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
}
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 16, 17, 18, 19, 22, 39, 48, 49, 50, 51 }
B grade: { 8, 9, 13, 32, 33, 43, 44, 45, 47 }
C grade: { 14, 15, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 40, 41, 42, 46
}
F grade: { }
2.1.3 Maple
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 29, 30, 40, 41
}
B grade: { 15, 22, 25, 26, 27, 28, 32, 33, 34, 35, 36, 37, 38, 39, 43, 44, 45, 47, 49, 50 }
C grade: { 31, 42, 46, 48 }
F grade: { 51 }
2.1.4 Maxima
A grade: { 1, 2, 3, 4, 5, 6, 7, 10, 11, 20, 21 }
B grade: { 8, 9, 16, 17 }
C grade: { 22 }
F grade: { 12, 13, 14, 15, 18, 19, 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 }
2.1.5 FriCAS
A grade: { 1, 2, 3, 4, 5, 6, 12, 15, 16, 17, 18, 19, 22, 27 }
B grade: { 7, 8, 9, 10, 11, 13, 14, 23, 24, 25, 26, 28, 35, 36, 37, 38, 39 }
C grade: { }
F grade: { 20, 21, 29, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51 }
2.1.6 Sympy
A grade: { 1, 2, 3, 4 }
B grade: { }
C grade: { }
F grade: { 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 }
2.1.7 Giac
A grade: { 1, 2, 3, 4, 12, 39 }
B grade: { }
C grade: { }
F grade: { 5, 6, 7, 8, 9, 10, 11, 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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51 }
2.1.8 Mupad
A grade: { }
B grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 39 }
C grade: { }
F grade: { 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51 }
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 \(\frac {\text {antiderivative leaf size}}{\text {optimal antiderivative leaf size}}\)
|
|
|
|
|
|
|
|
|
|
Problem 1 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 121 | 121 | 77 | 147 | 147 | 91 | 415 | 119 | 256 |
normalized size | 1 | 1.00 | 0.64 | 1.21 | 1.21 | 0.75 | 3.43 | 0.98 | 2.12 |
time (sec) |
N/A |
0.172 |
0.121 |
0.559 |
0.350 |
0.448 |
4.803 |
0.175 |
14.238 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 96 | 96 | 57 | 126 | 123 | 77 | 301 | 81 | 212 |
normalized size | 1 | 1.00 | 0.59 | 1.31 | 1.28 | 0.80 | 3.14 | 0.84 | 2.21 |
time (sec) |
N/A |
0.143 |
0.090 |
0.459 |
0.336 |
0.437 |
2.796 |
0.167 |
14.183 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 47 | 106 | 100 | 63 | 245 | 62 | 250 |
normalized size | 1 | 1.00 | 0.61 | 1.38 | 1.30 | 0.82 | 3.18 | 0.81 | 3.25 |
time (sec) |
N/A |
0.102 |
0.114 |
0.378 |
0.329 |
0.451 |
1.359 |
0.166 |
14.123 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 52 | 52 | 43 | 78 | 77 | 46 | 133 | 62 | 125 |
normalized size | 1 | 1.00 | 0.83 | 1.50 | 1.48 | 0.88 | 2.56 | 1.19 | 2.40 |
time (sec) |
N/A |
0.060 |
0.365 |
0.275 |
0.339 |
0.450 |
0.700 |
0.137 |
14.301 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 61 | 78 | 73 | 75 | 0 | 0 | 88 |
normalized size | 1 | 1.00 | 0.97 | 1.24 | 1.16 | 1.19 | 0.00 | 0.00 | 1.40 |
time (sec) |
N/A |
0.090 |
0.086 |
0.188 |
0.353 |
0.464 |
0.000 |
0.000 |
12.892 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 97 | 72 | 69 | 99 | 0 | 0 | 110 |
normalized size | 1 | 1.00 | 1.83 | 1.36 | 1.30 | 1.87 | 0.00 | 0.00 | 2.08 |
time (sec) |
N/A |
0.120 |
0.051 |
0.280 |
0.338 |
0.446 |
0.000 |
0.000 |
12.394 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 64 | 64 | 95 | 80 | 105 | 138 | 0 | 0 | 163 |
normalized size | 1 | 1.00 | 1.48 | 1.25 | 1.64 | 2.16 | 0.00 | 0.00 | 2.55 |
time (sec) |
N/A |
0.111 |
0.715 |
0.448 |
0.368 |
0.439 |
0.000 |
0.000 |
12.262 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
B |
B |
F(-1) |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 172 | 86 | 120 | 137 | 0 | 0 | 132 |
normalized size | 1 | 1.00 | 2.82 | 1.41 | 1.97 | 2.25 | 0.00 | 0.00 | 2.16 |
time (sec) |
N/A |
0.162 |
0.072 |
0.504 |
0.360 |
0.451 |
0.000 |
0.000 |
12.223 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
B |
B |
F(-1) |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 86 | 86 | 179 | 109 | 165 | 166 | 0 | 0 | 133 |
normalized size | 1 | 1.00 | 2.08 | 1.27 | 1.92 | 1.93 | 0.00 | 0.00 | 1.55 |
time (sec) |
N/A |
0.150 |
0.062 |
0.594 |
0.362 |
0.446 |
0.000 |
0.000 |
12.184 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F(-1) |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 105 | 105 | 204 | 132 | 187 | 201 | 0 | 0 | 244 |
normalized size | 1 | 1.00 | 1.94 | 1.26 | 1.78 | 1.91 | 0.00 | 0.00 | 2.32 |
time (sec) |
N/A |
0.162 |
0.053 |
0.604 |
0.360 |
0.470 |
0.000 |
0.000 |
12.369 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F(-1) |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 130 | 130 | 204 | 155 | 232 | 240 | 0 | 0 | 340 |
normalized size | 1 | 1.00 | 1.57 | 1.19 | 1.78 | 1.85 | 0.00 | 0.00 | 2.62 |
time (sec) |
N/A |
0.192 |
0.055 |
0.631 |
0.357 |
0.460 |
0.000 |
0.000 |
12.542 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 128 | 165 | 101 | 78 | 0 | 155 | 0 | 105 | -1 |
normalized size | 1 | 1.29 | 0.79 | 0.61 | 0.00 | 1.21 | 0.00 | 0.82 | -0.01 |
time (sec) |
N/A |
0.347 |
0.811 |
1.073 |
0.000 |
0.426 |
0.000 |
0.284 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 157 | 79 | 0 | 202 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.28 | 1.14 | 0.00 | 2.93 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.306 |
0.352 |
1.374 |
0.000 |
0.462 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 120 | 120 | 234 | 124 | 0 | 328 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.95 | 1.03 | 0.00 | 2.73 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.443 |
0.460 |
1.434 |
0.000 |
0.500 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
A |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 103 | 103 | 194 | 914 | 0 | 442 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.88 | 8.87 | 0.00 | 4.29 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.468 |
0.929 |
0.937 |
0.000 |
0.744 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 40 | 45 | 309 | 41 | 0 | 0 | 52 |
normalized size | 1 | 1.00 | 0.93 | 1.05 | 7.19 | 0.95 | 0.00 | 0.00 | 1.21 |
time (sec) |
N/A |
0.203 |
0.215 |
0.650 |
0.486 |
0.447 |
0.000 |
0.000 |
12.939 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 114 | 114 | 133 | 190 | 270 | 385 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.17 | 1.67 | 2.37 | 3.38 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.490 |
0.318 |
0.783 |
0.533 |
0.655 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 118 | 118 | 132 | 117 | 0 | 391 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.12 | 0.99 | 0.00 | 3.31 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.496 |
0.273 |
0.645 |
0.000 |
0.659 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 62 | 74 | 0 | 202 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.35 | 1.61 | 0.00 | 4.39 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.171 |
0.108 |
0.655 |
0.000 |
0.643 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 102 | 102 | 144 | 111 | 59 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.41 | 1.09 | 0.58 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.460 |
1.334 |
0.467 |
0.429 |
0.642 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 100 | 100 | 145 | 112 | 59 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.45 | 1.12 | 0.59 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.447 |
1.337 |
0.483 |
0.422 |
0.657 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
C |
A |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 63 | 111 | 190 | 193 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.37 | 2.41 | 4.13 | 4.20 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.194 |
0.182 |
0.442 |
0.612 |
0.589 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 105 | 105 | 746 | 120 | 0 | 781 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 7.10 | 1.14 | 0.00 | 7.44 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.289 |
5.647 |
1.465 |
0.000 |
0.952 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 165 | 165 | 331 | 208 | 0 | 1044 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.01 | 1.26 | 0.00 | 6.33 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.465 |
2.237 |
1.931 |
0.000 |
2.613 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 149 | 149 | 661 | 1025 | 0 | 3273 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 4.44 | 6.88 | 0.00 | 21.97 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.508 |
56.097 |
0.948 |
0.000 |
2.162 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 83 | 83 | 436 | 526 | 0 | 1303 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 5.25 | 6.34 | 0.00 | 15.70 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.228 |
54.718 |
0.557 |
0.000 |
1.346 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
A |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 166 | 166 | 61028 | 614 | 0 | 3048 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 367.64 | 3.70 | 0.00 | 18.36 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.515 |
39.396 |
0.751 |
0.000 |
2.018 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 168 | 168 | 99621 | 621 | 0 | 3175 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 592.98 | 3.70 | 0.00 | 18.90 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.536 |
40.367 |
0.557 |
0.000 |
2.466 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 238 | 238 | 611 | 593 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.57 | 2.49 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.503 |
6.636 |
4.934 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 246 | 246 | 625 | 587 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.54 | 2.39 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.484 |
6.664 |
4.918 |
0.000 |
0.578 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 267 | 267 | 13199 | 22962 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 49.43 | 86.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.500 |
34.483 |
1.392 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 116 | 116 | 4679 | 6804 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 40.34 | 58.66 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.204 |
40.247 |
0.707 |
0.000 |
0.486 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 252 | 252 | 5708 | 6817 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 22.65 | 27.05 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.511 |
33.617 |
0.912 |
0.000 |
0.471 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 256 | 256 | 1659 | 9043 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 6.48 | 35.32 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.520 |
10.107 |
0.737 |
0.000 |
0.460 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 123 | 123 | 567 | 4546 | 0 | 3539 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 4.61 | 36.96 | 0.00 | 28.77 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.459 |
3.169 |
0.829 |
0.000 |
1.416 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 367 | 231 | 0 | 1044 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 6.02 | 3.79 | 0.00 | 17.11 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.188 |
1.969 |
0.541 |
0.000 |
0.749 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 140 | 140 | 472502 | 340 | 0 | 2791 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 3375.01 | 2.43 | 0.00 | 19.94 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.502 |
35.100 |
0.519 |
0.000 |
0.973 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 140 | 140 | 309729 | 357 | 0 | 3005 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2212.35 | 2.55 | 0.00 | 21.46 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.471 |
34.492 |
0.405 |
0.000 |
1.067 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
B |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 181 | 181 | 178 | 606 | 0 | 2837 | 0 | 299 | 23933 |
normalized size | 1 | 1.00 | 0.98 | 3.35 | 0.00 | 15.67 | 0.00 | 1.65 | 132.23 |
time (sec) |
N/A |
0.509 |
1.120 |
0.553 |
0.000 |
113.592 |
0.000 |
0.227 |
27.407 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 154 | 154 | 179 | 190 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.16 | 1.23 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.487 |
3.766 |
2.066 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 146 | 146 | 203 | 254 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.39 | 1.74 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.488 |
4.063 |
1.901 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 254 | 254 | 23019 | 6196 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 90.63 | 24.39 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.520 |
29.796 |
0.842 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 250 | 250 | 8202 | 3291 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 32.81 | 13.16 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.526 |
29.281 |
0.591 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 114 | 114 | 3427 | 2924 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 30.06 | 25.65 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.207 |
29.176 |
0.827 |
0.000 |
1.860 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 246 | 246 | 4935 | 3690 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 20.06 | 15.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.524 |
30.056 |
0.595 |
0.000 |
10.586 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 254 | 254 | 23019 | 6052 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 90.63 | 23.83 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.505 |
30.551 |
0.993 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 114 | 114 | 3429 | 2856 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 30.08 | 25.05 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.206 |
28.767 |
0.853 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 391 | 391 | 274 | 269176 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.70 | 688.43 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.535 |
0.268 |
4.993 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 198 | 198 | 197 | 32723 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.99 | 165.27 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.196 |
0.146 |
1.059 |
0.000 |
1.016 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 398 | 398 | 374 | 26871 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.94 | 67.52 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.555 |
2.335 |
0.866 |
0.000 |
1.468 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 168 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.07 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.278 |
1.119 |
8.538 |
0.000 |
0.616 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
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 [29] had the largest ratio of
[.2727]
|
|
|
|
|
|
|
| Table 2.1:Rubi specific breakdown of results for each
integral
|
|
|
|
|
|
|
# |
grade |
|
|
normalized | antiderivative |
leaf size | |
|
\(\frac {\text {number of rules}}{\text {integrand leaf size}}\) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
A |
13 |
4 |
1.00 |
32 |
0.125 |
|
|
|
|
|
|
|
2 |
A |
11 |
4 |
1.00 |
32 |
0.125 |
|
|
|
|
|
|
|
3 |
A |
10 |
5 |
1.00 |
30 |
0.167 |
|
|
|
|
|
|
|
4 |
A |
4 |
4 |
1.00 |
24 |
0.167 |
|
|
|
|
|
|
|
5 |
A |
6 |
5 |
1.00 |
30 |
0.167 |
|
|
|
|
|
|
|
6 |
A |
8 |
7 |
1.00 |
32 |
0.219 |
|
|
|
|
|
|
|
7 |
A |
7 |
5 |
1.00 |
32 |
0.156 |
|
|
|
|
|
|
|
8 |
A |
6 |
6 |
1.00 |
32 |
0.188 |
|
|
|
|
|
|
|
9 |
A |
11 |
5 |
1.00 |
32 |
0.156 |
|
|
|
|
|
|
|
10 |
A |
11 |
4 |
1.00 |
32 |
0.125 |
|
|
|
|
|
|
|
11 |
A |
13 |
4 |
1.00 |
32 |
0.125 |
|
|
|
|
|
|
|
12 |
A |
5 |
5 |
1.29 |
34 |
0.147 |
|
|
|
|
|
|
|
13 |
A |
5 |
5 |
1.00 |
34 |
0.147 |
|
|
|
|
|
|
|
14 |
A |
8 |
7 |
1.00 |
34 |
0.206 |
|
|
|
|
|
|
|
15 |
A |
6 |
6 |
1.00 |
40 |
0.150 |
|
|
|
|
|
|
|
16 |
A |
3 |
3 |
1.00 |
40 |
0.075 |
|
|
|
|
|
|
|
17 |
A |
6 |
6 |
1.00 |
40 |
0.150 |
|
|
|
|
|
|
|
18 |
A |
6 |
6 |
1.00 |
40 |
0.150 |
|
|
|
|
|
|
|
19 |
A |
2 |
2 |
1.00 |
36 |
0.056 |
|
|
|
|
|
|
|
20 |
A |
6 |
6 |
1.00 |
36 |
0.167 |
|
|
|
|
|
|
|
21 |
A |
6 |
6 |
1.00 |
36 |
0.167 |
|
|
|
|
|
|
|
22 |
A |
3 |
3 |
1.00 |
36 |
0.083 |
|
|
|
|
|
|
|
23 |
A |
5 |
4 |
1.00 |
33 |
0.121 |
|
|
|
|
|
|
|
24 |
A |
8 |
6 |
1.00 |
33 |
0.182 |
|
|
|
|
|
|
|
25 |
A |
5 |
4 |
1.00 |
39 |
0.103 |
|
|
|
|
|
|
|
26 |
A |
2 |
2 |
1.00 |
39 |
0.051 |
|
|
|
|
|
|
|
27 |
A |
5 |
4 |
1.00 |
39 |
0.103 |
|
|
|
|
|
|
|
28 |
A |
5 |
4 |
1.00 |
39 |
0.103 |
|
|
|
|
|
|
|
29 |
A |
9 |
9 |
1.00 |
33 |
0.273 |
|
|
|
|
|
|
|
30 |
A |
9 |
9 |
1.00 |
33 |
0.273 |
|
|
|
|
|
|
|
31 |
A |
3 |
3 |
1.00 |
39 |
0.077 |
|
|
|
|
|
|
|
32 |
A |
1 |
1 |
1.00 |
39 |
0.026 |
|
|
|
|
|
|
|
33 |
A |
3 |
3 |
1.00 |
39 |
0.077 |
|
|
|
|
|
|
|
34 |
A |
3 |
3 |
1.00 |
39 |
0.077 |
|
|
|
|
|
|
|
35 |
A |
5 |
5 |
1.00 |
35 |
0.143 |
|
|
|
|
|
|
|
36 |
A |
2 |
2 |
1.00 |
35 |
0.057 |
|
|
|
|
|
|
|
37 |
A |
5 |
5 |
1.00 |
35 |
0.143 |
|
|
|
|
|
|
|
38 |
A |
5 |
5 |
1.00 |
35 |
0.143 |
|
|
|
|
|
|
|
39 |
A |
8 |
5 |
1.00 |
33 |
0.152 |
|
|
|
|
|
|
|
40 |
A |
5 |
3 |
1.00 |
33 |
0.091 |
|
|
|
|
|
|
|
41 |
A |
5 |
3 |
1.00 |
33 |
0.091 |
|
|
|
|
|
|
|
42 |
A |
3 |
3 |
1.00 |
39 |
0.077 |
|
|
|
|
|
|
|
43 |
A |
3 |
3 |
1.00 |
39 |
0.077 |
|
|
|
|
|
|
|
44 |
A |
1 |
1 |
1.00 |
39 |
0.026 |
|
|
|
|
|
|
|
45 |
A |
3 |
3 |
1.00 |
39 |
0.077 |
|
|
|
|
|
|
|
46 |
A |
3 |
3 |
1.00 |
39 |
0.077 |
|
|
|
|
|
|
|
47 |
A |
1 |
1 |
1.00 |
39 |
0.026 |
|
|
|
|
|
|
|
48 |
A |
3 |
3 |
1.00 |
35 |
0.086 |
|
|
|
|
|
|
|
49 |
A |
1 |
1 |
1.00 |
35 |
0.029 |
|
|
|
|
|
|
|
50 |
A |
3 |
3 |
1.00 |
35 |
0.086 |
|
|
|
|
|
|
|
51 |
A |
4 |
4 |
1.00 |
38 |
0.105 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|