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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 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 }
B grade: { 3 }
C grade: { 67, 68 }
F grade: { }
2.1.3 Maple
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, 30,
31, 32, 33, 34, 36, 38, 40, 41, 42, 44, 46, 50, 51, 52, 53, 57, 58, 61, 66 }
B grade: { 29, 59, 60, 64, 65 }
C grade: { 35, 47, 54, 55, 56 }
F grade: { 37, 39, 43, 45, 48, 49, 62, 63, 67, 68 }
2.1.4 Maxima
A grade: { 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 44, 46, 47, 49, 50, 52, 53, 57, 58, 59, 60, 64, 65, 66
B grade: { 1, 2, 4, 17, 22, 54, 55, 56, 61 }
C grade: { 28, 29 }
F grade: { 43, 45, 48, 51, 62, 63, 67, 68 }
2.1.5 FriCAS
A grade: { 1, 3, 4, 5, 7, 8, 10, 11, 12, 14, 15, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 38, 40, 41, 42,
44, 46, 53, 54, 56, 57, 58, 59, 60, 61, 64, 65, 66 }
B grade: { 2, 6, 9, 13, 16, 20, 21, 22, 30, 34, 36, 50, 51, 52, 55, 62, 63, 67, 68 }
C grade: { }
F grade: { 35, 37, 39, 43, 45, 47, 48, 49 }
2.1.6 Sympy
A grade: { 1, 3, 8, 10, 15, 17, 22, 23, 24, 26, 27, 28, 29, 33, 41, 57, 58, 59, 60, 61, 66 }
B grade: { }
C grade: { }
F grade: { 2, 4, 5, 6, 7, 9, 11, 12, 13, 14, 16, 18, 19, 20, 21, 25, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 62, 63, 64, 65, 67, 68 }
2.1.7 Giac
A grade: { 1, 2, 3, 4, 5, 9, 11, 12, 16, 17, 18, 19, 22, 27, 33, 41, 42, 44, 46, 57, 58, 66 }
B grade: { 7, 8, 10, 14, 15, 21, 23, 24, 59, 60, 61, 64, 65 }
C grade: { 54, 55, 56 }
F grade: { 6, 13, 20, 25, 26, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 43, 45, 47, 48, 49, 50, 51, 52, 53,
62, 63, 67, 68 }
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 |
A |
B |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 34 | 34 | 31 | 45 | 108 | 69 | 36 | 65 |
normalized size | 1 | 1. | 0.91 | 1.32 | 3.18 | 2.03 | 1.06 | 1.91 |
time (sec) |
N/A |
0.036 |
0.033 |
0.013 |
0.985 |
1.795 |
1.032 |
1.299 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 67 | 74 | 149 | 539 | 0 | 101 |
normalized size | 1 | 1. | 0.97 | 1.07 | 2.16 | 7.81 | 0. | 1.46 |
time (sec) |
N/A |
0.039 |
0.069 |
0.027 |
1.001 |
1.785 |
0. |
1.289 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 15 | 15 | 31 | 14 | 18 | 31 | 19 | 36 |
normalized size | 1 | 1. | 2.07 | 0.93 | 1.2 | 2.07 | 1.27 | 2.4 |
time (sec) |
N/A |
0.015 |
0.01 |
0.005 |
1.106 |
1.759 |
0.233 |
1.312 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 45 | 40 | 117 | 159 | 0 | 55 |
normalized size | 1 | 1. | 0.85 | 0.75 | 2.21 | 3. | 0. | 1.04 |
time (sec) |
N/A |
0.02 |
0.031 |
0.018 |
1.044 |
1.784 |
0. |
1.243 |
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Problem 5 |
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 | 25 | 25 | 23 | 27 | 32 | 104 | 0 | 32 |
normalized size | 1 | 1. | 0.92 | 1.08 | 1.28 | 4.16 | 0. | 1.28 |
time (sec) |
N/A |
0.036 |
0.014 |
0.013 |
1.184 |
1.674 |
0. |
1.266 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 70 | 70 | 74 | 529 | 0 | 0 |
normalized size | 1 | 1. | 1.06 | 1.06 | 1.12 | 8.02 | 0. | 0. |
time (sec) |
N/A |
0.035 |
0.066 |
0.021 |
1.088 |
1.728 |
0. |
0. |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 42 | 42 | 38 | 58 | 54 | 166 | 0 | 146 |
normalized size | 1 | 1. | 0.9 | 1.38 | 1.29 | 3.95 | 0. | 3.48 |
time (sec) |
N/A |
0.091 |
0.042 |
0.02 |
1.249 |
1.644 |
0. |
1.32 |
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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 |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 40 | 55 | 80 | 140 | 78 | 163 |
normalized size | 1 | 1. | 0.78 | 1.08 | 1.57 | 2.75 | 1.53 | 3.2 |
time (sec) |
N/A |
0.051 |
0.101 |
0.03 |
1.095 |
1.764 |
2.175 |
1.234 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 99 | 99 | 101 | 90 | 128 | 1125 | 0 | 131 |
normalized size | 1 | 1. | 1.02 | 0.91 | 1.29 | 11.36 | 0. | 1.32 |
time (sec) |
N/A |
0.088 |
0.218 |
0.044 |
1.642 |
1.764 |
0. |
1.399 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 27 | 34 | 51 | 66 | 60 | 73 |
normalized size | 1 | 1. | 0.87 | 1.1 | 1.65 | 2.13 | 1.94 | 2.35 |
time (sec) |
N/A |
0.028 |
0.022 |
0.009 |
1.038 |
1.813 |
0.546 |
1.301 |
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Problem 11 |
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 | 78 | 78 | 86 | 51 | 76 | 225 | 0 | 78 |
normalized size | 1 | 1. | 1.1 | 0.65 | 0.97 | 2.88 | 0. | 1. |
time (sec) |
N/A |
0.043 |
0.072 |
0.029 |
1.675 |
1.668 |
0. |
1.365 |
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Problem 12 |
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 | 37 | 37 | 33 | 34 | 42 | 138 | 0 | 47 |
normalized size | 1 | 1. | 0.89 | 0.92 | 1.14 | 3.73 | 0. | 1.27 |
time (sec) |
N/A |
0.061 |
0.02 |
0.033 |
1.318 |
1.848 |
0. |
1.302 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 88 | 88 | 94 | 86 | 82 | 1058 | 0 | 0 |
normalized size | 1 | 1. | 1.07 | 0.98 | 0.93 | 12.02 | 0. | 0. |
time (sec) |
N/A |
0.071 |
0.23 |
0.038 |
1.145 |
2.021 |
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 |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 57 | 57 | 46 | 69 | 49 | 216 | 0 | 170 |
normalized size | 1 | 1. | 0.81 | 1.21 | 0.86 | 3.79 | 0. | 2.98 |
time (sec) |
N/A |
0.126 |
0.089 |
0.035 |
1.109 |
2.002 |
0. |
1.296 |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 55 | 93 | 135 | 234 | 92 | 250 |
normalized size | 1 | 1. | 0.7 | 1.18 | 1.71 | 2.96 | 1.16 | 3.16 |
time (sec) |
N/A |
0.077 |
0.164 |
0.047 |
1.169 |
1.839 |
4.046 |
1.376 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 160 | 160 | 184 | 157 | 219 | 2429 | 0 | 224 |
normalized size | 1 | 1. | 1.15 | 0.98 | 1.37 | 15.18 | 0. | 1.4 |
time (sec) |
N/A |
0.143 |
0.303 |
0.065 |
1.729 |
1.818 |
0. |
1.266 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 33 | 33 | 33 | 28 | 84 | 97 | 44 | 76 |
normalized size | 1 | 1. | 1. | 0.85 | 2.55 | 2.94 | 1.33 | 2.3 |
time (sec) |
N/A |
0.03 |
0.008 |
0.008 |
1.143 |
1.7 |
1.026 |
1.324 |
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Problem 18 |
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 | 125 | 125 | 136 | 86 | 123 | 369 | 0 | 128 |
normalized size | 1 | 1. | 1.09 | 0.69 | 0.98 | 2.95 | 0. | 1.02 |
time (sec) |
N/A |
0.072 |
0.121 |
0.04 |
1.671 |
1.826 |
0. |
1.242 |
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Problem 19 |
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 | 49 | 55 | 68 | 231 | 0 | 68 |
normalized size | 1 | 1. | 0.89 | 1. | 1.24 | 4.2 | 0. | 1.24 |
time (sec) |
N/A |
0.094 |
0.033 |
0.052 |
1.32 |
1.663 |
0. |
1.274 |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 136 | 136 | 204 | 149 | 138 | 2395 | 0 | 0 |
normalized size | 1 | 1. | 1.5 | 1.1 | 1.01 | 17.61 | 0. | 0. |
time (sec) |
N/A |
0.114 |
0.355 |
0.06 |
1.352 |
1.8 |
0. |
0. |
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Problem 21 |
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 | 91 | 91 | 92 | 120 | 78 | 404 | 0 | 302 |
normalized size | 1 | 1. | 1.01 | 1.32 | 0.86 | 4.44 | 0. | 3.32 |
time (sec) |
N/A |
0.219 |
0.081 |
0.052 |
1.338 |
1.693 |
0. |
1.254 |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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|
|
|
grade |
A |
A |
A |
A |
B |
B |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 67 | 52 | 170 | 329 | 94 | 146 |
normalized size | 1 | 1. | 1. | 0.78 | 2.54 | 4.91 | 1.4 | 2.18 |
time (sec) |
N/A |
0.044 |
0.026 |
0.055 |
1.197 |
1.87 |
12.101 |
1.301 |
|
|
|
|
|
|
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|
|
Problem 23 |
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 | 8 | 8 | 8 | 7 | 8 | 20 | 5 | 20 |
normalized size | 1 | 1. | 1. | 0.88 | 1. | 2.5 | 0.62 | 2.5 |
time (sec) |
N/A |
0.011 |
0.002 |
0.006 |
1.112 |
1.746 |
0.334 |
1.3 |
|
|
|
|
|
|
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|
|
Problem 24 |
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 | 8 | 8 | 8 | 7 | 8 | 26 | 8 | 20 |
normalized size | 1 | 1. | 1. | 0.88 | 1. | 3.25 | 1. | 2.5 |
time (sec) |
N/A |
0.012 |
0.004 |
0.006 |
1.15 |
1.67 |
43.394 |
1.244 |
|
|
|
|
|
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|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 33 | 33 | 33 | 56 | 50 | 135 | 0 | 0 |
normalized size | 1 | 1. | 1. | 1.7 | 1.52 | 4.09 | 0. | 0. |
time (sec) |
N/A |
0.078 |
0.026 |
0.033 |
1.251 |
1.687 |
0. |
0. |
|
|
|
|
|
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|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 21 | 21 | 27 | 32 | 95 | 17 | 0 |
normalized size | 1 | 1. | 1. | 1.29 | 1.52 | 4.52 | 0.81 | 0. |
time (sec) |
N/A |
0.032 |
0.01 |
0.026 |
1.322 |
1.727 |
1.398 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
|
Problem 27 |
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 | 13 | 13 | 13 | 14 | 18 | 30 | 15 | 36 |
normalized size | 1 | 1. | 1. | 1.08 | 1.38 | 2.31 | 1.15 | 2.77 |
time (sec) |
N/A |
0.015 |
0.004 |
0.005 |
1.118 |
1.688 |
1.423 |
1.26 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
C |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 29 | 29 | 29 | 44 | 63 | 72 | 29 | 0 |
normalized size | 1 | 1. | 1. | 1.52 | 2.17 | 2.48 | 1. | 0. |
time (sec) |
N/A |
0.03 |
0.023 |
0.01 |
1.237 |
1.718 |
2.557 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
C |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 39 | 94 | 65 | 96 | 46 | 0 |
normalized size | 1 | 1. | 0.85 | 2.04 | 1.41 | 2.09 | 1. | 0. |
time (sec) |
N/A |
0.053 |
0.042 |
0.013 |
1.195 |
1.705 |
4.47 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 71 | 70 | 97 | 606 | 0 | 0 |
normalized size | 1 | 1. | 1.06 | 1.04 | 1.45 | 9.04 | 0. | 0. |
time (sec) |
N/A |
0.045 |
0.081 |
0.037 |
1.176 |
1.886 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 25 | 25 | 27 | 32 | 105 | 0 | 0 |
normalized size | 1 | 1. | 1. | 1.08 | 1.28 | 4.2 | 0. | 0. |
time (sec) |
N/A |
0.032 |
0.015 |
0.023 |
1.224 |
1.738 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 57 | 57 | 49 | 44 | 85 | 158 | 0 | 0 |
normalized size | 1 | 1. | 0.86 | 0.77 | 1.49 | 2.77 | 0. | 0. |
time (sec) |
N/A |
0.031 |
0.035 |
0.027 |
1.235 |
1.795 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
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 | 15 | 15 | 15 | 14 | 18 | 41 | 22 | 36 |
normalized size | 1 | 1. | 1. | 0.93 | 1.2 | 2.73 | 1.47 | 2.4 |
time (sec) |
N/A |
0.017 |
0.005 |
0.006 |
1.007 |
1.67 |
4.362 |
1.327 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 74 | 82 | 85 | 653 | 0 | 0 |
normalized size | 1 | 1. | 0.99 | 1.09 | 1.13 | 8.71 | 0. | 0. |
time (sec) |
N/A |
0.049 |
0.071 |
0.033 |
1.162 |
1.885 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 77 | 74 | 82 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.15 | 1.1 | 1.22 | 0. | 0. | 0. |
time (sec) |
N/A |
0.022 |
0.067 |
0.055 |
1.13 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 25 | 23 | 33 | 41 | 178 | 0 | 0 |
normalized size | 1 | 1. | 0.92 | 1.32 | 1.64 | 7.12 | 0. | 0. |
time (sec) |
N/A |
0.038 |
0.021 |
0.02 |
1.155 |
1.838 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 81 | 0 | 92 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.91 | 0. | 1.03 | 0. | 0. | 0. |
time (sec) |
N/A |
0.07 |
0.116 |
0.079 |
1.152 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 36 | 40 | 50 | 217 | 0 | 0 |
normalized size | 1 | 1. | 0.84 | 0.93 | 1.16 | 5.05 | 0. | 0. |
time (sec) |
N/A |
0.063 |
0.028 |
0.079 |
1.147 |
1.837 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 150 | 150 | 138 | 0 | 169 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.92 | 0. | 1.13 | 0. | 0. | 0. |
time (sec) |
N/A |
0.08 |
0.601 |
0.101 |
1.203 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 52 | 67 | 84 | 374 | 0 | 0 |
normalized size | 1 | 1. | 0.78 | 1. | 1.25 | 5.58 | 0. | 0. |
time (sec) |
N/A |
0.1 |
0.048 |
0.106 |
1.277 |
1.763 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
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 |
0.022 |
5.239 |
0.698 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
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 |
0.026 |
7.997 |
0.7 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
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 | 95 | 95 | 94 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.99 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.104 |
0.14 |
0.802 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.052 |
5.809 |
0.707 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 131 | 131 | 148 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.13 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.187 |
0.359 |
0.759 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.061 |
8.301 |
0.706 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 100 | 110 | 115 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.12 | 1.24 | 1.29 | 0. | 0. | 0. |
time (sec) |
N/A |
0.07 |
0.164 |
0.102 |
1.282 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
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|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 128 | 128 | 116 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.91 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.155 |
0.249 |
0.089 |
0. |
0. |
0. |
0. |
|
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Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 200 | 200 | 182 | 0 | 234 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.91 | 0. | 1.17 | 0. | 0. | 0. |
time (sec) |
N/A |
0.206 |
0.956 |
0.11 |
1.319 |
0. |
0. |
0. |
|
|
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Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 45 | 45 | 46 | 74 | 46 | 462 | 0 | 0 |
normalized size |
1 |
1. |
1.02 |
1.64 |
1.02 |
10.27 |
0. |
0. |
time (sec) |
N/A |
0.094 |
0.057 |
0.028 |
1.233 |
1.877 |
0. |
0. |
|
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Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 54 | 90 | 0 | 567 | 0 | 0 |
normalized size |
1 |
1. |
0.81 |
1.34 |
0. |
8.46 |
0. |
0. |
time (sec) |
N/A |
0.13 |
0.122 |
0.073 |
0. |
1.868 |
0. |
0. |
|
|
|
|
|
|
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Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 113 | 113 | 97 | 152 | 95 | 1045 | 0 | 0 |
normalized size |
1 |
1. |
0.86 |
1.35 |
0.84 |
9.25 |
0. |
0. |
time (sec) |
N/A |
0.225 |
0.152 |
0.102 |
1.332 |
1.851 |
0. |
0. |
|
|
|
|
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Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 71 | 71 | 60 | 54 | 93 | 333 | 0 | 0 |
normalized size | 1 | 1. | 0.85 | 0.76 | 1.31 | 4.69 | 0. | 0. |
time (sec) |
N/A |
0.045 |
0.697 |
0.053 |
1.199 |
1.834 |
0. |
0. |
|
|
|
|
|
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|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
B |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 113 | 113 | 62 | 136 | 1195 | 382 | 0 | 185 |
normalized size | 1 | 1. | 0.55 | 1.2 | 10.58 | 3.38 | 0. | 1.64 |
time (sec) |
N/A |
0.101 |
0.124 |
0.045 |
1.617 |
1.86 |
0. |
1.318 |
|
|
|
|
|
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|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
B |
B |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 39 | 66 | 945 | 319 | 0 | 134 |
normalized size | 1 | 1. | 0.72 | 1.22 | 17.5 | 5.91 | 0. | 2.48 |
time (sec) |
N/A |
0.053 |
0.029 |
0.033 |
1.555 |
1.97 |
0. |
1.306 |
|
|
|
|
|
|
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|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
B |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 37 | 37 | 25 | 36 | 698 | 143 | 0 | 53 |
normalized size | 1 | 1. | 0.68 | 0.97 | 18.86 | 3.86 | 0. | 1.43 |
time (sec) |
N/A |
0.017 |
0.004 |
0.031 |
1.608 |
1.722 |
0. |
1.218 |
|
|
|
|
|
|
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|
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 | 19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.036 |
1.473 |
0.033 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
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|
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|
Problem 58 |
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 |
0.04 |
10.197 |
0.047 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
Problem 59 |
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 | 346 | 346 | 104 | 831 | 656 | 251 | 269 | 2074 |
normalized size | 1 | 1. | 0.3 | 2.4 | 1.9 | 0.73 | 0.78 | 5.99 |
time (sec) |
N/A |
0.424 |
0.503 |
0.02 |
1.161 |
1.979 |
2.144 |
1.83 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 60 |
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 | 167 | 167 | 72 | 303 | 393 | 169 | 151 | 782 |
normalized size | 1 | 1. | 0.43 | 1.81 | 2.35 | 1.01 | 0.9 | 4.68 |
time (sec) |
N/A |
0.188 |
0.19 |
0.013 |
1.155 |
1.802 |
0.714 |
1.505 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 50 | 63 | 149 | 112 | 65 | 282 |
normalized size | 1 | 1. | 0.93 | 1.17 | 2.76 | 2.07 | 1.2 | 5.22 |
time (sec) |
N/A |
0.047 |
0.061 |
0.01 |
1.111 |
1.622 |
0.558 |
1.455 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 124 | 124 | 127 | 0 | 0 | 547 | 0 | 0 |
normalized size | 1 | 1. | 1.02 | 0. | 0. | 4.41 | 0. | 0. |
time (sec) |
N/A |
0.29 |
0.492 |
0.02 |
0. |
1.842 |
0. |
0. |
|
|
|
|
|
|
|
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|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 182 | 182 | 199 | 0 | 0 | 759 | 0 | 0 |
normalized size |
1 |
1. |
1.09 |
0. |
0. |
4.17 |
0. |
0. |
time (sec) |
N/A |
0.382 |
1.755 |
0.021 |
0. |
1.9 |
0. |
0. |
|
|
|
|
|
|
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|
Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 537 | 537 | 381 | 1815 | 867 | 468 | 0 | 2920 |
normalized size | 1 | 1. | 0.71 | 3.38 | 1.61 | 0.87 | 0. | 5.44 |
time (sec) |
N/A |
0.704 |
0.686 |
0.016 |
1.241 |
1.795 |
0. |
3.72 |
|
|
|
|
|
|
|
|
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|
|
|
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|
|
Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 261 | 261 | 118 | 659 | 498 | 296 | 0 | 954 |
normalized size | 1 | 1. | 0.45 | 2.52 | 1.91 | 1.13 | 0. | 3.66 |
time (sec) |
N/A |
0.319 |
0.375 |
0.013 |
1.119 |
1.785 |
0. |
2.309 |
|
|
|
|
|
|
|
|
|
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|
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|
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|
|
Problem 66 |
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 | 85 | 85 | 65 | 133 | 184 | 159 | 95 | 173 |
normalized size | 1 | 1. | 0.76 | 1.56 | 2.16 | 1.87 | 1.12 | 2.04 |
time (sec) |
N/A |
0.08 |
0.077 |
0.011 |
1.075 |
1.773 |
1.589 |
1.417 |
|
|
|
|
|
|
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|
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|
Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 232 | 232 | 231 | 0 | 0 | 1538 | 0 | 0 |
normalized size | 1 | 1. | 1. | 0. | 0. | 6.63 | 0. | 0. |
time (sec) |
N/A |
0.518 |
0.064 |
0.018 |
0. |
2.038 |
0. |
0. |
|
|
|
|
|
|
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|
Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F(-2) |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 329 | 329 | 211 | 0 | 0 | 2033 | 0 | 0 |
normalized size |
1 |
1. |
0.64 |
0. |
0. |
6.18 |
0. |
0. |
time (sec) |
N/A |
0.692 |
0.498 |
0.018 |
0. |
2.149 |
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 [54] had the largest ratio of [ 0.75
]
|
|
|
|
|
|
|
| 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 |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
2 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
3 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
4 |
A |
3 |
3 |
1. |
8 |
0.375 |
|
|
|
|
|
|
|
5 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
6 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
7 |
A |
5 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
8 |
A |
3 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
9 |
A |
6 |
5 |
1. |
14 |
0.357 |
|
|
|
|
|
|
|
10 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
11 |
A |
5 |
4 |
1. |
10 |
0.4 |
|
|
|
|
|
|
|
12 |
A |
5 |
4 |
1. |
14 |
0.286 |
|
|
|
|
|
|
|
13 |
A |
6 |
6 |
1. |
14 |
0.429 |
|
|
|
|
|
|
|
14 |
A |
7 |
6 |
1. |
14 |
0.429 |
|
|
|
|
|
|
|
15 |
A |
4 |
4 |
1. |
14 |
0.286 |
|
|
|
|
|
|
|
16 |
A |
10 |
5 |
1. |
14 |
0.357 |
|
|
|
|
|
|
|
17 |
A |
3 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
18 |
A |
8 |
4 |
1. |
10 |
0.4 |
|
|
|
|
|
|
|
19 |
A |
8 |
4 |
1. |
14 |
0.286 |
|
|
|
|
|
|
|
20 |
A |
9 |
5 |
1. |
14 |
0.357 |
|
|
|
|
|
|
|
|
21 |
A |
12 |
6 |
1. |
14 |
0.429 |
|
|
|
|
|
|
|
22 |
A |
3 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
23 |
A |
2 |
2 |
1. |
8 |
0.25 |
|
|
|
|
|
|
|
24 |
A |
2 |
2 |
1. |
8 |
0.25 |
|
|
|
|
|
|
|
25 |
A |
5 |
5 |
1. |
8 |
0.625 |
|
|
|
|
|
|
|
26 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
27 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
28 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
29 |
A |
4 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
30 |
A |
5 |
5 |
1. |
8 |
0.625 |
|
|
|
|
|
|
|
31 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
32 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
33 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
34 |
A |
5 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
35 |
A |
3 |
2 |
1. |
8 |
0.25 |
|
|
|
|
|
|
|
36 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
37 |
A |
5 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
38 |
A |
5 |
4 |
1. |
14 |
0.286 |
|
|
|
|
|
|
|
39 |
A |
8 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
40 |
A |
8 |
4 |
1. |
14 |
0.286 |
|
|
|
|
|
|
|
|
41 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
42 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
43 |
A |
3 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
44 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
45 |
A |
5 |
5 |
1. |
22 |
0.227 |
|
|
|
|
|
|
|
46 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
47 |
A |
3 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
48 |
A |
5 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
49 |
A |
8 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
50 |
A |
5 |
5 |
1. |
16 |
0.312 |
|
|
|
|
|
|
|
51 |
A |
7 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
52 |
A |
12 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
53 |
A |
4 |
4 |
1. |
18 |
0.222 |
|
|
|
|
|
|
|
54 |
A |
12 |
9 |
1. |
12 |
0.75 |
|
|
|
|
|
|
|
55 |
A |
8 |
7 |
1. |
10 |
0.7 |
|
|
|
|
|
|
|
56 |
A |
4 |
4 |
1. |
8 |
0.5 |
|
|
|
|
|
|
|
57 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
58 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
59 |
A |
16 |
4 |
1. |
18 |
0.222 |
|
|
|
|
|
|
|
60 |
A |
10 |
4 |
1. |
16 |
0.25 |
|
|
|
|
|
|
|
|
61 |
A |
4 |
4 |
1. |
14 |
0.286 |
|
|
|
|
|
|
|
62 |
A |
10 |
5 |
1. |
18 |
0.278 |
|
|
|
|
|
|
|
63 |
A |
11 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
64 |
A |
23 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
65 |
A |
13 |
5 |
1. |
16 |
0.312 |
|
|
|
|
|
|
|
66 |
A |
5 |
4 |
1. |
14 |
0.286 |
|
|
|
|
|
|
|
67 |
A |
13 |
5 |
1. |
18 |
0.278 |
|
|
|
|
|
|
|
68 |
A |
14 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|