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, 27, 29, 30, 31,
32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49 }
B grade: { }
C grade: { }
F grade: { 26, 28, 34}
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 15, 16, 17, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40,
41, 42, 43, 45, 47, 48, 49 }
B grade: { }
C grade: { 5, 12, 13, 14, 19, 21, 22, 26, 34, 44, 46 }
F grade: { 18, 20}
2.1.3 Maple
A grade: { 4, 5, 6, 7, 8, 13, 14, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 44, 45, 46, 47, 48, 49
}
B grade: { 1, 2, 3, 9, 10, 11, 21, 37, 38, 39, 40, 41, 42, 43 }
C grade: { 12, 15, 16, 17, 18, 19, 20, 34 }
F grade: { 29 }
2.1.4 Maxima
A grade: { 1, 2, 3, 4, 6, 7, 30, 36, 37, 38, 39
B grade: { 8, 9, 10, 11, 22, 40, 41, 42, 43 }
C grade: { }
F grade: { 5, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 44, 45, 46,
47, 48, 49 }
2.1.5 FriCAS
A grade: { 3, 4, 23, 24, 25, 30, 31 }
B grade: { 1, 2, 6, 7, 8, 27 }
C grade: { 32, 33 }
F grade: { 5, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 26, 28, 29, 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, 6, 23, 24, 25 }
B grade: { }
C grade: { }
F grade: { 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 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: { 4, 6, 23, 24, 25, 27, 30, 31, 32, 33 }
B grade: { 1, 2, 3, 7, 8, 28 }
C grade: { }
F grade: { 5, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 26, 29, 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 |
A |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 149 | 149 | 274 | 395 | 369 | 697 | 381 | 524 |
normalized size | 1 | 1. | 1.84 | 2.65 | 2.48 | 4.68 | 2.56 | 3.52 |
time (sec) |
N/A |
0.141 |
0.18 |
0.031 |
0.977 |
1.733 |
5.246 |
1.333 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
A |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 125 | 125 | 205 | 308 | 282 | 527 | 279 | 397 |
normalized size | 1 | 1. | 1.64 | 2.46 | 2.26 | 4.22 | 2.23 | 3.18 |
time (sec) |
N/A |
0.141 |
0.132 |
0.03 |
0.966 |
1.794 |
3.35 |
1.331 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 96 | 96 | 129 | 218 | 185 | 358 | 178 | 263 |
normalized size | 1 | 1. | 1.34 | 2.27 | 1.93 | 3.73 | 1.85 | 2.74 |
time (sec) |
N/A |
0.122 |
0.095 |
0.029 |
0.99 |
1.666 |
2.037 |
1.152 |
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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 | 84 | 84 | 96 | 92 | 112 | 228 | 92 | 157 |
normalized size | 1 | 1. | 1.14 | 1.1 | 1.33 | 2.71 | 1.1 | 1.87 |
time (sec) |
N/A |
0.076 |
0.01 |
0.029 |
0.973 |
1.649 |
1.184 |
1.163 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 114 | 114 | 257 | 148 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 2.25 | 1.3 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.081 |
0.245 |
0.132 |
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 |
A |
A |
A |
B |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 93 | 93 | 102 | 114 | 134 | 387 | 782 | 176 |
normalized size | 1 | 1. | 1.1 | 1.23 | 1.44 | 4.16 | 8.41 | 1.89 |
time (sec) |
N/A |
0.067 |
0.119 |
0.031 |
0.967 |
1.911 |
8.629 |
1.173 |
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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 | 130 | 130 | 133 | 154 | 257 | 933 | 0 | 791 |
normalized size |
1 |
1. |
1.02 |
1.18 |
1.98 |
7.18 |
0. |
6.08 |
time (sec) |
N/A |
0.129 |
0.15 |
0.039 |
0.971 |
2.371 |
0. |
1.742 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 175 | 175 | 173 | 223 | 458 | 1713 | 0 | 1511 |
normalized size |
1 |
1. |
0.99 |
1.27 |
2.62 |
9.79 |
0. |
8.63 |
time (sec) |
N/A |
0.188 |
0.263 |
0.046 |
1.012 |
4.121 |
0. |
3.094 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 359 | 359 | 506 | 1430 | 1056 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.41 | 3.98 | 2.94 | 0. | 0. | 0. |
time (sec) |
N/A |
0.534 |
0.887 |
0.058 |
1.799 |
0. |
0. |
0. |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 257 | 257 | 319 | 1050 | 707 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.24 | 4.09 | 2.75 | 0. | 0. | 0. |
time (sec) |
N/A |
0.409 |
0.636 |
0.049 |
1.795 |
0. |
0. |
0. |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 160 | 160 | 174 | 462 | 423 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.09 | 2.89 | 2.64 | 0. | 0. | 0. |
time (sec) |
N/A |
0.33 |
0.423 |
0.052 |
1.762 |
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 |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 188 | 188 | 938 | 1170 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 4.99 | 6.22 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.05 |
12.51 |
0.549 |
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 |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 321 | 321 | 317 | 605 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.99 | 1.88 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.313 |
4.383 |
0.117 |
0. |
0. |
0. |
0. |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 480 | 480 | 470 | 824 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.98 |
1.72 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.499 |
7.508 |
0.069 |
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 |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 614 | 614 | 830 | 6104 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.35 | 9.94 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
1.183 |
2.008 |
2.605 |
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 |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 387 | 387 | 591 | 4600 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.53 | 11.89 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.804 |
1.296 |
2.211 |
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 |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 244 | 244 | 331 | 12404 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.36 | 50.84 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.602 |
0.763 |
0.954 |
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 |
F |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 272 | 272 | 0 | 2367 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0. | 8.7 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.057 |
109.64 |
0.43 |
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 |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 517 | 517 | 1110 | 3497 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 2.15 | 6.76 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.525 |
15.401 |
0.613 |
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 |
F |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 953 | 953 | 0 | 53538 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0. |
56.18 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.027 |
87.225 |
3.486 |
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 |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 109 | 240 | 118 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1.63 | 3.58 | 1.76 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.07 |
0.276 |
0.04 |
0. |
0. |
0. |
0. |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 88 | 108 | 272 | 127 | 194 | 0 | 0 | 0 |
normalized size | 1 | 1.23 | 3.09 | 1.44 | 2.2 | 0. | 0. | 0. |
time (sec) |
N/A |
0.066 |
0.099 |
0.036 |
1.45 |
0. |
0. |
0. |
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|
|
|
|
Problem 23 |
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 | 182 | 220 | 254 | 306 | 0 | 1145 | 1083 | 423 |
normalized size |
1 |
1.21 |
1.4 |
1.68 |
0. |
6.29 |
5.95 |
2.32 |
time (sec) |
N/A |
0.223 |
0.284 |
0.031 |
0. |
2.339 |
45.228 |
2.399 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
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 | 158 | 158 | 170 | 223 | 0 | 918 | 1652 | 289 |
normalized size |
1 |
1. |
1.08 |
1.41 |
0. |
5.81 |
10.46 |
1.83 |
time (sec) |
N/A |
0.21 |
0.167 |
0.032 |
0. |
1.809 |
34.135 |
1.915 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
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 | 117 | 94 | 104 | 91 | 0 | 616 | 473 | 166 |
normalized size |
1 |
0.8 |
0.89 |
0.78 |
0. |
5.26 |
4.04 |
1.42 |
time (sec) |
N/A |
0.093 |
0.058 |
0.043 |
0. |
1.806 |
20.337 |
1.316 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
F |
C |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
N/A |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 325 | 0 | 285 | 362 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0.88 |
1.11 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.063 |
17.47 |
0.07 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 166 | 166 | 261 | 181 | 0 | 1350 | 0 | 366 |
normalized size |
1 |
1. |
1.57 |
1.09 |
0. |
8.13 |
0. |
2.2 |
time (sec) |
N/A |
0.279 |
0.356 |
0.034 |
0. |
26.882 |
0. |
1.174 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
F |
A |
A |
F(-2) |
F(-1) |
F(-1) |
B |
verified |
N/A |
N/A |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 226 | 0 | 379 | 310 | 0 | 0 | 0 | 684 |
normalized size |
1 |
0. |
1.68 |
1.37 |
0. |
0. |
0. |
3.03 |
time (sec) |
N/A |
0.066 |
0.712 |
0.04 |
0. |
0. |
0. |
89.474 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
NO |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1085 | 1216 | 684 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1.12 |
0.63 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.438 |
3.184 |
0.269 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.119 |
45.243 |
0.33 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-2) |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.369 |
42.476 |
0.457 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
C |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 336 | 332 | 299 | 500 | 0 | 20131 | 0 | 527 |
normalized size |
1 |
0.99 |
0.89 |
1.49 |
0. |
59.91 |
0. |
1.57 |
time (sec) |
N/A |
0.511 |
0.279 |
0.032 |
0. |
18.189 |
0. |
4.292 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
C |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 285 | 285 | 333 | 362 | 0 | 9110 | 0 | 432 |
normalized size |
1 |
1. |
1.17 |
1.27 |
0. |
31.96 |
0. |
1.52 |
time (sec) |
N/A |
0.452 |
0.102 |
0.031 |
0. |
14.72 |
0. |
2.18 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
F |
C |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
N/A |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 523 | 0 | 515 | 182 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0.98 |
0.35 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.062 |
99.21 |
0.171 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
F(-1) |
F(-1) |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 414 | 414 | 534 | 591 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.29 |
1.43 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.773 |
0.558 |
0.039 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 195 | 195 | 160 | 309 | 332 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.82 | 1.58 | 1.7 | 0. | 0. | 0. |
time (sec) |
N/A |
0.601 |
0.552 |
0.053 |
1.796 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 160 | 160 | 130 | 276 | 281 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.81 | 1.72 | 1.76 | 0. | 0. | 0. |
time (sec) |
N/A |
0.429 |
0.38 |
0.05 |
1.751 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 120 | 120 | 96 | 243 | 224 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.8 | 2.02 | 1.87 | 0. | 0. | 0. |
time (sec) |
N/A |
0.26 |
0.22 |
0.051 |
1.752 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 78 | 78 | 75 | 186 | 136 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.96 | 2.38 | 1.74 | 0. | 0. | 0. |
time (sec) |
N/A |
0.125 |
0.097 |
0.046 |
2.075 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 72 | 217 | 215 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.04 | 3.14 | 3.12 | 0. | 0. | 0. |
time (sec) |
N/A |
0.244 |
0.125 |
0.053 |
1.624 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 117 | 117 | 118 | 315 | 335 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.01 | 2.69 | 2.86 | 0. | 0. | 0. |
time (sec) |
N/A |
0.363 |
0.324 |
0.057 |
1.784 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 158 | 348 | 393 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.01 |
2.22 |
2.5 |
0. |
0. |
0. |
time (sec) |
N/A |
0.456 |
0.537 |
0.066 |
1.785 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 192 | 192 | 187 | 381 | 446 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.97 |
1.98 |
2.32 |
0. |
0. |
0. |
time (sec) |
N/A |
0.571 |
0.753 |
0.062 |
1.796 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 460 | 460 | 558 | 651 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.21 |
1.42 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.783 |
2.97 |
0.06 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 374 | 374 | 337 | 539 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.9 |
1.44 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.485 |
1.447 |
0.057 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 318 | 318 | 551 | 462 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.73 |
1.45 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.322 |
1.534 |
0.053 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 358 | 358 | 302 | 540 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.84 |
1.51 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.587 |
1.124 |
0.062 |
0. |
0. |
0. |
0. |
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Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 413 | 413 | 360 | 620 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.87 |
1.5 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.72 |
1.624 |
0.068 |
0. |
0. |
0. |
0. |
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Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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|
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|
|
grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 506 | 506 | 394 | 741 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.78 |
1.46 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.872 |
2.673 |
0.072 |
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 [ 2.167
]
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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 |
6 |
4 |
1. |
16 |
0.25 |
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2 |
A |
6 |
4 |
1. |
16 |
0.25 |
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3 |
A |
6 |
4 |
1. |
16 |
0.25 |
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4 |
A |
6 |
4 |
1. |
14 |
0.286 |
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5 |
A |
4 |
4 |
1. |
16 |
0.25 |
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6 |
A |
6 |
4 |
1. |
16 |
0.25 |
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7 |
A |
4 |
3 |
1. |
16 |
0.188 |
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8 |
A |
4 |
3 |
1. |
16 |
0.188 |
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9 |
A |
19 |
14 |
1. |
18 |
0.778 |
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10 |
A |
15 |
12 |
1. |
18 |
0.667 |
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11 |
A |
12 |
9 |
1. |
16 |
0.562 |
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12 |
A |
1 |
1 |
1. |
18 |
0.056 |
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13 |
A |
12 |
6 |
1. |
18 |
0.333 |
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14 |
A |
18 |
10 |
1. |
18 |
0.556 |
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15 |
A |
29 |
15 |
1. |
18 |
0.833 |
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16 |
A |
20 |
13 |
1. |
18 |
0.722 |
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17 |
A |
14 |
10 |
1. |
16 |
0.625 |
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18 |
A |
1 |
1 |
1. |
18 |
0.056 |
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19 |
A |
9 |
7 |
1. |
18 |
0.389 |
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20 |
A |
21 |
11 |
1. |
18 |
0.611 |
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21 |
A |
4 |
4 |
1.63 |
17 |
0.235 |
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22 |
A |
4 |
4 |
1.23 |
15 |
0.267 |
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23 |
A |
19 |
10 |
1.21 |
18 |
0.556 |
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24 |
A |
12 |
10 |
1. |
18 |
0.556 |
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25 |
A |
10 |
7 |
0.8 |
16 |
0.438 |
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26 |
F |
0 |
0 |
N/A |
0 |
N/A |
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27 |
A |
10 |
7 |
1. |
18 |
0.389 |
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28 |
F |
0 |
0 |
N/A |
0 |
N/A |
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29 |
A |
104 |
39 |
1.12 |
18 |
2.167 |
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30 |
A |
0 |
0 |
0. |
0 |
0. |
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31 |
A |
0 |
0 |
0. |
0 |
0. |
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32 |
A |
25 |
14 |
0.99 |
18 |
0.778 |
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33 |
A |
23 |
13 |
1. |
16 |
0.812 |
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34 |
F |
0 |
0 |
N/A |
0 |
N/A |
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35 |
A |
20 |
12 |
1. |
18 |
0.667 |
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36 |
A |
19 |
10 |
1. |
26 |
0.385 |
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37 |
A |
14 |
10 |
1. |
26 |
0.385 |
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38 |
A |
9 |
9 |
1. |
24 |
0.375 |
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39 |
A |
5 |
4 |
1. |
23 |
0.174 |
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40 |
A |
5 |
7 |
1. |
26 |
0.269 |
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41 |
A |
9 |
9 |
1. |
26 |
0.346 |
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42 |
A |
14 |
9 |
1. |
26 |
0.346 |
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43 |
A |
20 |
9 |
1. |
26 |
0.346 |
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44 |
A |
20 |
11 |
1. |
23 |
0.478 |
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45 |
A |
15 |
10 |
1. |
21 |
0.476 |
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46 |
A |
11 |
5 |
1. |
20 |
0.25 |
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47 |
A |
15 |
11 |
1. |
23 |
0.478 |
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48 |
A |
19 |
13 |
1. |
23 |
0.565 |
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49 |
A |
24 |
13 |
1. |
23 |
0.565 |
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