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: { 27, 28, 29, 30, 31, 36, 37, 38, 39, 40, 47, 48, 49, 50 }
B grade: { }
C 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, 32, 33, 34,
35, 41, 42, 43, 44, 45, 46, 51 }
F grade: { }
2.1.3 Maple
A grade: { 29, 32, 33, 36, 37, 38, 41, 42, 43 }
B grade: { 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 45, 46, 47, 48, 51
}
C grade: { 7, 16 }
F grade: { 25, 26, 30, 31, 34, 35, 39, 40, 44, 49, 50 }
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: { 27, 28, 29, 30, 31, 36, 37, 38, 39, 40 }
B grade: { 45, 46, 47, 48, 49, 50 }
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, 32, 33, 34,
35, 41, 42, 43, 44, 51 }
2.1.6 Sympy
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
}
2.1.7 Giac
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
}
2.2 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 |
C |
B |
F |
F(-1) |
F |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 975 | 975 | 690 | 17767396 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.71 |
18223. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
24.463 |
6.221 |
1.056 |
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 |
C |
B |
F |
F(-1) |
F |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 889 | 889 | 582 | 17247437 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.65 |
19400.9 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
23.995 |
4.913 |
0.432 |
0. |
0. |
0. |
0. |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
F |
F(-1) |
F |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 748 | 748 | 451 | 17766953 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.6 |
23752.6 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
23.63 |
2.34 |
0.401 |
0. |
0. |
0. |
0. |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
F |
F(-1) |
F |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 676 | 676 | 405 | 17247074 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.6 |
25513.4 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
26.571 |
0.535 |
0.45 |
0. |
0. |
0. |
0. |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
F |
F(-1) |
F |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 601 | 601 | 250 | 21947835 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.42 |
36518.9 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
23.288 |
0.278 |
0.405 |
0. |
0. |
0. |
0. |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
F(-2) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 574 | 574 | 228 | 17246812 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.4 |
30046.7 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
23.269 |
0.135 |
0.415 |
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 |
C |
C |
F(-2) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 571 | 571 | 223 | 53198 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.39 |
93.17 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
23.583 |
0.342 |
5.83 |
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 |
C |
B |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 612 | 612 | 261 | 1166066 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.43 |
1905.34 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
23.744 |
1.121 |
19.957 |
0. |
0. |
0. |
0. |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 690 | 690 | 289 | 2856003 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.42 |
4139.13 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
23.733 |
1.816 |
140.041 |
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 |
B |
F(-1) |
F(-1) |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 548 | 548 | 456 | 9581348 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.83 |
17484.2 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.085 |
6.12 |
0.35 |
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 |
C |
B |
F(-1) |
F(-1) |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 495 | 495 | 283 | 7491919 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.57 |
15135.2 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.82 |
2.524 |
0.305 |
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 |
B |
F |
F(-1) |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 383 | 383 | 252 | 9581048 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.66 |
25015.8 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.731 |
0.842 |
0.329 |
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 |
C |
B |
F |
F(-1) |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 352 | 352 | 228 | 7491708 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.65 |
21283.3 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.373 |
0.183 |
0.336 |
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 |
B |
F(-2) |
F(-1) |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 294 | 294 | 173 | 9338488 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.59 |
31763.6 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.286 |
0.093 |
0.326 |
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 |
B |
F(-2) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 298 | 298 | 173 | 7300213 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.58 |
24497.4 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.311 |
0.123 |
0.368 |
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 |
C |
F |
F(-1) |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 350 | 350 | 223 | 28513 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.64 |
81.47 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.701 |
0.245 |
1.877 |
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 |
B |
F |
F(-1) |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 395 | 395 | 264 | 675102 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.67 |
1709.12 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.765 |
1.402 |
18.206 |
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 |
B |
F(-1) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 500 | 500 | 315 | 1981914 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.63 |
3963.83 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.8 |
5.383 |
45.891 |
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 |
B |
F(-1) |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1190 | 1190 | 884 | 13067596 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.74 |
10981.2 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
6.507 |
23.276 |
0.326 |
0. |
0. |
0. |
0. |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
F(-1) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 864 | 864 | 611 | 13066795 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.71 |
15123.6 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
4.79 |
20.016 |
0.303 |
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 |
C |
B |
F(-2) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 686 | 686 | 286 | 13066412 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.42 |
19047.3 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
4.514 |
15.267 |
0.321 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-2) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 638 | 638 | 328 | 11847956 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.51 |
18570.5 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
3.95 |
4.812 |
0.291 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-2) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 635 | 635 | 318 | 13066372 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.5 |
20577. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
3.791 |
4.602 |
0.288 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-1) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 750 | 750 | 450 | 15825630 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.6 |
21100.8 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
4.578 |
4.348 |
122.368 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F(-1) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 829 | 829 | 583 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.7 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
4.939 |
6.166 |
180. |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F(-1) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1007 | 1007 | 786 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.78 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
4.928 |
6.172 |
180. |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-1) |
A |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 270 | 270 | 290 | 684 | 0 | 3421 | 0 | 0 |
normalized size |
1 |
1. |
1.07 |
2.53 |
0. |
12.67 |
0. |
0. |
time (sec) |
N/A |
0.613 |
5.826 |
0.25 |
0. |
36.87 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
A |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 209 | 209 | 208 | 467 | 0 | 2952 | 0 | 0 |
normalized size |
1 |
1. |
1. |
2.23 |
0. |
14.12 |
0. |
0. |
time (sec) |
N/A |
0.349 |
1.494 |
0.179 |
0. |
21.228 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 179 | 179 | 180 | 289 | 0 | 2641 | 0 | 0 |
normalized size |
1 |
1. |
1.01 |
1.61 |
0. |
14.75 |
0. |
0. |
time (sec) |
N/A |
0.224 |
0.279 |
0.186 |
0. |
16.188 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
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 | 203 | 203 | 192 | 0 | 0 | 5280 | 0 | 0 |
normalized size | 1 | 1. | 0.95 | 0. | 0. | 26.01 | 0. | 0. |
time (sec) |
N/A |
0.292 |
1.1 |
0.597 |
0. |
5.381 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
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 | 435 | 435 | 187 | 0 | 0 | 2934 | 0 | 0 |
normalized size | 1 | 1. | 0.43 | 0. | 0. | 6.74 | 0. | 0. |
time (sec) |
N/A |
0.544 |
1.156 |
0.509 |
0. |
19.086 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F |
F |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1254 | 1254 | 639 | 1945 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.51 |
1.55 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.993 |
29.37 |
0.212 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 829 | 829 | 428 | 1497 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.52 |
1.81 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.552 |
1.799 |
0.153 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 861 | 861 | 1258 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.46 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.585 |
26.903 |
0.497 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 943 | 943 | 1590 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.69 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.717 |
32.167 |
0.498 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-1) |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 182 | 182 | 173 | 240 | 0 | 2962 | 0 | 0 |
normalized size |
1 |
1. |
0.95 |
1.32 |
0. |
16.27 |
0. |
0. |
time (sec) |
N/A |
0.353 |
2.162 |
0.19 |
0. |
18.118 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 141 | 141 | 136 | 155 | 0 | 2450 | 0 | 0 |
normalized size | 1 | 1. | 0.96 | 1.1 | 0. | 17.38 | 0. | 0. |
time (sec) |
N/A |
0.21 |
0.236 |
0.189 |
0. |
13.274 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 79 | 102 | 0 | 733 | 0 | 0 |
normalized size | 1 | 1. | 1. | 1.29 | 0. | 9.28 | 0. | 0. |
time (sec) |
N/A |
0.114 |
0.111 |
0.19 |
0. |
3.86 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 142 | 142 | 136 | 0 | 0 | 2500 | 0 | 0 |
normalized size |
1 |
1. |
0.96 |
0. |
0. |
17.61 |
0. |
0. |
time (sec) |
N/A |
0.228 |
0.57 |
0.559 |
0. |
11.78 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-1) |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 249 | 249 | 188 | 0 | 0 | 3267 | 0 | 0 |
normalized size |
1 |
1. |
0.76 |
0. |
0. |
13.12 |
0. |
0. |
time (sec) |
N/A |
0.318 |
4.136 |
0.545 |
0. |
16.017 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-1) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 662 | 662 | 533 | 646 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.81 |
0.98 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.439 |
22.577 |
0.184 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 436 | 436 | 311 | 402 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.71 |
0.92 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.341 |
8.994 |
0.158 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 436 | 436 | 235 | 231 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.54 |
0.53 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.306 |
0.683 |
0.155 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F(-1) |
F(-1) |
F |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 707 | 707 | 683 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.97 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.728 |
27.892 |
0.537 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-1) |
B |
F |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 235 | 235 | 182725 | 826 | 0 | 8193 | 0 | 0 |
normalized size |
1 |
1. |
777.55 |
3.51 |
0. |
34.86 |
0. |
0. |
time (sec) |
N/A |
0.549 |
34.144 |
0.202 |
0. |
32.014 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
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 | 159 | 159 | 57597 | 601 | 0 | 2421 | 0 | 0 |
normalized size | 1 | 1. | 362.25 | 3.78 | 0. | 15.23 | 0. | 0. |
time (sec) |
N/A |
0.394 |
35.643 |
0.165 |
0. |
6.31 |
0. |
0. |
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Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 154 | 154 | 155 | 508 | 0 | 2384 | 0 | 0 |
normalized size | 1 | 1. | 1.01 | 3.3 | 0. | 15.48 | 0. | 0. |
time (sec) |
N/A |
0.285 |
2.836 |
0.159 |
0. |
5.54 |
0. |
0. |
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Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 155 | 155 | 156 | 417 | 0 | 2410 | 0 | 0 |
normalized size | 1 | 1. | 1.01 | 2.69 | 0. | 15.55 | 0. | 0. |
time (sec) |
N/A |
0.218 |
2.936 |
0.155 |
0. |
5.257 |
0. |
0. |
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Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-1) |
B |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 280 | 280 | 278 | 0 | 0 | 8429 | 0 | 0 |
normalized size |
1 |
1. |
0.99 |
0. |
0. |
30.1 |
0. |
0. |
time (sec) |
N/A |
0.387 |
3.295 |
0.446 |
0. |
28.113 |
0. |
0. |
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Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-1) |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 477 | 477 | 555 | 0 | 0 | 11061 | 0 | 0 |
normalized size |
1 |
1. |
1.16 |
0. |
0. |
23.19 |
0. |
0. |
time (sec) |
N/A |
0.552 |
6.069 |
0.449 |
0. |
35.949 |
0. |
0. |
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Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 981 | 981 | 831 | 3598 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.85 |
3.67 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.876 |
34.031 |
0.163 |
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 [1] had the largest ratio of [ 0.4242
]
|
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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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|
1 |
A |
21 |
14 |
1. |
33 |
0.424 |
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|
|
|
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2 |
A |
19 |
12 |
1. |
33 |
0.364 |
|
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|
|
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|
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3 |
A |
16 |
12 |
1. |
33 |
0.364 |
|
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|
|
|
|
|
4 |
A |
10 |
9 |
1. |
33 |
0.273 |
|
|
|
|
|
|
|
5 |
A |
10 |
9 |
1. |
31 |
0.29 |
|
|
|
|
|
|
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6 |
A |
9 |
7 |
1. |
24 |
0.292 |
|
|
|
|
|
|
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7 |
A |
18 |
13 |
1. |
31 |
0.419 |
|
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|
|
|
|
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8 |
A |
17 |
12 |
1. |
33 |
0.364 |
|
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|
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9 |
A |
21 |
14 |
1. |
33 |
0.424 |
|
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10 |
A |
15 |
10 |
1. |
33 |
0.303 |
|
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11 |
A |
14 |
9 |
1. |
33 |
0.273 |
|
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|
|
12 |
A |
11 |
8 |
1. |
33 |
0.242 |
|
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|
13 |
A |
9 |
7 |
1. |
33 |
0.212 |
|
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|
|
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|
|
14 |
A |
6 |
4 |
1. |
31 |
0.129 |
|
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|
|
|
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15 |
A |
6 |
3 |
1. |
24 |
0.125 |
|
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|
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16 |
A |
10 |
7 |
1. |
31 |
0.226 |
|
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|
|
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|
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17 |
A |
11 |
8 |
1. |
33 |
0.242 |
|
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18 |
A |
14 |
9 |
1. |
33 |
0.273 |
|
|
|
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|
|
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19 |
A |
20 |
12 |
1. |
33 |
0.364 |
|
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20 |
A |
14 |
11 |
1. |
33 |
0.333 |
|
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21 |
A |
10 |
7 |
1. |
33 |
0.212 |
|
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22 |
A |
7 |
5 |
1. |
33 |
0.152 |
|
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23 |
A |
7 |
5 |
1. |
31 |
0.161 |
|
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24 |
A |
13 |
10 |
1. |
31 |
0.323 |
|
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25 |
A |
13 |
10 |
1. |
33 |
0.303 |
|
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26 |
A |
18 |
12 |
1. |
33 |
0.364 |
|
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27 |
A |
9 |
8 |
1. |
35 |
0.229 |
|
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28 |
A |
8 |
7 |
1. |
35 |
0.2 |
|
|
|
|
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29 |
A |
8 |
7 |
1. |
33 |
0.212 |
|
|
|
|
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30 |
A |
10 |
7 |
1. |
33 |
0.212 |
|
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|
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|
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31 |
A |
22 |
9 |
1. |
35 |
0.257 |
|
|
|
|
|
|
|
32 |
A |
14 |
9 |
1. |
35 |
0.257 |
|
|
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|
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|
|
33 |
A |
8 |
6 |
1. |
26 |
0.231 |
|
|
|
|
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|
|
34 |
A |
9 |
8 |
1. |
35 |
0.229 |
|
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|
35 |
A |
10 |
8 |
1. |
35 |
0.229 |
|
|
|
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|
|
36 |
A |
8 |
7 |
1. |
35 |
0.2 |
|
|
|
|
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|
|
37 |
A |
7 |
6 |
1. |
35 |
0.171 |
|
|
|
|
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|
38 |
A |
4 |
4 |
1. |
33 |
0.121 |
|
|
|
|
|
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|
39 |
A |
8 |
5 |
1. |
33 |
0.152 |
|
|
|
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|
40 |
A |
11 |
6 |
1. |
35 |
0.171 |
|
|
|
|
|
|
|
|
41 |
A |
5 |
5 |
1. |
35 |
0.143 |
|
|
|
|
|
|
|
42 |
A |
4 |
4 |
1. |
35 |
0.114 |
|
|
|
|
|
|
|
43 |
A |
4 |
3 |
1. |
26 |
0.115 |
|
|
|
|
|
|
|
44 |
A |
7 |
7 |
1. |
35 |
0.2 |
|
|
|
|
|
|
|
45 |
A |
8 |
7 |
1. |
35 |
0.2 |
|
|
|
|
|
|
|
46 |
A |
6 |
6 |
1. |
35 |
0.171 |
|
|
|
|
|
|
|
47 |
A |
6 |
6 |
1. |
35 |
0.171 |
|
|
|
|
|
|
|
48 |
A |
6 |
6 |
1. |
33 |
0.182 |
|
|
|
|
|
|
|
49 |
A |
12 |
7 |
1. |
33 |
0.212 |
|
|
|
|
|
|
|
50 |
A |
16 |
8 |
1. |
35 |
0.229 |
|
|
|
|
|
|
|
51 |
A |
9 |
8 |
1. |
35 |
0.229 |
|
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