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, 71, 72 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 9, 11, 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, 59, 60, 61, 62, 64, 65,
66, 67, 68, 69, 70, 71, 72 }
B grade: { 4, 58, 63 }
C grade: { 5, 6, 7, 8, 10, 12 }
F grade: { }
2.1.3 Maple
A grade: { 1, 2, 3, 4, 5, 6, 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 54, 56, 57 }
B grade: { 7, 20, 21, 24, 25, 28, 29, 32, 33, 50, 51, 52, 53, 55 }
C grade: { }
F grade: { 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72
}
2.1.4 Maxima
A grade: { 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48 }
B grade: { 20, 21, 24, 25, 28, 29, 32, 33, 37, 41, 45, 49 }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61,
62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72 }
2.1.5 FriCAS
A grade: { 1, 2, 3, 5, 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 }
B grade: { 4, 6, 7 }
C grade: { }
F grade: { 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,
67, 68, 69, 70, 71, 72 }
2.1.6 Sympy
A grade: { 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 }
B grade: { }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61,
62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72 }
2.1.7 Giac
A grade: { 3, 4, 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 }
B grade: { 1, 2, 5, 6, 7 }
C grade: { }
F grade: { 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,
67, 68, 69, 70, 71, 72 }
2.1.8 Mupad
A grade: { }
B grade: { 4, 5, 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, 53, 54 }
C grade: { }
F grade: { 1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 50, 51, 52, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64,
65, 66, 67, 68, 69, 70, 71, 72 }
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 |
A |
F |
A |
F(-1) |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 119 | 119 | 154 | 75 | 0 | 140 | 0 | 240 | -1 |
normalized size | 1 | 1.00 | 1.29 | 0.63 | 0.00 | 1.18 | 0.00 | 2.02 | -0.01 |
time (sec) |
N/A |
0.071 |
0.724 |
0.152 |
0.000 |
0.429 |
0.000 |
0.977 |
0.000 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 117 | 65 | 0 | 115 | 0 | 174 | -1 |
normalized size | 1 | 1.00 | 1.31 | 0.73 | 0.00 | 1.29 | 0.00 | 1.96 | -0.01 |
time (sec) |
N/A |
0.047 |
0.328 |
0.125 |
0.000 |
0.425 |
0.000 |
2.431 |
0.000 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
F |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 89 | 53 | 0 | 76 | 0 | 101 | -1 |
normalized size | 1 | 1.00 | 1.51 | 0.90 | 0.00 | 1.29 | 0.00 | 1.71 | -0.02 |
time (sec) |
N/A |
0.029 |
0.148 |
0.125 |
0.000 |
0.476 |
0.000 |
0.974 |
0.000 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
B |
A |
F |
B |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 65 | 43 | 0 | 50 | 0 | 36 | 33 |
normalized size | 1 | 1.00 | 2.50 | 1.65 | 0.00 | 1.92 | 0.00 | 1.38 | 1.27 |
time (sec) |
N/A |
0.013 |
0.038 |
0.114 |
0.000 |
0.477 |
0.000 |
0.478 |
0.218 |
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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 |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 73 | 75 | 0 | 167 | 0 | 111 | 49 |
normalized size | 1 | 1.00 | 1.55 | 1.60 | 0.00 | 3.55 | 0.00 | 2.36 | 1.04 |
time (sec) |
N/A |
0.025 |
0.074 |
0.116 |
0.000 |
0.460 |
0.000 |
0.755 |
0.386 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
A |
F |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 108 | 125 | 0 | 252 | 0 | 293 | -1 |
normalized size | 1 | 1.00 | 1.40 | 1.62 | 0.00 | 3.27 | 0.00 | 3.81 | -0.01 |
time (sec) |
N/A |
0.042 |
0.178 |
0.134 |
0.000 |
0.458 |
0.000 |
1.464 |
0.000 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
B |
F |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 107 | 107 | 196 | 195 | 0 | 320 | 0 | 457 | -1 |
normalized size | 1 | 1.00 | 1.83 | 1.82 | 0.00 | 2.99 | 0.00 | 4.27 | -0.01 |
time (sec) |
N/A |
0.063 |
0.166 |
0.169 |
0.000 |
0.485 |
0.000 |
1.150 |
0.000 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 341 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 5.09 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.036 |
2.705 |
0.099 |
0.000 |
0.490 |
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 |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 124 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.88 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.034 |
0.222 |
0.213 |
0.000 |
0.454 |
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 |
C |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 270 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 4.09 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.031 |
2.615 |
0.161 |
0.000 |
0.461 |
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 |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 70 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.06 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.030 |
0.113 |
0.185 |
0.000 |
0.438 |
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 |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 604 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 9.15 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.033 |
6.107 |
0.165 |
0.000 |
0.429 |
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 |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 130 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.88 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.034 |
0.294 |
0.093 |
0.000 |
0.442 |
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 |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 90 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.22 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.038 |
0.193 |
0.489 |
0.000 |
0.442 |
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 |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 91 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.23 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.034 |
0.151 |
0.533 |
0.000 |
0.449 |
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 |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 90 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.50 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.018 |
0.113 |
0.414 |
0.000 |
0.429 |
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 |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 90 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.53 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.016 |
0.135 |
0.367 |
0.000 |
0.438 |
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 |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 56 | 20 | 26 | 26 | 46 | 49 | 40 |
normalized size | 1 | 1.00 | 1.81 | 0.65 | 0.84 | 0.84 | 1.48 | 1.58 | 1.29 |
time (sec) |
N/A |
0.013 |
0.027 |
0.049 |
0.802 |
0.435 |
0.743 |
1.984 |
4.728 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 91 | 92 | 93 | 59 | 388 | 95 | 83 |
normalized size | 1 | 1.00 | 1.62 | 1.64 | 1.66 | 1.05 | 6.93 | 1.70 | 1.48 |
time (sec) |
N/A |
0.034 |
0.133 |
0.092 |
0.512 |
0.443 |
2.062 |
2.004 |
4.524 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 81 | 81 | 113 | 184 | 173 | 94 | 918 | 121 | 111 |
normalized size | 1 | 1.00 | 1.40 | 2.27 | 2.14 | 1.16 | 11.33 | 1.49 | 1.37 |
time (sec) |
N/A |
0.061 |
0.329 |
0.099 |
0.542 |
0.433 |
4.721 |
1.184 |
4.603 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 133 | 272 | 253 | 130 | 1693 | 147 | 187 |
normalized size | 1 | 1.00 | 1.25 | 2.57 | 2.39 | 1.23 | 15.97 | 1.39 | 1.76 |
time (sec) |
N/A |
0.098 |
0.459 |
0.093 |
0.729 |
0.432 |
10.831 |
0.542 |
5.063 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
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 | 56 | 20 | 26 | 26 | 46 | 50 | 40 |
normalized size | 1 | 1.00 | 1.70 | 0.61 | 0.79 | 0.79 | 1.39 | 1.52 | 1.21 |
time (sec) |
N/A |
0.013 |
0.028 |
0.054 |
0.475 |
0.420 |
0.777 |
0.317 |
4.526 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 58 | 58 | 91 | 92 | 93 | 59 | 384 | 96 | 83 |
normalized size | 1 | 1.00 | 1.57 | 1.59 | 1.60 | 1.02 | 6.62 | 1.66 | 1.43 |
time (sec) |
N/A |
0.031 |
0.144 |
0.086 |
0.678 |
0.444 |
2.051 |
0.772 |
4.321 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 83 | 83 | 113 | 184 | 173 | 94 | 915 | 122 | 111 |
normalized size | 1 | 1.00 | 1.36 | 2.22 | 2.08 | 1.13 | 11.02 | 1.47 | 1.34 |
time (sec) |
N/A |
0.063 |
0.445 |
0.087 |
0.544 |
0.440 |
4.713 |
0.852 |
4.765 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 108 | 108 | 133 | 272 | 253 | 130 | 1690 | 148 | 187 |
normalized size | 1 | 1.00 | 1.23 | 2.52 | 2.34 | 1.20 | 15.65 | 1.37 | 1.73 |
time (sec) |
N/A |
0.096 |
0.480 |
0.095 |
0.547 |
0.482 |
10.702 |
1.308 |
5.085 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
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 | 56 | 20 | 26 | 26 | 48 | 50 | 40 |
normalized size | 1 | 1.00 | 1.70 | 0.61 | 0.79 | 0.79 | 1.45 | 1.52 | 1.21 |
time (sec) |
N/A |
0.012 |
0.024 |
0.051 |
0.770 |
0.442 |
0.734 |
0.470 |
4.497 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 58 | 58 | 91 | 92 | 93 | 59 | 384 | 96 | 83 |
normalized size | 1 | 1.00 | 1.57 | 1.59 | 1.60 | 1.02 | 6.62 | 1.66 | 1.43 |
time (sec) |
N/A |
0.032 |
0.024 |
0.092 |
0.524 |
0.444 |
2.021 |
0.511 |
0.002 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 83 | 83 | 113 | 184 | 173 | 94 | 915 | 122 | 112 |
normalized size | 1 | 1.00 | 1.36 | 2.22 | 2.08 | 1.13 | 11.02 | 1.47 | 1.35 |
time (sec) |
N/A |
0.063 |
0.288 |
0.099 |
0.470 |
0.429 |
4.729 |
0.329 |
4.645 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 108 | 108 | 133 | 272 | 253 | 130 | 1690 | 148 | 187 |
normalized size | 1 | 1.00 | 1.23 | 2.52 | 2.34 | 1.20 | 15.65 | 1.37 | 1.73 |
time (sec) |
N/A |
0.095 |
0.098 |
0.094 |
0.736 |
0.430 |
10.779 |
0.392 |
0.002 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
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 | 31 | 31 | 56 | 20 | 26 | 26 | 49 | 49 | 40 |
normalized size | 1 | 1.00 | 1.81 | 0.65 | 0.84 | 0.84 | 1.58 | 1.58 | 1.29 |
time (sec) |
N/A |
0.012 |
0.026 |
0.051 |
0.441 |
0.420 |
0.751 |
0.316 |
4.513 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
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 | 56 | 56 | 91 | 92 | 93 | 59 | 389 | 95 | 83 |
normalized size | 1 | 1.00 | 1.62 | 1.64 | 1.66 | 1.05 | 6.95 | 1.70 | 1.48 |
time (sec) |
N/A |
0.031 |
0.116 |
0.092 |
0.417 |
0.429 |
2.076 |
3.439 |
0.002 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 81 | 81 | 114 | 184 | 173 | 94 | 921 | 121 | 112 |
normalized size | 1 | 1.00 | 1.41 | 2.27 | 2.14 | 1.16 | 11.37 | 1.49 | 1.38 |
time (sec) |
N/A |
0.061 |
0.359 |
0.088 |
0.704 |
0.473 |
4.751 |
1.432 |
4.653 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 133 | 272 | 253 | 130 | 1695 | 147 | 187 |
normalized size | 1 | 1.00 | 1.25 | 2.57 | 2.39 | 1.23 | 15.99 | 1.39 | 1.76 |
time (sec) |
N/A |
0.090 |
0.326 |
0.093 |
0.545 |
0.449 |
10.876 |
2.472 |
0.002 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
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 | 63 | 63 | 63 | 38 | 49 | 46 | 42 | 36 | 19 |
normalized size | 1 | 1.00 | 1.00 | 0.60 | 0.78 | 0.73 | 0.67 | 0.57 | 0.30 |
time (sec) |
N/A |
0.023 |
0.026 |
0.052 |
0.448 |
0.425 |
0.658 |
4.984 |
4.614 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
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 | 88 | 88 | 126 | 76 | 115 | 88 | 466 | 81 | 64 |
normalized size | 1 | 1.00 | 1.43 | 0.86 | 1.31 | 1.00 | 5.30 | 0.92 | 0.73 |
time (sec) |
N/A |
0.048 |
0.207 |
0.097 |
0.319 |
0.422 |
1.685 |
0.990 |
4.562 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
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 | 113 | 113 | 180 | 114 | 195 | 133 | 1227 | 107 | 115 |
normalized size | 1 | 1.00 | 1.59 | 1.01 | 1.73 | 1.18 | 10.86 | 0.95 | 1.02 |
time (sec) |
N/A |
0.078 |
0.502 |
0.107 |
0.307 |
0.427 |
3.545 |
0.548 |
5.725 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 138 | 138 | 235 | 152 | 275 | 181 | 2356 | 133 | 168 |
normalized size | 1 | 1.00 | 1.70 | 1.10 | 1.99 | 1.31 | 17.07 | 0.96 | 1.22 |
time (sec) |
N/A |
0.111 |
0.891 |
0.096 |
0.363 |
0.447 |
7.996 |
0.424 |
7.043 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
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 | 65 | 65 | 65 | 38 | 49 | 46 | 42 | 36 | 19 |
normalized size | 1 | 1.00 | 1.00 | 0.58 | 0.75 | 0.71 | 0.65 | 0.55 | 0.29 |
time (sec) |
N/A |
0.023 |
0.025 |
0.056 |
0.424 |
0.425 |
0.662 |
0.663 |
4.539 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
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 | 90 | 90 | 130 | 76 | 115 | 88 | 462 | 81 | 64 |
normalized size | 1 | 1.00 | 1.44 | 0.84 | 1.28 | 0.98 | 5.13 | 0.90 | 0.71 |
time (sec) |
N/A |
0.046 |
0.215 |
0.096 |
0.455 |
0.430 |
1.685 |
0.404 |
4.569 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
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 | 115 | 115 | 184 | 114 | 195 | 133 | 1224 | 107 | 116 |
normalized size | 1 | 1.00 | 1.60 | 0.99 | 1.70 | 1.16 | 10.64 | 0.93 | 1.01 |
time (sec) |
N/A |
0.078 |
0.553 |
0.093 |
0.556 |
0.450 |
3.526 |
0.478 |
5.726 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 140 | 140 | 241 | 152 | 275 | 181 | 2353 | 133 | 168 |
normalized size | 1 | 1.00 | 1.72 | 1.09 | 1.96 | 1.29 | 16.81 | 0.95 | 1.20 |
time (sec) |
N/A |
0.115 |
0.943 |
0.099 |
0.470 |
0.430 |
7.688 |
2.187 |
7.118 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
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 | 65 | 65 | 65 | 38 | 49 | 46 | 42 | 36 | 19 |
normalized size | 1 | 1.00 | 1.00 | 0.58 | 0.75 | 0.71 | 0.65 | 0.55 | 0.29 |
time (sec) |
N/A |
0.022 |
0.021 |
0.056 |
0.482 |
0.456 |
0.633 |
1.865 |
4.514 |
|
|
|
|
|
|
|
|
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|
|
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|
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|
|
Problem 43 |
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 | 90 | 90 | 130 | 76 | 115 | 88 | 462 | 81 | 64 |
normalized size | 1 | 1.00 | 1.44 | 0.84 | 1.28 | 0.98 | 5.13 | 0.90 | 0.71 |
time (sec) |
N/A |
0.048 |
0.020 |
0.094 |
0.369 |
0.468 |
1.620 |
0.425 |
0.002 |
|
|
|
|
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|
Problem 44 |
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 | 115 | 115 | 183 | 114 | 195 | 133 | 1224 | 107 | 115 |
normalized size | 1 | 1.00 | 1.59 | 0.99 | 1.70 | 1.16 | 10.64 | 0.93 | 1.00 |
time (sec) |
N/A |
0.079 |
0.564 |
0.104 |
0.387 |
0.426 |
3.408 |
0.445 |
5.691 |
|
|
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|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 140 | 140 | 241 | 152 | 275 | 181 | 2353 | 133 | 168 |
normalized size | 1 | 1.00 | 1.72 | 1.09 | 1.96 | 1.29 | 16.81 | 0.95 | 1.20 |
time (sec) |
N/A |
0.116 |
0.026 |
0.099 |
0.513 |
0.446 |
8.008 |
0.380 |
0.002 |
|
|
|
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Problem 46 |
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 | 63 | 63 | 63 | 38 | 49 | 46 | 44 | 36 | 19 |
normalized size | 1 | 1.00 | 1.00 | 0.60 | 0.78 | 0.73 | 0.70 | 0.57 | 0.30 |
time (sec) |
N/A |
0.022 |
0.026 |
0.056 |
0.312 |
0.446 |
0.652 |
0.668 |
4.527 |
|
|
|
|
|
|
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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 | 88 | 88 | 126 | 76 | 115 | 88 | 468 | 81 | 64 |
normalized size | 1 | 1.00 | 1.43 | 0.86 | 1.31 | 1.00 | 5.32 | 0.92 | 0.73 |
time (sec) |
N/A |
0.043 |
0.135 |
0.101 |
0.379 |
0.429 |
1.652 |
0.564 |
0.002 |
|
|
|
|
|
|
|
|
|
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|
Problem 48 |
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 | 113 | 113 | 179 | 114 | 195 | 133 | 1229 | 107 | 116 |
normalized size | 1 | 1.00 | 1.58 | 1.01 | 1.73 | 1.18 | 10.88 | 0.95 | 1.03 |
time (sec) |
N/A |
0.077 |
0.504 |
0.101 |
0.545 |
0.430 |
3.448 |
0.791 |
5.710 |
|
|
|
|
|
|
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|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 138 | 138 | 235 | 152 | 275 | 181 | 2358 | 133 | 168 |
normalized size | 1 | 1.00 | 1.70 | 1.10 | 1.99 | 1.31 | 17.09 | 0.96 | 1.22 |
time (sec) |
N/A |
0.113 |
0.026 |
0.090 |
0.540 |
0.428 |
7.752 |
0.481 |
0.002 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 256 | 256 | 220 | 1040 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.86 | 4.06 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.382 |
1.063 |
0.298 |
0.000 |
0.444 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 207 | 207 | 185 | 890 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.89 | 4.30 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.260 |
0.986 |
0.220 |
0.000 |
0.433 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
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|
|
|
|
|
|
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|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 167 | 167 | 142 | 656 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.85 | 3.93 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.164 |
0.844 |
0.294 |
0.000 |
0.451 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 62 | 62 | 61 | 239 | 0 | 0 | 0 | 0 | 55 |
normalized size | 1 | 1.00 | 0.98 | 3.85 | 0.00 | 0.00 | 0.00 | 0.00 | 0.89 |
time (sec) |
N/A |
0.035 |
2.221 |
0.194 |
0.000 |
0.445 |
0.000 |
0.000 |
4.579 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 62 | 62 | 61 | 126 | 0 | 0 | 0 | 0 | 55 |
normalized size | 1 | 1.00 | 0.98 | 2.03 | 0.00 | 0.00 | 0.00 | 0.00 | 0.89 |
time (sec) |
N/A |
0.042 |
0.071 |
0.162 |
0.000 |
0.435 |
0.000 |
0.000 |
4.673 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 111 | 111 | 87 | 443 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.78 | 3.99 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.065 |
0.299 |
0.251 |
0.000 |
0.478 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 231 | 231 | 166 | 497 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.72 | 2.15 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.228 |
0.935 |
0.730 |
0.000 |
0.454 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 292 | 292 | 198 | 584 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.68 | 2.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.350 |
1.418 |
0.609 |
0.000 |
0.663 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 109 | 109 | 244 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.24 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.079 |
1.684 |
0.103 |
0.000 |
0.700 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 118 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.11 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.068 |
0.210 |
0.318 |
0.000 |
0.768 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 118 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.11 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.066 |
0.208 |
0.359 |
0.000 |
0.836 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 118 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.11 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.064 |
0.195 |
0.222 |
0.000 |
0.713 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 116 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.09 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.067 |
0.186 |
0.155 |
0.000 |
0.762 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 111 | 111 | 262 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.36 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.072 |
1.927 |
0.118 |
0.000 |
0.701 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 104 | 104 | 120 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.15 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.066 |
0.250 |
0.527 |
0.000 |
0.757 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 72 | 72 | 83 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.15 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.041 |
0.156 |
0.658 |
0.000 |
0.834 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 66 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 84 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.22 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.035 |
0.169 |
0.674 |
0.000 |
0.790 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 64 | 64 | 99 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.55 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.035 |
0.131 |
0.511 |
0.000 |
0.761 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 96 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.39 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.034 |
0.135 |
0.550 |
0.000 |
0.703 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 83 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.24 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.038 |
0.150 |
0.581 |
0.000 |
0.711 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 64 | 64 | 84 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.31 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.035 |
0.179 |
0.572 |
0.000 |
0.699 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 71 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 95 | 95 | 95 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.051 |
0.118 |
0.441 |
0.000 |
0.868 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 72 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 100 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.91 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.067 |
0.139 |
0.444 |
0.000 |
0.643 |
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 [36] had the largest ratio of
[.5000]
|
|
|
|
|
|
|
| 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 |
4 |
2 |
1.00 |
14 |
0.143 |
|
|
|
|
|
|
|
2 |
A |
3 |
2 |
1.00 |
14 |
0.143 |
|
|
|
|
|
|
|
3 |
A |
2 |
2 |
1.00 |
14 |
0.143 |
|
|
|
|
|
|
|
4 |
A |
1 |
1 |
1.00 |
14 |
0.071 |
|
|
|
|
|
|
|
5 |
A |
2 |
2 |
1.00 |
14 |
0.143 |
|
|
|
|
|
|
|
6 |
A |
3 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
7 |
A |
4 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
8 |
A |
2 |
2 |
1.00 |
14 |
0.143 |
|
|
|
|
|
|
|
9 |
A |
2 |
2 |
1.00 |
14 |
0.143 |
|
|
|
|
|
|
|
10 |
A |
2 |
2 |
1.00 |
14 |
0.143 |
|
|
|
|
|
|
|
11 |
A |
2 |
2 |
1.00 |
14 |
0.143 |
|
|
|
|
|
|
|
12 |
A |
2 |
2 |
1.00 |
14 |
0.143 |
|
|
|
|
|
|
|
13 |
A |
2 |
2 |
1.00 |
14 |
0.143 |
|
|
|
|
|
|
|
14 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
15 |
A |
2 |
2 |
1.00 |
13 |
0.154 |
|
|
|
|
|
|
|
16 |
A |
1 |
1 |
1.00 |
12 |
0.083 |
|
|
|
|
|
|
|
17 |
A |
1 |
1 |
1.00 |
12 |
0.083 |
|
|
|
|
|
|
|
18 |
A |
1 |
1 |
1.00 |
12 |
0.083 |
|
|
|
|
|
|
|
19 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
20 |
A |
4 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
21 |
A |
5 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
22 |
A |
1 |
1 |
1.00 |
12 |
0.083 |
|
|
|
|
|
|
|
23 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
24 |
A |
4 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
25 |
A |
5 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
26 |
A |
1 |
1 |
1.00 |
12 |
0.083 |
|
|
|
|
|
|
|
27 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
28 |
A |
4 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
29 |
A |
5 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
30 |
A |
1 |
1 |
1.00 |
12 |
0.083 |
|
|
|
|
|
|
|
31 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
32 |
A |
4 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
33 |
A |
5 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
34 |
A |
4 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
35 |
A |
6 |
5 |
1.00 |
12 |
0.417 |
|
|
|
|
|
|
|
36 |
A |
7 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
37 |
A |
8 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
38 |
A |
4 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
39 |
A |
6 |
5 |
1.00 |
12 |
0.417 |
|
|
|
|
|
|
|
40 |
A |
7 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
41 |
A |
8 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
42 |
A |
4 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
43 |
A |
6 |
5 |
1.00 |
12 |
0.417 |
|
|
|
|
|
|
|
44 |
A |
7 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
45 |
A |
8 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
46 |
A |
4 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
47 |
A |
6 |
5 |
1.00 |
12 |
0.417 |
|
|
|
|
|
|
|
48 |
A |
7 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
49 |
A |
8 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
50 |
A |
8 |
7 |
1.00 |
14 |
0.500 |
|
|
|
|
|
|
|
51 |
A |
7 |
7 |
1.00 |
14 |
0.500 |
|
|
|
|
|
|
|
52 |
A |
6 |
6 |
1.00 |
14 |
0.429 |
|
|
|
|
|
|
|
53 |
A |
2 |
2 |
1.00 |
14 |
0.143 |
|
|
|
|
|
|
|
54 |
A |
2 |
2 |
1.00 |
14 |
0.143 |
|
|
|
|
|
|
|
55 |
A |
4 |
4 |
1.00 |
14 |
0.286 |
|
|
|
|
|
|
|
56 |
A |
7 |
7 |
1.00 |
14 |
0.500 |
|
|
|
|
|
|
|
57 |
A |
8 |
7 |
1.00 |
14 |
0.500 |
|
|
|
|
|
|
|
58 |
A |
3 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
59 |
A |
3 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
60 |
A |
3 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
61 |
A |
3 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
62 |
A |
3 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
63 |
A |
3 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
64 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
65 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
66 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
67 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
68 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
69 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
70 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
71 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
72 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|