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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 5, 7, 10, 11, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 41, 42, 43, 44, 45, 46 }
B grade: { 6, 8, 12, 16, 17, 39, 40 }
C grade: { }
F grade: { 9}
2.1.3 Maple
A grade: { 4, 5, 9, 10, 13, 14, 15, 18, 19, 20, 21, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 44, 45, 46
}
B grade: { 1, 2, 3, 6, 7, 8, 11, 12, 16, 17, 24, 25, 26, 29, 30, 31, 36, 41 }
C grade: { }
F grade: { 34, 35, 39, 40 }
2.1.4 Maxima
A grade: { 9, 10, 19, 21, 23, 32, 33, 42, 44, 46
B grade: { 1, 2, 6, 7, 11, 12, 13, 16, 17, 18, 24, 25, 29, 30 }
C grade: { }
F grade: { 3, 4, 5, 8, 14, 15, 20, 22, 26, 27, 28, 31, 34, 35, 36, 37, 38, 39, 40, 41, 43, 45 }
2.1.5 FriCAS
A grade: { 4, 5, 9, 10, 13, 14, 15, 18, 19, 20, 21, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 44, 45, 46
}
B grade: { 3, 8, 12, 17, 26, 31, 36, 41 }
C grade: { 1, 2, 6, 7, 11, 16, 24, 25, 29, 30, 34, 35, 39, 40 }
F grade: { }
2.1.6 Sympy
A grade: { 4, 5, 9, 10, 14, 15, 19, 20, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 45, 46 }
B grade: { }
C grade: { }
F grade: { 1, 2, 3, 6, 7, 8, 11, 12, 13, 16, 17, 18, 21, 24, 25, 26, 29, 30, 31, 34, 35, 36, 39, 40, 41, 44
}
2.1.7 Giac
A grade: { 4, 5, 9, 10, 14, 15, 19, 20, 21, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 44, 45, 46 }
B grade: { 13, 18 }
C grade: { }
F grade: { 1, 2, 3, 6, 7, 8, 11, 12, 16, 17, 24, 25, 26, 29, 30, 31, 34, 35, 36, 39, 40, 41 }
2.2 Detailed conclusion table per each integral for all CAS systems
Detailed conclusion table per each integral is given by table below. The elapsed time is in
seconds. For failed result it is given as F(-1) if the failure was due to timeout. It is given as F(-2)
if the failure was due to an exception being raised, which could indicate a bug in the system. If
the failure was due to integral not being evaluated within the time limit, then it is given just an
F.
In this table,the column normalized size is defined as \(\frac{\text{antiderivative leaf size}}{\text{optimal antiderivative leaf size}}\)
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Problem 1 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 227 | 227 | 218 | 747 | 1250 | 2674 | 0 | 0 |
normalized size | 1 | 1. | 0.96 | 3.29 | 5.51 | 11.78 | 0. | 0. |
time (sec) |
N/A |
0.205 |
0.123 |
0.225 |
2.359 |
2.352 |
0. |
0. |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 151 | 431 | 689 | 1732 | 0 | 0 |
normalized size | 1 | 1. | 0.96 | 2.75 | 4.39 | 11.03 | 0. | 0. |
time (sec) |
N/A |
0.14 |
0.137 |
0.157 |
2.115 |
2.162 |
0. |
0. |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-1) |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 93 | 93 | 87 | 186 | 0 | 927 | 0 | 0 |
normalized size |
1 |
1. |
0.94 |
2. |
0. |
9.97 |
0. |
0. |
time (sec) |
N/A |
0.067 |
0.055 |
0.122 |
0. |
1.978 |
0. |
0. |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.027 |
6.488 |
0.197 |
0. |
0. |
0. |
0. |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
F(-1) |
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 |
0.026 |
6.2 |
0.211 |
0. |
0. |
0. |
0. |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 371 | 371 | 811 | 1506 | 4598 | 4317 | 0 | 0 |
normalized size | 1 | 1. | 2.19 | 4.06 | 12.39 | 11.64 | 0. | 0. |
time (sec) |
N/A |
0.443 |
8.837 |
0.312 |
3.588 |
3.173 |
0. |
0. |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 262 | 262 | 505 | 668 | 2304 | 2604 | 0 | 0 |
normalized size | 1 | 1. | 1.93 | 2.55 | 8.79 | 9.94 | 0. | 0. |
time (sec) |
N/A |
0.313 |
5.64 |
0.204 |
2.568 |
2.538 |
0. |
0. |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
B |
F(-1) |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 134 | 134 | 330 | 274 | 0 | 1378 | 0 | 0 |
normalized size |
1 |
1. |
2.46 |
2.04 |
0. |
10.28 |
0. |
0. |
time (sec) |
N/A |
0.115 |
5.423 |
0.137 |
0. |
2.194 |
0. |
0. |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
F |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.051 |
0. |
1.036 |
0. |
0. |
0. |
0. |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.048 |
28.806 |
1.433 |
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 |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 152 | 152 | 216 | 406 | 1732 | 1237 | 0 | 0 |
normalized size | 1 | 1. | 1.42 | 2.67 | 11.39 | 8.14 | 0. | 0. |
time (sec) |
N/A |
0.329 |
1.952 |
0.155 |
2.45 |
1.79 |
0. |
0. |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
B |
B |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 119 | 119 | 528 | 225 | 513 | 711 | 0 | 0 |
normalized size | 1 | 1. | 4.44 | 1.89 | 4.31 | 5.97 | 0. | 0. |
time (sec) |
N/A |
0.245 |
6.471 |
0.129 |
2.216 |
1.791 |
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 |
B |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 104 | 76 | 369 | 244 | 0 | 392 |
normalized size | 1 | 1. | 1.55 | 1.13 | 5.51 | 3.64 | 0. | 5.85 |
time (sec) |
N/A |
0.097 |
0.694 |
0.062 |
1.818 |
1.694 |
0. |
1.392 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.058 |
7.748 |
0.142 |
0. |
0. |
0. |
0. |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.052 |
5.914 |
0.182 |
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 |
B |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 288 | 288 | 1447 | 799 | 5750 | 2179 | 0 | 0 |
normalized size | 1 | 1. | 5.02 | 2.77 | 19.97 | 7.57 | 0. | 0. |
time (sec) |
N/A |
0.719 |
7.49 |
0.2 |
8.524 |
1.992 |
0. |
0. |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
B |
B |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 229 | 229 | 925 | 442 | 1395 | 1173 | 0 | 0 |
normalized size | 1 | 1. | 4.04 | 1.93 | 6.09 | 5.12 | 0. | 0. |
time (sec) |
N/A |
0.498 |
6.849 |
0.136 |
4.55 |
1.844 |
0. |
0. |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 140 | 140 | 172 | 138 | 1428 | 443 | 0 | 1215 |
normalized size | 1 | 1. | 1.23 | 0.99 | 10.2 | 3.16 | 0. | 8.68 |
time (sec) |
N/A |
0.196 |
1.603 |
0.074 |
1.943 |
1.672 |
0. |
1.631 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.055 |
15.517 |
2.137 |
0. |
0. |
0. |
0. |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.052 |
18.089 |
2.818 |
0. |
0. |
0. |
0. |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
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 |
0.047 |
0.791 |
0.259 |
0. |
0. |
0. |
0. |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.025 |
6.585 |
0.182 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.053 |
0.945 |
0.193 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 227 | 227 | 365 | 747 | 1250 | 2674 | 0 | 0 |
normalized size | 1 | 1. | 1.61 | 3.29 | 5.51 | 11.78 | 0. | 0. |
time (sec) |
N/A |
0.213 |
0.486 |
0.195 |
2.555 |
2.306 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 203 | 431 | 689 | 1732 | 0 | 0 |
normalized size | 1 | 1. | 1.29 | 2.75 | 4.39 | 11.03 | 0. | 0. |
time (sec) |
N/A |
0.141 |
0.25 |
0.144 |
2.234 |
2.179 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-1) |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 93 | 93 | 104 | 186 | 0 | 927 | 0 | 0 |
normalized size |
1 |
1. |
1.12 |
2. |
0. |
9.97 |
0. |
0. |
time (sec) |
N/A |
0.071 |
0.013 |
0.072 |
0. |
1.974 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
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 |
0.028 |
0.96 |
0.171 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
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 |
0.028 |
1.35 |
0.198 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 364 | 364 | 646 | 1478 | 4415 | 4317 | 0 | 0 |
normalized size | 1 | 1. | 1.77 | 4.06 | 12.13 | 11.86 | 0. | 0. |
time (sec) |
N/A |
0.455 |
2.596 |
0.297 |
3.62 |
3.219 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 257 | 257 | 356 | 651 | 2219 | 2604 | 0 | 0 |
normalized size | 1 | 1. | 1.39 | 2.53 | 8.63 | 10.13 | 0. | 0. |
time (sec) |
N/A |
0.308 |
1.434 |
0.199 |
2.637 |
2.605 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-1) |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 131 | 131 | 151 | 266 | 0 | 1378 | 0 | 0 |
normalized size |
1 |
1. |
1.15 |
2.03 |
0. |
10.52 |
0. |
0. |
time (sec) |
N/A |
0.12 |
0.502 |
0.135 |
0. |
2.357 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.051 |
43.047 |
0.989 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.049 |
32.362 |
1.416 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 526 | 526 | 449 | 0 | 0 | 5515 | 0 | 0 |
normalized size |
1 |
1. |
0.85 |
0. |
0. |
10.48 |
0. |
0. |
time (sec) |
N/A |
1.043 |
1.1 |
0.566 |
0. |
3.41 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 394 | 394 | 338 | 0 | 0 | 4018 | 0 | 0 |
normalized size |
1 |
1. |
0.86 |
0. |
0. |
10.2 |
0. |
0. |
time (sec) |
N/A |
0.867 |
0.77 |
0.441 |
0. |
2.766 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 257 | 257 | 214 | 516 | 0 | 2569 | 0 | 0 |
normalized size |
1 |
1. |
0.83 |
2.01 |
0. |
10. |
0. |
0. |
time (sec) |
N/A |
0.49 |
0.457 |
0.132 |
0. |
2.551 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.059 |
1.528 |
0.186 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.055 |
10.751 |
0.224 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F(-2) |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1523 | 1523 | 8174 | 0 | 0 | 14575 | 0 | 0 |
normalized size |
1 |
1. |
5.37 |
0. |
0. |
9.57 |
0. |
0. |
time (sec) |
N/A |
2.809 |
25.925 |
0.794 |
0. |
7.247 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F(-2) |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1117 | 1117 | 11147 | 0 | 0 | 9226 | 0 | 0 |
normalized size |
1 |
1. |
9.98 |
0. |
0. |
8.26 |
0. |
0. |
time (sec) |
N/A |
2.111 |
22.263 |
0.638 |
0. |
4.943 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 582 | 582 | 1037 | 1289 | 0 | 4639 | 0 | 0 |
normalized size |
1 |
1. |
1.78 |
2.21 |
0. |
7.97 |
0. |
0. |
time (sec) |
N/A |
1.047 |
9.793 |
0.194 |
0. |
3.356 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.058 |
28.11 |
1.746 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.055 |
42.487 |
2.641 |
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 | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.051 |
1.622 |
0.294 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.026 |
0.504 |
0.188 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.055 |
0.684 |
0.182 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
2.3 Detailed conclusion table specific for Rubi results
The following table is specific to Rubi. It gives additional statistics for each integral. the column
steps is the number of steps used by Rubi to obtain the antiderivative. The rules
column is the number of unique rules used. The integrand size column is the leaf size
of the integrand. Finally the ratio \(\frac{\text{number of rules}}{\text{integrand size}}\) is given. The larger this ratio is, the harder the
integral was to solve. In this test, problem number [40] had the largest ratio of [ 0.55
]
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|
| Table 2.1Rubi 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 |
11 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
2 |
A |
9 |
5 |
1. |
18 |
0.278 |
|
|
|
|
|
|
|
3 |
A |
7 |
4 |
1. |
16 |
0.25 |
|
|
|
|
|
|
|
4 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
5 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
6 |
A |
17 |
9 |
1. |
20 |
0.45 |
|
|
|
|
|
|
|
7 |
A |
14 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
8 |
A |
9 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
9 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
10 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
11 |
A |
9 |
8 |
1. |
20 |
0.4 |
|
|
|
|
|
|
|
12 |
A |
8 |
7 |
1. |
20 |
0.35 |
|
|
|
|
|
|
|
13 |
A |
5 |
4 |
1. |
18 |
0.222 |
|
|
|
|
|
|
|
14 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
15 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
16 |
A |
19 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
17 |
A |
17 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
18 |
A |
9 |
5 |
1. |
18 |
0.278 |
|
|
|
|
|
|
|
19 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
20 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
21 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
22 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
23 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
24 |
A |
11 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
25 |
A |
9 |
5 |
1. |
18 |
0.278 |
|
|
|
|
|
|
|
26 |
A |
7 |
4 |
1. |
16 |
0.25 |
|
|
|
|
|
|
|
27 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
28 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
29 |
A |
17 |
9 |
1. |
20 |
0.45 |
|
|
|
|
|
|
|
30 |
A |
14 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
31 |
A |
9 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
32 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
33 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
34 |
A |
14 |
8 |
1. |
20 |
0.4 |
|
|
|
|
|
|
|
35 |
A |
12 |
7 |
1. |
20 |
0.35 |
|
|
|
|
|
|
|
36 |
A |
10 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
37 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
38 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
39 |
A |
36 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
40 |
A |
30 |
11 |
1. |
20 |
0.55 |
|
|
|
|
|
|
|
|
41 |
A |
21 |
9 |
1. |
18 |
0.5 |
|
|
|
|
|
|
|
42 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
43 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
44 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
45 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
46 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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