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, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87,
88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98 }
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, 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, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87,
88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98 }
B grade: { }
C grade: { }
F grade: { }
2.1.3 Maple
A grade: { 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
48, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 90, 91, 92, 93, 94 }
B grade: { 64, 95, 96, 97, 98 }
C grade: { 55, 83, 84, 85, 86, 87, 88, 89 }
F grade: { 1, 19, 20, 21, 22, 23, 24, 25, 26, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50 }
2.1.4 Maxima
A grade: { 3, 4, 5, 6, 13, 14, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 48, 51, 52, 53, 54, 55, 56, 57, 58,
59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92,
93, 94, 95, 96, 97, 98 }
B grade: { 2, 12 }
C grade: { }
F grade: { 1, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 39, 40, 41, 42, 43, 44, 45, 46, 47,
49, 50, 78, 79, 80, 81, 82 }
2.1.5 FriCAS
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 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, 73, 74, 75, 76, 77, 78, 79, 80, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94,
95, 96, 97, 98 }
B grade: { 81, 82 }
C grade: { }
F grade: { 20, 21, 22, 25, 26 }
2.1.6 Sympy
A grade: { 2, 3, 4, 5, 6, 12, 13, 14, 27, 28, 29, 32, 35, 36, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63,
65, 66, 67, 68, 74, 75, 76, 77, 90, 91, 92, 93, 94 }
B grade: { }
C grade: { }
F grade: { 1, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 30, 31, 33, 34, 37, 38, 39, 40, 41,
42, 43, 44, 45, 46, 47, 48, 49, 50, 64, 69, 70, 71, 72, 73, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 95, 96,
97, 98 }
2.1.7 Giac
A grade: { 6, 27, 28, 29, 30, 31, 32, 33, 34, 43, 48, 56, 57, 58, 59, 60, 61, 62, 63, 74, 75, 76, 77, 90, 91, 92,
93, 94, 95, 96, 97, 98 }
B grade: { 39, 40, 41, 42, 78, 79, 80, 81 }
C grade: { 2, 3, 4, 5, 12, 13, 14, 51, 52, 53, 54, 65, 66, 67, 68 }
F grade: { 1, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 35, 36, 37, 38, 44, 45, 46, 47, 49,
50, 55, 64, 69, 70, 71, 72, 73, 82, 83, 84, 85, 86, 87, 88, 89 }
2.1.8 Mupad
A grade: { }
B grade: { 2, 3, 4, 5, 6, 12, 13, 14, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 48, 51, 52, 53, 54, 56, 57, 58,
59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93,
94 }
C grade: { }
F grade: { 1, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 39, 40, 41, 42, 43, 44, 45, 46, 47,
49, 50, 55, 64, 78, 79, 80, 81, 82, 95, 96, 97, 98 }
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 |
F |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 67 | 0 | 0 | 65 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.00 | 0.00 | 0.97 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.040 |
0.038 |
0.075 |
0.000 |
0.601 |
0.000 |
0.000 |
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 |
B |
A |
A |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 141 | 141 | 100 | 260 | 309 | 227 | 350 | 7867 | 260 |
normalized size | 1 | 1.00 | 0.71 | 1.84 | 2.19 | 1.61 | 2.48 | 55.79 | 1.84 |
time (sec) |
N/A |
0.108 |
0.126 |
0.012 |
0.504 |
0.653 |
0.275 |
1.671 |
3.681 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
A |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 78 | 165 | 206 | 147 | 231 | 4693 | 165 |
normalized size | 1 | 1.00 | 0.71 | 1.50 | 1.87 | 1.34 | 2.10 | 42.66 | 1.50 |
time (sec) |
N/A |
0.073 |
0.086 |
0.008 |
0.491 |
0.592 |
0.224 |
1.571 |
3.564 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
A |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 56 | 91 | 123 | 84 | 133 | 2494 | 91 |
normalized size | 1 | 1.00 | 0.71 | 1.15 | 1.56 | 1.06 | 1.68 | 31.57 | 1.15 |
time (sec) |
N/A |
0.043 |
0.068 |
0.007 |
0.504 |
0.673 |
0.181 |
0.910 |
3.507 |
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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 |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 34 | 38 | 60 | 38 | 60 | 1083 | 38 |
normalized size | 1 | 1.00 | 0.71 | 0.79 | 1.25 | 0.79 | 1.25 | 22.56 | 0.79 |
time (sec) |
N/A |
0.017 |
0.038 |
0.006 |
0.488 |
0.818 |
0.138 |
0.670 |
3.371 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 20 | 21 | 21 | 20 | 21 | 20 | 21 | 21 |
normalized size | 1 | 1.00 | 1.05 | 1.05 | 1.00 | 1.05 | 1.00 | 1.05 | 1.05 |
time (sec) |
N/A |
0.004 |
0.005 |
0.003 |
0.482 |
0.438 |
0.100 |
0.314 |
3.428 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 31 | 56 | 0 | 39 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 1.81 | 0.00 | 1.26 | 0.00 | 0.00 | -0.03 |
time (sec) |
N/A |
0.024 |
0.042 |
0.050 |
0.000 |
0.414 |
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 |
A |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 57 | 57 | 55 | 99 | 0 | 77 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.96 | 1.74 | 0.00 | 1.35 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.043 |
0.119 |
0.053 |
0.000 |
0.417 |
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 |
A |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 95 | 95 | 88 | 155 | 0 | 134 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.93 | 1.63 | 0.00 | 1.41 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.071 |
0.153 |
0.060 |
0.000 |
0.416 |
0.000 |
0.000 |
0.000 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 128 | 128 | 99 | 199 | 0 | 209 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.77 | 1.55 | 0.00 | 1.63 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.100 |
0.194 |
0.066 |
0.000 |
0.435 |
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 |
A |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 161 | 161 | 121 | 243 | 0 | 300 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.75 | 1.51 | 0.00 | 1.86 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.134 |
0.164 |
0.075 |
0.000 |
0.446 |
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 |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 141 | 141 | 100 | 260 | 309 | 227 | 350 | 7867 | 260 |
normalized size | 1 | 1.00 | 0.71 | 1.84 | 2.19 | 1.61 | 2.48 | 55.79 | 1.84 |
time (sec) |
N/A |
0.122 |
0.062 |
0.010 |
0.514 |
0.423 |
0.273 |
1.604 |
3.486 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
A |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 78 | 165 | 206 | 147 | 231 | 4693 | 165 |
normalized size | 1 | 1.00 | 0.71 | 1.50 | 1.87 | 1.34 | 2.10 | 42.66 | 1.50 |
time (sec) |
N/A |
0.086 |
0.079 |
0.011 |
0.495 |
0.424 |
0.221 |
1.150 |
3.429 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
A |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 56 | 91 | 123 | 84 | 133 | 2494 | 91 |
normalized size | 1 | 1.00 | 0.71 | 1.15 | 1.56 | 1.06 | 1.68 | 31.57 | 1.15 |
time (sec) |
N/A |
0.045 |
0.037 |
0.009 |
0.504 |
0.425 |
0.177 |
0.900 |
3.403 |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 57 | 57 | 55 | 99 | 0 | 77 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.96 | 1.74 | 0.00 | 1.35 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.043 |
0.061 |
0.066 |
0.000 |
0.439 |
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 |
A |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 95 | 95 | 88 | 155 | 0 | 134 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.93 | 1.63 | 0.00 | 1.41 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.098 |
0.050 |
0.080 |
0.000 |
0.450 |
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 |
A |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 128 | 128 | 99 | 199 | 0 | 209 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.77 | 1.55 | 0.00 | 1.63 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.134 |
0.044 |
0.095 |
0.000 |
0.468 |
0.000 |
0.000 |
0.000 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 161 | 161 | 121 | 243 | 0 | 300 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.75 | 1.51 | 0.00 | 1.86 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.180 |
0.032 |
0.117 |
0.000 |
0.451 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 72 | 72 | 72 | 0 | 0 | 68 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.00 | 0.00 | 0.94 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.054 |
0.012 |
0.072 |
0.000 |
0.431 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 20 |
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 | 71 | 71 | 71 | 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 |
0.070 |
0.015 |
0.173 |
0.000 |
0.433 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
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 | 71 | 71 | 71 | 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 |
0.060 |
0.014 |
0.120 |
0.000 |
0.456 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
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 | 71 | 71 | 71 | 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 |
0.057 |
0.014 |
0.143 |
0.000 |
0.430 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 67 | 0 | 0 | 65 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.00 | 0.00 | 0.97 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.022 |
0.012 |
0.001 |
0.000 |
0.439 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 69 | 0 | 0 | 67 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.00 | 0.00 | 0.97 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.024 |
0.014 |
0.073 |
0.000 |
0.445 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
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 | 73 | 73 | 73 | 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 |
0.057 |
0.011 |
0.140 |
0.000 |
0.430 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
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 | 73 | 73 | 73 | 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 |
0.060 |
0.012 |
0.122 |
0.000 |
0.448 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 15 | 15 | 15 | 14 | 13 | 13 | 15 | 13 | 13 |
normalized size | 1 | 1.00 | 1.00 | 0.93 | 0.87 | 0.87 | 1.00 | 0.87 | 0.87 |
time (sec) |
N/A |
0.003 |
0.004 |
0.006 |
0.432 |
0.409 |
0.094 |
0.261 |
3.538 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
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 | 15 | 15 | 15 | 16 | 15 | 15 | 15 | 15 | 15 |
normalized size | 1 | 1.00 | 1.00 | 1.07 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
time (sec) |
N/A |
0.003 |
0.003 |
0.005 |
0.429 |
0.420 |
0.099 |
0.264 |
3.442 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
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 | 19 | 19 | 19 | 14 | 13 | 13 | 10 | 13 | 11 |
normalized size | 1 | 1.00 | 1.00 | 0.74 | 0.68 | 0.68 | 0.53 | 0.68 | 0.58 |
time (sec) |
N/A |
0.005 |
0.005 |
0.013 |
0.428 |
0.404 |
0.093 |
0.362 |
0.092 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 131 | 131 | 36 | 99 | 24 | 89 | 0 | 94 | 82 |
normalized size | 1 | 1.00 | 0.27 | 0.76 | 0.18 | 0.68 | 0.00 | 0.72 | 0.63 |
time (sec) |
N/A |
0.147 |
0.007 |
0.032 |
0.640 |
0.471 |
0.000 |
0.308 |
3.433 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 108 | 108 | 36 | 87 | 24 | 77 | 0 | 82 | 72 |
normalized size | 1 | 1.00 | 0.33 | 0.81 | 0.22 | 0.71 | 0.00 | 0.76 | 0.67 |
time (sec) |
N/A |
0.094 |
0.007 |
0.020 |
0.698 |
0.425 |
0.000 |
0.483 |
3.464 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
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 | 85 | 85 | 36 | 75 | 24 | 65 | 37 | 70 | 75 |
normalized size | 1 | 1.00 | 0.42 | 0.88 | 0.28 | 0.76 | 0.44 | 0.82 | 0.88 |
time (sec) |
N/A |
0.066 |
0.006 |
0.019 |
0.651 |
0.439 |
130.971 |
0.425 |
3.417 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 62 | 62 | 30 | 66 | 24 | 51 | 0 | 58 | 55 |
normalized size | 1 | 1.00 | 0.48 | 1.06 | 0.39 | 0.82 | 0.00 | 0.94 | 0.89 |
time (sec) |
N/A |
0.044 |
0.009 |
0.019 |
0.642 |
0.430 |
0.000 |
0.334 |
3.411 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 30 | 27 | 29 | 34 | 0 | 28 | 32 |
normalized size | 1 | 1.00 | 0.79 | 0.71 | 0.76 | 0.89 | 0.00 | 0.74 | 0.84 |
time (sec) |
N/A |
0.025 |
0.007 |
0.019 |
0.637 |
0.417 |
0.000 |
0.275 |
3.497 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 38 | 64 | 24 | 44 | 34 | 0 | 42 |
normalized size | 1 | 1.00 | 0.70 | 1.19 | 0.44 | 0.81 | 0.63 | 0.00 | 0.78 |
time (sec) |
N/A |
0.045 |
0.016 |
0.021 |
0.633 |
0.445 |
4.509 |
0.000 |
3.488 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 49 | 72 | 24 | 58 | 39 | 0 | 61 |
normalized size | 1 | 1.00 | 0.64 | 0.94 | 0.31 | 0.75 | 0.51 | 0.00 | 0.79 |
time (sec) |
N/A |
0.065 |
0.039 |
0.022 |
0.643 |
0.441 |
53.166 |
0.000 |
3.505 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 100 | 100 | 61 | 84 | 24 | 74 | 0 | 0 | 80 |
normalized size | 1 | 1.00 | 0.61 | 0.84 | 0.24 | 0.74 | 0.00 | 0.00 | 0.80 |
time (sec) |
N/A |
0.089 |
0.052 |
0.022 |
0.641 |
0.437 |
0.000 |
0.000 |
3.453 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 123 | 123 | 73 | 96 | 24 | 86 | 0 | 0 | 99 |
normalized size | 1 | 1.00 | 0.59 | 0.78 | 0.20 | 0.70 | 0.00 | 0.00 | 0.80 |
time (sec) |
N/A |
0.108 |
0.064 |
0.023 |
0.640 |
0.426 |
0.000 |
0.000 |
3.525 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-1) |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 208 | 208 | 72 | 0 | 0 | 230 | 0 | 1091 | -1 |
normalized size | 1 | 1.00 | 0.35 | 0.00 | 0.00 | 1.11 | 0.00 | 5.25 | -0.00 |
time (sec) |
N/A |
0.266 |
0.028 |
0.073 |
0.000 |
0.436 |
0.000 |
1.754 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-1) |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 173 | 173 | 72 | 0 | 0 | 167 | 0 | 691 | -1 |
normalized size | 1 | 1.00 | 0.42 | 0.00 | 0.00 | 0.97 | 0.00 | 3.99 | -0.01 |
time (sec) |
N/A |
0.166 |
0.025 |
0.074 |
0.000 |
0.448 |
0.000 |
1.300 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 138 | 138 | 63 | 0 | 0 | 121 | 0 | 401 | -1 |
normalized size | 1 | 1.00 | 0.46 | 0.00 | 0.00 | 0.88 | 0.00 | 2.91 | -0.01 |
time (sec) |
N/A |
0.119 |
0.074 |
0.075 |
0.000 |
0.440 |
0.000 |
0.906 |
0.000 |
|
|
|
|
|
|
|
|
|
|
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|
|
|
|
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|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 105 | 105 | 63 | 0 | 0 | 90 | 0 | 198 | -1 |
normalized size | 1 | 1.00 | 0.60 | 0.00 | 0.00 | 0.86 | 0.00 | 1.89 | -0.01 |
time (sec) |
N/A |
0.077 |
0.039 |
0.076 |
0.000 |
0.439 |
0.000 |
0.523 |
0.000 |
|
|
|
|
|
|
|
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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 |
F |
F |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 72 | 72 | 63 | 0 | 0 | 64 | 0 | 58 | -1 |
normalized size | 1 | 1.00 | 0.88 | 0.00 | 0.00 | 0.89 | 0.00 | 0.81 | -0.01 |
time (sec) |
N/A |
0.045 |
0.029 |
0.072 |
0.000 |
0.426 |
0.000 |
0.390 |
0.000 |
|
|
|
|
|
|
|
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|
|
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|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 97 | 97 | 75 | 0 | 0 | 90 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.77 | 0.00 | 0.00 | 0.93 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.085 |
0.078 |
0.074 |
0.000 |
0.426 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
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|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 130 | 130 | 92 | 0 | 0 | 140 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.71 | 0.00 | 0.00 | 1.08 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.119 |
0.237 |
0.074 |
0.000 |
0.427 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 165 | 165 | 118 | 0 | 0 | 230 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.72 | 0.00 | 0.00 | 1.39 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.165 |
0.139 |
0.073 |
0.000 |
0.431 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 200 | 200 | 144 | 0 | 0 | 321 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.72 | 0.00 | 0.00 | 1.60 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.203 |
0.168 |
0.073 |
0.000 |
0.435 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 151 | 151 | 24 | 145 | 79 | 82 | 0 | 80 | 89 |
normalized size | 1 | 1.00 | 0.16 | 0.96 | 0.52 | 0.54 | 0.00 | 0.53 | 0.59 |
time (sec) |
N/A |
0.155 |
0.005 |
0.137 |
0.839 |
0.448 |
0.000 |
0.364 |
3.434 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 71 | 71 | 63 | 0 | 0 | 117 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.89 | 0.00 | 0.00 | 1.65 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.031 |
0.079 |
0.074 |
0.000 |
0.451 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 98 | 98 | 78 | 0 | 0 | 133 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.80 | 0.00 | 0.00 | 1.36 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.104 |
0.138 |
0.077 |
0.000 |
0.494 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 34 | 38 | 60 | 38 | 60 | 1083 | 38 |
normalized size | 1 | 1.00 | 0.71 | 0.79 | 1.25 | 0.79 | 1.25 | 22.56 | 0.79 |
time (sec) |
N/A |
0.017 |
0.029 |
0.006 |
0.539 |
0.414 |
0.145 |
0.649 |
0.002 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 135 | 135 | 56 | 80 | 117 | 74 | 116 | 2490 | 80 |
normalized size | 1 | 1.00 | 0.41 | 0.59 | 0.87 | 0.55 | 0.86 | 18.44 | 0.59 |
time (sec) |
N/A |
0.103 |
0.079 |
0.009 |
0.700 |
0.418 |
0.171 |
0.755 |
3.396 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 229 | 229 | 84 | 138 | 194 | 122 | 190 | 4287 | 138 |
normalized size | 1 | 1.00 | 0.37 | 0.60 | 0.85 | 0.53 | 0.83 | 18.72 | 0.60 |
time (sec) |
N/A |
0.191 |
0.116 |
0.007 |
0.848 |
0.428 |
0.219 |
0.939 |
3.542 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 348 | 348 | 117 | 212 | 291 | 182 | 284 | 7425 | 212 |
normalized size | 1 | 1.00 | 0.34 | 0.61 | 0.84 | 0.52 | 0.82 | 21.34 | 0.61 |
time (sec) |
N/A |
0.312 |
0.179 |
0.009 |
0.840 |
0.408 |
0.263 |
1.314 |
3.531 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 116 | 116 | 61 | 334 | 123 | 126 | 109 | 0 | -1 |
normalized size | 1 | 1.00 | 0.53 | 2.88 | 1.06 | 1.09 | 0.94 | 0.00 | -0.01 |
time (sec) |
N/A |
0.175 |
0.059 |
0.109 |
0.964 |
0.421 |
34.422 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
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 | 397 | 397 | 121 | 182 | 196 | 121 | 236 | 202 | 175 |
normalized size | 1 | 1.00 | 0.30 | 0.46 | 0.49 | 0.30 | 0.59 | 0.51 | 0.44 |
time (sec) |
N/A |
0.516 |
0.357 |
0.006 |
0.550 |
0.434 |
0.240 |
0.297 |
3.535 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
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 | 318 | 318 | 130 | 143 | 164 | 102 | 196 | 163 | 126 |
normalized size | 1 | 1.00 | 0.41 | 0.45 | 0.52 | 0.32 | 0.62 | 0.51 | 0.40 |
time (sec) |
N/A |
0.412 |
0.210 |
0.006 |
0.518 |
0.418 |
0.215 |
0.315 |
3.523 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
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 | 184 | 184 | 96 | 102 | 132 | 78 | 148 | 123 | 117 |
normalized size | 1 | 1.00 | 0.52 | 0.55 | 0.72 | 0.42 | 0.80 | 0.67 | 0.64 |
time (sec) |
N/A |
0.242 |
0.133 |
0.004 |
0.488 |
0.435 |
0.190 |
0.401 |
3.430 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
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 | 80 | 80 | 41 | 68 | 103 | 57 | 104 | 87 | 66 |
normalized size | 1 | 1.00 | 0.51 | 0.85 | 1.29 | 0.71 | 1.30 | 1.09 | 0.82 |
time (sec) |
N/A |
0.068 |
0.048 |
0.004 |
0.699 |
0.413 |
0.168 |
0.296 |
0.109 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
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 | 102 | 102 | 52 | 113 | 69 | 50 | 70 | 95 | 69 |
normalized size | 1 | 1.00 | 0.51 | 1.11 | 0.68 | 0.49 | 0.69 | 0.93 | 0.68 |
time (sec) |
N/A |
0.156 |
0.055 |
0.015 |
0.704 |
0.405 |
16.642 |
0.324 |
3.558 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
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 | 94 | 94 | 54 | 92 | 61 | 56 | 99 | 92 | 72 |
normalized size | 1 | 1.00 | 0.57 | 0.98 | 0.65 | 0.60 | 1.05 | 0.98 | 0.77 |
time (sec) |
N/A |
0.161 |
0.070 |
0.018 |
0.966 |
0.400 |
5.866 |
0.445 |
3.635 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
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 | 130 | 130 | 68 | 112 | 64 | 70 | 56 | 125 | 100 |
normalized size | 1 | 1.00 | 0.52 | 0.86 | 0.49 | 0.54 | 0.43 | 0.96 | 0.77 |
time (sec) |
N/A |
0.213 |
0.077 |
0.017 |
1.236 |
0.425 |
5.302 |
0.328 |
3.636 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
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 | 198 | 198 | 81 | 167 | 63 | 83 | 53 | 183 | 142 |
normalized size | 1 | 1.00 | 0.41 | 0.84 | 0.32 | 0.42 | 0.27 | 0.92 | 0.72 |
time (sec) |
N/A |
0.290 |
0.113 |
0.019 |
1.168 |
0.401 |
5.330 |
0.410 |
3.565 |
|
|
|
|
|
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Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 139 | 139 | 86 | 433 | 123 | 161 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.62 | 3.12 | 0.88 | 1.16 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.309 |
0.126 |
0.121 |
1.134 |
0.436 |
0.000 |
0.000 |
0.000 |
|
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Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
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|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 414 | 414 | 159 | 250 | 328 | 228 | 323 | 9953 | 249 |
normalized size | 1 | 1.00 | 0.38 | 0.60 | 0.79 | 0.55 | 0.78 | 24.04 | 0.60 |
time (sec) |
N/A |
0.672 |
0.125 |
0.011 |
0.797 |
0.417 |
0.283 |
1.524 |
3.646 |
|
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Problem 66 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 328 | 328 | 121 | 197 | 262 | 178 | 260 | 7061 | 196 |
normalized size | 1 | 1.00 | 0.37 | 0.60 | 0.80 | 0.54 | 0.79 | 21.53 | 0.60 |
time (sec) |
N/A |
0.532 |
0.246 |
0.010 |
0.717 |
0.416 |
0.281 |
1.345 |
3.559 |
|
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Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
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|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 242 | 242 | 91 | 144 | 196 | 132 | 199 | 4696 | 143 |
normalized size | 1 | 1.00 | 0.38 | 0.60 | 0.81 | 0.55 | 0.82 | 19.40 | 0.59 |
time (sec) |
N/A |
0.356 |
0.154 |
0.011 |
0.592 |
0.421 |
0.227 |
1.063 |
3.519 |
|
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Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 58 | 93 | 134 | 85 | 134 | 2747 | 92 |
normalized size | 1 | 1.00 | 0.68 | 1.09 | 1.58 | 1.00 | 1.58 | 32.32 | 1.08 |
time (sec) |
N/A |
0.120 |
0.072 |
0.008 |
0.609 |
0.414 |
0.179 |
0.934 |
3.443 |
|
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Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 96 | 96 | 54 | 126 | 87 | 75 | 0 | 0 | 80 |
normalized size | 1 | 1.00 | 0.56 | 1.31 | 0.91 | 0.78 | 0.00 | 0.00 | 0.83 |
time (sec) |
N/A |
0.258 |
0.129 |
0.062 |
1.045 |
0.421 |
0.000 |
0.000 |
3.584 |
|
|
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Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 58 | 135 | 68 | 83 | 0 | 0 | 89 |
normalized size | 1 | 1.00 | 0.68 | 1.59 | 0.80 | 0.98 | 0.00 | 0.00 | 1.05 |
time (sec) |
N/A |
0.277 |
0.155 |
0.071 |
0.918 |
0.413 |
0.000 |
0.000 |
3.588 |
|
|
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Problem 71 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 136 | 136 | 76 | 204 | 74 | 89 | 0 | 0 | 133 |
normalized size | 1 | 1.00 | 0.56 | 1.50 | 0.54 | 0.65 | 0.00 | 0.00 | 0.98 |
time (sec) |
N/A |
0.363 |
0.153 |
0.076 |
1.233 |
0.404 |
0.000 |
0.000 |
3.601 |
|
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Problem 72 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 217 | 217 | 116 | 290 | 85 | 137 | 0 | 0 | 202 |
normalized size | 1 | 1.00 | 0.53 | 1.34 | 0.39 | 0.63 | 0.00 | 0.00 | 0.93 |
time (sec) |
N/A |
0.459 |
0.229 |
0.079 |
1.048 |
0.417 |
0.000 |
0.000 |
3.660 |
|
|
|
|
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|
Problem 73 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 321 | 321 | 156 | 382 | 93 | 186 | 0 | 0 | 258 |
normalized size | 1 | 1.00 | 0.49 | 1.19 | 0.29 | 0.58 | 0.00 | 0.00 | 0.80 |
time (sec) |
N/A |
0.578 |
0.302 |
0.086 |
1.015 |
0.430 |
0.000 |
0.000 |
3.679 |
|
|
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Problem 74 |
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 | 754 | 754 | 458 | 1062 | 894 | 544 | 1445 | 1096 | 803 |
normalized size | 1 | 1.00 | 0.61 | 1.41 | 1.19 | 0.72 | 1.92 | 1.45 | 1.06 |
time (sec) |
N/A |
0.916 |
0.718 |
0.009 |
1.096 |
0.413 |
0.720 |
0.439 |
3.819 |
|
|
|
|
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|
Problem 75 |
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 | 495 | 495 | 320 | 640 | 599 | 354 | 899 | 674 | 537 |
normalized size | 1 | 1.00 | 0.65 | 1.29 | 1.21 | 0.72 | 1.82 | 1.36 | 1.08 |
time (sec) |
N/A |
0.636 |
0.470 |
0.008 |
0.867 |
0.411 |
0.496 |
0.447 |
3.663 |
|
|
|
|
|
|
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|
Problem 76 |
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 | 271 | 271 | 191 | 297 | 344 | 197 | 447 | 331 | 264 |
normalized size | 1 | 1.00 | 0.70 | 1.10 | 1.27 | 0.73 | 1.65 | 1.22 | 0.97 |
time (sec) |
N/A |
0.339 |
0.266 |
0.006 |
0.786 |
0.405 |
0.315 |
0.416 |
0.178 |
|
|
|
|
|
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Problem 77 |
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 | 102 | 102 | 50 | 108 | 149 | 83 | 158 | 132 | 120 |
normalized size | 1 | 1.00 | 0.49 | 1.06 | 1.46 | 0.81 | 1.55 | 1.29 | 1.18 |
time (sec) |
N/A |
0.096 |
0.066 |
0.006 |
0.613 |
0.400 |
0.276 |
0.373 |
0.110 |
|
|
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|
Problem 78 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 277 | 277 | 175 | 489 | 0 | 235 | 0 | 546 | -1 |
normalized size | 1 | 1.00 | 0.63 | 1.77 | 0.00 | 0.85 | 0.00 | 1.97 | -0.00 |
time (sec) |
N/A |
0.338 |
0.309 |
0.019 |
0.000 |
0.414 |
0.000 |
0.401 |
0.000 |
|
|
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|
Problem 79 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 258 | 258 | 163 | 406 | 0 | 353 | 0 | 2861 | -1 |
normalized size | 1 | 1.00 | 0.63 | 1.57 | 0.00 | 1.37 | 0.00 | 11.09 | -0.00 |
time (sec) |
N/A |
0.379 |
0.440 |
0.024 |
0.000 |
0.417 |
0.000 |
0.686 |
0.000 |
|
|
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|
Problem 80 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 294 | 294 | 267 | 418 | 0 | 550 | 0 | 1995 | -1 |
normalized size | 1 | 1.00 | 0.91 | 1.42 | 0.00 | 1.87 | 0.00 | 6.79 | -0.00 |
time (sec) |
N/A |
0.408 |
0.659 |
0.024 |
0.000 |
0.464 |
0.000 |
0.472 |
0.000 |
|
|
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|
Problem 81 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
B |
F(-1) |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 396 | 396 | 389 | 511 | 0 | 793 | 0 | 3178 | -1 |
normalized size | 1 | 1.00 | 0.98 | 1.29 | 0.00 | 2.00 | 0.00 | 8.03 | -0.00 |
time (sec) |
N/A |
0.521 |
0.817 |
0.023 |
0.000 |
0.450 |
0.000 |
0.530 |
0.000 |
|
|
|
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|
Problem 82 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
B |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 557 | 557 | 669 | 596 | 0 | 1084 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.20 | 1.07 | 0.00 | 1.95 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.684 |
0.740 |
0.024 |
0.000 |
0.509 |
0.000 |
0.000 |
0.000 |
|
|
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|
Problem 83 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
F(-1) |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 24 | 24 | 192 | 42 | 32 | 0 | 0 | 25 |
normalized size | 1 | 1.00 | 1.00 | 8.00 | 1.75 | 1.33 | 0.00 | 0.00 | 1.04 |
time (sec) |
N/A |
0.146 |
0.385 |
0.315 |
1.678 |
0.473 |
0.000 |
0.000 |
3.589 |
|
|
|
|
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|
Problem 84 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
F(-1) |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 22 | 23 | 198 | 42 | 25 | 0 | 0 | 23 |
normalized size | 1 | 1.00 | 1.05 | 9.00 | 1.91 | 1.14 | 0.00 | 0.00 | 1.05 |
time (sec) |
N/A |
0.131 |
0.311 |
0.213 |
1.627 |
0.658 |
0.000 |
0.000 |
3.499 |
|
|
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|
Problem 85 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
F(-1) |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 22 | 23 | 198 | 42 | 25 | 0 | 0 | 23 |
normalized size | 1 | 1.00 | 1.05 | 9.00 | 1.91 | 1.14 | 0.00 | 0.00 | 1.05 |
time (sec) |
N/A |
0.090 |
0.275 |
0.210 |
1.590 |
0.476 |
0.000 |
0.000 |
3.392 |
|
|
|
|
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|
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|
|
Problem 86 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 20 | 21 | 186 | 38 | 23 | 0 | 0 | 21 |
normalized size | 1 | 1.00 | 1.05 | 9.30 | 1.90 | 1.15 | 0.00 | 0.00 | 1.05 |
time (sec) |
N/A |
0.046 |
0.180 |
0.236 |
1.603 |
0.461 |
0.000 |
0.000 |
3.479 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 87 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 19 | 180 | 36 | 22 | 0 | 0 | 20 |
normalized size | 1 | 1.00 | 1.00 | 9.47 | 1.89 | 1.16 | 0.00 | 0.00 | 1.05 |
time (sec) |
N/A |
0.129 |
0.034 |
0.202 |
1.613 |
0.489 |
0.000 |
0.000 |
3.516 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 88 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 22 | 23 | 136 | 39 | 25 | 0 | 0 | 23 |
normalized size | 1 | 1.00 | 1.05 | 6.18 | 1.77 | 1.14 | 0.00 | 0.00 | 1.05 |
time (sec) |
N/A |
0.131 |
0.347 |
0.224 |
1.627 |
0.504 |
0.000 |
0.000 |
3.571 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 89 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
F(-1) |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 22 | 23 | 136 | 39 | 25 | 0 | 0 | 23 |
normalized size | 1 | 1.00 | 1.05 | 6.18 | 1.77 | 1.14 | 0.00 | 0.00 | 1.05 |
time (sec) |
N/A |
0.131 |
0.337 |
0.226 |
1.625 |
0.492 |
0.000 |
0.000 |
3.583 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 90 |
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 | 91 | 91 | 45 | 43 | 60 | 43 | 51 | 43 | 45 |
normalized size | 1 | 1.00 | 0.49 | 0.47 | 0.66 | 0.47 | 0.56 | 0.47 | 0.49 |
time (sec) |
N/A |
0.143 |
0.022 |
0.005 |
0.895 |
0.471 |
0.129 |
0.338 |
0.130 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 91 |
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 | 72 | 72 | 37 | 35 | 48 | 35 | 42 | 35 | 37 |
normalized size | 1 | 1.00 | 0.51 | 0.49 | 0.67 | 0.49 | 0.58 | 0.49 | 0.51 |
time (sec) |
N/A |
0.101 |
0.014 |
0.004 |
0.889 |
0.435 |
0.125 |
0.291 |
0.055 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 92 |
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 | 53 | 53 | 29 | 27 | 36 | 27 | 34 | 27 | 29 |
normalized size | 1 | 1.00 | 0.55 | 0.51 | 0.68 | 0.51 | 0.64 | 0.51 | 0.55 |
time (sec) |
N/A |
0.064 |
0.013 |
0.002 |
0.900 |
0.425 |
0.116 |
0.350 |
0.058 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 93 |
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 | 34 | 34 | 21 | 19 | 24 | 19 | 26 | 19 | 18 |
normalized size | 1 | 1.00 | 0.62 | 0.56 | 0.71 | 0.56 | 0.76 | 0.56 | 0.53 |
time (sec) |
N/A |
0.029 |
0.010 |
0.004 |
0.898 |
0.411 |
0.106 |
0.448 |
0.045 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 94 |
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 | 16 | 16 | 16 | 14 | 14 | 14 | 14 | 14 | 13 |
normalized size | 1 | 1.00 | 1.00 | 0.88 | 0.88 | 0.88 | 0.88 | 0.88 | 0.81 |
time (sec) |
N/A |
0.007 |
0.005 |
0.003 |
0.871 |
0.422 |
0.093 |
0.317 |
3.388 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 95 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 27 | 57 | 10 | 10 | 0 | 10 | -1 |
normalized size | 1 | 1.00 | 1.00 | 2.11 | 0.37 | 0.37 | 0.00 | 0.37 | -0.04 |
time (sec) |
N/A |
0.058 |
0.016 |
0.056 |
1.227 |
0.398 |
0.000 |
0.358 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 96 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 47 | 116 | 13 | 29 | 0 | 29 | -1 |
normalized size | 1 | 1.00 | 0.98 | 2.42 | 0.27 | 0.60 | 0.00 | 0.60 | -0.02 |
time (sec) |
N/A |
0.086 |
0.028 |
0.041 |
1.241 |
0.430 |
0.000 |
0.407 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 97 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 71 | 71 | 56 | 155 | 15 | 38 | 0 | 46 | -1 |
normalized size | 1 | 1.00 | 0.79 | 2.18 | 0.21 | 0.54 | 0.00 | 0.65 | -0.01 |
time (sec) |
N/A |
0.119 |
0.049 |
0.048 |
1.244 |
0.458 |
0.000 |
0.356 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 98 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 92 | 92 | 64 | 189 | 15 | 46 | 0 | 63 | -1 |
normalized size | 1 | 1.00 | 0.70 | 2.05 | 0.16 | 0.50 | 0.00 | 0.68 | -0.01 |
time (sec) |
N/A |
0.155 |
0.064 |
0.050 |
1.248 |
0.432 |
0.000 |
0.356 |
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 [48] had the largest ratio of
[.2500]
|
|
|
|
|
|
|
| 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 |
1 |
1 |
1.00 |
17 |
0.059 |
|
|
|
|
|
|
|
2 |
A |
5 |
2 |
1.00 |
17 |
0.118 |
|
|
|
|
|
|
|
3 |
A |
4 |
2 |
1.00 |
17 |
0.118 |
|
|
|
|
|
|
|
4 |
A |
3 |
2 |
1.00 |
17 |
0.118 |
|
|
|
|
|
|
|
5 |
A |
2 |
2 |
1.00 |
15 |
0.133 |
|
|
|
|
|
|
|
6 |
A |
1 |
1 |
1.00 |
9 |
0.111 |
|
|
|
|
|
|
|
7 |
A |
1 |
1 |
1.00 |
17 |
0.059 |
|
|
|
|
|
|
|
8 |
A |
2 |
2 |
1.00 |
17 |
0.118 |
|
|
|
|
|
|
|
9 |
A |
3 |
2 |
1.00 |
17 |
0.118 |
|
|
|
|
|
|
|
10 |
A |
4 |
2 |
1.00 |
17 |
0.118 |
|
|
|
|
|
|
|
11 |
A |
5 |
2 |
1.00 |
17 |
0.118 |
|
|
|
|
|
|
|
12 |
A |
6 |
3 |
1.00 |
48 |
0.062 |
|
|
|
|
|
|
|
13 |
A |
5 |
3 |
1.00 |
37 |
0.081 |
|
|
|
|
|
|
|
14 |
A |
4 |
3 |
1.00 |
26 |
0.115 |
|
|
|
|
|
|
|
15 |
A |
3 |
3 |
1.00 |
28 |
0.107 |
|
|
|
|
|
|
|
16 |
A |
4 |
3 |
1.00 |
39 |
0.077 |
|
|
|
|
|
|
|
17 |
A |
5 |
3 |
1.00 |
50 |
0.060 |
|
|
|
|
|
|
|
18 |
A |
6 |
3 |
1.00 |
61 |
0.049 |
|
|
|
|
|
|
|
19 |
A |
2 |
2 |
1.00 |
19 |
0.105 |
|
|
|
|
|
|
|
20 |
A |
2 |
2 |
1.00 |
50 |
0.040 |
|
|
|
|
|
|
|
21 |
A |
2 |
2 |
1.00 |
39 |
0.051 |
|
|
|
|
|
|
|
22 |
A |
2 |
2 |
1.00 |
28 |
0.071 |
|
|
|
|
|
|
|
23 |
A |
1 |
1 |
1.00 |
17 |
0.059 |
|
|
|
|
|
|
|
24 |
A |
1 |
1 |
1.00 |
19 |
0.053 |
|
|
|
|
|
|
|
25 |
A |
2 |
2 |
1.00 |
30 |
0.067 |
|
|
|
|
|
|
|
26 |
A |
2 |
2 |
1.00 |
41 |
0.049 |
|
|
|
|
|
|
|
27 |
A |
1 |
1 |
1.00 |
7 |
0.143 |
|
|
|
|
|
|
|
28 |
A |
1 |
1 |
1.00 |
7 |
0.143 |
|
|
|
|
|
|
|
29 |
A |
1 |
1 |
1.00 |
7 |
0.143 |
|
|
|
|
|
|
|
30 |
A |
6 |
3 |
1.00 |
13 |
0.231 |
|
|
|
|
|
|
|
31 |
A |
5 |
3 |
1.00 |
13 |
0.231 |
|
|
|
|
|
|
|
32 |
A |
4 |
3 |
1.00 |
13 |
0.231 |
|
|
|
|
|
|
|
33 |
A |
3 |
3 |
1.00 |
13 |
0.231 |
|
|
|
|
|
|
|
34 |
A |
2 |
2 |
1.00 |
13 |
0.154 |
|
|
|
|
|
|
|
35 |
A |
3 |
3 |
1.00 |
13 |
0.231 |
|
|
|
|
|
|
|
36 |
A |
4 |
3 |
1.00 |
13 |
0.231 |
|
|
|
|
|
|
|
37 |
A |
5 |
3 |
1.00 |
13 |
0.231 |
|
|
|
|
|
|
|
38 |
A |
6 |
3 |
1.00 |
13 |
0.231 |
|
|
|
|
|
|
|
39 |
A |
6 |
3 |
1.00 |
19 |
0.158 |
|
|
|
|
|
|
|
40 |
A |
5 |
3 |
1.00 |
19 |
0.158 |
|
|
|
|
|
|
|
41 |
A |
4 |
3 |
1.00 |
19 |
0.158 |
|
|
|
|
|
|
|
42 |
A |
3 |
3 |
1.00 |
19 |
0.158 |
|
|
|
|
|
|
|
43 |
A |
2 |
2 |
1.00 |
19 |
0.105 |
|
|
|
|
|
|
|
44 |
A |
3 |
3 |
1.00 |
19 |
0.158 |
|
|
|
|
|
|
|
45 |
A |
4 |
3 |
1.00 |
19 |
0.158 |
|
|
|
|
|
|
|
46 |
A |
5 |
3 |
1.00 |
19 |
0.158 |
|
|
|
|
|
|
|
47 |
A |
6 |
3 |
1.00 |
19 |
0.158 |
|
|
|
|
|
|
|
48 |
A |
9 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
49 |
A |
1 |
1 |
1.00 |
19 |
0.053 |
|
|
|
|
|
|
|
50 |
A |
2 |
2 |
1.00 |
21 |
0.095 |
|
|
|
|
|
|
|
51 |
A |
2 |
2 |
1.00 |
15 |
0.133 |
|
|
|
|
|
|
|
52 |
A |
8 |
3 |
1.00 |
20 |
0.150 |
|
|
|
|
|
|
|
53 |
A |
12 |
3 |
1.00 |
25 |
0.120 |
|
|
|
|
|
|
|
54 |
A |
17 |
3 |
1.00 |
30 |
0.100 |
|
|
|
|
|
|
|
55 |
A |
6 |
2 |
1.00 |
21 |
0.095 |
|
|
|
|
|
|
|
56 |
A |
24 |
3 |
1.00 |
21 |
0.143 |
|
|
|
|
|
|
|
57 |
A |
20 |
3 |
1.00 |
21 |
0.143 |
|
|
|
|
|
|
|
58 |
A |
11 |
3 |
1.00 |
19 |
0.158 |
|
|
|
|
|
|
|
59 |
A |
4 |
2 |
1.00 |
18 |
0.111 |
|
|
|
|
|
|
|
60 |
A |
9 |
4 |
1.00 |
21 |
0.190 |
|
|
|
|
|
|
|
61 |
A |
8 |
5 |
1.00 |
21 |
0.238 |
|
|
|
|
|
|
|
62 |
A |
9 |
4 |
1.00 |
21 |
0.190 |
|
|
|
|
|
|
|
63 |
A |
12 |
3 |
1.00 |
21 |
0.143 |
|
|
|
|
|
|
|
64 |
A |
5 |
2 |
1.00 |
22 |
0.091 |
|
|
|
|
|
|
|
65 |
A |
17 |
3 |
1.00 |
22 |
0.136 |
|
|
|
|
|
|
|
66 |
A |
14 |
3 |
1.00 |
22 |
0.136 |
|
|
|
|
|
|
|
67 |
A |
11 |
3 |
1.00 |
20 |
0.150 |
|
|
|
|
|
|
|
68 |
A |
4 |
3 |
1.00 |
19 |
0.158 |
|
|
|
|
|
|
|
69 |
A |
6 |
4 |
1.00 |
22 |
0.182 |
|
|
|
|
|
|
|
70 |
A |
6 |
4 |
1.00 |
22 |
0.182 |
|
|
|
|
|
|
|
71 |
A |
8 |
3 |
1.00 |
22 |
0.136 |
|
|
|
|
|
|
|
72 |
A |
11 |
3 |
1.00 |
22 |
0.136 |
|
|
|
|
|
|
|
73 |
A |
14 |
3 |
1.00 |
22 |
0.136 |
|
|
|
|
|
|
|
74 |
A |
28 |
3 |
1.00 |
25 |
0.120 |
|
|
|
|
|
|
|
75 |
A |
20 |
3 |
1.00 |
25 |
0.120 |
|
|
|
|
|
|
|
76 |
A |
13 |
3 |
1.00 |
23 |
0.130 |
|
|
|
|
|
|
|
77 |
A |
5 |
2 |
1.00 |
18 |
0.111 |
|
|
|
|
|
|
|
78 |
A |
13 |
4 |
1.00 |
25 |
0.160 |
|
|
|
|
|
|
|
79 |
A |
11 |
5 |
1.00 |
25 |
0.200 |
|
|
|
|
|
|
|
80 |
A |
11 |
5 |
1.00 |
25 |
0.200 |
|
|
|
|
|
|
|
81 |
A |
13 |
4 |
1.00 |
25 |
0.160 |
|
|
|
|
|
|
|
82 |
A |
17 |
3 |
1.00 |
25 |
0.120 |
|
|
|
|
|
|
|
83 |
A |
1 |
1 |
1.00 |
39 |
0.026 |
|
|
|
|
|
|
|
84 |
A |
1 |
1 |
1.00 |
38 |
0.026 |
|
|
|
|
|
|
|
85 |
A |
1 |
1 |
1.00 |
36 |
0.028 |
|
|
|
|
|
|
|
86 |
A |
1 |
1 |
1.00 |
35 |
0.029 |
|
|
|
|
|
|
|
87 |
A |
1 |
1 |
1.00 |
35 |
0.029 |
|
|
|
|
|
|
|
88 |
A |
1 |
1 |
1.00 |
38 |
0.026 |
|
|
|
|
|
|
|
89 |
A |
1 |
1 |
1.00 |
38 |
0.026 |
|
|
|
|
|
|
|
90 |
A |
5 |
2 |
1.00 |
15 |
0.133 |
|
|
|
|
|
|
|
91 |
A |
4 |
2 |
1.00 |
15 |
0.133 |
|
|
|
|
|
|
|
92 |
A |
3 |
2 |
1.00 |
15 |
0.133 |
|
|
|
|
|
|
|
93 |
A |
2 |
2 |
1.00 |
13 |
0.154 |
|
|
|
|
|
|
|
94 |
A |
1 |
1 |
1.00 |
11 |
0.091 |
|
|
|
|
|
|
|
95 |
A |
2 |
2 |
1.00 |
15 |
0.133 |
|
|
|
|
|
|
|
96 |
A |
3 |
3 |
1.00 |
15 |
0.200 |
|
|
|
|
|
|
|
97 |
A |
4 |
3 |
1.00 |
15 |
0.200 |
|
|
|
|
|
|
|
98 |
A |
5 |
3 |
1.00 |
15 |
0.200 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|