |
0.509 |
1.098 |
0.159 |
0. |
0. |
0. |
1.356 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-1) |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 154 | 154 | 179 | 190 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.16 |
1.23 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.487 |
3.733 |
1.569 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-1) |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 146 | 146 | 203 | 254 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.39 |
1.74 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.488 |
4.016 |
1.517 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 254 | 254 | 23019 | 6200 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
90.63 |
24.41 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.52 |
29.63 |
0.349 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 250 | 250 | 8202 | 3290 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 32.81 | 13.16 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.526 |
29.362 |
0.309 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 114 | 114 | 3427 | 2924 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 30.06 | 25.65 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.207 |
29.93 |
0.406 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 246 | 246 | 4935 | 3691 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
20.06 |
15. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.524 |
29.98 |
0.382 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 254 | 254 | 23019 | 6211 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
90.63 |
24.45 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.505 |
30.285 |
0.457 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 114 | 114 | 3429 | 2932 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
30.08 |
25.72 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.206 |
28.856 |
0.426 |
0. |
0. |
0. |
0. |
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Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 391 | 391 | 274 | 269228 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.7 |
688.56 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.535 |
0.287 |
3.937 |
0. |
0. |
0. |
0. |
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Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 198 | 198 | 195 | 32723 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.98 | 165.27 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.196 |
0.151 |
0.809 |
0. |
0. |
0. |
0. |
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Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 398 | 398 | 374 | 26864 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.94 | 67.5 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.555 |
2.34 |
0.73 |
0. |
0. |
0. |
0. |
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Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 168 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.07 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.278 |
1.059 |
4.027 |
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 [29] had the largest ratio of [ 0.2727
]
|
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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 |
13 |
4 |
1. |
32 |
0.125 |
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|
|
|
|
2 |
A |
11 |
4 |
1. |
32 |
0.125 |
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|
|
|
|
|
3 |
A |
10 |
5 |
1. |
30 |
0.167 |
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|
|
|
|
|
|
4 |
A |
4 |
4 |
1. |
24 |
0.167 |
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|
|
|
|
|
|
5 |
A |
6 |
5 |
1. |
30 |
0.167 |
|
|
|
|
|
|
|
6 |
A |
8 |
7 |
1. |
32 |
0.219 |
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|
|
|
|
|
|
7 |
A |
7 |
5 |
1. |
32 |
0.156 |
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|
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|
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8 |
A |
6 |
6 |
1. |
32 |
0.188 |
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9 |
A |
11 |
5 |
1. |
32 |
0.156 |
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10 |
A |
11 |
4 |
1. |
32 |
0.125 |
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11 |
A |
13 |
4 |
1. |
32 |
0.125 |
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|
12 |
A |
5 |
5 |
1.29 |
34 |
0.147 |
|
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|
13 |
A |
5 |
5 |
1. |
34 |
0.147 |
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14 |
A |
8 |
7 |
1. |
34 |
0.206 |
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15 |
A |
6 |
6 |
1. |
40 |
0.15 |
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16 |
A |
3 |
3 |
1. |
40 |
0.075 |
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17 |
A |
6 |
6 |
1. |
40 |
0.15 |
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18 |
A |
6 |
6 |
1. |
40 |
0.15 |
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19 |
A |
2 |
2 |
1. |
36 |
0.056 |
|
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20 |
A |
6 |
6 |
1. |
36 |
0.167 |
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21 |
A |
6 |
6 |
1. |
36 |
0.167 |
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22 |
A |
3 |
3 |
1. |
36 |
0.083 |
|
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23 |
A |
5 |
4 |
1. |
33 |
0.121 |
|
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24 |
A |
8 |
6 |
1. |
33 |
0.182 |
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25 |
A |
5 |
4 |
1. |
39 |
0.103 |
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26 |
A |
2 |
2 |
1. |
39 |
0.051 |
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|
27 |
A |
5 |
4 |
1. |
39 |
0.103 |
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28 |
A |
5 |
4 |
1. |
39 |
0.103 |
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29 |
A |
9 |
9 |
1. |
33 |
0.273 |
|
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30 |
A |
9 |
9 |
1. |
33 |
0.273 |
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31 |
A |
3 |
3 |
1. |
39 |
0.077 |
|
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|
32 |
A |
1 |
1 |
1. |
39 |
0.026 |
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|
33 |
A |
3 |
3 |
1. |
39 |
0.077 |
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34 |
A |
3 |
3 |
1. |
39 |
0.077 |
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|
35 |
A |
5 |
5 |
1. |
35 |
0.143 |
|
|
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|
|
36 |
A |
2 |
2 |
1. |
35 |
0.057 |
|
|
|
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|
37 |
A |
5 |
5 |
1. |
35 |
0.143 |
|
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|
38 |
A |
5 |
5 |
1. |
35 |
0.143 |
|
|
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|
39 |
A |
8 |
5 |
1. |
33 |
0.152 |
|
|
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|
40 |
A |
5 |
3 |
1. |
33 |
0.091 |
|
|
|
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|
41 |
A |
5 |
3 |
1. |
33 |
0.091 |
|
|
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|
|
42 |
A |
3 |
3 |
1. |
39 |
0.077 |
|
|
|
|
|
|
|
43 |
A |
3 |
3 |
1. |
39 |
0.077 |
|
|
|
|
|
|
|
44 |
A |
1 |
1 |
1. |
39 |
0.026 |
|
|
|
|
|
|
|
45 |
A |
3 |
3 |
1. |
39 |
0.077 |
|
|
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|
46 |
A |
3 |
3 |
1. |
39 |
0.077 |
|
|
|
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|
47 |
A |
1 |
1 |
1. |
39 |
0.026 |
|
|
|
|
|
|
|
48 |
A |
3 |
3 |
1. |
35 |
0.086 |
|
|
|
|
|
|
|
49 |
A |
1 |
1 |
1. |
35 |
0.029 |
|
|
|
|
|
|
|
50 |
A |
3 |
3 |
1. |
35 |
0.086 |
|
|
|
|
|
|
|
51 |
A |
4 |
4 |
1. |
38 |
0.105 |
|
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