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, 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 }
B grade: { }
C grade: { }
F grade: { 16, 17, 18, 19, 20}
2.1.2 Mathematica
A grade: { 1, 2, 4, 5, 6, 8, 9, 10, 14, 15, 21, 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, 51, 53, 54, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74,
76, 77 }
B grade: { 50, 63, 64 }
C grade: { 3, 7, 11, 12, 13, 18, 19, 55, 75 }
F grade: { 16, 17, 20, 22, 23, 52}
2.1.3 Maple
A grade: { 4, 5, 8, 9, 10, 13, 14, 15, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 35, 36, 37, 41, 42, 46, 47, 48,
49, 55, 56, 57, 60, 61, 62, 65, 66, 67, 71, 72, 76, 77 }
B grade: { 1, 2, 3, 6, 7, 11, 12, 32, 33, 34, 38, 39, 40, 43, 44, 45, 53, 54, 58, 59, 63, 64, 68, 69, 70, 73, 74, 75
}
C grade: { }
F grade: { 16, 17, 18, 19, 20, 50, 51, 52 }
2.1.4 Maxima
A grade: { 3, 4, 5, 9, 10, 14, 15, 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, 54, 56, 57, 61, 62, 65, 66, 67, 71, 72, 73, 74, 76, 77
B grade: { 1, 2, 6, 8, 11, 12, 53, 58, 59, 63, 64, 68, 69 }
C grade: { }
F grade: { 7, 13, 16, 17, 18, 19, 20, 48, 49, 50, 51, 52, 55, 60, 70, 75 }
2.1.5 FriCAS
A grade: { 4, 5, 9, 10, 14, 15, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 41, 42, 45, 46, 47, 48, 49, 50, 51, 52, 56,
57, 61, 62, 66, 67, 71, 72, 76, 77 }
B grade: { 8, 38, 39, 43, 44, 70, 75 }
C grade: { 1, 2, 3, 6, 7, 11, 12, 13, 53, 54, 55, 58, 59, 60, 63, 64, 65, 68, 69, 73, 74 }
F grade: { 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28 }
2.1.6 Sympy
A grade: { 4, 5, 8, 9, 10, 14, 15, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 49, 56, 57, 61, 62, 66, 67, 71, 72,
76, 77 }
B grade: { }
C grade: { }
F grade: { 1, 2, 3, 6, 7, 11, 12, 13, 16, 17, 18, 19, 20, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 50, 51, 52, 53, 54, 55, 58, 59, 60, 63, 64, 65, 68, 69, 70, 73, 74, 75 }
2.1.7 Giac
A grade: { 4, 5, 9, 10, 14, 15, 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, 56, 57, 61, 62, 66, 67, 71, 72, 76, 77 }
B grade: { 8 }
C grade: { }
F grade: { 1, 2, 3, 6, 7, 11, 12, 13, 16, 17, 18, 19, 20, 50, 51, 52, 53, 54, 55, 58, 59, 60, 63, 64, 65, 68, 69, 70,
73, 74, 75 }
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 |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 117 | 117 | 156 | 394 | 386 | 1310 | 0 | 0 |
normalized size | 1 | 1. | 1.33 | 3.37 | 3.3 | 11.2 | 0. | 0. |
time (sec) |
N/A |
0.187 |
1.675 |
0.112 |
1.741 |
1.802 |
0. |
0. |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 143 | 234 | 238 | 857 | 0 | 0 |
normalized size | 1 | 1. | 1.7 | 2.79 | 2.83 | 10.2 | 0. | 0. |
time (sec) |
N/A |
0.155 |
1.415 |
0.037 |
1.755 |
1.849 |
0. |
0. |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
A |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 57 | 57 | 211 | 109 | 105 | 489 | 0 | 0 |
normalized size | 1 | 1. | 3.7 | 1.91 | 1.84 | 8.58 | 0. | 0. |
time (sec) |
N/A |
0.09 |
4.008 |
0.029 |
1.854 |
1.764 |
0. |
0. |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.022 |
9.387 |
0.087 |
0. |
0. |
0. |
0. |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.022 |
16.403 |
0.073 |
0. |
0. |
0. |
0. |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 119 | 119 | 179 | 328 | 463 | 3497 | 0 | 0 |
normalized size | 1 | 1. | 1.5 | 2.76 | 3.89 | 29.39 | 0. | 0. |
time (sec) |
N/A |
0.21 |
2.021 |
0.041 |
1.822 |
1.903 |
0. |
0. |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
F |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 88 | 88 | 208 | 177 | 0 | 2051 | 0 | 0 |
normalized size | 1 | 1. | 2.36 | 2.01 | 0. | 23.31 | 0. | 0. |
time (sec) |
N/A |
0.137 |
4.606 |
0.036 |
0. |
1.95 |
0. |
0. |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 40 | 40 | 77 | 65 | 171 | 578 | 66 | 181 |
normalized size | 1 | 1. | 1.92 | 1.62 | 4.28 | 14.45 | 1.65 | 4.52 |
time (sec) |
N/A |
0.031 |
0.237 |
0.031 |
1.196 |
1.604 |
0.307 |
1.352 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.037 |
18.693 |
0.073 |
0. |
0. |
0. |
0. |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.035 |
19.143 |
0.089 |
0. |
0. |
0. |
0. |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 237 | 237 | 803 | 677 | 803 | 12265 | 0 | 0 |
normalized size | 1 | 1. | 3.39 | 2.86 | 3.39 | 51.75 | 0. | 0. |
time (sec) |
N/A |
0.392 |
12.421 |
0.088 |
1.938 |
2.799 |
0. |
0. |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 460 | 367 | 529 | 7159 | 0 | 0 |
normalized size | 1 | 1. | 2.93 | 2.34 | 3.37 | 45.6 | 0. | 0. |
time (sec) |
N/A |
0.248 |
7.876 |
0.059 |
1.89 |
2.638 |
0. |
0. |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
F |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 100 | 100 | 264 | 166 | 0 | 3710 | 0 | 0 |
normalized size | 1 | 1. | 2.64 | 1.66 | 0. | 37.1 | 0. | 0. |
time (sec) |
N/A |
0.137 |
6.137 |
0.039 |
0. |
2.421 |
0. |
0. |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.038 |
30.224 |
0.388 |
0. |
0. |
0. |
0. |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.037 |
22.399 |
0.485 |
0. |
0. |
0. |
0. |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
F |
F |
F |
F |
F(-2) |
F |
F(-2) |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1392 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.132 |
36.775 |
0.08 |
0. |
0. |
0. |
0. |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
F |
F |
F |
F |
F(-2) |
F |
F |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1363 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.106 |
27.257 |
0.059 |
0. |
0. |
0. |
0. |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
F |
C |
F |
F |
F(-2) |
F |
F |
verified |
N/A |
N/A |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1280 | 0 | 556 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0.43 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.029 |
5.117 |
0.102 |
0. |
0. |
0. |
0. |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
F |
C |
F |
F |
F(-2) |
F |
F |
verified |
N/A |
N/A |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1280 | 0 | 556 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0.43 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.031 |
4.247 |
0.086 |
0. |
0. |
0. |
0. |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
F |
F |
F |
F |
F(-2) |
F |
F |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1365 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.11 |
25.76 |
0.055 |
0. |
0. |
0. |
0. |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
F(-2) |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1340 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.149 |
27.283 |
0.076 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
F |
A |
A |
F(-2) |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.051 |
180.002 |
0.096 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
F |
A |
A |
F(-2) |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.055 |
180.003 |
0.101 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
F(-2) |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1342 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.155 |
32.934 |
0.077 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
F(-2) |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.069 |
26.753 |
0.089 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
F(-2) |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.054 |
1.939 |
0.104 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
F(-2) |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.059 |
2.237 |
0.12 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
F(-2) |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.072 |
23.916 |
0.086 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.03 |
13.806 |
0.046 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.03 |
9.495 |
0.034 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.017 |
7.147 |
0.052 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 169 | 169 | 244 | 929 | 251 | 640 | 0 | 261 |
normalized size | 1 | 1. | 1.44 | 5.5 | 1.49 | 3.79 | 0. | 1.54 |
time (sec) |
N/A |
0.195 |
0.401 |
0.046 |
1.319 |
2.195 |
0. |
1.194 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 122 | 122 | 169 | 446 | 170 | 419 | 0 | 166 |
normalized size | 1 | 1. | 1.39 | 3.66 | 1.39 | 3.43 | 0. | 1.36 |
time (sec) |
N/A |
0.118 |
0.26 |
0.04 |
1.223 |
2.141 |
0. |
1.181 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 81 | 165 | 100 | 239 | 0 | 90 |
normalized size | 1 | 1. | 1.09 | 2.23 | 1.35 | 3.23 | 0. | 1.22 |
time (sec) |
N/A |
0.054 |
0.27 |
0.036 |
1.148 |
2.156 |
0. |
1.15 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 122 | 61 | 65 | 165 | 0 | 66 |
normalized size | 1 | 1. | 0.78 | 0.39 | 0.41 | 1.05 | 0. | 0.42 |
time (sec) |
N/A |
0.295 |
0.249 |
0.157 |
1.742 |
2.182 |
0. |
1.2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 159 | 159 | 206 | 90 | 76 | 502 | 0 | 149 |
normalized size | 1 | 1. | 1.3 | 0.57 | 0.48 | 3.16 | 0. | 0.94 |
time (sec) |
N/A |
0.24 |
0.723 |
0.161 |
2.25 |
2.258 |
0. |
1.359 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 211 | 211 | 264 | 211 | 92 | 747 | 0 | 243 |
normalized size | 1 | 1. | 1.25 | 1. | 0.44 | 3.54 | 0. | 1.15 |
time (sec) |
N/A |
0.323 |
1.031 |
0.169 |
3.064 |
2.257 |
0. |
1.193 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 230 | 230 | 420 | 2226 | 400 | 1258 | 0 | 517 |
normalized size | 1 | 1. | 1.83 | 9.68 | 1.74 | 5.47 | 0. | 2.25 |
time (sec) |
N/A |
0.279 |
1.053 |
0.053 |
2.481 |
2.199 |
0. |
1.223 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 170 | 170 | 207 | 1080 | 257 | 813 | 0 | 306 |
normalized size | 1 | 1. | 1.22 | 6.35 | 1.51 | 4.78 | 0. | 1.8 |
time (sec) |
N/A |
0.189 |
0.872 |
0.044 |
2.007 |
2.131 |
0. |
1.211 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 133 | 133 | 114 | 404 | 143 | 455 | 0 | 147 |
normalized size | 1 | 1. | 0.86 | 3.04 | 1.08 | 3.42 | 0. | 1.11 |
time (sec) |
N/A |
0.131 |
0.496 |
0.043 |
1.556 |
2.307 |
0. |
1.228 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 297 | 297 | 199 | 106 | 109 | 302 | 0 | 105 |
normalized size | 1 | 1. | 0.67 | 0.36 | 0.37 | 1.02 | 0. | 0.35 |
time (sec) |
N/A |
0.77 |
0.444 |
0.3 |
3.498 |
2.094 |
0. |
1.235 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 420 | 420 | 442 | 163 | 135 | 1415 | 0 | 255 |
normalized size | 1 | 1. | 1.05 | 0.39 | 0.32 | 3.37 | 0. | 0.61 |
time (sec) |
N/A |
0.786 |
1.353 |
0.32 |
6.307 |
2.297 |
0. |
1.861 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 336 | 336 | 615 | 4207 | 549 | 1872 | 0 | 774 |
normalized size | 1 | 1. | 1.83 | 12.52 | 1.63 | 5.57 | 0. | 2.3 |
time (sec) |
N/A |
0.379 |
2.6 |
0.063 |
8.009 |
2.248 |
0. |
1.216 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 246 | 246 | 371 | 2016 | 344 | 1218 | 0 | 447 |
normalized size | 1 | 1. | 1.51 | 8.2 | 1.4 | 4.95 | 0. | 1.82 |
time (sec) |
N/A |
0.266 |
1.425 |
0.055 |
5.78 |
2.2 |
0. |
1.225 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 183 | 183 | 185 | 745 | 188 | 709 | 0 | 204 |
normalized size | 1 | 1. | 1.01 | 4.07 | 1.03 | 3.87 | 0. | 1.11 |
time (sec) |
N/A |
0.212 |
0.776 |
0.049 |
3.403 |
2.172 |
0. |
1.22 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 437 | 437 | 312 | 151 | 154 | 437 | 0 | 144 |
normalized size | 1 | 1. | 0.71 | 0.35 | 0.35 | 1. | 0. | 0.33 |
time (sec) |
N/A |
1.821 |
0.673 |
0.395 |
11.149 |
2.288 |
0. |
1.179 |
|
|
|
|
|
|
|
|
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|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 692 | 692 | 794 | 239 | 189 | 2703 | 0 | 358 |
normalized size | 1 | 1. | 1.15 | 0.35 | 0.27 | 3.91 | 0. | 0.52 |
time (sec) |
N/A |
1.812 |
2.973 |
0.401 |
24.648 |
2.382 |
0. |
2.198 |
|
|
|
|
|
|
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|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-2) |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.05 |
22.23 |
0.083 |
0. |
0. |
0. |
0. |
|
|
|
|
|
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|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.027 |
13.165 |
0.082 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
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|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 186 | 0 | 0 | 367 | 0 | 0 |
normalized size | 1 | 1. | 2.09 | 0. | 0. | 4.12 | 0. | 0. |
time (sec) |
N/A |
0.115 |
1.132 |
0.109 |
0. |
2.33 |
0. |
0. |
|
|
|
|
|
|
|
|
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|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 153 | 153 | 195 | 0 | 0 | 609 | 0 | 0 |
normalized size | 1 | 1. | 1.27 | 0. | 0. | 3.98 | 0. | 0. |
time (sec) |
N/A |
0.179 |
10.497 |
0.101 |
0. |
2.428 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
|
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|
|
|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F |
A |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 224 | 224 | 0 | 0 | 0 | 855 | 0 | 0 |
normalized size | 1 | 1. | 0. | 0. | 0. | 3.82 | 0. | 0. |
time (sec) |
N/A |
0.236 |
180.003 |
0.112 |
0. |
2.407 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 137 | 137 | 161 | 452 | 410 | 1438 | 0 | 0 |
normalized size | 1 | 1. | 1.18 | 3.3 | 2.99 | 10.5 | 0. | 0. |
time (sec) |
N/A |
0.265 |
2.091 |
0.063 |
1.422 |
2.666 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 103 | 103 | 147 | 272 | 242 | 942 | 0 | 0 |
normalized size | 1 | 1. | 1.43 | 2.64 | 2.35 | 9.15 | 0. | 0. |
time (sec) |
N/A |
0.213 |
1.713 |
0.05 |
1.29 |
2.464 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 227 | 129 | 0 | 536 | 0 | 0 |
normalized size | 1 | 1. | 3.03 | 1.72 | 0. | 7.15 | 0. | 0. |
time (sec) |
N/A |
0.128 |
4.051 |
0.044 |
0. |
2.446 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.031 |
4.135 |
0.122 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.029 |
10.955 |
0.121 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 277 | 277 | 508 | 873 | 844 | 8259 | 0 | 0 |
normalized size | 1 | 1. | 1.83 | 3.15 | 3.05 | 29.82 | 0. | 0. |
time (sec) |
N/A |
0.548 |
7.558 |
0.178 |
1.661 |
3.472 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 211 | 211 | 232 | 510 | 555 | 4886 | 0 | 0 |
normalized size | 1 | 1. | 1.1 | 2.42 | 2.63 | 23.16 | 0. | 0. |
time (sec) |
N/A |
0.396 |
5.513 |
0.087 |
1.469 |
2.903 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 127 | 127 | 192 | 221 | 0 | 2421 | 0 | 0 |
normalized size | 1 | 1. | 1.51 | 1.74 | 0. | 19.06 | 0. | 0. |
time (sec) |
N/A |
0.183 |
1.987 |
0.085 |
0. |
2.631 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.056 |
38.966 |
0.259 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.052 |
25.641 |
0.361 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 566 | 566 | 2010 | 1788 | 1751 | 27340 | 0 | 0 |
normalized size | 1 | 1. | 3.55 | 3.16 | 3.09 | 48.3 | 0. | 0. |
time (sec) |
N/A |
1.024 |
14.216 |
0.204 |
2.308 |
5.272 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 405 | 405 | 1142 | 1022 | 1173 | 15872 | 0 | 0 |
normalized size | 1 | 1. | 2.82 | 2.52 | 2.9 | 39.19 | 0. | 0. |
time (sec) |
N/A |
0.7 |
11.576 |
0.149 |
2.055 |
4.171 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 261 | 261 | 265 | 459 | 641 | 7571 | 0 | 0 |
normalized size | 1 | 1. | 1.02 | 1.76 | 2.46 | 29.01 | 0. | 0. |
time (sec) |
N/A |
0.35 |
3.43 |
0.105 |
1.94 |
3.166 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 66 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.055 |
60.239 |
0.392 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.053 |
52.265 |
0.563 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 212 | 212 | 239 | 1143 | 713 | 1821 | 0 | 0 |
normalized size | 1 | 1. | 1.13 | 5.39 | 3.36 | 8.59 | 0. | 0. |
time (sec) |
N/A |
0.341 |
3.411 |
0.203 |
1.683 |
2.455 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 191 | 720 | 455 | 1269 | 0 | 0 |
normalized size | 1 | 1. | 1.22 | 4.59 | 2.9 | 8.08 | 0. | 0. |
time (sec) |
N/A |
0.29 |
3.695 |
0.148 |
1.608 |
2.477 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 108 | 108 | 144 | 357 | 0 | 806 | 0 | 0 |
normalized size | 1 | 1. | 1.33 | 3.31 | 0. | 7.46 | 0. | 0. |
time (sec) |
N/A |
0.166 |
3.017 |
0.132 |
0. |
2.383 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 71 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.063 |
13.908 |
0.146 |
0. |
0. |
0. |
0. |
|
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Problem 72 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.059 |
22.358 |
0.153 |
0. |
0. |
0. |
0. |
|
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|
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|
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|
Problem 73 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 642 | 642 | 643 | 2662 | 1431 | 12852 | 0 | 0 |
normalized size | 1 | 1. | 1. | 4.15 | 2.23 | 20.02 | 0. | 0. |
time (sec) |
N/A |
2.212 |
12.306 |
0.293 |
2.676 |
3.701 |
0. |
0. |
|
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|
|
|
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|
Problem 74 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 476 | 476 | 506 | 1352 | 1017 | 7885 | 0 | 0 |
normalized size | 1 | 1. | 1.06 | 2.84 | 2.14 | 16.57 | 0. | 0. |
time (sec) |
N/A |
1.651 |
8.452 |
0.253 |
2.57 |
3.033 |
0. |
0. |
|
|
|
|
|
|
|
|
|
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|
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|
Problem 75 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 196 | 196 | 476 | 661 | 0 | 4065 | 0 | 0 |
normalized size | 1 | 1. | 2.43 | 3.37 | 0. | 20.74 | 0. | 0. |
time (sec) |
N/A |
0.286 |
6.655 |
0.224 |
0. |
2.847 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
|
|
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|
Problem 76 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.061 |
142.158 |
0.722 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
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|
|
|
|
|
|
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|
Problem 77 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.057 |
114.532 |
0.651 |
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 [11] had the largest ratio of [ 0.625
]
|
|
|
|
|
|
|
| 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 |
6 |
6 |
1. |
14 |
0.429 |
|
|
|
|
|
|
|
2 |
A |
5 |
5 |
1. |
14 |
0.357 |
|
|
|
|
|
|
|
3 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
4 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
5 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
6 |
A |
7 |
7 |
1. |
16 |
0.438 |
|
|
|
|
|
|
|
7 |
A |
6 |
6 |
1. |
16 |
0.375 |
|
|
|
|
|
|
|
8 |
A |
3 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
9 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
10 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
11 |
A |
13 |
10 |
1. |
16 |
0.625 |
|
|
|
|
|
|
|
12 |
A |
9 |
7 |
1. |
16 |
0.438 |
|
|
|
|
|
|
|
13 |
A |
7 |
7 |
1. |
14 |
0.5 |
|
|
|
|
|
|
|
14 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
15 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
16 |
F |
0 |
0 |
N/A |
0 |
N/A |
|
|
|
|
|
|
|
17 |
F |
0 |
0 |
N/A |
0 |
N/A |
|
|
|
|
|
|
|
18 |
F |
0 |
0 |
N/A |
0 |
N/A |
|
|
|
|
|
|
|
19 |
F |
0 |
0 |
N/A |
0 |
N/A |
|
|
|
|
|
|
|
20 |
F |
0 |
0 |
N/A |
0 |
N/A |
|
|
|
|
|
|
|
|
21 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
22 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
23 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
24 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
25 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
26 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
27 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
28 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
29 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
30 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
31 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
32 |
A |
5 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
33 |
A |
4 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
34 |
A |
3 |
3 |
1. |
18 |
0.167 |
|
|
|
|
|
|
|
35 |
A |
7 |
4 |
1. |
20 |
0.2 |
|
|
|
|
|
|
|
36 |
A |
7 |
4 |
1. |
20 |
0.2 |
|
|
|
|
|
|
|
37 |
A |
8 |
5 |
1. |
20 |
0.25 |
|
|
|
|
|
|
|
38 |
A |
10 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
39 |
A |
8 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
40 |
A |
7 |
3 |
1. |
18 |
0.167 |
|
|
|
|
|
|
|
|
41 |
A |
21 |
5 |
1. |
20 |
0.25 |
|
|
|
|
|
|
|
42 |
A |
24 |
7 |
1. |
20 |
0.35 |
|
|
|
|
|
|
|
43 |
A |
14 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
44 |
A |
11 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
45 |
A |
11 |
3 |
1. |
18 |
0.167 |
|
|
|
|
|
|
|
46 |
A |
53 |
7 |
1. |
20 |
0.35 |
|
|
|
|
|
|
|
47 |
A |
60 |
9 |
1. |
20 |
0.45 |
|
|
|
|
|
|
|
48 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
49 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
50 |
A |
2 |
2 |
1. |
20 |
0.1 |
|
|
|
|
|
|
|
51 |
A |
4 |
2 |
1. |
20 |
0.1 |
|
|
|
|
|
|
|
52 |
A |
5 |
2 |
1. |
20 |
0.1 |
|
|
|
|
|
|
|
53 |
A |
8 |
7 |
1. |
18 |
0.389 |
|
|
|
|
|
|
|
54 |
A |
7 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
55 |
A |
6 |
5 |
1. |
16 |
0.312 |
|
|
|
|
|
|
|
56 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
57 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
58 |
A |
15 |
9 |
1. |
20 |
0.45 |
|
|
|
|
|
|
|
59 |
A |
13 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
60 |
A |
9 |
7 |
1. |
18 |
0.389 |
|
|
|
|
|
|
|
|
61 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
62 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
63 |
A |
28 |
11 |
1. |
20 |
0.55 |
|
|
|
|
|
|
|
64 |
A |
22 |
11 |
1. |
20 |
0.55 |
|
|
|
|
|
|
|
65 |
A |
16 |
9 |
1. |
18 |
0.5 |
|
|
|
|
|
|
|
66 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
67 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
68 |
A |
6 |
6 |
1. |
20 |
0.3 |
|
|
|
|
|
|
|
69 |
A |
5 |
5 |
1. |
20 |
0.25 |
|
|
|
|
|
|
|
70 |
A |
4 |
4 |
1. |
18 |
0.222 |
|
|
|
|
|
|
|
71 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
72 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
73 |
A |
28 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
74 |
A |
24 |
11 |
1. |
20 |
0.55 |
|
|
|
|
|
|
|
75 |
A |
5 |
5 |
1. |
18 |
0.278 |
|
|
|
|
|
|
|
76 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
77 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|