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
}
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, 20, 22, 23, 24, 25, 26, 28, 29, 31, 32, 34, 36, 38,
42, 47, 48, 49, 50, 51, 52, 53 }
B grade: { 10, 17, 19, 27, 30, 33, 35, 39 }
C grade: { 21, 37, 40, 41, 43, 44, 45, 46 }
F grade: { }
2.1.3 Maple
A grade: { 5, 9, 10, 11, 12, 13, 14, 15, 50, 51, 52 }
B grade: { 1, 2, 3, 4, 6, 7, 8, 16, 17, 18, 19, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 34, 35, 38, 39, 40, 43, 44,
45, 48, 49 }
C grade: { 53 }
F grade: { 20, 21, 26, 27, 36, 37, 41, 42, 46, 47 }
2.1.4 Maxima
A grade: { 5, 11, 13, 15 }
B grade: { 1, 2, 3, 4, 6, 7, 8, 9, 50 }
C grade: { 10, 12, 14 }
F grade: { 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, 51, 52, 53 }
2.1.5 FriCAS
A grade: { 10, 11, 12, 14, 15 }
B grade: { 1, 2, 3, 4, 5, 6, 7, 9, 13, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45, 47, 48, 50, 51, 52, 53 }
C grade: { 29, 31, 49 }
F grade: { 8, 23, 46 }
2.1.6 Sympy
A grade: { 2, 3, 4, 5, 9 }
B grade: { 50 }
C grade: { }
F grade: { 1, 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, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53 }
2.1.7 Giac
A grade: { 5, 11, 15, 50 }
B grade: { 1, 2, 3, 4, 6, 7, 8, 9, 13, 19, 20, 26, 27, 28, 30, 35, 48 }
C grade: { 10, 12, 14, 29, 31, 49 }
F grade: { 16, 17, 18, 21, 22, 23, 24, 25, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 51, 52, 53
}
2.1.8 Mupad
A grade: { }
B grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 22, 24, 28, 29, 30, 32, 34, 35, 38, 40, 43, 45,
48, 49, 50 }
C grade: { }
F grade: { 17, 19, 20, 21, 23, 25, 26, 27, 31, 33, 36, 37, 39, 41, 42, 44, 46, 47, 51, 52, 53 }
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 |
B |
B |
B |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 160 | 160 | 231 | 472 | 624 | 2111 | 0 | 721 | 158 |
normalized size | 1 | 1.00 | 1.44 | 2.95 | 3.90 | 13.19 | 0.00 | 4.51 | 0.99 |
time (sec) |
N/A |
0.092 |
2.898 |
0.020 |
0.357 |
0.430 |
0.000 |
0.309 |
1.371 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
B |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 127 | 344 | 410 | 1164 | 264 | 447 | 111 |
normalized size | 1 | 1.00 | 1.15 | 3.13 | 3.73 | 10.58 | 2.40 | 4.06 | 1.01 |
time (sec) |
N/A |
0.068 |
2.083 |
0.026 |
0.336 |
0.461 |
119.402 |
0.232 |
1.315 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
B |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 100 | 235 | 239 | 557 | 170 | 241 | 72 |
normalized size | 1 | 1.00 | 1.35 | 3.18 | 3.23 | 7.53 | 2.30 | 3.26 | 0.97 |
time (sec) |
N/A |
0.047 |
1.449 |
0.020 |
0.324 |
0.453 |
27.591 |
0.195 |
1.281 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
B |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 65 | 144 | 114 | 197 | 102 | 103 | 41 |
normalized size | 1 | 1.00 | 1.51 | 3.35 | 2.65 | 4.58 | 2.37 | 2.40 | 0.95 |
time (sec) |
N/A |
0.033 |
0.507 |
0.020 |
0.317 |
0.503 |
5.795 |
0.158 |
0.114 |
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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 |
B |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 47 | 76 | 56 | 488 | 294 | 65 | 37 |
normalized size | 1 | 1.00 | 1.02 | 1.65 | 1.22 | 10.61 | 6.39 | 1.41 | 0.80 |
time (sec) |
N/A |
0.077 |
0.063 |
0.115 |
0.423 |
0.429 |
9.715 |
0.134 |
0.111 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
B |
B |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 97 | 172 | 207 | 1952 | 0 | 198 | 110 |
normalized size | 1 | 1.00 | 1.09 | 1.93 | 2.33 | 21.93 | 0.00 | 2.22 | 1.24 |
time (sec) |
N/A |
0.101 |
0.609 |
0.130 |
0.471 |
0.476 |
0.000 |
0.169 |
1.581 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
B |
B |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 142 | 142 | 147 | 352 | 509 | 7508 | 0 | 409 | 2719 |
normalized size | 1 | 1.00 | 1.04 | 2.48 | 3.58 | 52.87 | 0.00 | 2.88 | 19.15 |
time (sec) |
N/A |
0.160 |
0.361 |
0.125 |
0.558 |
0.669 |
0.000 |
0.217 |
1.913 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
B |
F(-1) |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 201 | 201 | 203 | 608 | 925 | 0 | 0 | 1356 | 3685 |
normalized size | 1 | 1.00 | 1.01 | 3.02 | 4.60 | 0.00 | 0.00 | 6.75 | 18.33 |
time (sec) |
N/A |
0.279 |
0.633 |
0.129 |
0.726 |
0.000 |
0.000 |
0.499 |
2.071 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
B |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 19 | 27 | 38 | 70 | 34 | 38 | 15 |
normalized size | 1 | 1.00 | 1.00 | 1.42 | 2.00 | 3.68 | 1.79 | 2.00 | 0.79 |
time (sec) |
N/A |
0.051 |
0.099 |
0.080 |
0.406 |
0.634 |
0.524 |
0.113 |
0.108 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
B |
A |
C |
A |
F |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 3 | 3 | 20 | 4 | 19 | 1 | 0 | 26 | 3 |
normalized size | 1 | 1.00 | 6.67 | 1.33 | 6.33 | 0.33 | 0.00 | 8.67 | 1.00 |
time (sec) |
N/A |
0.018 |
0.007 |
0.078 |
0.424 |
0.484 |
0.000 |
0.140 |
1.178 |
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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 | 14 | 14 | 18 | 15 | 17 | 17 | 0 | 23 | 14 |
normalized size | 1 | 1.00 | 1.29 | 1.07 | 1.21 | 1.21 | 0.00 | 1.64 | 1.00 |
time (sec) |
N/A |
0.021 |
0.008 |
0.139 |
0.423 |
0.444 |
0.000 |
0.118 |
1.371 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
C |
A |
F |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 24 | 41 | 21 | 49 | 1 | 0 | 60 | 20 |
normalized size | 1 | 1.00 | 1.71 | 0.88 | 2.04 | 0.04 | 0.00 | 2.50 | 0.83 |
time (sec) |
N/A |
0.022 |
0.061 |
0.072 |
0.421 |
0.431 |
0.000 |
0.130 |
1.215 |
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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 |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 40 | 28 | 46 | 211 | 0 | 52 | 27 |
normalized size | 1 | 1.00 | 1.29 | 0.90 | 1.48 | 6.81 | 0.00 | 1.68 | 0.87 |
time (sec) |
N/A |
0.023 |
0.040 |
0.128 |
0.413 |
0.415 |
0.000 |
0.122 |
1.196 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
C |
A |
F |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 13 | 13 | 13 | 14 | 11 | 1 | 0 | 24 | 18 |
normalized size | 1 | 1.00 | 1.00 | 1.08 | 0.85 | 0.08 | 0.00 | 1.85 | 1.38 |
time (sec) |
N/A |
0.022 |
0.008 |
0.060 |
0.419 |
0.407 |
0.000 |
0.120 |
1.343 |
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Problem 15 |
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 | 11 | 11 | 11 | 12 | 11 | 2 | 0 | 18 | 15 |
normalized size | 1 | 1.00 | 1.00 | 1.09 | 1.00 | 0.18 | 0.00 | 1.64 | 1.36 |
time (sec) |
N/A |
0.021 |
0.008 |
0.102 |
0.417 |
0.396 |
0.000 |
0.115 |
1.242 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
F |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 60 | 253 | 0 | 2367 | 0 | 0 | 66 |
normalized size | 1 | 1.00 | 0.95 | 4.02 | 0.00 | 37.57 | 0.00 | 0.00 | 1.05 |
time (sec) |
N/A |
0.123 |
0.186 |
0.140 |
0.000 |
0.569 |
0.000 |
0.000 |
3.078 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
B |
B |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 191 | 276 | 0 | 4877 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.25 | 3.25 | 0.00 | 57.38 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.128 |
0.846 |
0.101 |
0.000 |
0.656 |
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 |
B |
F |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 44 | 44 | 44 | 238 | 0 | 1551 | 0 | 0 | 51 |
normalized size | 1 | 1.00 | 1.00 | 5.41 | 0.00 | 35.25 | 0.00 | 0.00 | 1.16 |
time (sec) |
N/A |
0.077 |
0.032 |
0.088 |
0.000 |
0.483 |
0.000 |
0.000 |
1.623 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
B |
B |
F |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 137 | 238 | 0 | 3455 | 0 | 262 | -1 |
normalized size | 1 | 1.00 | 2.28 | 3.97 | 0.00 | 57.58 | 0.00 | 4.37 | -0.02 |
time (sec) |
N/A |
0.045 |
0.221 |
0.121 |
0.000 |
0.593 |
0.000 |
0.742 |
0.000 |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 56 | 0 | 0 | 3479 | 0 | 259 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.00 | 0.00 | 62.12 | 0.00 | 4.62 | -0.02 |
time (sec) |
N/A |
0.110 |
0.033 |
0.545 |
0.000 |
0.811 |
0.000 |
0.763 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 42 | 0 | 0 | 1539 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.88 | 0.00 | 0.00 | 32.06 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.089 |
0.092 |
0.474 |
0.000 |
0.473 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 82 | 82 | 86 | 593 | 0 | 4940 | 0 | 0 | 112 |
normalized size | 1 | 1.00 | 1.05 | 7.23 | 0.00 | 60.24 | 0.00 | 0.00 | 1.37 |
time (sec) |
N/A |
0.149 |
0.431 |
0.084 |
0.000 |
0.804 |
0.000 |
0.000 |
9.037 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 123 | 123 | 219 | 633 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.78 | 5.15 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.244 |
0.978 |
0.087 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 59 | 578 | 0 | 2362 | 0 | 0 | 64 |
normalized size | 1 | 1.00 | 0.94 | 9.17 | 0.00 | 37.49 | 0.00 | 0.00 | 1.02 |
time (sec) |
N/A |
0.102 |
0.164 |
0.069 |
0.000 |
0.581 |
0.000 |
0.000 |
3.396 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 88 | 88 | 161 | 578 | 0 | 5037 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.83 | 6.57 | 0.00 | 57.24 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.089 |
0.289 |
0.112 |
0.000 |
0.735 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 71 | 71 | 71 | 0 | 0 | 3949 | 0 | 470 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.00 | 0.00 | 55.62 | 0.00 | 6.62 | -0.01 |
time (sec) |
N/A |
0.139 |
0.062 |
0.449 |
0.000 |
0.678 |
0.000 |
7.812 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 180 | 0 | 0 | 4025 | 0 | 473 | -1 |
normalized size | 1 | 1.00 | 2.34 | 0.00 | 0.00 | 52.27 | 0.00 | 6.14 | -0.01 |
time (sec) |
N/A |
0.128 |
0.472 |
0.454 |
0.000 |
0.660 |
0.000 |
7.972 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 60 | 97 | 0 | 683 | 0 | 119 | 68 |
normalized size | 1 | 1.00 | 1.94 | 3.13 | 0.00 | 22.03 | 0.00 | 3.84 | 2.19 |
time (sec) |
N/A |
0.026 |
0.076 |
0.142 |
0.000 |
0.438 |
0.000 |
0.147 |
1.340 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
C |
F |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 45 | 45 | 62 | 142 | 0 | 228 | 0 | 124 | 43 |
normalized size | 1 | 1.00 | 1.38 | 3.16 | 0.00 | 5.07 | 0.00 | 2.76 | 0.96 |
time (sec) |
N/A |
0.034 |
0.051 |
0.149 |
0.000 |
0.457 |
0.000 |
0.196 |
1.299 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 50 | 50 | 116 | 158 | 0 | 1043 | 0 | 265 | 78 |
normalized size | 1 | 1.00 | 2.32 | 3.16 | 0.00 | 20.86 | 0.00 | 5.30 | 1.56 |
time (sec) |
N/A |
0.039 |
0.345 |
0.115 |
0.000 |
0.448 |
0.000 |
0.197 |
1.394 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
C |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 118 | 211 | 0 | 361 | 0 | 285 | -1 |
normalized size | 1 | 1.00 | 1.76 | 3.15 | 0.00 | 5.39 | 0.00 | 4.25 | -0.01 |
time (sec) |
N/A |
0.048 |
0.174 |
0.121 |
0.000 |
0.443 |
0.000 |
0.202 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 47 | 129 | 0 | 1576 | 0 | 0 | 39 |
normalized size | 1 | 1.00 | 1.00 | 2.74 | 0.00 | 33.53 | 0.00 | 0.00 | 0.83 |
time (sec) |
N/A |
0.110 |
0.103 |
0.118 |
0.000 |
0.534 |
0.000 |
0.000 |
1.689 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 134 | 137 | 0 | 3513 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.23 | 2.28 | 0.00 | 58.55 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.095 |
0.249 |
0.116 |
0.000 |
0.651 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 29 | 29 | 29 | 114 | 0 | 1298 | 0 | 0 | 23 |
normalized size | 1 | 1.00 | 1.00 | 3.93 | 0.00 | 44.76 | 0.00 | 0.00 | 0.79 |
time (sec) |
N/A |
0.065 |
0.015 |
0.099 |
0.000 |
0.478 |
0.000 |
0.000 |
1.743 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 77 | 114 | 0 | 1357 | 0 | 196 | 25 |
normalized size | 1 | 1.00 | 2.48 | 3.68 | 0.00 | 43.77 | 0.00 | 6.32 | 0.81 |
time (sec) |
N/A |
0.027 |
0.124 |
0.125 |
0.000 |
0.497 |
0.000 |
0.345 |
1.479 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 56 | 0 | 0 | 3397 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.00 | 0.00 | 60.66 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.109 |
0.049 |
0.500 |
0.000 |
0.644 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 132 | 0 | 0 | 1621 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.59 | 0.00 | 0.00 | 31.78 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.096 |
1.457 |
0.511 |
0.000 |
0.532 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 52 | 52 | 52 | 287 | 0 | 2541 | 0 | 0 | 45 |
normalized size | 1 | 1.00 | 1.00 | 5.52 | 0.00 | 48.87 | 0.00 | 0.00 | 0.87 |
time (sec) |
N/A |
0.124 |
0.131 |
0.093 |
0.000 |
0.587 |
0.000 |
0.000 |
2.003 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 109 | 289 | 0 | 2279 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.06 | 5.45 | 0.00 | 43.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.100 |
0.996 |
0.098 |
0.000 |
0.606 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 49 | 49 | 41 | 273 | 0 | 2299 | 0 | 0 | 41 |
normalized size | 1 | 1.00 | 0.84 | 5.57 | 0.00 | 46.92 | 0.00 | 0.00 | 0.84 |
time (sec) |
N/A |
0.087 |
0.042 |
0.081 |
0.000 |
0.589 |
0.000 |
0.000 |
1.938 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 78 | 78 | 70 | 0 | 0 | 6991 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.90 | 0.00 | 0.00 | 89.63 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.146 |
0.058 |
0.464 |
0.000 |
0.894 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 149 | 0 | 0 | 3931 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.75 | 0.00 | 0.00 | 46.25 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.155 |
6.007 |
0.478 |
0.000 |
0.822 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
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Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 63 | 435 | 0 | 6560 | 0 | 0 | 82 |
normalized size | 1 | 1.00 | 0.85 | 5.88 | 0.00 | 88.65 | 0.00 | 0.00 | 1.11 |
time (sec) |
N/A |
0.146 |
0.101 |
0.098 |
0.000 |
1.002 |
0.000 |
0.000 |
3.339 |
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Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 88 | 88 | 258 | 454 | 0 | 6591 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.93 | 5.16 | 0.00 | 74.90 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.128 |
7.092 |
0.097 |
0.000 |
1.018 |
0.000 |
0.000 |
0.000 |
|
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Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 70 | 70 | 43 | 420 | 0 | 5736 | 0 | 0 | 76 |
normalized size | 1 | 1.00 | 0.61 | 6.00 | 0.00 | 81.94 | 0.00 | 0.00 | 1.09 |
time (sec) |
N/A |
0.102 |
0.038 |
0.083 |
0.000 |
0.995 |
0.000 |
0.000 |
3.412 |
|
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Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 108 | 108 | 73 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.68 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.205 |
0.059 |
0.462 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
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Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 131 | 131 | 171 | 0 | 0 | 10729 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.31 | 0.00 | 0.00 | 81.90 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.236 |
6.753 |
0.489 |
0.000 |
1.905 |
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 |
B |
F |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 25 | 44 | 62 | 0 | 547 | 0 | 69 | 63 |
normalized size | 1 | 1.00 | 1.76 | 2.48 | 0.00 | 21.88 | 0.00 | 2.76 | 2.52 |
time (sec) |
N/A |
0.019 |
0.069 |
0.137 |
0.000 |
0.452 |
0.000 |
0.126 |
1.350 |
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Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
C |
F |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 46 | 66 | 0 | 175 | 0 | 73 | 22 |
normalized size | 1 | 1.00 | 1.70 | 2.44 | 0.00 | 6.48 | 0.00 | 2.70 | 0.81 |
time (sec) |
N/A |
0.020 |
0.040 |
0.135 |
0.000 |
0.452 |
0.000 |
0.155 |
1.254 |
|
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Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
B |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 40 | 41 | 73 | 95 | 102 | 25 | 38 |
normalized size | 1 | 1.00 | 1.05 | 1.08 | 1.92 | 2.50 | 2.68 | 0.66 | 1.00 |
time (sec) |
N/A |
0.067 |
0.079 |
0.079 |
0.413 |
0.450 |
0.982 |
0.131 |
0.093 |
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Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 86 | 116 | 0 | 5172 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.97 | 1.30 | 0.00 | 58.11 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.141 |
0.155 |
0.166 |
0.000 |
0.679 |
0.000 |
0.000 |
0.000 |
|
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Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 40 | 40 | 40 | 37 | 0 | 1290 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.92 | 0.00 | 32.25 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.076 |
0.016 |
0.141 |
0.000 |
0.683 |
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 |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 73 | 431 | 0 | 3938 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.99 | 5.82 | 0.00 | 53.22 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.120 |
0.497 |
0.124 |
0.000 |
0.826 |
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 [50] had the largest ratio of
[.6250]
|
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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 |
4 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
2 |
A |
4 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
3 |
A |
4 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
4 |
A |
4 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
5 |
A |
3 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
6 |
A |
5 |
5 |
1.00 |
14 |
0.357 |
|
|
|
|
|
|
|
7 |
A |
6 |
6 |
1.00 |
14 |
0.429 |
|
|
|
|
|
|
|
8 |
A |
7 |
6 |
1.00 |
14 |
0.429 |
|
|
|
|
|
|
|
9 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
10 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
11 |
A |
4 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
12 |
A |
4 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
13 |
A |
5 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
14 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
15 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
16 |
A |
6 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
17 |
A |
7 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
18 |
A |
5 |
5 |
1.00 |
15 |
0.333 |
|
|
|
|
|
|
|
19 |
A |
6 |
5 |
1.00 |
12 |
0.417 |
|
|
|
|
|
|
|
20 |
A |
7 |
5 |
1.00 |
15 |
0.333 |
|
|
|
|
|
|
|
21 |
A |
5 |
5 |
1.00 |
17 |
0.294 |
|
|
|
|
|
|
|
22 |
A |
7 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
23 |
A |
8 |
7 |
1.00 |
17 |
0.412 |
|
|
|
|
|
|
|
24 |
A |
6 |
5 |
1.00 |
15 |
0.333 |
|
|
|
|
|
|
|
25 |
A |
7 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
26 |
A |
8 |
6 |
1.00 |
15 |
0.400 |
|
|
|
|
|
|
|
27 |
A |
7 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
28 |
A |
5 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
29 |
A |
6 |
5 |
1.00 |
12 |
0.417 |
|
|
|
|
|
|
|
30 |
A |
6 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
31 |
A |
7 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
32 |
A |
5 |
5 |
1.00 |
17 |
0.294 |
|
|
|
|
|
|
|
33 |
A |
6 |
5 |
1.00 |
17 |
0.294 |
|
|
|
|
|
|
|
34 |
A |
4 |
4 |
1.00 |
15 |
0.267 |
|
|
|
|
|
|
|
35 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
36 |
A |
7 |
5 |
1.00 |
15 |
0.333 |
|
|
|
|
|
|
|
37 |
A |
5 |
5 |
1.00 |
17 |
0.294 |
|
|
|
|
|
|
|
38 |
A |
5 |
5 |
1.00 |
17 |
0.294 |
|
|
|
|
|
|
|
39 |
A |
4 |
4 |
1.00 |
17 |
0.235 |
|
|
|
|
|
|
|
40 |
A |
5 |
5 |
1.00 |
15 |
0.333 |
|
|
|
|
|
|
|
41 |
A |
8 |
6 |
1.00 |
15 |
0.400 |
|
|
|
|
|
|
|
42 |
A |
6 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
43 |
A |
6 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
44 |
A |
6 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
45 |
A |
6 |
5 |
1.00 |
15 |
0.333 |
|
|
|
|
|
|
|
46 |
A |
9 |
7 |
1.00 |
15 |
0.467 |
|
|
|
|
|
|
|
47 |
A |
7 |
7 |
1.00 |
17 |
0.412 |
|
|
|
|
|
|
|
48 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
49 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
50 |
A |
6 |
5 |
1.00 |
8 |
0.625 |
|
|
|
|
|
|
|
51 |
A |
8 |
7 |
1.00 |
15 |
0.467 |
|
|
|
|
|
|
|
52 |
A |
4 |
4 |
1.00 |
15 |
0.267 |
|
|
|
|
|
|
|
53 |
A |
6 |
6 |
1.00 |
15 |
0.400 |
|
|
|
|
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|