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, 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: { }
2.1.3 Maple
A grade: { 4, 50, 51 }
B grade: { 1, 2, 3, 8, 9, 10, 11, 15, 16, 17, 18 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 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 }
2.1.4 Maxima
A grade: {
B grade: { }
C grade: { }
F 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
}
2.1.5 FriCAS
A grade: { 4, 50, 51 }
B grade: { 1, 2, 3, 8, 9, 10, 11, 15, 16, 17, 18 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 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 }
2.1.6 Sympy
A grade: { 1, 2, 3, 4, 8, 9, 10, 11, 16, 17, 18, 50, 51 }
B grade: { }
C grade: { 5, 6, 12, 19, 22, 23, 24, 25, 26, 32, 33, 40 }
F grade: { 7, 13, 14, 15, 20, 21, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49
}
2.1.7 Giac
A grade: { 50, 51 }
B grade: { 1, 2, 3, 4, 8, 9, 10, 11, 15, 16, 17, 18 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 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 }
2.2 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 |
F(-2) |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 189 | 189 | 151 | 1229 | 0 | 2070 | 6156 | 2306 |
normalized size |
1 |
1. |
0.8 |
6.5 |
0. |
10.95 |
32.57 |
12.2 |
time (sec) |
N/A |
0.227 |
0.253 |
0.008 |
0. |
1.647 |
6.376 |
1.232 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-2) |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 144 | 144 | 113 | 711 | 0 | 1224 | 3373 | 1362 |
normalized size |
1 |
1. |
0.78 |
4.94 |
0. |
8.5 |
23.42 |
9.46 |
time (sec) |
N/A |
0.121 |
0.126 |
0.006 |
0. |
1.629 |
3.745 |
1.29 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-2) |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 97 | 97 | 73 | 321 | 0 | 566 | 1515 | 645 |
normalized size |
1 |
1. |
0.75 |
3.31 |
0. |
5.84 |
15.62 |
6.65 |
time (sec) |
N/A |
0.061 |
0.06 |
0.006 |
0. |
1.502 |
1.931 |
1.161 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-2) |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 43 | 111 | 0 | 216 | 459 | 225 |
normalized size |
1 |
1. |
0.72 |
1.85 |
0. |
3.6 |
7.65 |
3.75 |
time (sec) |
N/A |
0.031 |
0.04 |
0.003 |
0. |
1.552 |
0.906 |
1.209 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 118 | 118 | 93 | 0 | 0 | 0 | 428 | 0 |
normalized size | 1 | 1. | 0.79 | 0. | 0. | 0. | 3.63 | 0. |
time (sec) |
N/A |
0.098 |
0.107 |
0.036 |
0. |
0. |
12.676 |
0. |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 171 | 171 | 108 | 0 | 0 | 0 | 2076 | 0 |
normalized size | 1 | 1. | 0.63 | 0. | 0. | 0. | 12.14 | 0. |
time (sec) |
N/A |
0.241 |
0.12 |
0.041 |
0. |
0. |
82.721 |
0. |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 209 | 209 | 133 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.64 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.296 |
0.128 |
0.056 |
0. |
0. |
0. |
0. |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-2) |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 292 | 292 | 247 | 2443 | 0 | 3826 | 12199 | 4432 |
normalized size |
1 |
1. |
0.85 |
8.37 |
0. |
13.1 |
41.78 |
15.18 |
time (sec) |
N/A |
0.287 |
0.444 |
0.008 |
0. |
1.757 |
11.652 |
1.332 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-2) |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 216 | 216 | 178 | 1471 | 0 | 2369 | 7019 | 2714 |
normalized size |
1 |
1. |
0.82 |
6.81 |
0. |
10.97 |
32.5 |
12.56 |
time (sec) |
N/A |
0.247 |
0.278 |
0.007 |
0. |
1.697 |
7.196 |
1.23 |
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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) |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 144 | 144 | 113 | 711 | 0 | 1168 | 3373 | 1362 |
normalized size |
1 |
1. |
0.78 |
4.94 |
0. |
8.11 |
23.42 |
9.46 |
time (sec) |
N/A |
0.134 |
0.155 |
0.006 |
0. |
1.652 |
3.728 |
1.273 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-2) |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 91 | 91 | 67 | 263 | 0 | 495 | 1137 | 513 |
normalized size |
1 |
1. |
0.74 |
2.89 |
0. |
5.44 |
12.49 |
5.64 |
time (sec) |
N/A |
0.069 |
0.05 |
0.006 |
0. |
1.582 |
1.798 |
1.197 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 178 | 178 | 146 | 0 | 0 | 0 | 666 | 0 |
normalized size | 1 | 1. | 0.82 | 0. | 0. | 0. | 3.74 | 0. |
time (sec) |
N/A |
0.189 |
0.2 |
0.049 |
0. |
0. |
32.282 |
0. |
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Problem 13 |
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 | 247 | 247 | 156 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.63 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.442 |
0.174 |
0.057 |
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 |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 292 | 292 | 165 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.57 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.408 |
0.17 |
0.068 |
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 |
A |
B |
F(-2) |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 379 | 379 | 327 | 3953 | 0 | 5917 | 0 | 7066 |
normalized size |
1 |
1. |
0.86 |
10.43 |
0. |
15.61 |
0. |
18.64 |
time (sec) |
N/A |
0.401 |
0.695 |
0.01 |
0. |
1.931 |
0. |
1.369 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-2) |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 284 | 284 | 239 | 2443 | 0 | 3831 | 12199 | 4432 |
normalized size |
1 |
1. |
0.84 |
8.6 |
0. |
13.49 |
42.95 |
15.61 |
time (sec) |
N/A |
0.284 |
0.392 |
0.007 |
0. |
1.841 |
12.308 |
1.318 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-2) |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 189 | 189 | 151 | 1229 | 0 | 1976 | 6156 | 2306 |
normalized size |
1 |
1. |
0.8 |
6.5 |
0. |
10.46 |
32.57 |
12.2 |
time (sec) |
N/A |
0.176 |
0.247 |
0.006 |
0. |
1.664 |
6.834 |
1.725 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-2) |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 121 | 121 | 90 | 475 | 0 | 871 | 2220 | 909 |
normalized size |
1 |
1. |
0.74 |
3.93 |
0. |
7.2 |
18.35 |
7.51 |
time (sec) |
N/A |
0.079 |
0.07 |
0.007 |
0. |
1.611 |
3.215 |
1.189 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 258 | 258 | 217 | 0 | 0 | 0 | 911 | 0 |
normalized size | 1 | 1. | 0.84 | 0. | 0. | 0. | 3.53 | 0. |
time (sec) |
N/A |
0.282 |
0.346 |
0.036 |
0. |
0. |
54.634 |
0. |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 347 | 347 | 209 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.6 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.661 |
0.313 |
0.047 |
0. |
0. |
0. |
0. |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 480 | 480 | 218 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.45 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.073 |
0.301 |
0.055 |
0. |
0. |
0. |
0. |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 363 | 363 | 315 | 0 | 0 | 0 | 1132 | 0 |
normalized size | 1 | 1. | 0.87 | 0. | 0. | 0. | 3.12 | 0. |
time (sec) |
N/A |
0.371 |
0.496 |
0.042 |
0. |
0. |
103.325 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 260 | 260 | 219 | 0 | 0 | 0 | 911 | 0 |
normalized size | 1 | 1. | 0.84 | 0. | 0. | 0. | 3.5 | 0. |
time (sec) |
N/A |
0.295 |
0.313 |
0.037 |
0. |
0. |
53.971 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 180 | 180 | 147 | 0 | 0 | 0 | 666 | 0 |
normalized size | 1 | 1. | 0.82 | 0. | 0. | 0. | 3.7 | 0. |
time (sec) |
N/A |
0.188 |
0.205 |
0.049 |
0. |
0. |
35.276 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
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|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 120 | 120 | 93 | 0 | 0 | 0 | 428 | 0 |
normalized size | 1 | 1. | 0.78 | 0. | 0. | 0. | 3.57 | 0. |
time (sec) |
N/A |
0.096 |
0.099 |
0.036 |
0. |
0. |
12.9 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 56 | 0 | 0 | 0 | 204 | 0 |
normalized size | 1 | 1. | 0.73 | 0. | 0. | 0. | 2.65 | 0. |
time (sec) |
N/A |
0.038 |
0.048 |
0.033 |
0. |
0. |
5.305 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
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 | 125 | 125 | 100 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.8 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.137 |
0.102 |
0.055 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
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 | 206 | 206 | 149 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.72 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.385 |
0.164 |
0.075 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
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 | 342 | 342 | 195 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.57 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.775 |
0.214 |
0.063 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
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 | 340 | 340 | 212 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.62 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.716 |
0.342 |
0.046 |
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 |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 246 | 246 | 158 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.64 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.405 |
0.223 |
0.055 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 171 | 171 | 108 | 0 | 0 | 0 | 2076 | 0 |
normalized size | 1 | 1. | 0.63 | 0. | 0. | 0. | 12.14 | 0. |
time (sec) |
N/A |
0.227 |
0.127 |
0.04 |
0. |
0. |
85.735 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 103 | 103 | 81 | 0 | 0 | 0 | 954 | 0 |
normalized size | 1 | 1. | 0.79 | 0. | 0. | 0. | 9.26 | 0. |
time (sec) |
N/A |
0.047 |
0.06 |
0.039 |
0. |
0. |
39.524 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
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 | 205 | 205 | 147 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.72 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.383 |
0.171 |
0.069 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
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 | 304 | 304 | 207 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.68 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.802 |
0.271 |
0.056 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
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 | 491 | 491 | 265 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.54 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.425 |
0.352 |
0.075 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
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 | 433 | 433 | 222 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.51 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.138 |
0.356 |
0.052 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
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 | 292 | 292 | 169 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.58 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.407 |
0.186 |
0.067 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
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 | 208 | 208 | 133 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.64 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.298 |
0.139 |
0.054 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 103 | 103 | 81 | 0 | 0 | 0 | 3172 | 0 |
normalized size | 1 | 1. | 0.79 | 0. | 0. | 0. | 30.8 | 0. |
time (sec) |
N/A |
0.047 |
0.061 |
0.05 |
0. |
0. |
144.324 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
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 | 333 | 333 | 197 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.59 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.719 |
0.207 |
0.06 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
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 | 452 | 452 | 266 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.59 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.338 |
0.374 |
0.074 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
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 | 665 | 665 | 329 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.49 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.179 |
0.499 |
0.076 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
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 | 1059 | 1047 | 248 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0.99 |
0.23 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.512 |
0.446 |
0.072 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
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 | 495 | 464 | 198 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0.94 |
0.4 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.749 |
0.249 |
0.067 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
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 | 253 | 238 | 147 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0.94 |
0.58 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.227 |
0.117 |
0.053 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
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 | 162 | 162 | 118 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.73 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.159 |
0.195 |
0.06 |
0. |
0. |
0. |
0. |
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Problem 48 |
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 | 295 | 295 | 128 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.43 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.413 |
0.214 |
0.065 |
0. |
0. |
0. |
0. |
|
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Problem 49 |
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 | 483 | 483 | 128 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.27 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.039 |
0.384 |
0.081 |
0. |
0. |
0. |
0. |
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Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 91 | 112 | 0 | 537 | 97 | 153 |
normalized size |
1 |
1. |
1.08 |
1.33 |
0. |
6.39 |
1.15 |
1.82 |
time (sec) |
N/A |
0.077 |
0.121 |
0.008 |
0. |
1.64 |
35.64 |
1.174 |
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Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 91 | 112 | 0 | 518 | 97 | 153 |
normalized size |
1 |
1. |
1.08 |
1.33 |
0. |
6.17 |
1.15 |
1.82 |
time (sec) |
N/A |
0.077 |
0.132 |
0.009 |
0. |
1.632 |
35.814 |
1.168 |
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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 [48] had the largest ratio of [ 0.1935
]
|
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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 |
2 |
1 |
1. |
29 |
0.034 |
|
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|
|
|
|
|
2 |
A |
2 |
1 |
1. |
29 |
0.034 |
|
|
|
|
|
|
|
3 |
A |
2 |
1 |
1. |
27 |
0.037 |
|
|
|
|
|
|
|
4 |
A |
2 |
1 |
1. |
20 |
0.05 |
|
|
|
|
|
|
|
5 |
A |
3 |
2 |
1. |
29 |
0.069 |
|
|
|
|
|
|
|
6 |
A |
3 |
3 |
1. |
29 |
0.103 |
|
|
|
|
|
|
|
7 |
A |
3 |
3 |
1. |
29 |
0.103 |
|
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|
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|
8 |
A |
2 |
1 |
1. |
31 |
0.032 |
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9 |
A |
2 |
1 |
1. |
31 |
0.032 |
|
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10 |
A |
2 |
1 |
1. |
29 |
0.034 |
|
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|
11 |
A |
2 |
1 |
1. |
22 |
0.045 |
|
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|
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|
12 |
A |
3 |
2 |
1. |
31 |
0.065 |
|
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|
|
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|
13 |
A |
4 |
3 |
1. |
31 |
0.097 |
|
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|
|
|
|
|
14 |
A |
4 |
3 |
1. |
31 |
0.097 |
|
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15 |
A |
2 |
1 |
1. |
31 |
0.032 |
|
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16 |
A |
2 |
1 |
1. |
31 |
0.032 |
|
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17 |
A |
2 |
1 |
1. |
29 |
0.034 |
|
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18 |
A |
2 |
1 |
1. |
22 |
0.045 |
|
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19 |
A |
3 |
2 |
1. |
31 |
0.065 |
|
|
|
|
|
|
|
20 |
A |
4 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
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21 |
A |
5 |
3 |
1. |
31 |
0.097 |
|
|
|
|
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22 |
A |
3 |
2 |
1. |
31 |
0.065 |
|
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23 |
A |
3 |
2 |
1. |
31 |
0.065 |
|
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24 |
A |
3 |
2 |
1. |
31 |
0.065 |
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25 |
A |
3 |
2 |
1. |
29 |
0.069 |
|
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26 |
A |
2 |
2 |
1. |
22 |
0.091 |
|
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|
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|
27 |
A |
4 |
2 |
1. |
31 |
0.065 |
|
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|
|
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|
28 |
A |
5 |
3 |
1. |
31 |
0.097 |
|
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|
29 |
A |
6 |
3 |
1. |
31 |
0.097 |
|
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|
30 |
A |
4 |
3 |
1. |
31 |
0.097 |
|
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|
|
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|
|
31 |
A |
4 |
3 |
1. |
31 |
0.097 |
|
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|
32 |
A |
3 |
3 |
1. |
29 |
0.103 |
|
|
|
|
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|
|
33 |
A |
2 |
2 |
1. |
22 |
0.091 |
|
|
|
|
|
|
|
34 |
A |
5 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
35 |
A |
6 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
36 |
A |
7 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
37 |
A |
5 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
38 |
A |
4 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
39 |
A |
3 |
3 |
1. |
29 |
0.103 |
|
|
|
|
|
|
|
40 |
A |
2 |
2 |
1. |
22 |
0.091 |
|
|
|
|
|
|
|
|
41 |
A |
6 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
42 |
A |
7 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
43 |
A |
8 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
44 |
A |
6 |
4 |
0.99 |
31 |
0.129 |
|
|
|
|
|
|
|
45 |
A |
5 |
4 |
0.94 |
31 |
0.129 |
|
|
|
|
|
|
|
46 |
A |
4 |
4 |
0.94 |
29 |
0.138 |
|
|
|
|
|
|
|
47 |
A |
6 |
5 |
1. |
31 |
0.161 |
|
|
|
|
|
|
|
48 |
A |
7 |
6 |
1. |
31 |
0.194 |
|
|
|
|
|
|
|
49 |
A |
8 |
6 |
1. |
31 |
0.194 |
|
|
|
|
|
|
|
50 |
A |
5 |
5 |
1. |
29 |
0.172 |
|
|
|
|
|
|
|
51 |
A |
5 |
5 |
1. |
29 |
0.172 |
|
|
|
|
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|