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, 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 }
B grade: { 35, 36, 37 }
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, 28, 29, 30,
31, 32, 33, 34, 37, 38, 39, 40, 43, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60 }
B grade: { 27, 35, 36, 41, 42, 44, 45, 46, 47, 48, 54 }
C grade: { 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78 }
F grade: { }
2.1.3 Maple
A grade: { 23, 28, 29, 30 }
B grade: { 20, 21, 22, 24, 25, 26, 27, 31, 32, 33, 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
}
C grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 34, 35, 36, 37, 38, 39
}
F grade: { }
2.1.4 Maxima
A grade: { 1, 2, 3, 4, 8, 9, 10, 11, 15, 16, 17, 18, 19, 34, 36, 37, 38, 39
B grade: { 35 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 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 }
2.1.5 FriCAS
A grade: { 1, 2, 3, 4, 8, 9, 10, 11, 15, 16, 17, 18, 19, 20, 21, 22, 23, 27, 28, 29, 30, 34, 35, 36, 37, 38, 39, 41,
42, 43, 47, 48, 49, 54, 55, 56 }
B grade: { 5, 6, 7, 12, 13, 14, 40 }
C grade: { }
F grade: { 24, 25, 26, 31, 32, 33, 44, 45, 46, 50, 51, 52, 53, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
70, 71, 72, 73, 74, 75, 76, 77, 78 }
2.1.6 Sympy
A grade: { }
B grade: { }
C grade: { 10, 11, 15, 16, 17, 18, 19, 29, 30, 34, 35, 36, 37, 38, 39 }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 20, 21, 22, 23, 24, 25, 26, 27, 28, 31, 32, 33, 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 }
2.1.7 Giac
A grade: { 4, 8, 9, 10, 11, 15, 16, 34, 35, 36, 37, 47, 48, 49, 54, 55, 56 }
B grade: { 1, 2, 3, 26, 33, 38, 39, 40, 41, 42, 43, 44, 45, 46, 50, 51, 57, 58 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 52, 53, 59, 60, 61, 62,
63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78 }
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 |
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grade |
A |
A |
A |
C |
A |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 415 | 415 | 355 | 959 | 644 | 888 | 0 | 1122 |
normalized size |
1 |
1. |
0.86 |
2.31 |
1.55 |
2.14 |
0. |
2.7 |
time (sec) |
N/A |
0.673 |
0.494 |
0.029 |
1.994 |
1.1 |
0. |
3.147 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
A |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 286 | 286 | 244 | 652 | 459 | 617 | 0 | 772 |
normalized size |
1 |
1. |
0.85 |
2.28 |
1.6 |
2.16 |
0. |
2.7 |
time (sec) |
N/A |
0.563 |
0.326 |
0.013 |
3.756 |
1.104 |
0. |
1.654 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
A |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 168 | 170 | 141 | 377 | 263 | 386 | 0 | 429 |
normalized size |
1 |
1.01 |
0.84 |
2.24 |
1.57 |
2.3 |
0. |
2.55 |
time (sec) |
N/A |
0.25 |
0.17 |
0.01 |
3.295 |
1.086 |
0. |
3.044 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 95 | 95 | 71 | 185 | 154 | 224 | 0 | 198 |
normalized size |
1 |
1. |
0.75 |
1.95 |
1.62 |
2.36 |
0. |
2.08 |
time (sec) |
N/A |
0.073 |
0.062 |
0.011 |
4.046 |
1.042 |
0. |
1.896 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 122 | 122 | 117 | 373 | 0 | 1019 | 0 | 0 |
normalized size |
1 |
1. |
0.96 |
3.06 |
0. |
8.35 |
0. |
0. |
time (sec) |
N/A |
0.311 |
0.145 |
0.04 |
0. |
29.882 |
0. |
0. |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F(-2) |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 163 | 163 | 211 | 899 | 0 | 2082 | 0 | 0 |
normalized size |
1 |
1. |
1.29 |
5.52 |
0. |
12.77 |
0. |
0. |
time (sec) |
N/A |
0.331 |
0.442 |
0.041 |
0. |
118.982 |
0. |
0. |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F(-2) |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 248 | 248 | 273 | 1449 | 0 | 3105 | 0 | 0 |
normalized size |
1 |
1. |
1.1 |
5.84 |
0. |
12.52 |
0. |
0. |
time (sec) |
N/A |
0.355 |
0.396 |
0.045 |
0. |
1.448 |
0. |
0. |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 340 | 340 | 241 | 643 | 524 | 644 | 0 | 551 |
normalized size |
1 |
1. |
0.71 |
1.89 |
1.54 |
1.89 |
0. |
1.62 |
time (sec) |
N/A |
0.633 |
0.37 |
0.025 |
3.199 |
1.143 |
0. |
2.271 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 228 | 228 | 160 | 423 | 356 | 435 | 0 | 352 |
normalized size |
1 |
1. |
0.7 |
1.86 |
1.56 |
1.91 |
0. |
1.54 |
time (sec) |
N/A |
0.493 |
0.207 |
0.024 |
4.342 |
1.143 |
0. |
2.824 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
A |
A |
C |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 130 | 133 | 88 | 235 | 205 | 267 | 617 | 186 |
normalized size | 1 | 1.02 | 0.68 | 1.81 | 1.58 | 2.05 | 4.75 | 1.43 |
time (sec) |
N/A |
0.23 |
0.101 |
0.018 |
3.604 |
1.087 |
112.876 |
1.983 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
A |
A |
C |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 45 | 117 | 105 | 167 | 282 | 97 |
normalized size | 1 | 1. | 0.71 | 1.86 | 1.67 | 2.65 | 4.48 | 1.54 |
time (sec) |
N/A |
0.061 |
0.034 |
0.016 |
3.776 |
1.019 |
20.572 |
3.103 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 122 | 122 | 117 | 373 | 0 | 1019 | 0 | 0 |
normalized size |
1 |
1. |
0.96 |
3.06 |
0. |
8.35 |
0. |
0. |
time (sec) |
N/A |
0.283 |
0.127 |
0. |
0. |
30.013 |
0. |
0. |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F(-2) |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 163 | 163 | 211 | 899 | 0 | 2082 | 0 | 0 |
normalized size |
1 |
1. |
1.29 |
5.52 |
0. |
12.77 |
0. |
0. |
time (sec) |
N/A |
0.295 |
0.413 |
0. |
0. |
117.112 |
0. |
0. |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F(-2) |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 248 | 248 | 273 | 1449 | 0 | 3105 | 0 | 0 |
normalized size |
1 |
1. |
1.1 |
5.84 |
0. |
12.52 |
0. |
0. |
time (sec) |
N/A |
0.329 |
0.179 |
0. |
0. |
1.43 |
0. |
0. |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
A |
A |
C |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 57 | 139 | 134 | 189 | 313 | 123 |
normalized size | 1 | 1. | 0.72 | 1.76 | 1.7 | 2.39 | 3.96 | 1.56 |
time (sec) |
N/A |
0.139 |
0.061 |
0. |
4.966 |
1.142 |
46.387 |
2.264 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
A |
A |
C |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 45 | 117 | 105 | 167 | 282 | 97 |
normalized size | 1 | 1. | 0.71 | 1.86 | 1.67 | 2.65 | 4.48 | 1.54 |
time (sec) |
N/A |
0.061 |
0.032 |
0. |
2.397 |
1.043 |
20.862 |
1.866 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
A |
A |
C |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 48 | 96 | 89 | 196 | 245 | 0 |
normalized size |
1 |
1. |
1. |
2. |
1.85 |
4.08 |
5.1 |
0. |
time (sec) |
N/A |
0.183 |
0.052 |
0. |
4.226 |
1.183 |
28.239 |
0. |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
A |
A |
C |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 48 | 97 | 89 | 201 | 221 | 0 |
normalized size |
1 |
1. |
1. |
2.02 |
1.85 |
4.19 |
4.6 |
0. |
time (sec) |
N/A |
0.176 |
0.056 |
0. |
3.266 |
1.147 |
27.753 |
0. |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
A |
A |
C |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 71 | 71 | 56 | 108 | 132 | 154 | 218 | 0 |
normalized size |
1 |
1. |
0.79 |
1.52 |
1.86 |
2.17 |
3.07 |
0. |
time (sec) |
N/A |
0.184 |
0.047 |
0. |
3.974 |
1.018 |
34.289 |
0. |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-2) |
A |
F(-1) |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 591 | 584 | 427 | 1446 | 0 | 2147 | 0 | 0 |
normalized size |
1 |
0.99 |
0.72 |
2.45 |
0. |
3.63 |
0. |
0. |
time (sec) |
N/A |
1.517 |
1.408 |
0.038 |
0. |
1.45 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
A |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 451 | 450 | 311 | 987 | 0 | 1517 | 0 | 0 |
normalized size |
1 |
1. |
0.69 |
2.19 |
0. |
3.36 |
0. |
0. |
time (sec) |
N/A |
1.01 |
0.995 |
0.017 |
0. |
1.307 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
A |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 300 | 297 | 200 | 588 | 0 | 980 | 0 | 0 |
normalized size |
1 |
0.99 |
0.67 |
1.96 |
0. |
3.27 |
0. |
0. |
time (sec) |
N/A |
0.446 |
0.646 |
0.013 |
0. |
1.274 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 221 | 221 | 142 | 287 | 0 | 602 | 0 | 0 |
normalized size |
1 |
1. |
0.64 |
1.3 |
0. |
2.72 |
0. |
0. |
time (sec) |
N/A |
0.147 |
0.38 |
0.011 |
0. |
1.148 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F(-1) |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 278 | 278 | 225 | 503 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.81 |
1.81 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.49 |
0.786 |
0.054 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F(-1) |
F(-2) |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 322 | 322 | 309 | 1200 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.96 |
3.73 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.579 |
1.033 |
0.046 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F(-1) |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 363 | 361 | 492 | 1848 | 0 | 0 | 0 | 2238 |
normalized size |
1 |
0.99 |
1.36 |
5.09 |
0. |
0. |
0. |
6.17 |
time (sec) |
N/A |
0.677 |
1.916 |
0.052 |
0. |
0. |
0. |
16.905 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F(-2) |
A |
F(-1) |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 501 | 496 | 1107 | 965 | 0 | 1538 | 0 | 0 |
normalized size |
1 |
0.99 |
2.21 |
1.93 |
0. |
3.07 |
0. |
0. |
time (sec) |
N/A |
1.281 |
6.542 |
0.028 |
0. |
2.118 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
F(-1) |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 368 | 369 | 555 | 635 | 0 | 1065 | 0 | 0 |
normalized size |
1 |
1. |
1.51 |
1.73 |
0. |
2.89 |
0. |
0. |
time (sec) |
N/A |
0.875 |
3.814 |
0.025 |
0. |
1.913 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
C |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 246 | 249 | 390 | 365 | 0 | 689 | 736 | 0 |
normalized size |
1 |
1.01 |
1.59 |
1.48 |
0. |
2.8 |
2.99 |
0. |
time (sec) |
N/A |
0.4 |
1.624 |
0.022 |
0. |
1.701 |
138.497 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
C |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 177 | 177 | 169 | 180 | 0 | 460 | 338 | 0 |
normalized size |
1 |
1. |
0.95 |
1.02 |
0. |
2.6 |
1.91 |
0. |
time (sec) |
N/A |
0.124 |
0.414 |
0.018 |
0. |
1.643 |
25.875 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F(-1) |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 278 | 278 | 225 | 503 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.81 |
1.81 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.464 |
0.729 |
0. |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F(-1) |
F(-2) |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 322 | 322 | 309 | 1200 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.96 |
3.73 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.53 |
0.969 |
0. |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F(-1) |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 363 | 361 | 492 | 1848 | 0 | 0 | 0 | 2238 |
normalized size |
1 |
0.99 |
1.36 |
5.09 |
0. |
0. |
0. |
6.17 |
time (sec) |
N/A |
0.588 |
1.836 |
0. |
0. |
0. |
0. |
10.678 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
C |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 151 | 149 | 137 | 147 | 176 | 308 | 130 |
normalized size | 1 | 1.74 | 1.71 | 1.57 | 1.69 | 2.02 | 3.54 | 1.49 |
time (sec) |
N/A |
0.146 |
0.344 |
0. |
2.139 |
1.624 |
44.26 |
2.21 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
B |
B |
C |
B |
A |
C |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 52 | 135 | 126 | 120 | 142 | 150 | 277 | 104 |
normalized size | 1 | 2.6 | 2.42 | 2.31 | 2.73 | 2.88 | 5.33 | 2. |
time (sec) |
N/A |
0.071 |
0.214 |
0. |
2.84 |
1.633 |
21.362 |
2.518 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
B |
B |
C |
A |
A |
C |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 55 | 135 | 128 | 95 | 86 | 184 | 240 | 96 |
normalized size | 1 | 2.45 | 2.33 | 1.73 | 1.56 | 3.35 | 4.36 | 1.75 |
time (sec) |
N/A |
0.185 |
0.407 |
0. |
2.306 |
1.56 |
26.976 |
2.134 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
B |
A |
C |
A |
A |
C |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 55 | 135 | 89 | 96 | 86 | 203 | 216 | 112 |
normalized size | 1 | 2.45 | 1.62 | 1.75 | 1.56 | 3.69 | 3.93 | 2.04 |
time (sec) |
N/A |
0.18 |
0.167 |
0. |
3.561 |
1.1 |
28.173 |
2.621 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 83 | 129 | 82 | 103 | 88 | 173 | 212 | 196 |
normalized size | 1 | 1.55 | 0.99 | 1.24 | 1.06 | 2.08 | 2.55 | 2.36 |
time (sec) |
N/A |
0.191 |
0.117 |
0. |
4.02 |
1.173 |
33.628 |
1.987 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 116 | 171 | 94 | 123 | 119 | 216 | 219 | 266 |
normalized size | 1 | 1.47 | 0.81 | 1.06 | 1.03 | 1.86 | 1.89 | 2.29 |
time (sec) |
N/A |
0.217 |
0.116 |
0. |
3.643 |
1.031 |
58.436 |
2.363 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
B |
F(-1) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 199 | 242 | 336 | 1095 | 0 | 2485 | 0 | 817 |
normalized size |
1 |
1.22 |
1.69 |
5.5 |
0. |
12.49 |
0. |
4.11 |
time (sec) |
N/A |
0.328 |
0.813 |
0.052 |
0. |
1.248 |
0. |
3.459 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F(-2) |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1348 | 1345 | 3599 | 6728 | 0 | 6765 | 0 | 3560 |
normalized size |
1 |
1. |
2.67 |
4.99 |
0. |
5.02 |
0. |
2.64 |
time (sec) |
N/A |
2.366 |
7.098 |
0.046 |
0. |
10.902 |
0. |
3.682 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F(-2) |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 721 | 719 | 2722 | 3571 | 0 | 3633 | 0 | 2006 |
normalized size |
1 |
1. |
3.78 |
4.95 |
0. |
5.04 |
0. |
2.78 |
time (sec) |
N/A |
0.963 |
6.483 |
0.021 |
0. |
3.786 |
0. |
3.573 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 330 | 330 | 306 | 1431 | 0 | 1843 | 0 | 856 |
normalized size |
1 |
1. |
0.93 |
4.34 |
0. |
5.58 |
0. |
2.59 |
time (sec) |
N/A |
0.298 |
1.849 |
0.016 |
0. |
1.543 |
0. |
1.583 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F(-2) |
F(-1) |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 450 | 453 | 1944 | 4227 | 0 | 0 | 0 | 1461 |
normalized size |
1 |
1.01 |
4.32 |
9.39 |
0. |
0. |
0. |
3.25 |
time (sec) |
N/A |
1.369 |
6.199 |
0.044 |
0. |
0. |
0. |
3.323 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F(-2) |
F(-1) |
F(-1) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 521 | 521 | 2665 | 5051 | 0 | 0 | 0 | 2140 |
normalized size |
1 |
1. |
5.12 |
9.69 |
0. |
0. |
0. |
4.11 |
time (sec) |
N/A |
1.696 |
6.274 |
0.042 |
0. |
0. |
0. |
13.147 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F(-2) |
F(-1) |
F(-1) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 658 | 657 | 2157 | 12065 | 0 | 0 | 0 | 11268 |
normalized size |
1 |
1. |
3.28 |
18.34 |
0. |
0. |
0. |
17.12 |
time (sec) |
N/A |
2.68 |
6.449 |
0.063 |
0. |
0. |
0. |
38.844 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F(-2) |
A |
F(-1) |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1032 | 1032 | 3220 | 3958 | 0 | 4766 | 0 | 2032 |
normalized size |
1 |
1. |
3.12 |
3.84 |
0. |
4.62 |
0. |
1.97 |
time (sec) |
N/A |
1.788 |
6.672 |
0.042 |
0. |
43.625 |
0. |
1.977 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F(-2) |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 540 | 540 | 2402 | 2002 | 0 | 2503 | 0 | 994 |
normalized size |
1 |
1. |
4.45 |
3.71 |
0. |
4.64 |
0. |
1.84 |
time (sec) |
N/A |
0.713 |
6.327 |
0.026 |
0. |
9.366 |
0. |
1.391 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 246 | 246 | 225 | 763 | 0 | 1277 | 0 | 425 |
normalized size |
1 |
1. |
0.91 |
3.1 |
0. |
5.19 |
0. |
1.73 |
time (sec) |
N/A |
0.23 |
1.07 |
0.018 |
0. |
2.511 |
0. |
2.662 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F(-1) |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 290 | 290 | 465 | 1822 | 0 | 0 | 0 | 797 |
normalized size |
1 |
1. |
1.6 |
6.28 |
0. |
0. |
0. |
2.75 |
time (sec) |
N/A |
0.672 |
3.86 |
0.033 |
0. |
0. |
0. |
1.809 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F(-1) |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 364 | 364 | 417 | 3670 | 0 | 0 | 0 | 1874 |
normalized size |
1 |
1. |
1.15 |
10.08 |
0. |
0. |
0. |
5.15 |
time (sec) |
N/A |
1.097 |
2.924 |
0.042 |
0. |
0. |
0. |
12.504 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F(-1) |
F(-1) |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 484 | 484 | 535 | 9100 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.11 |
18.8 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.563 |
6.305 |
0.088 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F(-1) |
F(-1) |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 685 | 685 | 739 | 15990 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.08 |
23.34 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.778 |
6.331 |
0.148 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F(-2) |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 718 | 715 | 2195 | 2528 | 0 | 3170 | 0 | 1284 |
normalized size |
1 |
1. |
3.06 |
3.52 |
0. |
4.42 |
0. |
1.79 |
time (sec) |
N/A |
1.336 |
6.518 |
0.04 |
0. |
14.866 |
0. |
4.73 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 371 | 369 | 379 | 1199 | 0 | 1631 | 0 | 603 |
normalized size |
1 |
0.99 |
1.02 |
3.23 |
0. |
4.4 |
0. |
1.63 |
time (sec) |
N/A |
0.509 |
2.018 |
0.03 |
0. |
4.981 |
0. |
3.439 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 164 | 164 | 173 | 425 | 0 | 879 | 0 | 262 |
normalized size |
1 |
1. |
1.05 |
2.59 |
0. |
5.36 |
0. |
1.6 |
time (sec) |
N/A |
0.149 |
0.766 |
0.02 |
0. |
2.514 |
0. |
3.686 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F(-1) |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 188 | 188 | 304 | 746 | 0 | 0 | 0 | 467 |
normalized size |
1 |
1. |
1.62 |
3.97 |
0. |
0. |
0. |
2.48 |
time (sec) |
N/A |
0.341 |
1. |
0.031 |
0. |
0. |
0. |
1.632 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F(-1) |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 254 | 254 | 324 | 2973 | 0 | 0 | 0 | 1831 |
normalized size |
1 |
1. |
1.28 |
11.7 |
0. |
0. |
0. |
7.21 |
time (sec) |
N/A |
0.638 |
2.011 |
0.052 |
0. |
0. |
0. |
11.154 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F(-1) |
F(-2) |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 424 | 424 | 513 | 7119 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.21 |
16.79 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.967 |
2.734 |
0.107 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F(-1) |
F(-2) |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 826 | 826 | 800 | 18802 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.97 |
22.76 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.433 |
6.081 |
0.246 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1182 | 1154 | 11933 | 14778 | 0 | 0 | 0 | 0 |
normalized size | 1 | 0.98 | 10.1 | 12.5 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
4.166 |
17.717 |
0.088 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 774 | 769 | 917 | 10271 | 0 | 0 | 0 | 0 |
normalized size | 1 | 0.99 | 1.18 | 13.27 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
2.23 |
13.329 |
0.042 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 706 | 706 | 633 | 6257 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.9 | 8.86 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
1.845 |
8.114 |
0.051 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 687 | 687 | 938 | 16177 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.37 |
23.55 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.903 |
13.409 |
0.096 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 964 | 964 | 9529 | 34395 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
9.88 |
35.68 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
3.116 |
16.847 |
0.209 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 66 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1716 | 1716 | 15719 | 68345 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
9.16 |
39.83 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
7.045 |
19.426 |
0.36 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1235 | 1235 | 12483 | 15857 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
10.11 |
12.84 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
4.395 |
18.448 |
0.056 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 766 | 766 | 922 | 9544 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.2 | 12.46 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
2.061 |
13.015 |
0.043 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 527 | 527 | 562 | 6049 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.07 | 11.48 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.98 |
9.696 |
0.032 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 540 | 540 | 551 | 4732 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.02 |
8.76 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.111 |
6.814 |
0.043 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 71 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 597 | 596 | 724 | 13614 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.21 |
22.8 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.359 |
11.974 |
0.094 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 72 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1034 | 1034 | 9186 | 33007 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
8.88 |
31.92 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
3.16 |
16.589 |
0.213 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 73 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 838 | 831 | 1000 | 10546 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0.99 |
1.19 |
12.58 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.167 |
13.841 |
0.051 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 74 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 528 | 524 | 615 | 6174 | 0 | 0 | 0 | 0 |
normalized size | 1 | 0.99 | 1.16 | 11.69 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
1.028 |
8.052 |
0.035 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 75 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 387 | 384 | 418 | 2497 | 0 | 0 | 0 | 0 |
normalized size | 1 | 0.99 | 1.08 | 6.45 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.505 |
5.836 |
0.028 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 76 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 422 | 422 | 477 | 3984 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.13 |
9.44 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.691 |
5.566 |
0.045 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 77 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 642 | 642 | 699 | 12981 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.09 |
20.22 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.517 |
10.931 |
0.122 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 78 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1116 | 1116 | 8844 | 34102 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
7.92 |
30.56 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
3.342 |
16.579 |
0.297 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
2.3 Detailed conclusion table specific for Rubi results
The following table is specific to Rubi. It gives additional statistics for each integral. the column
steps is the number of steps used by Rubi to obtain the antiderivative. The rules
column is the number of unique rules used. The integrand size column is the leaf size
of the integrand. Finally the ratio \(\frac{\text{number of rules}}{\text{integrand size}}\) is given. The larger this ratio is, the harder the
integral was to solve. In this test, problem number [36] had the largest ratio of [ 0.25
]
|
|
|
|
|
|
|
| Table 2.1Rubi specific breakdown of results for each
integral
|
|
|
|
|
|
|
# |
grade |
|
|
normalized | antiderivative |
leaf size | |
|
\(\frac{\text{number of rules}}{\text{integrand leaf size}}\) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
A |
7 |
6 |
1. |
37 |
0.162 |
|
|
|
|
|
|
|
2 |
A |
6 |
6 |
1. |
37 |
0.162 |
|
|
|
|
|
|
|
3 |
A |
5 |
5 |
1.01 |
35 |
0.143 |
|
|
|
|
|
|
|
4 |
A |
5 |
5 |
1. |
30 |
0.167 |
|
|
|
|
|
|
|
5 |
A |
6 |
6 |
1. |
37 |
0.162 |
|
|
|
|
|
|
|
6 |
A |
6 |
6 |
1. |
37 |
0.162 |
|
|
|
|
|
|
|
7 |
A |
5 |
5 |
1. |
37 |
0.135 |
|
|
|
|
|
|
|
8 |
A |
6 |
5 |
1. |
37 |
0.135 |
|
|
|
|
|
|
|
9 |
A |
5 |
5 |
1. |
37 |
0.135 |
|
|
|
|
|
|
|
10 |
A |
4 |
4 |
1.02 |
35 |
0.114 |
|
|
|
|
|
|
|
11 |
A |
4 |
4 |
1. |
30 |
0.133 |
|
|
|
|
|
|
|
12 |
A |
6 |
6 |
1. |
37 |
0.162 |
|
|
|
|
|
|
|
13 |
A |
6 |
6 |
1. |
37 |
0.162 |
|
|
|
|
|
|
|
14 |
A |
5 |
5 |
1. |
37 |
0.135 |
|
|
|
|
|
|
|
15 |
A |
4 |
4 |
1. |
31 |
0.129 |
|
|
|
|
|
|
|
16 |
A |
4 |
4 |
1. |
30 |
0.133 |
|
|
|
|
|
|
|
17 |
A |
7 |
7 |
1. |
33 |
0.212 |
|
|
|
|
|
|
|
18 |
A |
7 |
7 |
1. |
33 |
0.212 |
|
|
|
|
|
|
|
19 |
A |
6 |
6 |
1. |
33 |
0.182 |
|
|
|
|
|
|
|
20 |
A |
8 |
7 |
0.99 |
40 |
0.175 |
|
|
|
|
|
|
|
|
21 |
A |
7 |
7 |
1. |
40 |
0.175 |
|
|
|
|
|
|
|
22 |
A |
6 |
6 |
0.99 |
38 |
0.158 |
|
|
|
|
|
|
|
23 |
A |
6 |
6 |
1. |
33 |
0.182 |
|
|
|
|
|
|
|
24 |
A |
7 |
7 |
1. |
40 |
0.175 |
|
|
|
|
|
|
|
25 |
A |
7 |
7 |
1. |
40 |
0.175 |
|
|
|
|
|
|
|
26 |
A |
5 |
5 |
0.99 |
40 |
0.125 |
|
|
|
|
|
|
|
27 |
A |
7 |
6 |
0.99 |
40 |
0.15 |
|
|
|
|
|
|
|
28 |
A |
6 |
6 |
1. |
40 |
0.15 |
|
|
|
|
|
|
|
29 |
A |
5 |
5 |
1.01 |
38 |
0.132 |
|
|
|
|
|
|
|
30 |
A |
5 |
5 |
1. |
33 |
0.152 |
|
|
|
|
|
|
|
31 |
A |
7 |
7 |
1. |
40 |
0.175 |
|
|
|
|
|
|
|
32 |
A |
7 |
7 |
1. |
40 |
0.175 |
|
|
|
|
|
|
|
33 |
A |
5 |
5 |
0.99 |
40 |
0.125 |
|
|
|
|
|
|
|
34 |
A |
5 |
5 |
1.74 |
30 |
0.167 |
|
|
|
|
|
|
|
35 |
B |
5 |
5 |
2.6 |
29 |
0.172 |
|
|
|
|
|
|
|
36 |
B |
8 |
8 |
2.45 |
32 |
0.25 |
|
|
|
|
|
|
|
37 |
B |
8 |
8 |
2.45 |
32 |
0.25 |
|
|
|
|
|
|
|
38 |
A |
6 |
6 |
1.55 |
32 |
0.188 |
|
|
|
|
|
|
|
39 |
A |
7 |
7 |
1.47 |
32 |
0.219 |
|
|
|
|
|
|
|
40 |
A |
5 |
5 |
1.22 |
32 |
0.156 |
|
|
|
|
|
|
|
|
41 |
A |
8 |
7 |
1. |
36 |
0.194 |
|
|
|
|
|
|
|
42 |
A |
7 |
6 |
1. |
34 |
0.176 |
|
|
|
|
|
|
|
43 |
A |
7 |
6 |
1. |
29 |
0.207 |
|
|
|
|
|
|
|
44 |
A |
9 |
8 |
1.01 |
36 |
0.222 |
|
|
|
|
|
|
|
45 |
A |
9 |
8 |
1. |
36 |
0.222 |
|
|
|
|
|
|
|
46 |
A |
9 |
9 |
1. |
36 |
0.25 |
|
|
|
|
|
|
|
47 |
A |
7 |
7 |
1. |
36 |
0.194 |
|
|
|
|
|
|
|
48 |
A |
6 |
6 |
1. |
34 |
0.176 |
|
|
|
|
|
|
|
49 |
A |
6 |
6 |
1. |
29 |
0.207 |
|
|
|
|
|
|
|
50 |
A |
8 |
8 |
1. |
36 |
0.222 |
|
|
|
|
|
|
|
51 |
A |
8 |
8 |
1. |
36 |
0.222 |
|
|
|
|
|
|
|
52 |
A |
8 |
8 |
1. |
36 |
0.222 |
|
|
|
|
|
|
|
53 |
A |
6 |
6 |
1. |
36 |
0.167 |
|
|
|
|
|
|
|
54 |
A |
6 |
6 |
1. |
36 |
0.167 |
|
|
|
|
|
|
|
55 |
A |
5 |
5 |
0.99 |
34 |
0.147 |
|
|
|
|
|
|
|
56 |
A |
5 |
5 |
1. |
29 |
0.172 |
|
|
|
|
|
|
|
57 |
A |
7 |
7 |
1. |
36 |
0.194 |
|
|
|
|
|
|
|
58 |
A |
7 |
7 |
1. |
36 |
0.194 |
|
|
|
|
|
|
|
59 |
A |
5 |
5 |
1. |
36 |
0.139 |
|
|
|
|
|
|
|
60 |
A |
6 |
5 |
1. |
36 |
0.139 |
|
|
|
|
|
|
|
|
61 |
A |
10 |
7 |
0.98 |
38 |
0.184 |
|
|
|
|
|
|
|
62 |
A |
9 |
7 |
0.99 |
38 |
0.184 |
|
|
|
|
|
|
|
63 |
A |
9 |
7 |
1. |
38 |
0.184 |
|
|
|
|
|
|
|
64 |
A |
9 |
8 |
1. |
38 |
0.21 |
|
|
|
|
|
|
|
65 |
A |
9 |
7 |
1. |
38 |
0.184 |
|
|
|
|
|
|
|
66 |
A |
10 |
8 |
1. |
38 |
0.21 |
|
|
|
|
|
|
|
67 |
A |
10 |
7 |
1. |
38 |
0.184 |
|
|
|
|
|
|
|
68 |
A |
9 |
7 |
1. |
38 |
0.184 |
|
|
|
|
|
|
|
69 |
A |
8 |
7 |
1. |
38 |
0.184 |
|
|
|
|
|
|
|
70 |
A |
8 |
7 |
1. |
38 |
0.184 |
|
|
|
|
|
|
|
71 |
A |
8 |
7 |
1. |
38 |
0.184 |
|
|
|
|
|
|
|
72 |
A |
9 |
8 |
1. |
38 |
0.21 |
|
|
|
|
|
|
|
73 |
A |
9 |
7 |
0.99 |
38 |
0.184 |
|
|
|
|
|
|
|
74 |
A |
8 |
7 |
0.99 |
38 |
0.184 |
|
|
|
|
|
|
|
75 |
A |
7 |
6 |
0.99 |
38 |
0.158 |
|
|
|
|
|
|
|
76 |
A |
7 |
6 |
1. |
38 |
0.158 |
|
|
|
|
|
|
|
77 |
A |
8 |
7 |
1. |
38 |
0.184 |
|
|
|
|
|
|
|
78 |
A |
9 |
7 |
1. |
38 |
0.184 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|