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, 23, 24, 25, 27, 28, 30, 31, 32, 33, 35, 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, 26, 29, 34, 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, 28 }
C grade: { 32, 33 }
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 }
2.1.6 Sympy
A grade: { 1, 2, 3, 4, 6, 7, 8, 23, 24, 25 }
B grade: { }
C grade: { }
F grade: { 5, 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: { 23, 24, 25, 27, 30, 31, 32, 33, 35 }
B grade: { 1, 2, 3, 4, 6, 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, 36, 37, 38, 39, 40, 41, 42, 43, 44,
45, 46, 47, 48, 49 }
2.1.8 Mupad
A grade: { 30, 31 }
B grade: { 1, 2, 3, 4, 6, 7, 8, 23, 24, 25, 27, 28, 32, 33, 35 }
C grade: { }
F grade: { 5, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 26, 29, 34, 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 |
Mupad |
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grade |
A |
A |
A |
B |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 149 | 149 | 274 | 395 | 273 | 322 | 381 | 2351 | 272 |
normalized size | 1 | 1.00 | 1.84 | 2.65 | 1.83 | 2.16 | 2.56 | 15.78 | 1.83 |
time (sec) |
N/A |
0.141 |
0.189 |
0.034 |
0.334 |
0.493 |
2.961 |
0.233 |
1.696 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 125 | 125 | 205 | 308 | 209 | 244 | 279 | 1375 | 197 |
normalized size | 1 | 1.00 | 1.64 | 2.46 | 1.67 | 1.95 | 2.23 | 11.00 | 1.58 |
time (sec) |
N/A |
0.141 |
0.135 |
0.031 |
0.326 |
0.800 |
1.982 |
0.190 |
1.150 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
A |
A |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 96 | 96 | 129 | 218 | 137 | 163 | 178 | 785 | 127 |
normalized size | 1 | 1.00 | 1.34 | 2.27 | 1.43 | 1.70 | 1.85 | 8.18 | 1.32 |
time (sec) |
N/A |
0.122 |
0.099 |
0.032 |
0.312 |
0.658 |
1.264 |
0.390 |
0.941 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 96 | 92 | 83 | 98 | 92 | 292 | 67 |
normalized size | 1 | 1.00 | 1.14 | 1.10 | 0.99 | 1.17 | 1.10 | 3.48 | 0.80 |
time (sec) |
N/A |
0.076 |
0.011 |
0.029 |
0.316 |
0.604 |
0.674 |
0.206 |
0.827 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 114 | 114 | 257 | 148 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.25 | 1.30 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.081 |
0.258 |
0.089 |
0.000 |
0.768 |
0.000 |
0.000 |
0.000 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 93 | 93 | 102 | 114 | 99 | 182 | 690 | 374 | 115 |
normalized size | 1 | 1.00 | 1.10 | 1.23 | 1.06 | 1.96 | 7.42 | 4.02 | 1.24 |
time (sec) |
N/A |
0.067 |
0.123 |
0.036 |
0.317 |
0.944 |
3.618 |
0.181 |
3.596 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 130 | 130 | 133 | 154 | 190 | 454 | 3216 | 1197 | 427 |
normalized size | 1 | 1.00 | 1.02 | 1.18 | 1.46 | 3.49 | 24.74 | 9.21 | 3.28 |
time (sec) |
N/A |
0.129 |
0.156 |
0.039 |
0.332 |
0.854 |
7.126 |
0.195 |
3.998 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
B |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 175 | 175 | 173 | 223 | 339 | 859 | 10946 | 3320 | 418 |
normalized size | 1 | 1.00 | 0.99 | 1.27 | 1.94 | 4.91 | 62.55 | 18.97 | 2.39 |
time (sec) |
N/A |
0.188 |
0.271 |
0.040 |
0.336 |
1.705 |
12.914 |
0.238 |
2.295 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 359 | 359 | 506 | 1430 | 782 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.41 | 3.98 | 2.18 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.534 |
0.931 |
0.069 |
0.562 |
1.111 |
0.000 |
0.000 |
0.000 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 257 | 257 | 319 | 1050 | 524 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.24 | 4.09 | 2.04 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.409 |
0.655 |
0.066 |
0.565 |
1.736 |
0.000 |
0.000 |
0.000 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 160 | 160 | 174 | 462 | 313 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.09 | 2.89 | 1.96 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.330 |
0.440 |
0.058 |
0.557 |
0.581 |
0.000 |
0.000 |
0.000 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 188 | 188 | 759 | 1170 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 4.04 | 6.22 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.050 |
10.613 |
0.743 |
0.000 |
0.552 |
0.000 |
0.000 |
0.000 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 321 | 321 | 317 | 605 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.99 | 1.88 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.313 |
4.600 |
0.072 |
0.000 |
0.970 |
0.000 |
0.000 |
0.000 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 480 | 480 | 470 | 824 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.98 | 1.72 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.499 |
7.064 |
0.079 |
0.000 |
0.608 |
0.000 |
0.000 |
0.000 |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 614 | 614 | 830 | 6104 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.35 | 9.94 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.183 |
2.111 |
11.293 |
0.000 |
1.217 |
0.000 |
0.000 |
0.000 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 387 | 387 | 591 | 4600 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.53 | 11.89 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.804 |
1.314 |
6.212 |
0.000 |
0.815 |
0.000 |
0.000 |
0.000 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 244 | 244 | 331 | 12404 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.36 | 50.84 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.602 |
0.746 |
1.246 |
0.000 |
0.740 |
0.000 |
0.000 |
0.000 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
F |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 272 | 272 | 0 | 2367 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.00 | 8.70 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.057 |
98.024 |
0.655 |
0.000 |
0.486 |
0.000 |
0.000 |
0.000 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 517 | 517 | 813 | 3497 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.57 | 6.76 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.525 |
13.851 |
0.888 |
0.000 |
0.601 |
0.000 |
0.000 |
0.000 |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
F |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 953 | 953 | 0 | 53538 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.00 | 56.18 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.027 |
85.894 |
3.862 |
0.000 |
0.638 |
0.000 |
0.000 |
0.000 |
|
|
|
|
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|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 109 | 240 | 118 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.63 | 3.58 | 1.76 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.070 |
0.278 |
0.043 |
0.000 |
0.941 |
0.000 |
0.000 |
0.000 |
|
|
|
|
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|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 88 | 108 | 272 | 127 | 144 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.23 | 3.09 | 1.44 | 1.64 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.066 |
0.100 |
0.043 |
0.417 |
0.698 |
0.000 |
0.000 |
0.000 |
|
|
|
|
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|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 182 | 220 | 254 | 306 | 240 | 519 | 622 | 290 | 823 |
normalized size | 1 | 1.21 | 1.40 | 1.68 | 1.32 | 2.85 | 3.42 | 1.59 | 4.52 |
time (sec) |
N/A |
0.223 |
0.277 |
0.036 |
0.424 |
0.947 |
18.880 |
66.415 |
2.045 |
|
|
|
|
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Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 158 | 158 | 170 | 223 | 171 | 401 | 2907 | 194 | 309 |
normalized size | 1 | 1.00 | 1.08 | 1.41 | 1.08 | 2.54 | 18.40 | 1.23 | 1.96 |
time (sec) |
N/A |
0.210 |
0.171 |
0.031 |
0.410 |
0.684 |
15.808 |
1.038 |
1.385 |
|
|
|
|
|
|
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|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 117 | 94 | 104 | 91 | 95 | 249 | 294 | 143 | 242 |
normalized size | 1 | 0.80 | 0.89 | 0.78 | 0.81 | 2.13 | 2.51 | 1.22 | 2.07 |
time (sec) |
N/A |
0.093 |
0.055 |
0.030 |
0.415 |
1.342 |
10.478 |
0.365 |
1.310 |
|
|
|
|
|
|
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|
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|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
F |
C |
A |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
N/A |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 325 | 0 | 285 | 362 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.88 | 1.11 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.063 |
17.365 |
0.072 |
0.000 |
0.702 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 166 | 166 | 261 | 181 | 172 | 634 | 0 | 271 | 727 |
normalized size | 1 | 1.00 | 1.57 | 1.09 | 1.04 | 3.82 | 0.00 | 1.63 | 4.38 |
time (sec) |
N/A |
0.279 |
0.364 |
0.038 |
0.413 |
8.201 |
0.000 |
0.226 |
2.598 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
F |
A |
A |
A |
B |
F(-1) |
B |
B |
verified |
N/A |
N/A |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 226 | 0 | 379 | 310 | 311 | 1639 | 0 | 507 | 2016 |
normalized size | 1 | 0.00 | 1.68 | 1.37 | 1.38 | 7.25 | 0.00 | 2.24 | 8.92 |
time (sec) |
N/A |
0.066 |
0.681 |
0.044 |
0.424 |
48.701 |
0.000 |
41.457 |
5.280 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
NO |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1085 | 1216 | 684 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.12 | 0.63 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
2.438 |
3.002 |
0.421 |
0.000 |
0.458 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.119 |
44.302 |
0.352 |
0.000 |
0.808 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.369 |
42.011 |
0.612 |
0.000 |
0.914 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
C |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 336 | 332 | 299 | 500 | 295 | 9282 | 0 | 383 | 1081 |
normalized size | 1 | 0.99 | 0.89 | 1.49 | 0.88 | 27.62 | 0.00 | 1.14 | 3.22 |
time (sec) |
N/A |
0.511 |
0.280 |
0.035 |
0.418 |
2.842 |
0.000 |
4.759 |
1.819 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
C |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 285 | 285 | 333 | 362 | 248 | 3928 | 0 | 304 | 621 |
normalized size | 1 | 1.00 | 1.17 | 1.27 | 0.87 | 13.78 | 0.00 | 1.07 | 2.18 |
time (sec) |
N/A |
0.452 |
0.104 |
0.031 |
0.422 |
2.098 |
0.000 |
2.178 |
1.033 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
F |
C |
C |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
N/A |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 523 | 0 | 515 | 182 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.98 | 0.35 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.062 |
100.895 |
0.150 |
0.000 |
0.589 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
F(-1) |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 414 | 414 | 534 | 591 | 394 | 0 | 0 | 595 | 2638 |
normalized size | 1 | 1.00 | 1.29 | 1.43 | 0.95 | 0.00 | 0.00 | 1.44 | 6.37 |
time (sec) |
N/A |
0.773 |
0.520 |
0.043 |
0.427 |
0.000 |
0.000 |
28.280 |
1.405 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 195 | 195 | 160 | 309 | 246 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.82 | 1.58 | 1.26 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.601 |
0.571 |
0.057 |
0.558 |
1.107 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 160 | 160 | 130 | 276 | 208 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.81 | 1.72 | 1.30 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.429 |
0.391 |
0.056 |
0.556 |
0.567 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 120 | 120 | 96 | 243 | 166 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.80 | 2.02 | 1.38 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.260 |
0.225 |
0.056 |
0.553 |
0.947 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 78 | 78 | 75 | 186 | 101 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.96 | 2.38 | 1.29 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.125 |
0.098 |
0.051 |
0.597 |
0.649 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 72 | 217 | 159 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.04 | 3.14 | 2.30 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.244 |
0.129 |
0.059 |
0.523 |
0.725 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 117 | 117 | 118 | 315 | 248 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.01 | 2.69 | 2.12 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.363 |
0.336 |
0.067 |
0.565 |
1.048 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 158 | 348 | 291 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.01 | 2.22 | 1.85 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.456 |
0.570 |
0.071 |
0.569 |
1.215 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 192 | 192 | 187 | 381 | 330 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.97 | 1.98 | 1.72 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.571 |
0.809 |
0.071 |
0.563 |
0.959 |
0.000 |
0.000 |
0.000 |
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Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
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|
grade |
A |
A |
C |
A |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 460 | 460 | 558 | 651 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.21 | 1.42 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.783 |
2.934 |
0.097 |
0.000 |
0.834 |
0.000 |
0.000 |
0.000 |
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Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 374 | 374 | 337 | 539 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.90 | 1.44 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.485 |
1.461 |
0.065 |
0.000 |
0.779 |
0.000 |
0.000 |
0.000 |
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Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 318 | 318 | 432 | 462 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.36 | 1.45 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.322 |
1.617 |
0.062 |
0.000 |
0.610 |
0.000 |
0.000 |
0.000 |
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Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 358 | 358 | 302 | 540 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.84 | 1.51 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.587 |
1.149 |
0.070 |
0.000 |
0.710 |
0.000 |
0.000 |
0.000 |
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Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
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|
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|
grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 413 | 413 | 360 | 620 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.87 | 1.50 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.720 |
1.644 |
0.074 |
0.000 |
0.667 |
0.000 |
0.000 |
0.000 |
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Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
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|
|
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|
|
|
grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 506 | 506 | 394 | 741 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.78 | 1.46 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.872 |
2.741 |
0.081 |
0.000 |
0.702 |
0.000 |
0.000 |
0.000 |
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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 2.1:Rubi 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 |
1.00 |
16 |
0.250 |
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2 |
A |
6 |
4 |
1.00 |
16 |
0.250 |
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3 |
A |
6 |
4 |
1.00 |
16 |
0.250 |
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4 |
A |
6 |
4 |
1.00 |
14 |
0.286 |
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5 |
A |
4 |
4 |
1.00 |
16 |
0.250 |
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6 |
A |
6 |
4 |
1.00 |
16 |
0.250 |
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7 |
A |
4 |
3 |
1.00 |
16 |
0.188 |
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8 |
A |
4 |
3 |
1.00 |
16 |
0.188 |
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9 |
A |
19 |
14 |
1.00 |
18 |
0.778 |
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10 |
A |
15 |
12 |
1.00 |
18 |
0.667 |
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11 |
A |
12 |
9 |
1.00 |
16 |
0.562 |
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12 |
A |
1 |
1 |
1.00 |
18 |
0.056 |
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13 |
A |
12 |
6 |
1.00 |
18 |
0.333 |
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14 |
A |
18 |
10 |
1.00 |
18 |
0.556 |
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15 |
A |
29 |
15 |
1.00 |
18 |
0.833 |
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16 |
A |
20 |
13 |
1.00 |
18 |
0.722 |
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17 |
A |
14 |
10 |
1.00 |
16 |
0.625 |
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18 |
A |
1 |
1 |
1.00 |
18 |
0.056 |
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19 |
A |
9 |
7 |
1.00 |
18 |
0.389 |
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20 |
A |
21 |
11 |
1.00 |
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 |
1.00 |
18 |
0.556 |
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25 |
A |
10 |
7 |
0.80 |
16 |
0.438 |
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26 |
F |
0 |
0 |
N/A |
0 |
N/A |
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27 |
A |
10 |
7 |
1.00 |
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 |
0.00 |
0 |
0.000 |
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31 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
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32 |
A |
25 |
14 |
0.99 |
18 |
0.778 |
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33 |
A |
23 |
13 |
1.00 |
16 |
0.812 |
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34 |
F |
0 |
0 |
N/A |
0 |
N/A |
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35 |
A |
20 |
12 |
1.00 |
18 |
0.667 |
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36 |
A |
19 |
10 |
1.00 |
26 |
0.385 |
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37 |
A |
14 |
10 |
1.00 |
26 |
0.385 |
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38 |
A |
9 |
9 |
1.00 |
24 |
0.375 |
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39 |
A |
5 |
4 |
1.00 |
23 |
0.174 |
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40 |
A |
5 |
7 |
1.00 |
26 |
0.269 |
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41 |
A |
9 |
9 |
1.00 |
26 |
0.346 |
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42 |
A |
14 |
9 |
1.00 |
26 |
0.346 |
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43 |
A |
20 |
9 |
1.00 |
26 |
0.346 |
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44 |
A |
20 |
11 |
1.00 |
23 |
0.478 |
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45 |
A |
15 |
10 |
1.00 |
21 |
0.476 |
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46 |
A |
11 |
5 |
1.00 |
20 |
0.250 |
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47 |
A |
15 |
11 |
1.00 |
23 |
0.478 |
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48 |
A |
19 |
13 |
1.00 |
23 |
0.565 |
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49 |
A |
24 |
13 |
1.00 |
23 |
0.565 |
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