Chapter 2
detailed summary tables of results
2.1 List of integrals sorted by grade for each CAS
2.1.1 Rubi
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 27, 28, 32, 33, 35, 37, 38,
41, 42, 43, 47, 48, 52, 53, 54, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70 }
B grade: { 31, 36, 55, 56 }
C grade: { 6, 24, 25, 26, 29, 30, 44, 45, 46, 49, 50, 51, 57 }
F grade: { 34, 39, 40 }
2.1.3 Maple
A grade: { 5, 6, 12, 14, 23, 25, 26, 27, 28, 29, 30, 33, 35, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 54, 55, 63,
64, 65, 66, 67, 68, 69, 70 }
B grade: { 1, 2, 3, 4, 7, 8, 9, 11, 13, 16, 19, 21, 22, 24, 31, 32, 38, 53, 60, 61, 62 }
C grade: { 10, 15, 17, 18, 20, 34, 36, 37, 39, 40, 52, 56, 57, 58, 59 }
F grade: { 41 }
2.1.4 Maxima
A grade: { 5, 24, 25, 26, 27, 29, 30, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 55, 60, 65, 66, 69, 70
}
B grade: { 1, 2, 3, 6, 7, 9, 12, 14, 21, 22, 53, 54, 56, 61 }
C grade: { }
F grade: { 4, 8, 10, 11, 13, 15, 16, 17, 18, 19, 20, 23, 28, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 52, 57, 58,
59, 62, 63, 64, 67, 68 }
2.1.5 FriCAS
A grade: { 3, 5, 6, 9, 12, 24, 25, 26, 27, 29, 42, 43, 44, 45, 46, 47, 49, 50, 51, 65, 66, 69, 70 }
B grade: { 1, 2, 7, 14, 30 }
C grade: { }
F grade: { 4, 8, 10, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 28, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 48,
52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68 }
2.1.6 Sympy
A grade: { 1, 2, 3, 5, 6, 7, 9, 12, 23, 24, 25, 26, 27, 44, 45, 46, 47, 65, 66, 69 }
B grade: { 49, 50, 51 }
C grade: { }
F grade: { 4, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 48, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68, 70 }
2.1.7 Giac
A grade: { 27, 42, 43, 47, 65, 66 }
B grade: { }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62,
63, 64, 67, 68, 69, 70 }
2.1.8 Mupad
A grade: { 23, 42, 43, 65, 66, 69, 70 }
B grade: { 1, 2, 3, 5, 6, 7, 9, 12, 14, 21, 24, 25, 26, 27, 29, 30, 44, 45, 46, 47, 49, 50, 51 }
C grade: { }
F grade: { 4, 8, 10, 11, 13, 15, 16, 17, 18, 19, 20, 22, 28, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 48, 52, 53,
54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68 }
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 |
B |
B |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 72 | 72 | 56 | 225 | 370 | 151 | 231 | 0 | 371 |
normalized size | 1 | 1.00 | 0.78 | 3.12 | 5.14 | 2.10 | 3.21 | 0.00 | 5.15 |
time (sec) |
N/A |
0.251 |
0.022 |
0.038 |
0.419 |
0.511 |
4.669 |
0.000 |
0.654 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
B |
B |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 54 | 161 | 238 | 129 | 182 | 0 | 144 |
normalized size | 1 | 1.00 | 0.81 | 2.40 | 3.55 | 1.93 | 2.72 | 0.00 | 2.15 |
time (sec) |
N/A |
0.055 |
0.023 |
0.040 |
0.421 |
0.656 |
3.929 |
0.000 |
1.170 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
B |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 40 | 92 | 120 | 60 | 95 | 0 | 73 |
normalized size | 1 | 1.00 | 0.83 | 1.92 | 2.50 | 1.25 | 1.98 | 0.00 | 1.52 |
time (sec) |
N/A |
0.030 |
0.016 |
0.038 |
0.420 |
0.577 |
2.014 |
0.000 |
1.454 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 52 | 132 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.83 | 2.10 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.058 |
0.018 |
0.058 |
0.000 |
0.539 |
0.000 |
0.000 |
0.000 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 50 | 73 | 92 | 75 | 284 | 0 | 88 |
normalized size | 1 | 1.00 | 0.82 | 1.20 | 1.51 | 1.23 | 4.66 | 0.00 | 1.44 |
time (sec) |
N/A |
0.047 |
0.027 |
0.044 |
0.316 |
0.512 |
6.886 |
0.000 |
0.665 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
A |
B |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 51 | 71 | 120 | 70 | 314 | 0 | 103 |
normalized size | 1 | 1.00 | 0.81 | 1.13 | 1.90 | 1.11 | 4.98 | 0.00 | 1.63 |
time (sec) |
N/A |
0.045 |
0.022 |
0.043 |
0.424 |
0.565 |
12.389 |
0.000 |
0.754 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
B |
B |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 216 | 543 | 597 | 337 | 583 | 0 | 633 |
normalized size | 1 | 1.00 | 1.38 | 3.46 | 3.80 | 2.15 | 3.71 | 0.00 | 4.03 |
time (sec) |
N/A |
0.221 |
0.120 |
0.057 |
1.802 |
0.647 |
11.420 |
0.000 |
3.263 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 183 | 183 | 163 | 593 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.89 | 3.24 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.221 |
0.431 |
0.203 |
0.000 |
0.671 |
0.000 |
0.000 |
0.000 |
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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 |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 95 | 95 | 107 | 220 | 218 | 150 | 240 | 0 | 216 |
normalized size | 1 | 1.00 | 1.13 | 2.32 | 2.29 | 1.58 | 2.53 | 0.00 | 2.27 |
time (sec) |
N/A |
0.119 |
0.076 |
0.056 |
1.488 |
0.721 |
4.303 |
0.000 |
1.611 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 183 | 183 | 170 | 1433 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.93 | 7.83 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.339 |
0.079 |
0.775 |
0.000 |
0.415 |
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 |
F(-1) |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 119 | 119 | 135 | 471 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.13 | 3.96 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.187 |
0.229 |
0.172 |
0.000 |
0.452 |
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 |
A |
A |
B |
A |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 117 | 117 | 194 | 182 | 268 | 209 | 1144 | 0 | 232 |
normalized size | 1 | 1.00 | 1.66 | 1.56 | 2.29 | 1.79 | 9.78 | 0.00 | 1.98 |
time (sec) |
N/A |
0.153 |
0.132 |
0.057 |
0.538 |
0.436 |
21.956 |
0.000 |
2.857 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
F(-1) |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 194 | 194 | 163 | 547 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.84 | 2.82 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.257 |
0.750 |
0.143 |
0.000 |
0.472 |
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 |
A |
B |
B |
F(-1) |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 170 | 170 | 245 | 242 | 534 | 448 | 0 | 0 | 438 |
normalized size | 1 | 1.00 | 1.44 | 1.42 | 3.14 | 2.64 | 0.00 | 0.00 | 2.58 |
time (sec) |
N/A |
0.227 |
0.351 |
0.065 |
0.511 |
0.475 |
0.000 |
0.000 |
3.645 |
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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 | 271 | 271 | 349 | 3242 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.29 | 11.96 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.439 |
0.766 |
2.247 |
0.000 |
0.469 |
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 |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 164 | 164 | 196 | 567 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.20 | 3.46 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.243 |
0.522 |
0.145 |
0.000 |
0.455 |
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(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 279 | 279 | 252 | 2894 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.90 | 10.37 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.458 |
0.153 |
0.280 |
0.000 |
0.463 |
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 |
A |
C |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 163 | 163 | 263 | 2696 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.61 | 16.54 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.300 |
0.690 |
0.442 |
0.000 |
0.459 |
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 |
A |
B |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 180 | 180 | 225 | 631 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.25 | 3.51 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.319 |
0.307 |
0.159 |
0.000 |
0.445 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
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|
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|
|
|
|
|
|
|
|
|
Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F(-1) |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 287 | 287 | 360 | 7083 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.25 | 24.68 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.500 |
1.225 |
1.626 |
0.000 |
0.444 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
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|
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|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
F |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 31 | 68 | 44 | 0 | 0 | 0 | 25 |
normalized size | 1 | 1.00 | 1.00 | 2.19 | 1.42 | 0.00 | 0.00 | 0.00 | 0.81 |
time (sec) |
N/A |
0.038 |
0.004 |
0.062 |
0.456 |
0.461 |
0.000 |
0.000 |
0.081 |
|
|
|
|
|
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|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 41 | 34 | 98 | 123 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.83 | 2.39 | 3.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.045 |
0.008 |
0.061 |
0.486 |
0.494 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
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|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-2) |
F(-2) |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 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 | -0.05 |
time (sec) |
N/A |
0.017 |
6.110 |
3.232 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
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|
|
|
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|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 233 | 233 | 157 | 494 | 346 | 317 | 654 | 0 | 787 |
normalized size | 1 | 1.00 | 0.67 | 2.12 | 1.48 | 1.36 | 2.81 | 0.00 | 3.38 |
time (sec) |
N/A |
0.381 |
0.298 |
0.047 |
0.428 |
0.476 |
25.363 |
0.000 |
1.019 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 155 | 155 | 118 | 283 | 220 | 199 | 376 | 0 | 411 |
normalized size | 1 | 1.00 | 0.76 | 1.83 | 1.42 | 1.28 | 2.43 | 0.00 | 2.65 |
time (sec) |
N/A |
0.190 |
0.160 |
0.054 |
0.427 |
0.468 |
10.655 |
0.000 |
0.782 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 97 | 97 | 163 | 146 | 116 | 103 | 177 | 0 | 136 |
normalized size | 1 | 1.00 | 1.68 | 1.51 | 1.20 | 1.06 | 1.82 | 0.00 | 1.40 |
time (sec) |
N/A |
0.112 |
0.065 |
0.059 |
0.420 |
0.456 |
4.545 |
0.000 |
1.804 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
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 | 38 | 38 | 49 | 42 | 36 | 48 | 51 | 36 | 49 |
normalized size | 1 | 1.00 | 1.29 | 1.11 | 0.95 | 1.26 | 1.34 | 0.95 | 1.29 |
time (sec) |
N/A |
0.018 |
0.016 |
0.036 |
0.317 |
0.430 |
0.366 |
0.106 |
1.103 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
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 | 162 | 162 | 160 | 224 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.99 | 1.38 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.150 |
0.122 |
0.099 |
0.000 |
0.441 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
F(-1) |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 151 | 151 | 121 | 205 | 177 | 190 | 0 | 0 | 127 |
normalized size | 1 | 1.00 | 0.80 | 1.36 | 1.17 | 1.26 | 0.00 | 0.00 | 0.84 |
time (sec) |
N/A |
0.121 |
0.206 |
0.049 |
0.417 |
0.695 |
0.000 |
0.000 |
1.829 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
B |
F(-1) |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 227 | 227 | 175 | 438 | 409 | 682 | 0 | 0 | 399 |
normalized size | 1 | 1.00 | 0.77 | 1.93 | 1.80 | 3.00 | 0.00 | 0.00 | 1.76 |
time (sec) |
N/A |
0.303 |
0.795 |
0.051 |
0.427 |
2.129 |
0.000 |
0.000 |
7.534 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 382 | 382 | 801 | 1622 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.10 | 4.25 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.574 |
4.230 |
0.146 |
0.000 |
0.419 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 222 | 222 | 264 | 748 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.19 | 3.37 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.373 |
0.455 |
0.135 |
0.000 |
0.427 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
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 | 102 | 102 | 109 | 180 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.07 | 1.76 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.107 |
0.095 |
0.333 |
0.000 |
0.437 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
C |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 261 | 261 | 0 | 2149 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.00 | 8.23 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.165 |
6.510 |
2.312 |
0.000 |
0.403 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 568 | 568 | 419 | 1087 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.74 | 1.91 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.351 |
7.492 |
0.148 |
0.000 |
0.471 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 564 | 564 | 1844 | 6682 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 3.27 | 11.85 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.937 |
9.978 |
10.281 |
0.000 |
0.452 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 337 | 337 | 592 | 16362 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.76 | 48.55 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.632 |
0.819 |
2.493 |
0.000 |
0.433 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 143 | 143 | 266 | 359 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.86 | 2.51 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.214 |
0.154 |
0.421 |
0.000 |
0.437 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
C |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 372 | 372 | 0 | 4389 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.00 | 11.80 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.203 |
8.947 |
0.691 |
0.000 |
0.545 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
C |
F |
F |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1233 | 1233 | 0 | 4764 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.00 | 3.86 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
2.311 |
17.431 |
1.798 |
0.000 |
0.474 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-1) |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 177 | 177 | 162 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.92 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.246 |
0.393 |
1.958 |
0.000 |
0.430 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
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 |
0.058 |
4.995 |
1.479 |
0.000 |
0.411 |
0.000 |
0.000 |
0.000 |
|
|
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|
Problem 43 |
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 |
0.056 |
0.507 |
1.434 |
0.000 |
0.422 |
0.000 |
0.000 |
0.000 |
|
|
|
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Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 95 | 132 | 104 | 87 | 155 | 0 | 133 |
normalized size | 1 | 1.00 | 0.90 | 1.25 | 0.98 | 0.82 | 1.46 | 0.00 | 1.25 |
time (sec) |
N/A |
0.111 |
0.077 |
0.040 |
0.409 |
0.414 |
1.580 |
0.000 |
0.594 |
|
|
|
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Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 114 | 95 | 85 | 66 | 117 | 0 | 102 |
normalized size | 1 | 1.00 | 1.44 | 1.20 | 1.08 | 0.84 | 1.48 | 0.00 | 1.29 |
time (sec) |
N/A |
0.092 |
0.057 |
0.043 |
0.412 |
0.411 |
1.154 |
0.000 |
0.860 |
|
|
|
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Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 90 | 66 | 68 | 52 | 78 | 0 | 61 |
normalized size | 1 | 1.00 | 1.50 | 1.10 | 1.13 | 0.87 | 1.30 | 0.00 | 1.02 |
time (sec) |
N/A |
0.055 |
0.036 |
0.038 |
0.410 |
0.407 |
0.643 |
0.000 |
0.970 |
|
|
|
|
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|
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Problem 47 |
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 | 33 | 33 | 39 | 36 | 31 | 39 | 46 | 31 | 42 |
normalized size | 1 | 1.00 | 1.18 | 1.09 | 0.94 | 1.18 | 1.39 | 0.94 | 1.27 |
time (sec) |
N/A |
0.011 |
0.016 |
0.035 |
0.309 |
0.400 |
0.393 |
0.126 |
0.446 |
|
|
|
|
|
|
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|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 120 | 120 | 171 | 103 | 134 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.42 | 0.86 | 1.12 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.106 |
0.009 |
0.058 |
0.470 |
0.393 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
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|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 62 | 62 | 67 | 63 | 77 | 57 | 168 | 0 | 63 |
normalized size | 1 | 1.00 | 1.08 | 1.02 | 1.24 | 0.92 | 2.71 | 0.00 | 1.02 |
time (sec) |
N/A |
0.039 |
0.062 |
0.046 |
0.411 |
0.414 |
1.886 |
0.000 |
1.042 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 96 | 96 | 92 | 105 | 112 | 95 | 382 | 0 | 232 |
normalized size | 1 | 1.00 | 0.96 | 1.09 | 1.17 | 0.99 | 3.98 | 0.00 | 2.42 |
time (sec) |
N/A |
0.083 |
0.105 |
0.047 |
0.409 |
0.498 |
2.920 |
0.000 |
1.220 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 129 | 129 | 128 | 162 | 165 | 135 | 760 | 0 | 288 |
normalized size | 1 | 1.00 | 0.99 | 1.26 | 1.28 | 1.05 | 5.89 | 0.00 | 2.23 |
time (sec) |
N/A |
0.114 |
0.149 |
0.048 |
0.415 |
0.449 |
4.656 |
0.000 |
1.048 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 863 | 863 | 701 | 631 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.81 | 0.73 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.207 |
0.913 |
1.312 |
0.000 |
0.628 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 543 | 543 | 409 | 2192 | 8520 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.75 | 4.04 | 15.69 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.607 |
0.362 |
1.286 |
5.931 |
0.440 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 152 | 152 | 231 | 198 | 284 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.52 | 1.30 | 1.87 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.138 |
0.023 |
0.062 |
0.530 |
0.443 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 244 | 244 | 771 | 317 | 284 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 3.16 | 1.30 | 1.16 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.239 |
11.805 |
0.067 |
0.552 |
0.458 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
C |
B |
F |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 668 | 668 | 1536 | 53434 | 8518 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.30 | 79.99 | 12.75 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.855 |
25.833 |
2.698 |
1.349 |
0.411 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 933 | 933 | 933 | 682 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.73 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.367 |
7.384 |
1.235 |
0.000 |
0.530 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 673 | 673 | 604 | 344 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.90 | 0.51 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.898 |
0.596 |
0.333 |
0.000 |
0.477 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 770 | 770 | 666 | 377 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.86 | 0.49 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.982 |
0.827 |
0.314 |
0.000 |
0.585 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 274 | 274 | 283 | 833 | 328 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.03 | 3.04 | 1.20 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.271 |
0.060 |
1.056 |
0.546 |
0.430 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 543 | 543 | 409 | 2192 | 14300 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.75 | 4.04 | 26.34 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.600 |
0.400 |
1.244 |
2.840 |
0.427 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 367 | 367 | 443 | 4743 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.21 | 12.92 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.673 |
0.463 |
1.622 |
0.000 |
0.462 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
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 | 132 | 132 | 97 | 143 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.73 | 1.08 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.096 |
0.117 |
0.684 |
0.000 |
0.439 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 64 |
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 | 216 | 216 | 125 | 176 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.58 | 0.81 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.162 |
0.069 |
0.927 |
0.000 |
0.436 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 163 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 7.09 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.038 |
0.422 |
1.473 |
0.000 |
0.424 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 66 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 0 | 165 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 6.60 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.051 |
0.100 |
1.376 |
0.000 |
0.448 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 187 | 187 | 145 | 187 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.78 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.216 |
0.724 |
3.160 |
0.000 |
0.452 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 281 | 281 | 189 | 222 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.67 | 0.79 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.331 |
0.162 |
4.530 |
0.000 |
0.470 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 30 | 0 | 181 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 6.03 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.03 |
time (sec) |
N/A |
0.131 |
1.577 |
4.682 |
0.000 |
0.441 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
F(-1) |
A |
verified |
N/A |
N/A |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 32 | 0 | 225 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 7.03 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.03 |
time (sec) |
N/A |
0.179 |
0.800 |
4.702 |
0.000 |
0.457 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
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 [35] had the largest ratio of
[1.250]
|
|
|
|
|
|
|
| Table 2.1:Rubi specific breakdown of results for each
integral
|
|
|
|
|
|
|
# |
grade |
|
|
normalized | antiderivative |
leaf size | |
|
\(\frac {\text {number of rules}}{\text {integrand leaf size}}\) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
A |
6 |
5 |
1.00 |
21 |
0.238 |
|
|
|
|
|
|
|
2 |
A |
6 |
5 |
1.00 |
21 |
0.238 |
|
|
|
|
|
|
|
3 |
A |
5 |
5 |
1.00 |
19 |
0.263 |
|
|
|
|
|
|
|
4 |
A |
5 |
4 |
1.00 |
21 |
0.190 |
|
|
|
|
|
|
|
5 |
A |
7 |
7 |
1.00 |
21 |
0.333 |
|
|
|
|
|
|
|
6 |
A |
5 |
5 |
1.00 |
21 |
0.238 |
|
|
|
|
|
|
|
7 |
A |
13 |
9 |
1.00 |
23 |
0.391 |
|
|
|
|
|
|
|
8 |
A |
11 |
10 |
1.00 |
23 |
0.435 |
|
|
|
|
|
|
|
9 |
A |
8 |
7 |
1.00 |
21 |
0.333 |
|
|
|
|
|
|
|
10 |
A |
8 |
7 |
1.00 |
23 |
0.304 |
|
|
|
|
|
|
|
11 |
A |
6 |
6 |
1.00 |
23 |
0.261 |
|
|
|
|
|
|
|
12 |
A |
10 |
9 |
1.00 |
23 |
0.391 |
|
|
|
|
|
|
|
13 |
A |
10 |
9 |
1.00 |
23 |
0.391 |
|
|
|
|
|
|
|
14 |
A |
15 |
10 |
1.00 |
23 |
0.435 |
|
|
|
|
|
|
|
15 |
A |
14 |
11 |
1.00 |
23 |
0.478 |
|
|
|
|
|
|
|
16 |
A |
10 |
10 |
1.00 |
21 |
0.476 |
|
|
|
|
|
|
|
17 |
A |
10 |
8 |
1.00 |
23 |
0.348 |
|
|
|
|
|
|
|
18 |
A |
7 |
8 |
1.00 |
23 |
0.348 |
|
|
|
|
|
|
|
19 |
A |
9 |
8 |
1.00 |
23 |
0.348 |
|
|
|
|
|
|
|
20 |
A |
16 |
13 |
1.00 |
23 |
0.565 |
|
|
|
|
|
|
|
21 |
A |
5 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
22 |
A |
5 |
4 |
1.00 |
19 |
0.210 |
|
|
|
|
|
|
|
23 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
24 |
A |
7 |
6 |
1.00 |
18 |
0.333 |
|
|
|
|
|
|
|
25 |
A |
7 |
6 |
1.00 |
18 |
0.333 |
|
|
|
|
|
|
|
26 |
A |
7 |
6 |
1.00 |
16 |
0.375 |
|
|
|
|
|
|
|
27 |
A |
4 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
28 |
A |
5 |
5 |
1.00 |
18 |
0.278 |
|
|
|
|
|
|
|
29 |
A |
8 |
8 |
1.00 |
18 |
0.444 |
|
|
|
|
|
|
|
30 |
A |
9 |
8 |
1.00 |
18 |
0.444 |
|
|
|
|
|
|
|
31 |
A |
16 |
13 |
1.00 |
20 |
0.650 |
|
|
|
|
|
|
|
32 |
A |
13 |
10 |
1.00 |
18 |
0.556 |
|
|
|
|
|
|
|
33 |
A |
6 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
34 |
A |
2 |
2 |
1.00 |
20 |
0.100 |
|
|
|
|
|
|
|
35 |
A |
25 |
25 |
1.00 |
20 |
1.250 |
|
|
|
|
|
|
|
36 |
A |
21 |
14 |
1.00 |
20 |
0.700 |
|
|
|
|
|
|
|
37 |
A |
15 |
11 |
1.00 |
18 |
0.611 |
|
|
|
|
|
|
|
38 |
A |
6 |
7 |
1.00 |
12 |
0.583 |
|
|
|
|
|
|
|
39 |
A |
2 |
2 |
1.00 |
20 |
0.100 |
|
|
|
|
|
|
|
40 |
A |
35 |
22 |
1.00 |
20 |
1.100 |
|
|
|
|
|
|
|
41 |
A |
6 |
4 |
1.00 |
18 |
0.222 |
|
|
|
|
|
|
|
42 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
43 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
44 |
A |
7 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
45 |
A |
7 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
46 |
A |
7 |
6 |
1.00 |
8 |
0.750 |
|
|
|
|
|
|
|
47 |
A |
3 |
3 |
1.00 |
6 |
0.500 |
|
|
|
|
|
|
|
48 |
A |
5 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
49 |
A |
7 |
7 |
1.00 |
10 |
0.700 |
|
|
|
|
|
|
|
50 |
A |
8 |
7 |
1.00 |
10 |
0.700 |
|
|
|
|
|
|
|
51 |
A |
8 |
7 |
1.00 |
10 |
0.700 |
|
|
|
|
|
|
|
52 |
A |
23 |
5 |
1.00 |
16 |
0.312 |
|
|
|
|
|
|
|
53 |
A |
17 |
5 |
1.00 |
16 |
0.312 |
|
|
|
|
|
|
|
54 |
A |
5 |
5 |
1.00 |
14 |
0.357 |
|
|
|
|
|
|
|
55 |
A |
15 |
7 |
1.00 |
16 |
0.438 |
|
|
|
|
|
|
|
56 |
A |
25 |
7 |
1.00 |
16 |
0.438 |
|
|
|
|
|
|
|
57 |
A |
31 |
7 |
1.00 |
16 |
0.438 |
|
|
|
|
|
|
|
58 |
A |
31 |
13 |
1.00 |
18 |
0.722 |
|
|
|
|
|
|
|
59 |
A |
37 |
16 |
1.00 |
18 |
0.889 |
|
|
|
|
|
|
|
60 |
A |
17 |
5 |
1.00 |
14 |
0.357 |
|
|
|
|
|
|
|
61 |
A |
17 |
5 |
1.00 |
16 |
0.312 |
|
|
|
|
|
|
|
62 |
A |
12 |
8 |
1.00 |
19 |
0.421 |
|
|
|
|
|
|
|
63 |
A |
2 |
2 |
1.00 |
28 |
0.071 |
|
|
|
|
|
|
|
64 |
A |
3 |
3 |
1.00 |
33 |
0.091 |
|
|
|
|
|
|
|
65 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
66 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
67 |
A |
4 |
4 |
1.00 |
35 |
0.114 |
|
|
|
|
|
|
|
68 |
A |
5 |
5 |
1.00 |
40 |
0.125 |
|
|
|
|
|
|
|
69 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
70 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|