Chapter 2
detailed summary tables of results
2.1 List of integrals sorted by grade for each CAS
2.1.1 Rubi
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 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
}
B grade: { 32, 34 }
C grade: { 17, 19, 21, 23, 65, 66, 67, 68, 69, 70 }
F grade: { }
2.1.3 Maple
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64 }
B grade: { 24, 26, 27, 28, 35, 36 }
C grade: { 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 47, 48, 49, 50, 51, 52, 53, 65, 66, 67, 68, 69, 70
}
F grade: { }
2.1.4 Maxima
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43,
44, 45, 46, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64 }
B grade: { 14, 15, 25, 26, 41 }
C grade: { }
F grade: { 16, 17, 18, 19, 20, 21, 22, 23, 24, 47, 48, 49, 50, 51, 52, 53, 65, 66, 67, 68, 69, 70
}
2.1.5 FriCAS
A grade: { 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
39, 42, 43, 44, 45, 46, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64 }
B grade: { 7, 40, 41, 58 }
C grade: { }
F grade: { 16, 17, 18, 19, 20, 21, 22, 23, 47, 48, 49, 50, 51, 52, 53, 65, 66, 67, 68, 69, 70 }
2.1.6 Sympy
A grade: { 9, 32, 33 }
B grade: { }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 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 }
2.1.7 Giac
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 46
}
B grade: { 24, 37, 38, 39, 40, 41, 42, 43, 44, 45, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64 }
C grade: { }
F grade: { 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 47, 48, 49, 50, 51, 52, 53, 65, 66, 67, 68, 69, 70
}
2.1.8 Mupad
A grade: { }
B grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64 }
C grade: { }
F grade: { 16, 17, 18, 19, 20, 21, 22, 23, 47, 48, 49, 50, 51, 52, 53, 65, 66, 67, 68, 69, 70 }
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 |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 87 | 81 | 78 | 60 | 74 | 0 | 79 | 56 |
normalized size | 1 | 1.00 | 0.93 | 0.90 | 0.69 | 0.85 | 0.00 | 0.91 | 0.64 |
time (sec) |
N/A |
0.056 |
0.345 |
1.264 |
0.338 |
0.403 |
0.000 |
1.985 |
2.438 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 98 | 98 | 75 | 138 | 126 | 114 | 0 | 121 | 102 |
normalized size | 1 | 1.00 | 0.77 | 1.41 | 1.29 | 1.16 | 0.00 | 1.23 | 1.04 |
time (sec) |
N/A |
0.063 |
0.388 |
1.442 |
0.338 |
0.903 |
0.000 |
0.785 |
2.485 |
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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 |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 65 | 65 | 61 | 58 | 43 | 56 | 0 | 57 | 42 |
normalized size | 1 | 1.00 | 0.94 | 0.89 | 0.66 | 0.86 | 0.00 | 0.88 | 0.65 |
time (sec) |
N/A |
0.044 |
0.217 |
1.291 |
0.348 |
0.805 |
0.000 |
0.240 |
2.319 |
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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 |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 70 | 70 | 54 | 98 | 97 | 95 | 0 | 98 | 77 |
normalized size | 1 | 1.00 | 0.77 | 1.40 | 1.39 | 1.36 | 0.00 | 1.40 | 1.10 |
time (sec) |
N/A |
0.047 |
0.143 |
1.102 |
0.352 |
0.863 |
0.000 |
0.437 |
2.447 |
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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 |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 36 | 35 | 34 | 37 | 0 | 34 | 28 |
normalized size | 1 | 1.00 | 0.84 | 0.81 | 0.79 | 0.86 | 0.00 | 0.79 | 0.65 |
time (sec) |
N/A |
0.040 |
0.113 |
1.342 |
0.342 |
0.891 |
0.000 |
1.091 |
2.367 |
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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 |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 40 | 40 | 48 | 59 | 58 | 72 | 0 | 60 | 41 |
normalized size | 1 | 1.00 | 1.20 | 1.48 | 1.45 | 1.80 | 0.00 | 1.50 | 1.02 |
time (sec) |
N/A |
0.025 |
0.015 |
0.895 |
0.339 |
0.634 |
0.000 |
0.243 |
2.415 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
B |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 15 | 15 | 15 | 16 | 15 | 31 | 0 | 15 | 17 |
normalized size | 1 | 1.00 | 1.00 | 1.07 | 1.00 | 2.07 | 0.00 | 1.00 | 1.13 |
time (sec) |
N/A |
0.013 |
0.004 |
0.979 |
0.418 |
0.441 |
0.000 |
0.258 |
2.345 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 24 | 35 | 32 | 38 | 40 | 0 | 40 | 22 |
normalized size | 1 | 1.00 | 1.46 | 1.33 | 1.58 | 1.67 | 0.00 | 1.67 | 0.92 |
time (sec) |
N/A |
0.027 |
0.017 |
0.936 |
0.340 |
0.462 |
0.000 |
0.215 |
0.056 |
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Problem 9 |
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 | 33 | 37 | 37 | 28 | 51 | 37 | 25 |
normalized size | 1 | 1.00 | 1.06 | 1.19 | 1.19 | 0.90 | 1.65 | 1.19 | 0.81 |
time (sec) |
N/A |
0.030 |
0.037 |
1.075 |
0.435 |
0.441 |
5.945 |
0.203 |
2.358 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 30 | 30 | 50 | 33 | 27 | 28 | 0 | 34 | 28 |
normalized size | 1 | 1.00 | 1.67 | 1.10 | 0.90 | 0.93 | 0.00 | 1.13 | 0.93 |
time (sec) |
N/A |
0.047 |
0.020 |
1.360 |
0.477 |
0.425 |
0.000 |
0.242 |
0.041 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 45 | 65 | 73 | 49 | 0 | 73 | 67 |
normalized size | 1 | 1.00 | 0.74 | 1.07 | 1.20 | 0.80 | 0.00 | 1.20 | 1.10 |
time (sec) |
N/A |
0.044 |
0.097 |
1.549 |
0.474 |
0.429 |
0.000 |
0.776 |
2.443 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 50 | 50 | 71 | 54 | 43 | 45 | 0 | 57 | 43 |
normalized size | 1 | 1.00 | 1.42 | 1.08 | 0.86 | 0.90 | 0.00 | 1.14 | 0.86 |
time (sec) |
N/A |
0.068 |
0.024 |
1.497 |
0.366 |
0.437 |
0.000 |
0.203 |
2.368 |
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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 |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 68 | 86 | 103 | 68 | 0 | 96 | 91 |
normalized size | 1 | 1.00 | 0.76 | 0.97 | 1.16 | 0.76 | 0.00 | 1.08 | 1.02 |
time (sec) |
N/A |
0.059 |
0.112 |
1.947 |
0.426 |
0.427 |
0.000 |
0.210 |
2.942 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
B |
A |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 26 | 512 | 304 | 33 | 0 | 0 | 42 |
normalized size | 1 | 1.00 | 1.00 | 19.69 | 11.69 | 1.27 | 0.00 | 0.00 | 1.62 |
time (sec) |
N/A |
0.040 |
0.352 |
2.066 |
0.642 |
0.480 |
0.000 |
0.000 |
2.856 |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
B |
A |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 25 | 25 | 510 | 296 | 33 | 0 | 0 | 41 |
normalized size | 1 | 1.00 | 1.00 | 20.40 | 11.84 | 1.32 | 0.00 | 0.00 | 1.64 |
time (sec) |
N/A |
0.041 |
0.338 |
1.415 |
0.835 |
0.436 |
0.000 |
0.000 |
2.675 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 84 | 251 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.76 | 2.28 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.081 |
1.382 |
1.450 |
0.000 |
0.440 |
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 |
C |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 184 | 670 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.67 | 6.09 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.083 |
1.436 |
1.417 |
0.000 |
0.434 |
0.000 |
0.000 |
0.000 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 72 | 72 | 58 | 201 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.81 | 2.79 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.054 |
0.406 |
1.721 |
0.000 |
0.422 |
0.000 |
0.000 |
0.000 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 68 | 68 | 126 | 588 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.85 | 8.65 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.057 |
0.761 |
1.805 |
0.000 |
0.422 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
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|
|
|
|
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|
|
|
Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 66 | 198 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.88 | 2.64 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.064 |
0.316 |
1.701 |
0.000 |
0.428 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
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|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 133 | 614 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.73 | 7.97 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.064 |
1.312 |
1.433 |
0.000 |
0.419 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
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|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 112 | 112 | 79 | 241 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.71 | 2.15 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.090 |
0.747 |
1.759 |
0.000 |
0.427 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
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|
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|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 112 | 112 | 143 | 636 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.28 | 5.68 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.086 |
1.556 |
1.566 |
0.000 |
0.443 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 21 | 21 | 41 | 0 | 19 | 0 | 47 | 21 |
normalized size | 1 | 1.00 | 1.00 | 1.95 | 0.00 | 0.90 | 0.00 | 2.24 | 1.00 |
time (sec) |
N/A |
0.023 |
0.175 |
2.576 |
0.000 |
0.410 |
0.000 |
0.361 |
0.210 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
B |
A |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 21 | 21 | 506 | 283 | 29 | 0 | 0 | 37 |
normalized size | 1 | 1.00 | 1.00 | 24.10 | 13.48 | 1.38 | 0.00 | 0.00 | 1.76 |
time (sec) |
N/A |
0.029 |
0.329 |
1.533 |
0.756 |
0.421 |
0.000 |
0.000 |
2.755 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 19 | 70 | 42 | 19 | 0 | 24 | 42 |
normalized size | 1 | 1.00 | 1.00 | 3.68 | 2.21 | 1.00 | 0.00 | 1.26 | 2.21 |
time (sec) |
N/A |
0.024 |
0.069 |
1.564 |
0.630 |
0.415 |
0.000 |
0.278 |
2.456 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 19 | 56 | 30 | 19 | 0 | 33 | 19 |
normalized size | 1 | 1.00 | 1.00 | 2.95 | 1.58 | 1.00 | 0.00 | 1.74 | 1.00 |
time (sec) |
N/A |
0.023 |
0.037 |
1.174 |
0.328 |
0.431 |
0.000 |
0.239 |
2.387 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 19 | 47 | 33 | 19 | 0 | 25 | 23 |
normalized size | 1 | 1.00 | 1.00 | 2.47 | 1.74 | 1.00 | 0.00 | 1.32 | 1.21 |
time (sec) |
N/A |
0.023 |
0.038 |
1.289 |
0.324 |
0.407 |
0.000 |
0.281 |
2.375 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
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 | 19 | 19 | 19 | 34 | 20 | 19 | 0 | 22 | 21 |
normalized size | 1 | 1.00 | 1.00 | 1.79 | 1.05 | 1.00 | 0.00 | 1.16 | 1.11 |
time (sec) |
N/A |
0.023 |
0.063 |
1.083 |
0.335 |
0.443 |
0.000 |
0.440 |
2.399 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
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 | 17 | 17 | 17 | 18 | 22 | 19 | 0 | 26 | 22 |
normalized size | 1 | 1.00 | 1.00 | 1.06 | 1.29 | 1.12 | 0.00 | 1.53 | 1.29 |
time (sec) |
N/A |
0.015 |
0.017 |
0.941 |
0.365 |
0.433 |
0.000 |
0.246 |
0.058 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
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 | 11 | 11 | 11 | 12 | 11 | 19 | 0 | 12 | 11 |
normalized size | 1 | 1.00 | 1.00 | 1.09 | 1.00 | 1.73 | 0.00 | 1.09 | 1.00 |
time (sec) |
N/A |
0.013 |
0.005 |
0.847 |
0.361 |
0.406 |
0.000 |
0.174 |
2.417 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 11 | 11 | 23 | 12 | 11 | 11 | 14 | 12 | 11 |
normalized size | 1 | 1.00 | 2.09 | 1.09 | 1.00 | 1.00 | 1.27 | 1.09 | 1.00 |
time (sec) |
N/A |
0.005 |
0.008 |
0.201 |
0.380 |
0.413 |
0.113 |
0.343 |
0.020 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
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 | 17 | 17 | 33 | 18 | 23 | 17 | 49 | 15 | 14 |
normalized size | 1 | 1.00 | 1.94 | 1.06 | 1.35 | 1.00 | 2.88 | 0.88 | 0.82 |
time (sec) |
N/A |
0.023 |
0.015 |
1.033 |
0.341 |
0.416 |
8.215 |
0.338 |
2.378 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 46 | 32 | 21 | 19 | 0 | 23 | 22 |
normalized size | 1 | 1.00 | 2.42 | 1.68 | 1.11 | 1.00 | 0.00 | 1.21 | 1.16 |
time (sec) |
N/A |
0.024 |
0.041 |
1.724 |
0.663 |
0.562 |
0.000 |
0.187 |
0.036 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 31 | 45 | 33 | 19 | 0 | 25 | 19 |
normalized size | 1 | 1.00 | 1.63 | 2.37 | 1.74 | 1.00 | 0.00 | 1.32 | 1.00 |
time (sec) |
N/A |
0.024 |
0.031 |
1.452 |
0.337 |
0.413 |
0.000 |
0.639 |
2.348 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 38 | 52 | 30 | 19 | 0 | 33 | 23 |
normalized size | 1 | 1.00 | 2.00 | 2.74 | 1.58 | 1.00 | 0.00 | 1.74 | 1.21 |
time (sec) |
N/A |
0.025 |
0.033 |
2.008 |
0.352 |
0.424 |
0.000 |
0.322 |
2.423 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 76 | 92 | 95 | 99 | 0 | 164 | 153 |
normalized size | 1 | 1.00 | 0.89 | 1.08 | 1.12 | 1.16 | 0.00 | 1.93 | 1.80 |
time (sec) |
N/A |
0.072 |
0.276 |
1.043 |
0.891 |
0.448 |
0.000 |
0.504 |
4.941 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 60 | 72 | 70 | 88 | 0 | 122 | 109 |
normalized size | 1 | 1.00 | 0.95 | 1.14 | 1.11 | 1.40 | 0.00 | 1.94 | 1.73 |
time (sec) |
N/A |
0.058 |
0.164 |
1.221 |
0.322 |
0.437 |
0.000 |
0.491 |
4.186 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 47 | 51 | 58 | 74 | 0 | 107 | 85 |
normalized size | 1 | 1.00 | 1.00 | 1.09 | 1.23 | 1.57 | 0.00 | 2.28 | 1.81 |
time (sec) |
N/A |
0.043 |
0.019 |
1.134 |
0.329 |
0.429 |
0.000 |
0.247 |
3.044 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 24 | 24 | 32 | 31 | 60 | 0 | 57 | 47 |
normalized size | 1 | 1.00 | 1.00 | 1.33 | 1.29 | 2.50 | 0.00 | 2.38 | 1.96 |
time (sec) |
N/A |
0.018 |
0.013 |
0.939 |
0.368 |
0.453 |
0.000 |
1.354 |
2.411 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 16 | 16 | 16 | 30 | 37 | 36 | 0 | 43 | 57 |
normalized size | 1 | 1.00 | 1.00 | 1.88 | 2.31 | 2.25 | 0.00 | 2.69 | 3.56 |
time (sec) |
N/A |
0.029 |
0.004 |
0.808 |
0.400 |
0.441 |
0.000 |
0.219 |
2.428 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 15 | 15 | 26 | 21 | 20 | 17 | 0 | 39 | 17 |
normalized size | 1 | 1.00 | 1.73 | 1.40 | 1.33 | 1.13 | 0.00 | 2.60 | 1.13 |
time (sec) |
N/A |
0.032 |
0.009 |
0.939 |
0.398 |
0.458 |
0.000 |
0.232 |
2.411 |
|
|
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Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 35 | 38 | 34 | 29 | 0 | 82 | 31 |
normalized size | 1 | 1.00 | 0.92 | 1.00 | 0.89 | 0.76 | 0.00 | 2.16 | 0.82 |
time (sec) |
N/A |
0.042 |
0.069 |
1.134 |
0.519 |
0.422 |
0.000 |
0.221 |
2.453 |
|
|
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Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 57 | 49 | 46 | 42 | 0 | 98 | 55 |
normalized size | 1 | 1.00 | 1.06 | 0.91 | 0.85 | 0.78 | 0.00 | 1.81 | 1.02 |
time (sec) |
N/A |
0.051 |
0.069 |
1.422 |
0.323 |
0.424 |
0.000 |
0.269 |
2.511 |
|
|
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Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 76 | 76 | 73 | 60 | 57 | 53 | 0 | 140 | 75 |
normalized size | 1 | 1.00 | 0.96 | 0.79 | 0.75 | 0.70 | 0.00 | 1.84 | 0.99 |
time (sec) |
N/A |
0.065 |
0.133 |
1.429 |
0.421 |
0.425 |
0.000 |
0.225 |
2.504 |
|
|
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|
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Problem 46 |
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 | 92 | 92 | 89 | 70 | 69 | 64 | 0 | 154 | 113 |
normalized size | 1 | 1.00 | 0.97 | 0.76 | 0.75 | 0.70 | 0.00 | 1.67 | 1.23 |
time (sec) |
N/A |
0.076 |
0.118 |
1.898 |
0.351 |
0.427 |
0.000 |
0.585 |
6.313 |
|
|
|
|
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Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 169 | 169 | 102 | 526 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.60 | 3.11 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.144 |
0.506 |
1.367 |
0.000 |
0.436 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
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|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 135 | 135 | 90 | 508 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.67 | 3.76 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.122 |
0.305 |
1.794 |
0.000 |
0.460 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
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|
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|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 109 | 109 | 73 | 456 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.67 | 4.18 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.101 |
0.205 |
1.534 |
0.000 |
0.432 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
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|
|
|
|
|
|
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|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 57 | 450 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.67 | 5.29 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.092 |
0.091 |
1.669 |
0.000 |
0.452 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
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|
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|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 116 | 116 | 81 | 470 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.70 | 4.05 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.107 |
0.184 |
1.347 |
0.000 |
0.431 |
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 |
C |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 147 | 147 | 91 | 482 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.62 | 3.28 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.135 |
0.580 |
2.161 |
0.000 |
0.428 |
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 |
C |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 176 | 176 | 104 | 493 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.59 | 2.80 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.144 |
0.695 |
1.520 |
0.000 |
0.439 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
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|
|
|
|
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|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 122 | 122 | 80 | 144 | 127 | 122 | 0 | 246 | 197 |
normalized size | 1 | 1.00 | 0.66 | 1.18 | 1.04 | 1.00 | 0.00 | 2.02 | 1.61 |
time (sec) |
N/A |
0.099 |
0.653 |
1.710 |
0.351 |
0.451 |
0.000 |
0.275 |
5.166 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 97 | 97 | 71 | 130 | 139 | 117 | 0 | 230 | 160 |
normalized size | 1 | 1.00 | 0.73 | 1.34 | 1.43 | 1.21 | 0.00 | 2.37 | 1.65 |
time (sec) |
N/A |
0.084 |
0.261 |
1.371 |
0.360 |
0.450 |
0.000 |
0.298 |
5.032 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 78 | 78 | 51 | 83 | 79 | 94 | 0 | 162 | 123 |
normalized size | 1 | 1.00 | 0.65 | 1.06 | 1.01 | 1.21 | 0.00 | 2.08 | 1.58 |
time (sec) |
N/A |
0.081 |
0.235 |
1.523 |
0.339 |
0.426 |
0.000 |
0.282 |
4.607 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 59 | 70 | 75 | 82 | 0 | 115 | 89 |
normalized size | 1 | 1.00 | 1.16 | 1.37 | 1.47 | 1.61 | 0.00 | 2.25 | 1.75 |
time (sec) |
N/A |
0.051 |
0.019 |
1.088 |
0.435 |
0.427 |
0.000 |
0.255 |
3.141 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 27 | 35 | 34 | 71 | 0 | 60 | 161 |
normalized size | 1 | 1.00 | 1.00 | 1.30 | 1.26 | 2.63 | 0.00 | 2.22 | 5.96 |
time (sec) |
N/A |
0.019 |
0.007 |
0.945 |
0.390 |
0.426 |
0.000 |
0.400 |
2.538 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 38 | 41 | 46 | 45 | 0 | 70 | 68 |
normalized size | 1 | 1.00 | 1.41 | 1.52 | 1.70 | 1.67 | 0.00 | 2.59 | 2.52 |
time (sec) |
N/A |
0.050 |
0.029 |
0.777 |
0.458 |
0.439 |
0.000 |
0.228 |
2.515 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 42 | 42 | 55 | 45 | 42 | 33 | 0 | 86 | 34 |
normalized size | 1 | 1.00 | 1.31 | 1.07 | 1.00 | 0.79 | 0.00 | 2.05 | 0.81 |
time (sec) |
N/A |
0.059 |
0.061 |
0.903 |
0.399 |
0.461 |
0.000 |
0.219 |
2.503 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 53 | 57 | 55 | 45 | 0 | 138 | 66 |
normalized size | 1 | 1.00 | 0.95 | 1.02 | 0.98 | 0.80 | 0.00 | 2.46 | 1.18 |
time (sec) |
N/A |
0.084 |
0.143 |
1.359 |
0.346 |
0.424 |
0.000 |
0.242 |
2.535 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 88 | 88 | 70 | 84 | 77 | 65 | 0 | 200 | 81 |
normalized size | 1 | 1.00 | 0.80 | 0.95 | 0.88 | 0.74 | 0.00 | 2.27 | 0.92 |
time (sec) |
N/A |
0.083 |
0.200 |
1.729 |
0.664 |
0.442 |
0.000 |
0.206 |
2.527 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 98 | 98 | 87 | 89 | 89 | 73 | 0 | 222 | 104 |
normalized size | 1 | 1.00 | 0.89 | 0.91 | 0.91 | 0.74 | 0.00 | 2.27 | 1.06 |
time (sec) |
N/A |
0.115 |
0.221 |
1.693 |
0.350 |
0.425 |
0.000 |
0.209 |
2.580 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 132 | 132 | 102 | 115 | 115 | 93 | 0 | 284 | 126 |
normalized size | 1 | 1.00 | 0.77 | 0.87 | 0.87 | 0.70 | 0.00 | 2.15 | 0.95 |
time (sec) |
N/A |
0.099 |
0.348 |
2.173 |
0.366 |
0.467 |
0.000 |
0.245 |
2.667 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 178 | 178 | 640 | 832 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 3.60 | 4.67 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.161 |
6.584 |
1.403 |
0.000 |
0.448 |
0.000 |
0.000 |
0.000 |
|
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Problem 66 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 136 | 136 | 249 | 647 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.83 | 4.76 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.128 |
2.015 |
1.797 |
0.000 |
0.573 |
0.000 |
0.000 |
0.000 |
|
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|
Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 156 | 719 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.42 | 6.54 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.111 |
1.510 |
1.569 |
0.000 |
0.460 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
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|
|
Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 117 | 117 | 179 | 602 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.53 | 5.15 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.134 |
1.487 |
1.740 |
0.000 |
0.437 |
0.000 |
0.000 |
0.000 |
|
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|
Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 150 | 150 | 169 | 766 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.13 | 5.11 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.154 |
2.255 |
1.358 |
0.000 |
0.442 |
0.000 |
0.000 |
0.000 |
|
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|
Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 185 | 185 | 313 | 645 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.69 | 3.49 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.159 |
4.468 |
1.716 |
0.000 |
0.458 |
0.000 |
0.000 |
0.000 |
|
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|
2.3 Detailed conclusion table specific for Rubi results
The following table is specific to Rubi. It gives additional statistics for each integral. the column
steps is the number of steps used by Rubi to obtain the antiderivative. The rules
column is the number of unique rules used. The integrand size column is the leaf
size of the integrand. Finally the ratio \(\frac {\text {number of rules}}{\text {integrand size}}\) is given. The larger this ratio is, the harder
the integral was to solve. In this test, problem number [39] had the largest ratio of
[.2308]
|
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|
| Table 2.1:Rubi specific breakdown of results for each
integral
|
|
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|
|
|
|
# |
grade |
|
|
normalized | antiderivative |
leaf size | |
|
\(\frac {\text {number of rules}}{\text {integrand leaf size}}\) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
A |
3 |
2 |
1.00 |
21 |
0.095 |
|
|
|
|
|
|
|
2 |
A |
4 |
3 |
1.00 |
21 |
0.143 |
|
|
|
|
|
|
|
3 |
A |
3 |
2 |
1.00 |
21 |
0.095 |
|
|
|
|
|
|
|
4 |
A |
3 |
3 |
1.00 |
21 |
0.143 |
|
|
|
|
|
|
|
5 |
A |
3 |
3 |
1.00 |
21 |
0.143 |
|
|
|
|
|
|
|
6 |
A |
2 |
2 |
1.00 |
19 |
0.105 |
|
|
|
|
|
|
|
7 |
A |
3 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
8 |
A |
2 |
2 |
1.00 |
19 |
0.105 |
|
|
|
|
|
|
|
9 |
A |
2 |
2 |
1.00 |
21 |
0.095 |
|
|
|
|
|
|
|
10 |
A |
3 |
2 |
1.00 |
21 |
0.095 |
|
|
|
|
|
|
|
11 |
A |
3 |
3 |
1.00 |
21 |
0.143 |
|
|
|
|
|
|
|
12 |
A |
4 |
3 |
1.00 |
21 |
0.143 |
|
|
|
|
|
|
|
13 |
A |
4 |
3 |
1.00 |
21 |
0.143 |
|
|
|
|
|
|
|
14 |
A |
1 |
1 |
1.00 |
29 |
0.034 |
|
|
|
|
|
|
|
15 |
A |
1 |
1 |
1.00 |
28 |
0.036 |
|
|
|
|
|
|
|
16 |
A |
4 |
4 |
1.00 |
25 |
0.160 |
|
|
|
|
|
|
|
17 |
A |
4 |
4 |
1.00 |
25 |
0.160 |
|
|
|
|
|
|
|
18 |
A |
3 |
3 |
1.00 |
25 |
0.120 |
|
|
|
|
|
|
|
19 |
A |
3 |
3 |
1.00 |
25 |
0.120 |
|
|
|
|
|
|
|
20 |
A |
3 |
3 |
1.00 |
25 |
0.120 |
|
|
|
|
|
|
|
21 |
A |
3 |
3 |
1.00 |
25 |
0.120 |
|
|
|
|
|
|
|
22 |
A |
4 |
4 |
1.00 |
25 |
0.160 |
|
|
|
|
|
|
|
23 |
A |
4 |
4 |
1.00 |
25 |
0.160 |
|
|
|
|
|
|
|
24 |
A |
1 |
1 |
1.00 |
23 |
0.043 |
|
|
|
|
|
|
|
25 |
A |
1 |
1 |
1.00 |
24 |
0.042 |
|
|
|
|
|
|
|
26 |
A |
1 |
1 |
1.00 |
21 |
0.048 |
|
|
|
|
|
|
|
27 |
A |
1 |
1 |
1.00 |
21 |
0.048 |
|
|
|
|
|
|
|
28 |
A |
1 |
1 |
1.00 |
21 |
0.048 |
|
|
|
|
|
|
|
29 |
A |
1 |
1 |
1.00 |
21 |
0.048 |
|
|
|
|
|
|
|
30 |
A |
1 |
1 |
1.00 |
19 |
0.053 |
|
|
|
|
|
|
|
31 |
A |
2 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
32 |
A |
1 |
1 |
1.00 |
8 |
0.125 |
|
|
|
|
|
|
|
33 |
A |
1 |
1 |
1.00 |
19 |
0.053 |
|
|
|
|
|
|
|
34 |
A |
1 |
1 |
1.00 |
21 |
0.048 |
|
|
|
|
|
|
|
35 |
A |
1 |
1 |
1.00 |
21 |
0.048 |
|
|
|
|
|
|
|
36 |
A |
1 |
1 |
1.00 |
21 |
0.048 |
|
|
|
|
|
|
|
37 |
A |
7 |
5 |
1.00 |
28 |
0.179 |
|
|
|
|
|
|
|
38 |
A |
6 |
5 |
1.00 |
28 |
0.179 |
|
|
|
|
|
|
|
39 |
A |
6 |
6 |
1.00 |
26 |
0.231 |
|
|
|
|
|
|
|
40 |
A |
4 |
3 |
1.00 |
19 |
0.158 |
|
|
|
|
|
|
|
41 |
A |
4 |
4 |
1.00 |
26 |
0.154 |
|
|
|
|
|
|
|
42 |
A |
3 |
3 |
1.00 |
28 |
0.107 |
|
|
|
|
|
|
|
43 |
A |
5 |
5 |
1.00 |
28 |
0.179 |
|
|
|
|
|
|
|
44 |
A |
6 |
5 |
1.00 |
28 |
0.179 |
|
|
|
|
|
|
|
45 |
A |
7 |
5 |
1.00 |
28 |
0.179 |
|
|
|
|
|
|
|
46 |
A |
7 |
5 |
1.00 |
28 |
0.179 |
|
|
|
|
|
|
|
47 |
A |
10 |
7 |
1.00 |
32 |
0.219 |
|
|
|
|
|
|
|
48 |
A |
9 |
7 |
1.00 |
32 |
0.219 |
|
|
|
|
|
|
|
49 |
A |
8 |
7 |
1.00 |
32 |
0.219 |
|
|
|
|
|
|
|
50 |
A |
7 |
6 |
1.00 |
32 |
0.188 |
|
|
|
|
|
|
|
51 |
A |
8 |
7 |
1.00 |
32 |
0.219 |
|
|
|
|
|
|
|
52 |
A |
9 |
7 |
1.00 |
32 |
0.219 |
|
|
|
|
|
|
|
53 |
A |
10 |
7 |
1.00 |
32 |
0.219 |
|
|
|
|
|
|
|
54 |
A |
7 |
5 |
1.00 |
29 |
0.172 |
|
|
|
|
|
|
|
55 |
A |
6 |
5 |
1.00 |
29 |
0.172 |
|
|
|
|
|
|
|
56 |
A |
6 |
6 |
1.00 |
29 |
0.207 |
|
|
|
|
|
|
|
57 |
A |
5 |
5 |
1.00 |
27 |
0.185 |
|
|
|
|
|
|
|
58 |
A |
4 |
3 |
1.00 |
20 |
0.150 |
|
|
|
|
|
|
|
59 |
A |
4 |
4 |
1.00 |
27 |
0.148 |
|
|
|
|
|
|
|
60 |
A |
4 |
4 |
1.00 |
29 |
0.138 |
|
|
|
|
|
|
|
61 |
A |
6 |
5 |
1.00 |
29 |
0.172 |
|
|
|
|
|
|
|
62 |
A |
6 |
5 |
1.00 |
29 |
0.172 |
|
|
|
|
|
|
|
63 |
A |
8 |
6 |
1.00 |
29 |
0.207 |
|
|
|
|
|
|
|
64 |
A |
7 |
5 |
1.00 |
29 |
0.172 |
|
|
|
|
|
|
|
65 |
A |
8 |
6 |
1.00 |
33 |
0.182 |
|
|
|
|
|
|
|
66 |
A |
7 |
6 |
1.00 |
33 |
0.182 |
|
|
|
|
|
|
|
67 |
A |
6 |
5 |
1.00 |
33 |
0.152 |
|
|
|
|
|
|
|
68 |
A |
6 |
5 |
1.00 |
33 |
0.152 |
|
|
|
|
|
|
|
69 |
A |
7 |
6 |
1.00 |
33 |
0.182 |
|
|
|
|
|
|
|
70 |
A |
8 |
6 |
1.00 |
33 |
0.182 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|