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, 6, 7, 8, 9, 10, 11, 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, 43, 44, 46, 47, 48, 49, 50 }
B grade: { 4, 42 }
C grade: { 5, 12, 13 }
F grade: { 3, 45 }
2.1.2 Mathematica
A grade: { 1, 2, 4, 6, 7, 10, 11, 14, 15, 16, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 40, 42, 43, 44, 45, 46, 48, 49, 50 }
B grade: { 3, 17, 21, 41 }
C grade: { 5, 8, 9, 12, 13, 37, 38, 39, 47 }
F grade: { }
2.1.3 Maple
A grade: { 2, 6, 7, 9, 19, 24, 27, 32, 34, 36, 37, 43, 48 }
B grade: { 1, 3, 4, 8, 40, 42, 44, 47, 49, 50 }
C grade: { 5, 12, 18, 28, 31, 33, 35, 38, 39, 46 }
F grade: { 10, 11, 13, 14, 15, 16, 17, 20, 21, 22, 23, 25, 26, 29, 30, 41, 45 }
2.1.4 Maxima
A grade: { 1, 2, 3, 9, 16, 18, 19, 20, 24, 25, 27, 28, 31, 32, 33, 34, 35, 36, 37, 46, 48 }
B grade: { 7, 40, 43 }
C grade: { }
F grade: { 4, 5, 6, 8, 10, 11, 12, 13, 14, 15, 17, 21, 22, 23, 26, 29, 30, 38, 39, 41, 42, 44, 45, 47, 49, 50
}
2.1.5 FriCAS
A grade: { 1, 2, 3, 5, 9, 10, 11, 12, 13, 14, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
39, 46, 48 }
B grade: { 4, 6, 7, 15, 16, 17, 21, 22, 37, 38, 40, 41, 42, 43, 44, 47, 49, 50 }
C grade: { }
F grade: { 8, 45 }
2.1.6 Sympy
A grade: { 1, 2, 10, 19, 20, 24, 25, 31, 33, 34, 35, 36, 46, 48 }
B grade: { 28, 37, 40 }
C grade: { 16, 18 }
F grade: { 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 17, 21, 22, 23, 26, 27, 29, 30, 32, 38, 39, 41, 42, 43, 44, 45,
47, 49, 50 }
2.1.7 Giac
A grade: { 2, 6, 10, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 33, 34, 35, 36, 40, 43, 47, 48, 49, 50
}
B grade: { 1, 4, 7, 8, 12, 13, 27, 29, 30, 31, 32, 37, 41, 42, 44 }
C grade: { }
F grade: { 3, 5, 9, 11, 23, 38, 39, 45, 46 }
2.1.8 Mupad
A grade: { }
B grade: { 4, 9, 12, 19, 24, 30, 37, 40, 46, 47, 48, 50 }
C grade: { }
F grade: { 1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34,
35, 36, 38, 39, 41, 42, 43, 44, 45, 49 }
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 |
Mupad |
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grade |
A |
A |
A |
B |
A |
A |
A |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 52 | 93 | 58 | 54 | 102 | 272 | -1 |
normalized size | 1 | 1.00 | 1.02 | 1.82 | 1.14 | 1.06 | 2.00 | 5.33 | -0.02 |
time (sec) |
N/A |
0.029 |
0.024 |
0.062 |
0.961 |
0.463 |
7.755 |
1.152 |
0.000 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 17 | 17 | 17 | 16 | 15 | 15 | 12 | 15 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.94 | 0.88 | 0.88 | 0.71 | 0.88 | -0.06 |
time (sec) |
N/A |
0.028 |
0.006 |
0.077 |
0.958 |
0.417 |
0.210 |
0.852 |
0.000 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
F |
B |
B |
A |
A |
F |
F |
F |
verified |
N/A |
N/A |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 0 | 205 | 251 | 4 | 49 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 2.97 | 3.64 | 0.06 | 0.71 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.153 |
0.666 |
0.967 |
3.854 |
43.869 |
0.000 |
0.000 |
0.000 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
B |
A |
B |
F |
B |
F(-1) |
B |
B |
verified |
N/A |
NO |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 97 | 332 | 194 | 426 | 0 | 402 | 0 | 235 | 666 |
normalized size | 1 | 3.42 | 2.00 | 4.39 | 0.00 | 4.14 | 0.00 | 2.42 | 6.87 |
time (sec) |
N/A |
0.669 |
0.404 |
0.142 |
0.000 |
0.458 |
0.000 |
1.245 |
1.573 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
C |
C |
C |
F |
A |
F(-1) |
F |
F |
verified |
N/A |
NO |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 45 | 289 | 159 | 312 | 0 | 33 | 0 | 0 | -1 |
normalized size | 1 | 6.42 | 3.53 | 6.93 | 0.00 | 0.73 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.491 |
2.205 |
0.604 |
0.000 |
0.536 |
0.000 |
0.000 |
0.000 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
F |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 74 | 64 | 0 | 88 | 0 | 79 | -1 |
normalized size | 1 | 1.00 | 1.32 | 1.14 | 0.00 | 1.57 | 0.00 | 1.41 | -0.02 |
time (sec) |
N/A |
0.069 |
0.119 |
0.079 |
0.000 |
0.551 |
0.000 |
1.202 |
0.000 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
B |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 15 | 15 | 15 | 12 | 27 | 30 | 0 | 28 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.80 | 1.80 | 2.00 | 0.00 | 1.87 | -0.07 |
time (sec) |
N/A |
0.031 |
0.015 |
0.079 |
0.422 |
0.505 |
0.000 |
1.229 |
0.000 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
B |
F(-2) |
F(-1) |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 137 | 137 | 99 | 1605 | 0 | 0 | 0 | 495 | -1 |
normalized size | 1 | 1.00 | 0.72 | 11.72 | 0.00 | 0.00 | 0.00 | 3.61 | -0.01 |
time (sec) |
N/A |
0.180 |
10.296 |
0.234 |
0.000 |
0.000 |
0.000 |
1.436 |
0.000 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
A |
A |
A |
F |
F |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 41 | 283 | 42 | 60 | 32 | 0 | 0 | 60 |
normalized size | 1 | 1.00 | 6.90 | 1.02 | 1.46 | 0.78 | 0.00 | 0.00 | 1.46 |
time (sec) |
N/A |
0.022 |
4.377 |
0.070 |
1.012 |
0.482 |
0.000 |
0.000 |
0.417 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
F |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 44 | 62 | 38 | 0 | 0 | 28 | 56 | 46 | -1 |
normalized size | 1 | 1.41 | 0.86 | 0.00 | 0.00 | 0.64 | 1.27 | 1.05 | -0.02 |
time (sec) |
N/A |
0.676 |
0.172 |
0.100 |
0.000 |
0.470 |
2.168 |
1.472 |
0.000 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
F |
A |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 68 | 68 | 64 | 0 | 0 | 43 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.94 | 0.00 | 0.00 | 0.63 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.146 |
0.054 |
0.092 |
0.000 |
0.436 |
0.000 |
0.000 |
0.000 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
C |
C |
C |
F |
A |
F(-1) |
B |
B |
verified |
N/A |
NO |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 141 | 269 | 1822 | 439 | 0 | 191 | 0 | 364 | 661 |
normalized size | 1 | 1.91 | 12.92 | 3.11 | 0.00 | 1.35 | 0.00 | 2.58 | 4.69 |
time (sec) |
N/A |
0.746 |
3.796 |
0.096 |
0.000 |
0.489 |
0.000 |
1.467 |
1.373 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
C |
C |
F |
F |
A |
F(-1) |
B |
F |
verified |
N/A |
NO |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 152 | 286 | 2180 | 0 | 0 | 200 | 0 | 373 | -1 |
normalized size | 1 | 1.88 | 14.34 | 0.00 | 0.00 | 1.32 | 0.00 | 2.45 | -0.01 |
time (sec) |
N/A |
0.457 |
4.953 |
0.103 |
0.000 |
0.502 |
0.000 |
1.568 |
0.000 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F(-2) |
F |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 45 | 51 | 44 | 0 | 0 | 63 | 0 | 57 | -1 |
normalized size | 1 | 1.13 | 0.98 | 0.00 | 0.00 | 1.40 | 0.00 | 1.27 | -0.02 |
time (sec) |
N/A |
0.121 |
0.045 |
180.000 |
0.000 |
0.484 |
0.000 |
1.295 |
0.000 |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
F |
B |
F(-1) |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 34 | 34 | 64 | 0 | 0 | 62 | 0 | 36 | -1 |
normalized size | 1 | 1.00 | 1.88 | 0.00 | 0.00 | 1.82 | 0.00 | 1.06 | -0.03 |
time (sec) |
N/A |
0.035 |
0.101 |
0.101 |
0.000 |
0.424 |
0.000 |
1.383 |
0.000 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
A |
B |
C |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 22 | 22 | 0 | 18 | 56 | 78 | 18 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.00 | 0.82 | 2.55 | 3.55 | 0.82 | -0.05 |
time (sec) |
N/A |
0.020 |
0.028 |
0.191 |
0.962 |
0.450 |
14.501 |
1.057 |
0.000 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
B |
F |
F |
B |
F(-1) |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 32 | 32 | 89 | 0 | 0 | 58 | 0 | 36 | -1 |
normalized size | 1 | 1.00 | 2.78 | 0.00 | 0.00 | 1.81 | 0.00 | 1.12 | -0.03 |
time (sec) |
N/A |
0.041 |
0.088 |
0.096 |
0.000 |
0.426 |
0.000 |
1.119 |
0.000 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
C |
A |
A |
C |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 43 | 89 | 41 | 32 | 37 | 62 | -1 |
normalized size | 1 | 1.00 | 1.00 | 2.07 | 0.95 | 0.74 | 0.86 | 1.44 | -0.02 |
time (sec) |
N/A |
0.045 |
0.025 |
0.138 |
0.956 |
0.442 |
159.336 |
1.161 |
0.000 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 28 | 28 | 21 | 34 | 34 | 48 | 35 | 174 |
normalized size | 1 | 1.00 | 1.00 | 0.75 | 1.21 | 1.21 | 1.71 | 1.25 | 6.21 |
time (sec) |
N/A |
0.009 |
0.005 |
0.024 |
0.413 |
0.417 |
1.172 |
1.011 |
0.135 |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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|
grade |
A |
A |
A |
F |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 26 | 0 | 22 | 22 | 20 | 22 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.00 | 0.85 | 0.85 | 0.77 | 0.85 | -0.04 |
time (sec) |
N/A |
0.036 |
0.022 |
0.115 |
0.552 |
0.425 |
9.120 |
1.096 |
0.000 |
|
|
|
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|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 85 | 0 | 0 | 138 | 0 | 38 | -1 |
normalized size | 1 | 1.00 | 2.24 | 0.00 | 0.00 | 3.63 | 0.00 | 1.00 | -0.03 |
time (sec) |
N/A |
0.052 |
0.089 |
0.409 |
0.000 |
0.446 |
0.000 |
1.101 |
0.000 |
|
|
|
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|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 70 | 94 | 88 | 0 | 0 | 110 | 0 | 52 | -1 |
normalized size | 1 | 1.34 | 1.26 | 0.00 | 0.00 | 1.57 | 0.00 | 0.74 | -0.01 |
time (sec) |
N/A |
0.132 |
0.101 |
0.740 |
0.000 |
0.484 |
0.000 |
1.217 |
0.000 |
|
|
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|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 58 | 0 | 0 | 48 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.00 | 0.00 | 0.83 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.142 |
0.041 |
0.170 |
0.000 |
0.469 |
0.000 |
0.000 |
0.000 |
|
|
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|
Problem 24 |
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 | 55 | 55 | 41 | 37 | 36 | 35 | 31 | 36 | 27 |
normalized size | 1 | 1.00 | 0.75 | 0.67 | 0.65 | 0.64 | 0.56 | 0.65 | 0.49 |
time (sec) |
N/A |
0.053 |
0.023 |
0.006 |
0.415 |
0.448 |
5.992 |
1.032 |
0.340 |
|
|
|
|
|
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|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 26 | 0 | 22 | 22 | 20 | 22 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.00 | 0.85 | 0.85 | 0.77 | 0.85 | -0.04 |
time (sec) |
N/A |
0.033 |
0.020 |
0.084 |
1.059 |
0.421 |
4.328 |
1.001 |
0.000 |
|
|
|
|
|
|
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|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 78 | 78 | 119 | 0 | 0 | 115 | 0 | 122 | -1 |
normalized size | 1 | 1.00 | 1.53 | 0.00 | 0.00 | 1.47 | 0.00 | 1.56 | -0.01 |
time (sec) |
N/A |
0.271 |
0.064 |
0.096 |
0.000 |
0.445 |
0.000 |
1.303 |
0.000 |
|
|
|
|
|
|
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|
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|
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|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 39 | 39 | 25 | 35 | 35 | 39 | 0 | 73 | -1 |
normalized size | 1 | 1.00 | 0.64 | 0.90 | 0.90 | 1.00 | 0.00 | 1.87 | -0.03 |
time (sec) |
N/A |
0.048 |
0.033 |
0.109 |
0.964 |
0.445 |
0.000 |
1.082 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
B |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 17 | 17 | 17 | 54 | 15 | 23 | 29 | 23 | -1 |
normalized size | 1 | 1.00 | 1.00 | 3.18 | 0.88 | 1.35 | 1.71 | 1.35 | -0.06 |
time (sec) |
N/A |
0.028 |
0.016 |
0.172 |
0.952 |
0.463 |
2.330 |
0.927 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 57 | 57 | 77 | 0 | 0 | 81 | 0 | 104 | -1 |
normalized size | 1 | 1.00 | 1.35 | 0.00 | 0.00 | 1.42 | 0.00 | 1.82 | -0.02 |
time (sec) |
N/A |
0.080 |
0.069 |
0.335 |
0.000 |
0.473 |
0.000 |
1.094 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 45 | 45 | 45 | 0 | 0 | 69 | 0 | 108 | 37 |
normalized size | 1 | 1.00 | 1.00 | 0.00 | 0.00 | 1.53 | 0.00 | 2.40 | 0.82 |
time (sec) |
N/A |
0.042 |
0.036 |
0.227 |
0.000 |
0.453 |
0.000 |
1.021 |
0.030 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
A |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 29 | 29 | 33 | 56 | 22 | 47 | 19 | 54 | -1 |
normalized size | 1 | 1.00 | 1.14 | 1.93 | 0.76 | 1.62 | 0.66 | 1.86 | -0.03 |
time (sec) |
N/A |
0.048 |
0.026 |
0.173 |
0.953 |
0.442 |
7.264 |
1.104 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 21 | 21 | 20 | 19 | 22 | 0 | 40 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.95 | 0.90 | 1.05 | 0.00 | 1.90 | -0.05 |
time (sec) |
N/A |
0.044 |
0.015 |
0.074 |
0.945 |
0.440 |
0.000 |
1.035 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 34 | 34 | 27 | 119 | 27 | 27 | 29 | 28 | -1 |
normalized size | 1 | 1.00 | 0.79 | 3.50 | 0.79 | 0.79 | 0.85 | 0.82 | -0.03 |
time (sec) |
N/A |
0.034 |
0.020 |
0.138 |
0.973 |
0.437 |
2.737 |
1.011 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 33 | 33 | 21 | 29 | 29 | 33 | 26 | 58 | -1 |
normalized size | 1 | 1.00 | 0.64 | 0.88 | 0.88 | 1.00 | 0.79 | 1.76 | -0.03 |
time (sec) |
N/A |
0.042 |
0.025 |
0.101 |
0.970 |
0.439 |
5.247 |
0.975 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 25 | 35 | 97 | 15 | 15 | 22 | 22 | -1 |
normalized size | 1 | 1.00 | 1.40 | 3.88 | 0.60 | 0.60 | 0.88 | 0.88 | -0.04 |
time (sec) |
N/A |
0.029 |
0.043 |
0.512 |
0.465 |
0.441 |
19.712 |
1.094 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 34 | 34 | 40 | 39 | 25 | 41 | 41 | 44 | -1 |
normalized size | 1 | 1.00 | 1.18 | 1.15 | 0.74 | 1.21 | 1.21 | 1.29 | -0.03 |
time (sec) |
N/A |
0.035 |
0.018 |
0.101 |
0.962 |
0.422 |
4.342 |
1.115 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
B |
B |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 16 | 16 | 46 | 14 | 24 | 29 | 46 | 27 | 13 |
normalized size | 1 | 1.00 | 2.88 | 0.88 | 1.50 | 1.81 | 2.88 | 1.69 | 0.81 |
time (sec) |
N/A |
0.018 |
0.053 |
0.028 |
0.955 |
0.437 |
22.867 |
1.002 |
0.206 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 23 | 36 | 112 | 0 | 42 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.57 | 4.87 | 0.00 | 1.83 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.032 |
0.101 |
0.086 |
0.000 |
0.459 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 23 | 40 | 112 | 0 | 18 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.74 | 4.87 | 0.00 | 0.78 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.032 |
0.096 |
0.025 |
0.000 |
0.455 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 22 | 39 | 54 | 82 | 93 | 97 | 36 | 55 |
normalized size | 1 | 1.00 | 1.77 | 2.45 | 3.73 | 4.23 | 4.41 | 1.64 | 2.50 |
time (sec) |
N/A |
0.071 |
0.051 |
0.168 |
0.971 |
0.453 |
1.367 |
1.131 |
0.381 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 42 | 42 | 87 | 0 | 0 | 146 | 0 | 94 | -1 |
normalized size | 1 | 1.00 | 2.07 | 0.00 | 0.00 | 3.48 | 0.00 | 2.24 | -0.02 |
time (sec) |
N/A |
0.153 |
0.094 |
0.207 |
0.000 |
0.489 |
0.000 |
1.370 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
B |
A |
B |
F |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 59 | 45 | 91 | 0 | 54 | 0 | 92 | -1 |
normalized size | 1 | 2.11 | 1.61 | 3.25 | 0.00 | 1.93 | 0.00 | 3.29 | -0.04 |
time (sec) |
N/A |
0.184 |
0.037 |
0.250 |
0.000 |
0.491 |
0.000 |
2.218 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 34 | 34 | 55 | 37 | 565 | 186 | 0 | 50 | -1 |
normalized size | 1 | 1.00 | 1.62 | 1.09 | 16.62 | 5.47 | 0.00 | 1.47 | -0.03 |
time (sec) |
N/A |
0.045 |
0.073 |
0.060 |
1.348 |
0.441 |
0.000 |
1.065 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F(-1) |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 39 | 50 | 65 | 67 | 0 | 63 | 0 | 67 | -1 |
normalized size | 1 | 1.28 | 1.67 | 1.72 | 0.00 | 1.62 | 0.00 | 1.72 | -0.03 |
time (sec) |
N/A |
0.046 |
0.088 |
0.409 |
0.000 |
0.518 |
0.000 |
2.347 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
F |
A |
F |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
N/A |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 337 | 0 | 552 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 1.64 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.794 |
2.182 |
0.261 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 58 | 3134 | 67 | 52 | 87 | 0 | 78 |
normalized size | 1 | 1.00 | 0.75 | 40.70 | 0.87 | 0.68 | 1.13 | 0.00 | 1.01 |
time (sec) |
N/A |
0.220 |
0.022 |
1.159 |
0.978 |
0.470 |
2.491 |
0.000 |
0.257 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 120 | 120 | 136 | 508 | 0 | 287 | 0 | 92 | 413 |
normalized size | 1 | 1.00 | 1.13 | 4.23 | 0.00 | 2.39 | 0.00 | 0.77 | 3.44 |
time (sec) |
N/A |
0.131 |
0.313 |
0.053 |
0.000 |
0.485 |
0.000 |
1.151 |
1.086 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
NO |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 37 | 39 | 28 | 26 | 22 | 29 | 27 | 40 |
normalized size | 1 | 1.19 | 1.26 | 0.90 | 0.84 | 0.71 | 0.94 | 0.87 | 1.29 |
time (sec) |
N/A |
0.009 |
0.391 |
0.020 |
1.126 |
0.493 |
89.157 |
1.062 |
0.797 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 29 | 29 | 29 | 138 | 0 | 60 | 0 | 34 | -1 |
normalized size | 1 | 1.00 | 1.00 | 4.76 | 0.00 | 2.07 | 0.00 | 1.17 | -0.03 |
time (sec) |
N/A |
0.024 |
0.012 |
0.124 |
0.000 |
0.504 |
0.000 |
1.332 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 106 | 198 | 0 | 164 | 0 | 111 | 455 |
normalized size | 1 | 1.00 | 1.00 | 1.87 | 0.00 | 1.55 | 0.00 | 1.05 | 4.29 |
time (sec) |
N/A |
0.110 |
0.183 |
0.088 |
0.000 |
0.512 |
0.000 |
1.458 |
1.160 |
|
|
|
|
|
|
|
|
|
|
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 [1] had the largest ratio of
[1.400]
|
|
|
|
|
|
|
| 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 |
8 |
7 |
1.00 |
5 |
1.400 |
|
|
|
|
|
|
|
2 |
A |
2 |
2 |
1.00 |
15 |
0.133 |
|
|
|
|
|
|
|
3 |
F |
0 |
0 |
N/A |
0 |
N/A |
|
|
|
|
|
|
|
4 |
B |
32 |
13 |
3.42 |
14 |
0.929 |
|
|
|
|
|
|
|
5 |
C |
5 |
4 |
6.42 |
19 |
0.210 |
|
|
|
|
|
|
|
6 |
A |
7 |
7 |
1.00 |
13 |
0.538 |
|
|
|
|
|
|
|
7 |
A |
4 |
4 |
1.00 |
13 |
0.308 |
|
|
|
|
|
|
|
8 |
A |
9 |
7 |
1.00 |
14 |
0.500 |
|
|
|
|
|
|
|
9 |
A |
7 |
7 |
1.00 |
12 |
0.583 |
|
|
|
|
|
|
|
10 |
A |
5 |
5 |
1.41 |
19 |
0.263 |
|
|
|
|
|
|
|
11 |
A |
7 |
8 |
1.00 |
29 |
0.276 |
|
|
|
|
|
|
|
12 |
C |
40 |
15 |
1.91 |
14 |
1.071 |
|
|
|
|
|
|
|
13 |
C |
32 |
16 |
1.88 |
27 |
0.593 |
|
|
|
|
|
|
|
14 |
A |
9 |
11 |
1.13 |
18 |
0.611 |
|
|
|
|
|
|
|
15 |
A |
3 |
4 |
1.00 |
24 |
0.167 |
|
|
|
|
|
|
|
16 |
A |
3 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
17 |
A |
3 |
4 |
1.00 |
22 |
0.182 |
|
|
|
|
|
|
|
18 |
A |
4 |
4 |
1.00 |
15 |
0.267 |
|
|
|
|
|
|
|
19 |
A |
4 |
4 |
1.00 |
13 |
0.308 |
|
|
|
|
|
|
|
20 |
A |
2 |
3 |
1.00 |
23 |
0.130 |
|
|
|
|
|
|
|
21 |
A |
5 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
22 |
A |
7 |
8 |
1.34 |
15 |
0.533 |
|
|
|
|
|
|
|
23 |
A |
4 |
9 |
1.00 |
25 |
0.360 |
|
|
|
|
|
|
|
24 |
A |
5 |
5 |
1.00 |
27 |
0.185 |
|
|
|
|
|
|
|
25 |
A |
2 |
3 |
1.00 |
23 |
0.130 |
|
|
|
|
|
|
|
26 |
A |
18 |
11 |
1.00 |
27 |
0.407 |
|
|
|
|
|
|
|
27 |
A |
3 |
3 |
1.00 |
17 |
0.176 |
|
|
|
|
|
|
|
28 |
A |
2 |
2 |
1.00 |
13 |
0.154 |
|
|
|
|
|
|
|
29 |
A |
7 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
30 |
A |
5 |
5 |
1.00 |
15 |
0.333 |
|
|
|
|
|
|
|
31 |
A |
4 |
4 |
1.00 |
15 |
0.267 |
|
|
|
|
|
|
|
32 |
A |
2 |
2 |
1.00 |
17 |
0.118 |
|
|
|
|
|
|
|
33 |
A |
5 |
5 |
1.00 |
13 |
0.385 |
|
|
|
|
|
|
|
34 |
A |
3 |
3 |
1.00 |
15 |
0.200 |
|
|
|
|
|
|
|
35 |
A |
2 |
2 |
1.00 |
13 |
0.154 |
|
|
|
|
|
|
|
36 |
A |
5 |
5 |
1.00 |
13 |
0.385 |
|
|
|
|
|
|
|
37 |
A |
2 |
2 |
1.00 |
11 |
0.182 |
|
|
|
|
|
|
|
38 |
A |
2 |
2 |
1.00 |
24 |
0.083 |
|
|
|
|
|
|
|
39 |
A |
2 |
2 |
1.00 |
24 |
0.083 |
|
|
|
|
|
|
|
40 |
A |
4 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
41 |
A |
6 |
7 |
1.00 |
12 |
0.583 |
|
|
|
|
|
|
|
42 |
B |
5 |
5 |
2.11 |
13 |
0.385 |
|
|
|
|
|
|
|
43 |
A |
4 |
4 |
1.00 |
13 |
0.308 |
|
|
|
|
|
|
|
44 |
A |
4 |
4 |
1.28 |
15 |
0.267 |
|
|
|
|
|
|
|
45 |
F |
0 |
0 |
N/A |
0 |
N/A |
|
|
|
|
|
|
|
46 |
A |
13 |
10 |
1.00 |
12 |
0.833 |
|
|
|
|
|
|
|
47 |
A |
12 |
8 |
1.00 |
12 |
0.667 |
|
|
|
|
|
|
|
48 |
A |
6 |
6 |
1.19 |
18 |
0.333 |
|
|
|
|
|
|
|
49 |
A |
4 |
4 |
1.00 |
14 |
0.286 |
|
|
|
|
|
|
|
50 |
A |
6 |
6 |
1.00 |
14 |
0.429 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|