Chapter 2
detailed summary tables of results
2.1 List of integrals sorted by grade for each CAS
2.1.1 Rubi
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 2, 4, 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, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62,
63, 64, 65, 66, 67, 68, 69, 70 }
B grade: { 1, 3, 5, 41, 42 }
C grade: { }
F grade: { }
2.1.3 Maple
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 61, 62, 63,
64, 65, 66, 67 }
B grade: { 39, 40, 41, 42, 43, 51 }
C grade: { 17, 18, 19, 20, 21, 22, 23, 24, 55, 56, 57, 58, 59, 60 }
F grade: { 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 68, 69, 70 }
2.1.4 Maxima
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 43, 44, 45, 46, 50, 51, 61, 62, 63, 64, 65, 66, 67
B grade: { 39, 40, 41, 42, 47, 48, 49 }
C grade: { }
F grade: { 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, 52, 53, 54, 55, 56, 57, 58, 59, 60, 68, 69, 70 }
2.1.5 FriCAS
A grade: { 2, 4, 6, 8, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 61, 62, 63, 64, 65, 66, 67 }
B grade: { 1, 3, 5, 7, 39, 40, 41, 42 }
C grade: { }
F grade: { 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, 55, 56, 57, 58, 59, 60, 68, 69, 70 }
2.1.6 Sympy
A grade: { 1, 43, 44, 45, 51, 52, 53 }
B grade: { }
C grade: { }
F grade: { 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, 46, 47, 48, 49, 50, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,
66, 67, 68, 69, 70 }
2.1.7 Giac
A grade: { 1, 2, 4, 6, 8, 43, 45, 46, 50, 52, 53, 54, 61, 62, 63, 64 }
B grade: { 3, 5, 7, 39, 40, 41, 42, 44, 47, 48, 49, 51 }
C grade: { }
F grade: { 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, 55, 56, 57, 58, 59, 60, 65, 66, 67, 68, 69, 70 }
2.2 Detailed conclusion table per each integral for all CAS systems
Detailed conclusion table per each integral is given by table below. The elapsed time is in
seconds. For failed result it is given as F(-1) if the failure was due to timeout. It is given as F(-2)
if the failure was due to an exception being raised, which could indicate a bug in the system. If
the failure was due to integral not being evaluated within the time limit, then it is given just an
F.
In this table,the column normalized size is defined as \(\frac{\text{antiderivative leaf size}}{\text{optimal antiderivative leaf size}}\)
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Problem 1 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
A |
B |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 12 | 12 | 38 | 20 | 26 | 93 | 37 | 20 |
normalized size | 1 | 1. | 3.17 | 1.67 | 2.17 | 7.75 | 3.08 | 1.67 |
time (sec) |
N/A |
0.004 |
0.018 |
0.012 |
1.055 |
0.474 |
2.515 |
1.301 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 11 | 11 | 11 | 12 | 18 | 43 | 0 | 18 |
normalized size | 1 | 1. | 1. | 1.09 | 1.64 | 3.91 | 0. | 1.64 |
time (sec) |
N/A |
0.008 |
0.008 |
0.011 |
1.017 |
0.448 |
0. |
1.282 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
A |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 34 | 34 | 75 | 40 | 62 | 201 | 0 | 124 |
normalized size | 1 | 1. | 2.21 | 1.18 | 1.82 | 5.91 | 0. | 3.65 |
time (sec) |
N/A |
0.014 |
0.013 |
0.022 |
1.042 |
0.485 |
0. |
1.291 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 35 | 23 | 34 | 108 | 0 | 34 |
normalized size | 1 | 1. | 1.3 | 0.85 | 1.26 | 4. | 0. | 1.26 |
time (sec) |
N/A |
0.011 |
0.013 |
0.018 |
1.057 |
0.45 |
0. |
1.254 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
A |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 55 | 55 | 113 | 59 | 96 | 313 | 0 | 186 |
normalized size | 1 | 1. | 2.05 | 1.07 | 1.75 | 5.69 | 0. | 3.38 |
time (sec) |
N/A |
0.026 |
0.016 |
0.021 |
1.066 |
0.492 |
0. |
1.288 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 42 | 42 | 56 | 33 | 47 | 166 | 0 | 47 |
normalized size | 1 | 1. | 1.33 | 0.79 | 1.12 | 3.95 | 0. | 1.12 |
time (sec) |
N/A |
0.014 |
0.016 |
0.02 |
1.013 |
0.452 |
0. |
1.261 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 76 | 76 | 151 | 78 | 123 | 424 | 0 | 246 |
normalized size | 1 | 1. | 1.99 | 1.03 | 1.62 | 5.58 | 0. | 3.24 |
time (sec) |
N/A |
0.036 |
0.014 |
0.022 |
1.041 |
0.498 |
0. |
1.294 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 55 | 55 | 77 | 43 | 61 | 223 | 0 | 61 |
normalized size | 1 | 1. | 1.4 | 0.78 | 1.11 | 4.05 | 0. | 1.11 |
time (sec) |
N/A |
0.017 |
0.011 |
0.02 |
1.011 |
0.461 |
0. |
1.229 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 90 | 90 | 63 | 160 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.7 |
1.78 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.037 |
0.201 |
1.171 |
0. |
0. |
0. |
0. |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 50 | 88 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.75 |
1.31 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.026 |
0.081 |
1.2 |
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 |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 49 | 132 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.78 | 2.1 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.024 |
0.058 |
1.344 |
0. |
0. |
0. |
0. |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 41 | 40 | 69 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.98 | 1.68 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.017 |
0.034 |
0.836 |
0. |
0. |
0. |
0. |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 41 | 40 | 91 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.98 | 2.22 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.017 |
0.039 |
1.095 |
0. |
0. |
0. |
0. |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 53 | 88 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.79 | 1.31 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.027 |
0.057 |
1.323 |
0. |
0. |
0. |
0. |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 60 | 142 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.9 | 2.12 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.026 |
0.107 |
1.229 |
0. |
0. |
0. |
0. |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 90 | 90 | 65 | 104 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.72 |
1.16 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.038 |
0.166 |
1.198 |
0. |
0. |
0. |
0. |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 103 | 103 | 67 | 1029 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.65 |
9.99 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.053 |
0.196 |
0.335 |
0. |
0. |
0. |
0. |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 55 | 319 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.73 |
4.25 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.031 |
0.187 |
0.241 |
0. |
0. |
0. |
0. |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 71 | 71 | 54 | 520 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.76 | 7.32 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.032 |
0.127 |
0.219 |
0. |
0. |
0. |
0. |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 42 | 165 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.98 | 3.84 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.018 |
0.036 |
0.196 |
0. |
0. |
0. |
0. |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 42 | 521 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.98 | 12.12 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.018 |
0.032 |
0.255 |
0. |
0. |
0. |
0. |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 63 | 189 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.82 | 2.45 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.033 |
0.082 |
0.215 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 60 | 547 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.78 | 7.1 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.032 |
0.143 |
0.252 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 105 | 105 | 70 | 216 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.67 |
2.06 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.049 |
0.144 |
0.275 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 54 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.06 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.022 |
0.078 |
0.104 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 51 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1. | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.022 |
0.059 |
0.145 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 51 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.96 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.022 |
0.054 |
0.166 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 51 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.96 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.021 |
0.048 |
0.143 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 51 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.96 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.021 |
0.093 |
0.142 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 68 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.28 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.021 |
0.142 |
0.09 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 57 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.06 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.03 |
0.047 |
0.112 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 59 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.09 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.027 |
0.068 |
0.146 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 59 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.05 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.026 |
0.07 |
0.263 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 59 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.05 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.026 |
0.066 |
0.142 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 59 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.05 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.027 |
0.064 |
0.288 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 76 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.36 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.027 |
0.156 |
0.093 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 59 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.86 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.026 |
0.092 |
0.641 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 72 | 72 | 65 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.9 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.03 |
0.076 |
0.619 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 50 | 50 | 92 | 101 | 2253 | 302 | 0 | 174 |
normalized size |
1 |
1. |
1.84 |
2.02 |
45.06 |
6.04 |
0. |
3.48 |
time (sec) |
N/A |
0.017 |
0.258 |
0.119 |
1.969 |
0.487 |
0. |
1.291 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 36 | 36 | 72 | 78 | 1173 | 227 | 0 | 135 |
normalized size |
1 |
1. |
2. |
2.17 |
32.58 |
6.31 |
0. |
3.75 |
time (sec) |
N/A |
0.012 |
0.243 |
0.17 |
1.767 |
0.481 |
0. |
1.239 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 22 | 51 | 52 | 405 | 150 | 0 | 93 |
normalized size | 1 | 1. | 2.32 | 2.36 | 18.41 | 6.82 | 0. | 4.23 |
time (sec) |
N/A |
0.009 |
0.087 |
0.086 |
1.674 |
0.473 |
0. |
1.308 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 5 | 5 | 28 | 31 | 47 | 77 | 0 | 16 |
normalized size | 1 | 1. | 5.6 | 6.2 | 9.4 | 15.4 | 0. | 3.2 |
time (sec) |
N/A |
0.007 |
0.013 |
0.181 |
1.646 |
0.471 |
0. |
1.245 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 12 | 12 | 12 | 26 | 14 | 12 | 12 | 15 |
normalized size | 1 | 1. | 1. | 2.17 | 1.17 | 1. | 1. | 1.25 |
time (sec) |
N/A |
0.009 |
0.007 |
0.127 |
1.502 |
0.463 |
0.476 |
1.329 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 29 | 29 | 23 | 30 | 15 | 31 | 29 | 59 |
normalized size | 1 | 1. | 0.79 | 1.03 | 0.52 | 1.07 | 1. | 2.03 |
time (sec) |
N/A |
0.011 |
0.017 |
0.082 |
1.635 |
0.462 |
1.478 |
1.311 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 31 | 38 | 23 | 53 | 46 | 82 |
normalized size | 1 | 1. | 0.72 | 0.88 | 0.53 | 1.23 | 1.07 | 1.91 |
time (sec) |
N/A |
0.015 |
0.027 |
0.077 |
1.665 |
0.471 |
19.723 |
1.302 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 57 | 57 | 37 | 44 | 31 | 66 | 0 | 105 |
normalized size |
1 |
1. |
0.65 |
0.77 |
0.54 |
1.16 |
0. |
1.84 |
time (sec) |
N/A |
0.019 |
0.041 |
0.08 |
1.639 |
0.472 |
0. |
1.239 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 61 | 102 | 2947 | 286 | 0 | 221 |
normalized size |
1 |
1. |
0.73 |
1.21 |
35.08 |
3.4 |
0. |
2.63 |
time (sec) |
N/A |
0.04 |
0.324 |
0.116 |
14.647 |
0.508 |
0. |
1.431 |
|
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|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 65 | 65 | 51 | 79 | 1503 | 219 | 0 | 167 |
normalized size |
1 |
1. |
0.78 |
1.22 |
23.12 |
3.37 |
0. |
2.57 |
time (sec) |
N/A |
0.029 |
0.217 |
0.088 |
2.664 |
0.496 |
0. |
1.286 |
|
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Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 39 | 53 | 429 | 136 | 0 | 97 |
normalized size | 1 | 1. | 0.85 | 1.15 | 9.33 | 2.96 | 0. | 2.11 |
time (sec) |
N/A |
0.022 |
0.072 |
0.078 |
1.691 |
0.487 |
0. |
1.322 |
|
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Problem 50 |
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 | 26 | 26 | 30 | 32 | 32 | 188 | 0 | 18 |
normalized size | 1 | 1. | 1.15 | 1.23 | 1.23 | 7.23 | 0. | 0.69 |
time (sec) |
N/A |
0.016 |
0.016 |
0.126 |
1.667 |
0.494 |
0. |
1.239 |
|
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Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 14 | 14 | 14 | 27 | 18 | 57 | 17 | 46 |
normalized size | 1 | 1. | 1. | 1.93 | 1.29 | 4.07 | 1.21 | 3.29 |
time (sec) |
N/A |
0.013 |
0.006 |
0.132 |
1.497 |
0.469 |
0.789 |
1.282 |
|
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Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 36 | 36 | 27 | 31 | 0 | 84 | 39 | 58 |
normalized size | 1 | 1. | 0.75 | 0.86 | 0. | 2.33 | 1.08 | 1.61 |
time (sec) |
N/A |
0.018 |
0.021 |
0.076 |
0. |
0.471 |
3.578 |
1.385 |
|
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Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 55 | 55 | 36 | 39 | 0 | 109 | 61 | 76 |
normalized size | 1 | 1. | 0.65 | 0.71 | 0. | 1.98 | 1.11 | 1.38 |
time (sec) |
N/A |
0.027 |
0.034 |
0.081 |
0. |
0.481 |
32.76 |
1.449 |
|
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Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 42 | 45 | 0 | 127 | 0 | 93 |
normalized size |
1 |
1. |
0.57 |
0.61 |
0. |
1.72 |
0. |
1.26 |
time (sec) |
N/A |
0.036 |
0.045 |
0.076 |
0. |
0.489 |
0. |
1.48 |
|
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Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 123 | 123 | 58 | 1313 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.47 |
10.67 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.06 |
0.189 |
0.222 |
0. |
0. |
0. |
0. |
|
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Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 71 | 71 | 46 | 372 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.65 | 5.24 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.038 |
0.113 |
0.319 |
0. |
0. |
0. |
0. |
|
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Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 46 | 343 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.96 | 7.15 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.027 |
0.041 |
0.278 |
0. |
0. |
0. |
0. |
|
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Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 50 | 50 | 38 | 125 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.76 | 2.5 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.03 |
0.049 |
0.25 |
0. |
0. |
0. |
0. |
|
|
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Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 52 | 349 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.66 | 4.42 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.038 |
0.095 |
0.235 |
0. |
0. |
0. |
0. |
|
|
|
|
|
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|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 123 | 123 | 63 | 158 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.51 | 1.28 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.061 |
0.147 |
0.264 |
0. |
0. |
0. |
0. |
|
|
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|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 164 | 164 | 59 | 53 | 89 | 360 | 0 | 93 |
normalized size |
1 |
1. |
0.36 |
0.32 |
0.54 |
2.2 |
0. |
0.57 |
time (sec) |
N/A |
0.035 |
0.058 |
0.273 |
1.5 |
0.547 |
0. |
1.298 |
|
|
|
|
|
|
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|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 118 | 118 | 47 | 41 | 65 | 252 | 0 | 69 |
normalized size |
1 |
1. |
0.4 |
0.35 |
0.55 |
2.14 |
0. |
0.58 |
time (sec) |
N/A |
0.028 |
0.036 |
0.175 |
1.499 |
0.493 |
0. |
1.276 |
|
|
|
|
|
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|
Problem 63 |
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 | 62 | 62 | 33 | 29 | 41 | 150 | 0 | 31 |
normalized size | 1 | 1. | 0.53 | 0.47 | 0.66 | 2.42 | 0. | 0.5 |
time (sec) |
N/A |
0.021 |
0.02 |
0.096 |
1.511 |
0.471 |
0. |
1.296 |
|
|
|
|
|
|
|
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|
Problem 64 |
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 | 16 | 16 | 16 | 18 | 12 | 70 | 0 | 12 |
normalized size | 1 | 1. | 1. | 1.12 | 0.75 | 4.38 | 0. | 0.75 |
time (sec) |
N/A |
0.015 |
0.006 |
0.138 |
1.528 |
0.466 |
0. |
1.351 |
|
|
|
|
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|
Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 36 | 36 | 25 | 24 | 34 | 122 | 0 | 0 |
normalized size |
1 |
1. |
0.69 |
0.67 |
0.94 |
3.39 |
0. |
0. |
time (sec) |
N/A |
0.016 |
0.025 |
0.163 |
1.497 |
0.489 |
0. |
0. |
|
|
|
|
|
|
|
|
|
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|
Problem 66 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 86 | 86 | 38 | 41 | 78 | 178 | 0 | 0 |
normalized size |
1 |
1. |
0.44 |
0.48 |
0.91 |
2.07 |
0. |
0. |
time (sec) |
N/A |
0.029 |
0.043 |
0.205 |
1.524 |
0.495 |
0. |
0. |
|
|
|
|
|
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|
Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 132 | 132 | 55 | 57 | 119 | 236 | 0 | 0 |
normalized size |
1 |
1. |
0.42 |
0.43 |
0.9 |
1.79 |
0. |
0. |
time (sec) |
N/A |
0.045 |
0.095 |
0.245 |
1.535 |
0.524 |
0. |
0. |
|
|
|
|
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|
Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 80 | 80 | 71 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.89 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.05 |
0.115 |
0.765 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
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|
Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 82 | 82 | 73 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.89 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.049 |
0.088 |
0.343 |
0. |
0. |
0. |
0. |
|
|
|
|
|
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|
Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 91 | 91 | 77 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.85 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.045 |
0.141 |
1.082 |
0. |
0. |
0. |
0. |
|
|
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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 [47] had the largest ratio of [ 0.4
]
|
|
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|
|
|
|
| Table 2.1Rubi specific breakdown of results for each
integral
|
|
|
|
|
|
|
# |
grade |
|
|
normalized | antiderivative |
leaf size | |
|
\(\frac{\text{number of rules}}{\text{integrand leaf size}}\) |
|
|
|
|
|
|
|
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|
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|
|
|
|
|
1 |
A |
1 |
1 |
1. |
6 |
0.167 |
|
|
|
|
|
|
|
2 |
A |
2 |
2 |
1. |
8 |
0.25 |
|
|
|
|
|
|
|
3 |
A |
2 |
2 |
1. |
8 |
0.25 |
|
|
|
|
|
|
|
4 |
A |
2 |
1 |
1. |
8 |
0.125 |
|
|
|
|
|
|
|
5 |
A |
3 |
2 |
1. |
8 |
0.25 |
|
|
|
|
|
|
|
6 |
A |
2 |
1 |
1. |
8 |
0.125 |
|
|
|
|
|
|
|
7 |
A |
4 |
2 |
1. |
8 |
0.25 |
|
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|
|
8 |
A |
2 |
1 |
1. |
8 |
0.125 |
|
|
|
|
|
|
|
9 |
A |
4 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
10 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
11 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
12 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
13 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
14 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
15 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
16 |
A |
4 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
17 |
A |
4 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
18 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
19 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
20 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
|
21 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
22 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
23 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
24 |
A |
4 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
25 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
26 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
27 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
28 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
29 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
30 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
31 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
32 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
33 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
34 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
35 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
36 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
37 |
A |
2 |
2 |
1. |
8 |
0.25 |
|
|
|
|
|
|
|
38 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
39 |
A |
5 |
3 |
1. |
8 |
0.375 |
|
|
|
|
|
|
|
40 |
A |
4 |
3 |
1. |
8 |
0.375 |
|
|
|
|
|
|
|
|
41 |
A |
3 |
3 |
1. |
8 |
0.375 |
|
|
|
|
|
|
|
42 |
A |
2 |
2 |
1. |
8 |
0.25 |
|
|
|
|
|
|
|
43 |
A |
2 |
2 |
1. |
8 |
0.25 |
|
|
|
|
|
|
|
44 |
A |
3 |
3 |
1. |
8 |
0.375 |
|
|
|
|
|
|
|
45 |
A |
4 |
3 |
1. |
8 |
0.375 |
|
|
|
|
|
|
|
46 |
A |
5 |
3 |
1. |
8 |
0.375 |
|
|
|
|
|
|
|
47 |
A |
6 |
4 |
1. |
10 |
0.4 |
|
|
|
|
|
|
|
48 |
A |
5 |
4 |
1. |
10 |
0.4 |
|
|
|
|
|
|
|
49 |
A |
4 |
4 |
1. |
10 |
0.4 |
|
|
|
|
|
|
|
50 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
51 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
52 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
53 |
A |
4 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
54 |
A |
5 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
55 |
A |
7 |
4 |
1. |
10 |
0.4 |
|
|
|
|
|
|
|
56 |
A |
5 |
4 |
1. |
10 |
0.4 |
|
|
|
|
|
|
|
57 |
A |
4 |
4 |
1. |
10 |
0.4 |
|
|
|
|
|
|
|
58 |
A |
4 |
4 |
1. |
10 |
0.4 |
|
|
|
|
|
|
|
59 |
A |
5 |
4 |
1. |
10 |
0.4 |
|
|
|
|
|
|
|
60 |
A |
7 |
4 |
1. |
10 |
0.4 |
|
|
|
|
|
|
|
|
61 |
A |
3 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
62 |
A |
3 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
63 |
A |
3 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
64 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
65 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
66 |
A |
5 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
67 |
A |
7 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
68 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
69 |
A |
3 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
70 |
A |
3 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|