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, 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: { }
2.1.3 Maple
A grade: { }
B grade: { }
C grade: { 1, 2, 3, 4, 8, 9, 10, 11, 15, 16, 17, 18 }
F grade: { 5, 6, 7, 12, 13, 14, 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 }
2.1.4 Maxima
A grade: {
B grade: { }
C grade: { }
F 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 }
2.1.5 FriCAS
A grade: { }
B grade: { 1, 2, 3, 4, 8, 9, 10, 11, 15, 16, 17, 18 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 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 }
2.1.6 Sympy
A grade: { }
B grade: { }
C grade: { 5, 6, 12, 19, 22, 23, 24, 25, 31, 32 }
F grade: { 1, 2, 3, 4, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45, 46 }
2.1.7 Giac
A grade: { }
B grade: { 1, 2, 3, 4, 8, 9, 10, 11, 16, 17, 18 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 15, 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 }
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 |
C |
F(-2) |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 210 | 210 | 172 | 4972 | 0 | 6700 | 0 | 9351 |
normalized size |
1 |
1. |
0.82 |
23.68 |
0. |
31.9 |
0. |
44.53 |
time (sec) |
N/A |
0.27 |
0.605 |
0.152 |
0. |
1.364 |
0. |
1.669 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 160 | 160 | 129 | 2410 | 0 | 3420 | 0 | 4610 |
normalized size |
1 |
1. |
0.81 |
15.06 |
0. |
21.38 |
0. |
28.81 |
time (sec) |
N/A |
0.176 |
0.311 |
0.075 |
0. |
1.212 |
0. |
1.128 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 108 | 108 | 84 | 891 | 0 | 1354 | 0 | 1742 |
normalized size |
1 |
1. |
0.78 |
8.25 |
0. |
12.54 |
0. |
16.13 |
time (sec) |
N/A |
0.084 |
0.147 |
0.056 |
0. |
1.135 |
0. |
1.123 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 49 | 262 | 0 | 463 | 0 | 441 |
normalized size |
1 |
1. |
0.74 |
3.97 |
0. |
7.02 |
0. |
6.68 |
time (sec) |
N/A |
0.04 |
0.059 |
0.09 |
0. |
1.081 |
0. |
1.257 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 120 | 120 | 95 | 0 | 0 | 0 | 666 | 0 |
normalized size | 1 | 1. | 0.79 | 0. | 0. | 0. | 5.55 | 0. |
time (sec) |
N/A |
0.12 |
0.151 |
0.397 |
0. |
0. |
10.615 |
0. |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 177 | 177 | 110 | 0 | 0 | 0 | 4129 | 0 |
normalized size | 1 | 1. | 0.62 | 0. | 0. | 0. | 23.33 | 0. |
time (sec) |
N/A |
0.258 |
0.153 |
0.379 |
0. |
0. |
52.69 |
0. |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 228 | 228 | 136 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.6 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.273 |
0.162 |
0.354 |
0. |
0. |
0. |
0. |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 318 | 318 | 273 | 11389 | 0 | 14151 | 0 | 20733 |
normalized size |
1 |
1. |
0.86 |
35.81 |
0. |
44.5 |
0. |
65.2 |
time (sec) |
N/A |
0.411 |
1.107 |
0.175 |
0. |
1.796 |
0. |
1.432 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 237 | 237 | 199 | 5908 | 0 | 7675 | 0 | 10939 |
normalized size |
1 |
1. |
0.84 |
24.93 |
0. |
32.38 |
0. |
46.16 |
time (sec) |
N/A |
0.31 |
0.544 |
0.116 |
0. |
1.459 |
0. |
1.294 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 160 | 160 | 129 | 2410 | 0 | 3268 | 0 | 4610 |
normalized size |
1 |
1. |
0.81 |
15.06 |
0. |
20.42 |
0. |
28.81 |
time (sec) |
N/A |
0.172 |
0.343 |
0.076 |
0. |
1.248 |
0. |
1.19 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 102 | 102 | 78 | 732 | 0 | 1208 | 0 | 1381 |
normalized size |
1 |
1. |
0.76 |
7.18 |
0. |
11.84 |
0. |
13.54 |
time (sec) |
N/A |
0.076 |
0.11 |
0.052 |
0. |
1.12 |
0. |
1.09 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 185 | 185 | 153 | 0 | 0 | 0 | 1085 | 0 |
normalized size | 1 | 1. | 0.83 | 0. | 0. | 0. | 5.86 | 0. |
time (sec) |
N/A |
0.226 |
0.26 |
0.505 |
0. |
0. |
27.206 |
0. |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 268 | 268 | 159 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.59 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.669 |
0.246 |
0.5 |
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 |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 322 | 322 | 168 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.52 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.544 |
0.24 |
0.498 |
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 |
C |
F(-2) |
B |
F(-1) |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 410 | 410 | 358 | 20937 | 0 | 24423 | 0 | 0 |
normalized size |
1 |
1. |
0.87 |
51.07 |
0. |
59.57 |
0. |
0. |
time (sec) |
N/A |
0.624 |
0.943 |
0.266 |
0. |
2.391 |
0. |
0. |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 310 | 310 | 265 | 11389 | 0 | 14151 | 0 | 20733 |
normalized size |
1 |
1. |
0.85 |
36.74 |
0. |
45.65 |
0. |
66.88 |
time (sec) |
N/A |
0.414 |
1.136 |
0.179 |
0. |
1.78 |
0. |
1.463 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 210 | 210 | 172 | 4972 | 0 | 6376 | 0 | 9351 |
normalized size |
1 |
1. |
0.82 |
23.68 |
0. |
30.36 |
0. |
44.53 |
time (sec) |
N/A |
0.258 |
0.518 |
0.111 |
0. |
1.404 |
0. |
1.267 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F(-2) |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 137 | 137 | 106 | 1609 | 0 | 2437 | 0 | 3075 |
normalized size |
1 |
1. |
0.77 |
11.74 |
0. |
17.79 |
0. |
22.45 |
time (sec) |
N/A |
0.111 |
0.15 |
0.067 |
0. |
1.164 |
0. |
1.158 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 270 | 270 | 229 | 0 | 0 | 0 | 1503 | 0 |
normalized size | 1 | 1. | 0.85 | 0. | 0. | 0. | 5.57 | 0. |
time (sec) |
N/A |
0.389 |
0.521 |
0.487 |
0. |
0. |
56.133 |
0. |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 394 | 389 | 217 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0.99 |
0.55 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.024 |
0.443 |
0.492 |
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 |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 380 | 380 | 332 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.87 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.621 |
0.896 |
0.492 |
0. |
0. |
0. |
0. |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 272 | 272 | 231 | 0 | 0 | 0 | 1503 | 0 |
normalized size | 1 | 1. | 0.85 | 0. | 0. | 0. | 5.53 | 0. |
time (sec) |
N/A |
0.398 |
0.53 |
0.493 |
0. |
0. |
56.49 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 187 | 187 | 154 | 0 | 0 | 0 | 1085 | 0 |
normalized size | 1 | 1. | 0.82 | 0. | 0. | 0. | 5.8 | 0. |
time (sec) |
N/A |
0.254 |
0.289 |
0.491 |
0. |
0. |
26.428 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 122 | 122 | 95 | 0 | 0 | 0 | 666 | 0 |
normalized size | 1 | 1. | 0.78 | 0. | 0. | 0. | 5.46 | 0. |
time (sec) |
N/A |
0.126 |
0.147 |
0.347 |
0. |
0. |
9.554 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 78 | 78 | 57 | 0 | 0 | 0 | 284 | 0 |
normalized size | 1 | 1. | 0.73 | 0. | 0. | 0. | 3.64 | 0. |
time (sec) |
N/A |
0.04 |
0.065 |
0.36 |
0. |
0. |
3.17 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
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 | 127 | 127 | 102 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.8 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.144 |
0.128 |
0.676 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
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 | 212 | 212 | 152 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.72 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.529 |
0.202 |
0.671 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
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 | 407 | 407 | 199 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.49 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.247 |
0.264 |
0.677 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
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 | 386 | 381 | 220 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0.99 |
0.57 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.135 |
0.522 |
0.509 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
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 | 267 | 267 | 161 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.6 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.675 |
0.304 |
0.54 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 178 | 178 | 110 | 0 | 0 | 0 | 4129 | 0 |
normalized size | 1 | 1. | 0.62 | 0. | 0. | 0. | 23.2 | 0. |
time (sec) |
N/A |
0.253 |
0.159 |
0.349 |
0. |
0. |
47.948 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 107 | 107 | 83 | 0 | 0 | 0 | 1897 | 0 |
normalized size | 1 | 1. | 0.78 | 0. | 0. | 0. | 17.73 | 0. |
time (sec) |
N/A |
0.056 |
0.071 |
0.358 |
0. |
0. |
12.835 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
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 | 211 | 211 | 150 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.71 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.521 |
0.208 |
0.667 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
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 | 315 | 315 | 209 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.66 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.097 |
0.317 |
0.716 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
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 | 567 | 567 | 270 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.48 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.341 |
0.455 |
0.731 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
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 | 322 | 322 | 172 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.53 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.557 |
0.28 |
0.542 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
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 | 228 | 228 | 136 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.6 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.281 |
0.184 |
0.347 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
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 | 112 | 112 | 83 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.74 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.056 |
0.073 |
0.366 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
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 | 366 | 366 | 201 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.55 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.217 |
0.267 |
0.698 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
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 | 482 | 482 | 271 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.56 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.071 |
0.463 |
0.678 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
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 | 211 | 211 | 162 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.77 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.246 |
0.44 |
1.056 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
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 | 271 | 255 | 164 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0.94 |
0.61 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.327 |
0.208 |
0.403 |
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 |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 164 | 164 | 138 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.84 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.179 |
0.252 |
0.676 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
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 | 304 | 304 | 138 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.45 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.536 |
0.348 |
0.669 |
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 |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 139 | 139 | 124 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.89 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.115 |
0.155 |
0.795 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
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 | 139 | 167 | 124 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1.2 |
0.89 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.118 |
0.052 |
0.8 |
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 [44] had the largest ratio of [ 0.1935
]
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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 |
12 |
3 |
1. |
29 |
0.103 |
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|
|
|
|
|
|
2 |
A |
10 |
3 |
1. |
29 |
0.103 |
|
|
|
|
|
|
|
3 |
A |
8 |
3 |
1. |
27 |
0.111 |
|
|
|
|
|
|
|
4 |
A |
6 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
5 |
A |
5 |
4 |
1. |
29 |
0.138 |
|
|
|
|
|
|
|
6 |
A |
3 |
3 |
1. |
29 |
0.103 |
|
|
|
|
|
|
|
7 |
A |
3 |
3 |
1. |
29 |
0.103 |
|
|
|
|
|
|
|
8 |
A |
14 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
9 |
A |
12 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
10 |
A |
10 |
3 |
1. |
29 |
0.103 |
|
|
|
|
|
|
|
11 |
A |
8 |
3 |
1. |
22 |
0.136 |
|
|
|
|
|
|
|
12 |
A |
7 |
4 |
1. |
31 |
0.129 |
|
|
|
|
|
|
|
13 |
A |
6 |
5 |
1. |
31 |
0.161 |
|
|
|
|
|
|
|
14 |
A |
4 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
15 |
A |
16 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
16 |
A |
14 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
17 |
A |
12 |
3 |
1. |
29 |
0.103 |
|
|
|
|
|
|
|
18 |
A |
10 |
3 |
1. |
22 |
0.136 |
|
|
|
|
|
|
|
19 |
A |
9 |
4 |
1. |
31 |
0.129 |
|
|
|
|
|
|
|
20 |
A |
8 |
5 |
0.99 |
31 |
0.161 |
|
|
|
|
|
|
|
|
21 |
A |
11 |
4 |
1. |
31 |
0.129 |
|
|
|
|
|
|
|
22 |
A |
9 |
4 |
1. |
31 |
0.129 |
|
|
|
|
|
|
|
23 |
A |
7 |
4 |
1. |
31 |
0.129 |
|
|
|
|
|
|
|
24 |
A |
5 |
4 |
1. |
29 |
0.138 |
|
|
|
|
|
|
|
25 |
A |
2 |
2 |
1. |
22 |
0.091 |
|
|
|
|
|
|
|
26 |
A |
4 |
2 |
1. |
31 |
0.065 |
|
|
|
|
|
|
|
27 |
A |
5 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
28 |
A |
6 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
29 |
A |
8 |
5 |
0.99 |
31 |
0.161 |
|
|
|
|
|
|
|
30 |
A |
6 |
5 |
1. |
31 |
0.161 |
|
|
|
|
|
|
|
31 |
A |
3 |
3 |
1. |
29 |
0.103 |
|
|
|
|
|
|
|
32 |
A |
2 |
2 |
1. |
22 |
0.091 |
|
|
|
|
|
|
|
33 |
A |
5 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
34 |
A |
6 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
35 |
A |
7 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
36 |
A |
4 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
37 |
A |
3 |
3 |
1. |
29 |
0.103 |
|
|
|
|
|
|
|
38 |
A |
2 |
2 |
1. |
22 |
0.091 |
|
|
|
|
|
|
|
39 |
A |
6 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
40 |
A |
7 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
|
41 |
A |
7 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
42 |
A |
4 |
4 |
0.94 |
29 |
0.138 |
|
|
|
|
|
|
|
43 |
A |
6 |
5 |
1. |
31 |
0.161 |
|
|
|
|
|
|
|
44 |
A |
7 |
6 |
1. |
31 |
0.194 |
|
|
|
|
|
|
|
45 |
A |
4 |
4 |
1. |
47 |
0.085 |
|
|
|
|
|
|
|
46 |
A |
4 |
4 |
1.2 |
55 |
0.073 |
|
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