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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46 }
B grade: { }
C grade: { }
F grade: { }
2.1.3 Maple
A grade: { }
B grade: { }
C grade: { 1, 2, 3, 4, 8, 9, 10, 11, 15, 16, 17, 18 }
F grade: { 5, 6, 7, 12, 13, 14, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46 }
2.1.4 Maxima
A grade: { 3, 4, 11, 18 }
B grade: { 1, 2, 8, 9, 10, 15, 16, 17 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46 }
2.1.5 FriCAS
A grade: { }
B grade: { 1, 2, 3, 4, 8, 9, 10, 11, 15, 16, 17, 18 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46 }
2.1.6 Sympy
A grade: { 3, 4, 11 }
B grade: { }
C grade: { 5, 6, 12, 19, 21, 22, 23, 24, 25, 31, 32 }
F grade: { 1, 2, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 20, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46 }
2.1.7 Giac
A grade: { }
B grade: { 1, 2, 3, 4, 8, 9, 10, 11, 16, 17, 18 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 15, 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 }
2.1.8 Mupad
A grade: { }
B grade: { 1, 2, 3, 4, 8, 9, 10, 11, 15, 16, 17, 18 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46 }
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 |
C |
B |
B |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 210 | 210 | 172 | 4972 | 464 | 3073 | 0 | 6927 | 1089 |
normalized size | 1 | 1.00 | 0.82 | 23.68 | 2.21 | 14.63 | 0.00 | 32.99 | 5.19 |
time (sec) |
N/A |
0.270 |
0.994 |
0.211 |
0.936 |
0.805 |
0.000 |
1.191 |
5.644 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
B |
B |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 160 | 160 | 129 | 2410 | 332 | 1524 | 0 | 3415 | 588 |
normalized size | 1 | 1.00 | 0.81 | 15.06 | 2.08 | 9.52 | 0.00 | 21.34 | 3.68 |
time (sec) |
N/A |
0.176 |
0.525 |
0.138 |
0.861 |
0.672 |
0.000 |
0.809 |
5.231 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 108 | 108 | 84 | 891 | 200 | 562 | 8500 | 1290 | 271 |
normalized size | 1 | 1.00 | 0.78 | 8.25 | 1.85 | 5.20 | 78.70 | 11.94 | 2.51 |
time (sec) |
N/A |
0.084 |
0.247 |
0.109 |
0.697 |
0.663 |
88.266 |
1.796 |
4.958 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 49 | 262 | 91 | 185 | 1698 | 327 | 91 |
normalized size | 1 | 1.00 | 0.74 | 3.97 | 1.38 | 2.80 | 25.73 | 4.95 | 1.38 |
time (sec) |
N/A |
0.040 |
0.067 |
0.117 |
0.598 |
0.722 |
29.325 |
0.455 |
4.830 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 120 | 120 | 95 | 0 | 0 | 0 | 666 | 0 | -1 |
normalized size | 1 | 1.00 | 0.79 | 0.00 | 0.00 | 0.00 | 5.55 | 0.00 | -0.01 |
time (sec) |
N/A |
0.120 |
0.218 |
0.722 |
0.000 |
0.633 |
10.144 |
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 |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 177 | 177 | 110 | 0 | 0 | 0 | 4129 | 0 | -1 |
normalized size | 1 | 1.00 | 0.62 | 0.00 | 0.00 | 0.00 | 23.33 | 0.00 | -0.01 |
time (sec) |
N/A |
0.258 |
0.238 |
0.669 |
0.000 |
0.537 |
48.650 |
0.000 |
0.000 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 228 | 228 | 136 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.60 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.273 |
0.190 |
0.664 |
0.000 |
0.699 |
0.000 |
0.000 |
0.000 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
B |
B |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 318 | 318 | 273 | 11389 | 748 | 6638 | 0 | 15358 | 1882 |
normalized size | 1 | 1.00 | 0.86 | 35.81 | 2.35 | 20.87 | 0.00 | 48.30 | 5.92 |
time (sec) |
N/A |
0.411 |
1.478 |
0.226 |
1.119 |
0.795 |
0.000 |
2.013 |
6.348 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
B |
B |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 237 | 237 | 199 | 5908 | 540 | 3515 | 0 | 8103 | 1119 |
normalized size | 1 | 1.00 | 0.84 | 24.93 | 2.28 | 14.83 | 0.00 | 34.19 | 4.72 |
time (sec) |
N/A |
0.310 |
0.605 |
0.171 |
0.905 |
0.797 |
0.000 |
2.006 |
5.591 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
B |
B |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 160 | 160 | 129 | 2410 | 332 | 1426 | 0 | 3415 | 588 |
normalized size | 1 | 1.00 | 0.81 | 15.06 | 2.08 | 8.91 | 0.00 | 21.34 | 3.68 |
time (sec) |
N/A |
0.172 |
0.313 |
0.144 |
0.782 |
0.749 |
0.000 |
0.818 |
5.196 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 102 | 102 | 78 | 732 | 155 | 527 | 6399 | 1023 | 265 |
normalized size | 1 | 1.00 | 0.76 | 7.18 | 1.52 | 5.17 | 62.74 | 10.03 | 2.60 |
time (sec) |
N/A |
0.076 |
0.175 |
0.109 |
0.634 |
0.609 |
74.164 |
0.582 |
5.113 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 185 | 185 | 153 | 0 | 0 | 0 | 1085 | 0 | -1 |
normalized size | 1 | 1.00 | 0.83 | 0.00 | 0.00 | 0.00 | 5.86 | 0.00 | -0.01 |
time (sec) |
N/A |
0.226 |
0.339 |
0.857 |
0.000 |
0.558 |
23.456 |
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 |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 268 | 268 | 159 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.59 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.669 |
0.318 |
0.821 |
0.000 |
0.669 |
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 |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 322 | 322 | 168 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.52 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.544 |
0.312 |
0.813 |
0.000 |
0.591 |
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 |
B |
B |
F(-1) |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 410 | 410 | 358 | 20937 | 1032 | 11628 | 0 | 0 | 2949 |
normalized size | 1 | 1.00 | 0.87 | 51.07 | 2.52 | 28.36 | 0.00 | 0.00 | 7.19 |
time (sec) |
N/A |
0.624 |
1.430 |
0.344 |
1.467 |
1.140 |
0.000 |
0.000 |
7.495 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
B |
B |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 310 | 310 | 265 | 11389 | 748 | 6557 | 0 | 15358 | 1882 |
normalized size | 1 | 1.00 | 0.85 | 36.74 | 2.41 | 21.15 | 0.00 | 49.54 | 6.07 |
time (sec) |
N/A |
0.414 |
1.520 |
0.249 |
0.923 |
0.819 |
0.000 |
1.303 |
6.414 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
B |
B |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 210 | 210 | 172 | 4972 | 464 | 2833 | 0 | 6927 | 1089 |
normalized size | 1 | 1.00 | 0.82 | 23.68 | 2.21 | 13.49 | 0.00 | 32.99 | 5.19 |
time (sec) |
N/A |
0.258 |
0.753 |
0.178 |
1.008 |
0.811 |
0.000 |
1.057 |
5.653 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
A |
B |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 137 | 137 | 106 | 1609 | 219 | 1104 | 0 | 2278 | 563 |
normalized size | 1 | 1.00 | 0.77 | 11.74 | 1.60 | 8.06 | 0.00 | 16.63 | 4.11 |
time (sec) |
N/A |
0.111 |
0.177 |
0.126 |
0.845 |
0.725 |
0.000 |
0.750 |
5.307 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 270 | 270 | 229 | 0 | 0 | 0 | 1503 | 0 | -1 |
normalized size | 1 | 1.00 | 0.85 | 0.00 | 0.00 | 0.00 | 5.57 | 0.00 | -0.00 |
time (sec) |
N/A |
0.389 |
0.661 |
0.842 |
0.000 |
0.612 |
45.806 |
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 |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 394 | 389 | 217 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.99 | 0.55 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.024 |
0.571 |
0.842 |
0.000 |
0.553 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 380 | 380 | 332 | 0 | 0 | 0 | 1921 | 0 | -1 |
normalized size | 1 | 1.00 | 0.87 | 0.00 | 0.00 | 0.00 | 5.06 | 0.00 | -0.00 |
time (sec) |
N/A |
0.621 |
1.195 |
0.918 |
0.000 |
0.643 |
80.302 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 272 | 272 | 231 | 0 | 0 | 0 | 1503 | 0 | -1 |
normalized size | 1 | 1.00 | 0.85 | 0.00 | 0.00 | 0.00 | 5.53 | 0.00 | -0.00 |
time (sec) |
N/A |
0.398 |
0.764 |
0.872 |
0.000 |
0.646 |
45.412 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 187 | 187 | 154 | 0 | 0 | 0 | 1085 | 0 | -1 |
normalized size | 1 | 1.00 | 0.82 | 0.00 | 0.00 | 0.00 | 5.80 | 0.00 | -0.01 |
time (sec) |
N/A |
0.254 |
0.365 |
0.821 |
0.000 |
0.614 |
23.350 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 122 | 122 | 95 | 0 | 0 | 0 | 666 | 0 | -1 |
normalized size | 1 | 1.00 | 0.78 | 0.00 | 0.00 | 0.00 | 5.46 | 0.00 | -0.01 |
time (sec) |
N/A |
0.126 |
0.166 |
0.662 |
0.000 |
0.613 |
10.355 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 78 | 78 | 57 | 0 | 0 | 0 | 284 | 0 | -1 |
normalized size | 1 | 1.00 | 0.73 | 0.00 | 0.00 | 0.00 | 3.64 | 0.00 | -0.01 |
time (sec) |
N/A |
0.040 |
0.072 |
0.702 |
0.000 |
0.758 |
4.089 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-2) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 127 | 127 | 102 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.80 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.144 |
0.149 |
0.991 |
0.000 |
0.765 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-2) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 212 | 212 | 152 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.72 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.529 |
0.266 |
1.023 |
0.000 |
0.606 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-2) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 407 | 407 | 199 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.49 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.247 |
0.344 |
1.023 |
0.000 |
0.937 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 386 | 381 | 220 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.99 | 0.57 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.135 |
0.674 |
0.879 |
0.000 |
0.623 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 267 | 267 | 161 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.60 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.675 |
0.348 |
0.878 |
0.000 |
0.683 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 178 | 178 | 110 | 0 | 0 | 0 | 4129 | 0 | -1 |
normalized size | 1 | 1.00 | 0.62 | 0.00 | 0.00 | 0.00 | 23.20 | 0.00 | -0.01 |
time (sec) |
N/A |
0.253 |
0.189 |
0.697 |
0.000 |
0.650 |
48.847 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 107 | 107 | 83 | 0 | 0 | 0 | 1897 | 0 | -1 |
normalized size | 1 | 1.00 | 0.78 | 0.00 | 0.00 | 0.00 | 17.73 | 0.00 | -0.01 |
time (sec) |
N/A |
0.056 |
0.109 |
0.661 |
0.000 |
0.814 |
17.416 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-2) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 211 | 211 | 150 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.71 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.521 |
0.252 |
0.998 |
0.000 |
0.648 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-2) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 315 | 315 | 209 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.66 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.097 |
0.363 |
1.088 |
0.000 |
0.880 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 567 | 567 | 270 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.48 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
2.341 |
0.534 |
1.031 |
0.000 |
1.014 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 322 | 322 | 172 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.53 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.557 |
0.322 |
0.876 |
0.000 |
0.584 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 228 | 228 | 136 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.60 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.281 |
0.206 |
0.726 |
0.000 |
0.587 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 112 | 112 | 83 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.74 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.056 |
0.124 |
0.675 |
0.000 |
0.572 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-2) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 366 | 366 | 201 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.55 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.217 |
0.312 |
1.033 |
0.000 |
0.731 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 482 | 482 | 271 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.56 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
2.071 |
0.845 |
1.022 |
0.000 |
1.263 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-2) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 211 | 211 | 162 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.77 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.246 |
0.589 |
1.451 |
0.000 |
0.714 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 271 | 255 | 164 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.94 | 0.61 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.327 |
0.286 |
0.679 |
0.000 |
0.842 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
F |
F |
F(-2) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 164 | 164 | 138 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.84 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.179 |
0.281 |
1.087 |
0.000 |
0.594 |
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 |
A |
F |
F |
F |
F(-2) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 304 | 304 | 138 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.45 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.536 |
0.532 |
0.992 |
0.000 |
0.867 |
0.000 |
0.000 |
0.000 |
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Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 139 | 139 | 124 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.89 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.115 |
0.183 |
1.064 |
0.000 |
0.608 |
0.000 |
0.000 |
0.000 |
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Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 139 | 167 | 124 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.20 | 0.89 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.118 |
0.062 |
1.119 |
0.000 |
0.717 |
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 [44] had the largest ratio of
[.1935]
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| Table 2.1:Rubi specific breakdown of results for each
integral
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# |
grade |
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normalized | antiderivative |
leaf size | |
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\(\frac {\text {number of rules}}{\text {integrand leaf size}}\) |
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1 |
A |
12 |
3 |
1.00 |
29 |
0.103 |
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2 |
A |
10 |
3 |
1.00 |
29 |
0.103 |
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3 |
A |
8 |
3 |
1.00 |
27 |
0.111 |
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4 |
A |
6 |
3 |
1.00 |
20 |
0.150 |
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5 |
A |
5 |
4 |
1.00 |
29 |
0.138 |
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6 |
A |
3 |
3 |
1.00 |
29 |
0.103 |
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7 |
A |
3 |
3 |
1.00 |
29 |
0.103 |
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8 |
A |
14 |
3 |
1.00 |
31 |
0.097 |
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9 |
A |
12 |
3 |
1.00 |
31 |
0.097 |
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10 |
A |
10 |
3 |
1.00 |
29 |
0.103 |
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11 |
A |
8 |
3 |
1.00 |
22 |
0.136 |
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12 |
A |
7 |
4 |
1.00 |
31 |
0.129 |
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13 |
A |
6 |
5 |
1.00 |
31 |
0.161 |
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14 |
A |
4 |
3 |
1.00 |
31 |
0.097 |
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15 |
A |
16 |
3 |
1.00 |
31 |
0.097 |
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16 |
A |
14 |
3 |
1.00 |
31 |
0.097 |
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17 |
A |
12 |
3 |
1.00 |
29 |
0.103 |
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18 |
A |
10 |
3 |
1.00 |
22 |
0.136 |
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19 |
A |
9 |
4 |
1.00 |
31 |
0.129 |
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20 |
A |
8 |
5 |
0.99 |
31 |
0.161 |
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21 |
A |
11 |
4 |
1.00 |
31 |
0.129 |
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22 |
A |
9 |
4 |
1.00 |
31 |
0.129 |
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23 |
A |
7 |
4 |
1.00 |
31 |
0.129 |
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24 |
A |
5 |
4 |
1.00 |
29 |
0.138 |
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25 |
A |
2 |
2 |
1.00 |
22 |
0.091 |
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26 |
A |
4 |
2 |
1.00 |
31 |
0.065 |
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27 |
A |
5 |
3 |
1.00 |
31 |
0.097 |
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28 |
A |
6 |
3 |
1.00 |
31 |
0.097 |
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29 |
A |
8 |
5 |
0.99 |
31 |
0.161 |
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30 |
A |
6 |
5 |
1.00 |
31 |
0.161 |
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31 |
A |
3 |
3 |
1.00 |
29 |
0.103 |
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32 |
A |
2 |
2 |
1.00 |
22 |
0.091 |
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33 |
A |
5 |
3 |
1.00 |
31 |
0.097 |
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34 |
A |
6 |
3 |
1.00 |
31 |
0.097 |
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35 |
A |
7 |
3 |
1.00 |
31 |
0.097 |
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36 |
A |
4 |
3 |
1.00 |
31 |
0.097 |
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37 |
A |
3 |
3 |
1.00 |
29 |
0.103 |
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38 |
A |
2 |
2 |
1.00 |
22 |
0.091 |
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39 |
A |
6 |
3 |
1.00 |
31 |
0.097 |
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40 |
A |
7 |
3 |
1.00 |
31 |
0.097 |
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41 |
A |
7 |
3 |
1.00 |
31 |
0.097 |
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42 |
A |
4 |
4 |
0.94 |
29 |
0.138 |
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43 |
A |
6 |
5 |
1.00 |
31 |
0.161 |
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44 |
A |
7 |
6 |
1.00 |
31 |
0.194 |
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45 |
A |
4 |
4 |
1.00 |
47 |
0.085 |
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46 |
A |
4 |
4 |
1.20 |
55 |
0.073 |
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