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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 17, 18, 19, 20, 22, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37,
41, 42, 43, 44, 45, 46, 47, 48, 49 }
B grade: { 13, 21, 32, 38, 39, 40 }
C grade: { 16, 23, 24, 25, 26 }
F grade: { }
2.1.3 Maple
A grade: { 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 20, 21, 27, 28, 29, 30, 32, 33, 34, 35, 38, 40, 49
}
B grade: { 10, 11, 17, 18, 19, 22, 23, 24, 25, 26 }
C grade: { }
F grade: { 1, 31, 36, 37, 39, 41, 42, 43, 44, 45, 46, 47, 48 }
2.1.4 Maxima
A grade: { 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 21, 38, 39 }
B grade: { 8 }
C grade: { }
F grade: { 1, 6, 12, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 41, 42,
43, 44, 45, 46, 47, 48, 49 }
2.1.5 FriCAS
A grade: { 2, 3, 4, 5, 7, 8, 9, 10, 11, 14, 15, 17, 18, 19, 20, 24, 25, 26, 45, 46, 47, 48 }
B grade: { 13, 21, 23, 38, 39 }
C grade: { }
F grade: { 1, 6, 12, 16, 22, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 41, 42, 43, 44, 49 }
2.1.6 Sympy
A grade: { 9, 10, 11, 15 }
B grade: { }
C grade: { 2, 3, 4, 5, 7, 8, 13, 14 }
F grade: { 1, 6, 12, 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 }
2.1.7 Giac
A grade: { 7, 8, 9, 10, 11, 14, 15, 20, 23, 39, 45, 49 }
B grade: { 2, 3, 4, 5, 13, 17, 18, 19, 21, 24, 25, 26, 38 }
C grade: { }
F grade: { 1, 6, 12, 16, 22, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 41, 42, 43, 44, 46, 47, 48
}
2.1.8 Mupad
A grade: { }
B grade: { 1, 5, 6, 7, 10, 11, 12, 13, 21, 38, 39, 40 }
C grade: { }
F grade: { 2, 3, 4, 8, 9, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
41, 42, 43, 44, 45, 46, 47, 48, 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 |
F |
F |
F |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 62 | 62 | 56 | 0 | 0 | 0 | 0 | 0 | 58 |
normalized size | 1 | 1.00 | 0.90 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.94 |
time (sec) |
N/A |
0.090 |
0.043 |
0.153 |
0.000 |
0.584 |
0.000 |
0.000 |
0.827 |
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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 |
C |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 40 | 43 | 66 | 32 | 119 | 152 | -1 |
normalized size | 1 | 1.00 | 0.69 | 0.74 | 1.14 | 0.55 | 2.05 | 2.62 | -0.02 |
time (sec) |
N/A |
0.020 |
0.028 |
0.055 |
0.347 |
0.921 |
64.531 |
0.147 |
0.000 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
C |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 35 | 38 | 52 | 27 | 90 | 116 | -1 |
normalized size | 1 | 1.00 | 0.74 | 0.81 | 1.11 | 0.57 | 1.91 | 2.47 | -0.02 |
time (sec) |
N/A |
0.019 |
0.021 |
0.052 |
0.334 |
1.915 |
23.353 |
0.152 |
0.000 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
C |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 36 | 36 | 27 | 31 | 38 | 20 | 58 | 80 | -1 |
normalized size | 1 | 1.00 | 0.75 | 0.86 | 1.06 | 0.56 | 1.61 | 2.22 | -0.03 |
time (sec) |
N/A |
0.012 |
0.022 |
0.050 |
0.338 |
1.988 |
8.121 |
0.133 |
0.000 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
C |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 16 | 16 | 16 | 24 | 20 | 12 | 29 | 41 | 20 |
normalized size | 1 | 1.00 | 1.00 | 1.50 | 1.25 | 0.75 | 1.81 | 2.56 | 1.25 |
time (sec) |
N/A |
0.004 |
0.004 |
0.050 |
0.335 |
1.862 |
3.708 |
0.148 |
1.057 |
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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 |
F |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 54 | 105 | 0 | 0 | 0 | 0 | 42 |
normalized size | 1 | 1.00 | 0.96 | 1.88 | 0.00 | 0.00 | 0.00 | 0.00 | 0.75 |
time (sec) |
N/A |
0.083 |
0.032 |
0.114 |
0.000 |
0.812 |
0.000 |
0.000 |
0.847 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
C |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 32 | 46 | 51 | 21 | 78 | 32 | 28 |
normalized size | 1 | 1.00 | 0.84 | 1.21 | 1.34 | 0.55 | 2.05 | 0.84 | 0.74 |
time (sec) |
N/A |
0.015 |
0.025 |
0.052 |
0.475 |
0.857 |
17.552 |
0.142 |
0.689 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
B |
A |
C |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 55 | 57 | 80 | 30 | 146 | 59 | -1 |
normalized size | 1 | 1.00 | 1.02 | 1.06 | 1.48 | 0.56 | 2.70 | 1.09 | -0.02 |
time (sec) |
N/A |
0.020 |
0.048 |
0.051 |
0.459 |
1.723 |
48.391 |
0.127 |
0.000 |
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Problem 9 |
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 | 56 | 56 | 42 | 66 | 54 | 39 | 51 | 68 | -1 |
normalized size | 1 | 1.00 | 0.75 | 1.18 | 0.96 | 0.70 | 0.91 | 1.21 | -0.02 |
time (sec) |
N/A |
0.045 |
0.035 |
0.060 |
0.485 |
1.048 |
0.446 |
0.142 |
0.000 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 44 | 93 | 46 | 38 | 41 | 48 | 38 |
normalized size | 1 | 1.00 | 0.94 | 1.98 | 0.98 | 0.81 | 0.87 | 1.02 | 0.81 |
time (sec) |
N/A |
0.020 |
0.025 |
0.060 |
0.446 |
0.795 |
0.205 |
0.141 |
0.612 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 25 | 25 | 52 | 23 | 26 | 22 | 26 | 23 |
normalized size | 1 | 1.00 | 1.00 | 2.08 | 0.92 | 1.04 | 0.88 | 1.04 | 0.92 |
time (sec) |
N/A |
0.010 |
0.012 |
0.060 |
0.457 |
0.920 |
0.137 |
0.129 |
0.098 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
F |
F |
F |
F |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 54 | 125 | 0 | 0 | 0 | 0 | 49 |
normalized size | 1 | 1.00 | 0.92 | 2.12 | 0.00 | 0.00 | 0.00 | 0.00 | 0.83 |
time (sec) |
N/A |
0.064 |
0.036 |
0.116 |
0.000 |
0.957 |
0.000 |
0.000 |
0.608 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
B |
A |
A |
B |
C |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 32 | 32 | 93 | 42 | 52 | 65 | 27 | 61 | 30 |
normalized size | 1 | 1.00 | 2.91 | 1.31 | 1.62 | 2.03 | 0.84 | 1.91 | 0.94 |
time (sec) |
N/A |
0.031 |
0.152 |
0.048 |
0.335 |
0.967 |
2.031 |
0.126 |
0.590 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
C |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 35 | 54 | 32 | 38 | 51 | 61 | -1 |
normalized size | 1 | 1.00 | 0.92 | 1.42 | 0.84 | 1.00 | 1.34 | 1.61 | -0.03 |
time (sec) |
N/A |
0.022 |
0.028 |
0.059 |
0.467 |
0.464 |
1.164 |
0.147 |
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 |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 69 | 98 | 64 | 92 | 99 | 80 | -1 |
normalized size | 1 | 1.00 | 1.15 | 1.63 | 1.07 | 1.53 | 1.65 | 1.33 | -0.02 |
time (sec) |
N/A |
0.042 |
0.057 |
0.060 |
0.461 |
0.641 |
2.335 |
0.140 |
0.000 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
A |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 63 | 166 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.91 | 2.41 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.094 |
0.094 |
0.121 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 197 | 197 | 174 | 507 | 0 | 151 | 0 | 408 | -1 |
normalized size | 1 | 1.00 | 0.88 | 2.57 | 0.00 | 0.77 | 0.00 | 2.07 | -0.01 |
time (sec) |
N/A |
0.232 |
0.225 |
0.066 |
0.000 |
0.730 |
0.000 |
0.196 |
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 |
B |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 155 | 155 | 149 | 360 | 0 | 129 | 0 | 300 | -1 |
normalized size | 1 | 1.00 | 0.96 | 2.32 | 0.00 | 0.83 | 0.00 | 1.94 | -0.01 |
time (sec) |
N/A |
0.137 |
0.288 |
0.060 |
0.000 |
0.768 |
0.000 |
0.171 |
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 |
B |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 116 | 116 | 129 | 272 | 0 | 117 | 0 | 203 | -1 |
normalized size | 1 | 1.00 | 1.11 | 2.34 | 0.00 | 1.01 | 0.00 | 1.75 | -0.01 |
time (sec) |
N/A |
0.089 |
0.191 |
0.061 |
0.000 |
0.699 |
0.000 |
0.186 |
0.000 |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
F |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 110 | 127 | 0 | 102 | 0 | 134 | -1 |
normalized size | 1 | 1.00 | 1.39 | 1.61 | 0.00 | 1.29 | 0.00 | 1.70 | -0.01 |
time (sec) |
N/A |
0.053 |
0.131 |
0.057 |
0.000 |
0.638 |
0.000 |
0.156 |
0.000 |
|
|
|
|
|
|
|
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|
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|
|
|
|
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|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 36 | 36 | 114 | 50 | 55 | 75 | 0 | 81 | 33 |
normalized size | 1 | 1.00 | 3.17 | 1.39 | 1.53 | 2.08 | 0.00 | 2.25 | 0.92 |
time (sec) |
N/A |
0.024 |
0.116 |
0.052 |
0.332 |
0.552 |
0.000 |
0.165 |
0.884 |
|
|
|
|
|
|
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|
|
|
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|
|
|
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|
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|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 210 | 210 | 375 | 607 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.79 | 2.89 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.300 |
0.440 |
1.133 |
0.000 |
0.587 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
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|
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|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 115 | 154 | 0 | 278 | 0 | 96 | -1 |
normalized size | 1 | 1.00 | 1.67 | 2.23 | 0.00 | 4.03 | 0.00 | 1.39 | -0.01 |
time (sec) |
N/A |
0.096 |
0.416 |
0.063 |
0.000 |
0.674 |
0.000 |
0.179 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 123 | 123 | 199 | 453 | 0 | 428 | 0 | 217 | -1 |
normalized size | 1 | 1.00 | 1.62 | 3.68 | 0.00 | 3.48 | 0.00 | 1.76 | -0.01 |
time (sec) |
N/A |
0.194 |
0.770 |
0.068 |
0.000 |
0.638 |
0.000 |
0.225 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 180 | 180 | 241 | 759 | 0 | 548 | 0 | 450 | -1 |
normalized size | 1 | 1.00 | 1.34 | 4.22 | 0.00 | 3.04 | 0.00 | 2.50 | -0.01 |
time (sec) |
N/A |
0.297 |
0.523 |
0.073 |
0.000 |
0.669 |
0.000 |
0.218 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 239 | 239 | 307 | 1172 | 0 | 673 | 0 | 841 | -1 |
normalized size | 1 | 1.00 | 1.28 | 4.90 | 0.00 | 2.82 | 0.00 | 3.52 | -0.00 |
time (sec) |
N/A |
0.457 |
0.536 |
0.074 |
0.000 |
0.910 |
0.000 |
0.253 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 366 | 366 | 420 | 769 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.15 | 2.10 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.305 |
5.326 |
6.967 |
0.000 |
0.611 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 272 | 272 | 314 | 545 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.15 | 2.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.231 |
4.741 |
1.755 |
0.000 |
0.746 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 145 | 145 | 213 | 250 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.47 | 1.72 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.134 |
0.815 |
0.342 |
0.000 |
0.784 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 86 | 86 | 99 | 167 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.15 | 1.94 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.064 |
0.132 |
0.241 |
0.000 |
0.541 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 324 | 324 | 408 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.26 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.493 |
0.346 |
1.915 |
0.000 |
0.634 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
F(-1) |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 254 | 254 | 802 | 297 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 3.16 | 1.17 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.416 |
3.183 |
1.381 |
0.000 |
0.925 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 464 | 464 | 656 | 809 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.41 | 1.74 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.402 |
8.888 |
2.720 |
0.000 |
0.669 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 264 | 264 | 314 | 481 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.19 | 1.82 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.256 |
0.802 |
1.618 |
0.000 |
0.726 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 140 | 140 | 162 | 247 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.16 | 1.76 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.102 |
0.109 |
0.632 |
0.000 |
0.758 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 448 | 448 | 554 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.24 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.542 |
0.328 |
2.084 |
0.000 |
0.706 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 378 | 378 | 289 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.76 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.617 |
0.510 |
2.473 |
0.000 |
0.725 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
B |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 127 | 65 | 63 | 88 | 0 | 91 | 44 |
normalized size | 1 | 1.00 | 2.65 | 1.35 | 1.31 | 1.83 | 0.00 | 1.90 | 0.92 |
time (sec) |
N/A |
0.063 |
0.246 |
0.109 |
0.360 |
0.742 |
0.000 |
0.246 |
1.034 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
A |
B |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 130 | 0 | 66 | 94 | 0 | 74 | 42 |
normalized size | 1 | 1.00 | 2.71 | 0.00 | 1.38 | 1.96 | 0.00 | 1.54 | 0.88 |
time (sec) |
N/A |
0.070 |
0.288 |
0.085 |
0.390 |
0.726 |
0.000 |
0.263 |
0.726 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
F(-1) |
F(-2) |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 280 | 199 | 0 | 0 | 0 | 0 | 91 |
normalized size | 1 | 1.00 | 3.29 | 2.34 | 0.00 | 0.00 | 0.00 | 0.00 | 1.07 |
time (sec) |
N/A |
0.086 |
0.726 |
0.121 |
0.000 |
0.000 |
0.000 |
0.000 |
1.247 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 95 | 95 | 79 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.83 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.121 |
0.454 |
0.056 |
0.000 |
0.748 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 87 | 101 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.16 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.106 |
0.263 |
0.056 |
0.000 |
0.736 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 87 | 54 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.62 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.094 |
0.092 |
0.053 |
0.000 |
0.610 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
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|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 75 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.74 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.059 |
0.052 |
0.056 |
0.000 |
0.575 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
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|
Problem 45 |
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 | 39 | 39 | 33 | 0 | 0 | 23 | 0 | 39 | -1 |
normalized size | 1 | 1.00 | 0.85 | 0.00 | 0.00 | 0.59 | 0.00 | 1.00 | -0.03 |
time (sec) |
N/A |
0.030 |
0.038 |
0.055 |
0.000 |
0.711 |
0.000 |
0.169 |
0.000 |
|
|
|
|
|
|
|
|
|
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|
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|
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|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 41 | 30 | 0 | 0 | 32 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.73 | 0.00 | 0.00 | 0.78 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.043 |
0.045 |
0.056 |
0.000 |
0.796 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
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|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 54 | 0 | 0 | 41 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.64 | 0.00 | 0.00 | 0.49 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.067 |
0.135 |
0.055 |
0.000 |
0.848 |
0.000 |
0.000 |
0.000 |
|
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|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 81 | 81 | 50 | 0 | 0 | 51 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.62 | 0.00 | 0.00 | 0.63 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.068 |
0.124 |
0.053 |
0.000 |
0.822 |
0.000 |
0.000 |
0.000 |
|
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Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 59 | 158 | 0 | 0 | 0 | 114 | -1 |
normalized size | 1 | 1.00 | 0.86 | 2.29 | 0.00 | 0.00 | 0.00 | 1.65 | -0.01 |
time (sec) |
N/A |
0.095 |
0.055 |
0.358 |
0.000 |
0.774 |
0.000 |
0.560 |
0.000 |
|
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|
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 [34] had the largest ratio of
[1.200]
|
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|
| Table 2.1:Rubi 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 |
7 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
2 |
A |
4 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
3 |
A |
4 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
4 |
A |
4 |
3 |
1.00 |
8 |
0.375 |
|
|
|
|
|
|
|
5 |
A |
3 |
3 |
1.00 |
6 |
0.500 |
|
|
|
|
|
|
|
6 |
A |
7 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
7 |
A |
5 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
8 |
A |
6 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
9 |
A |
5 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
10 |
A |
4 |
4 |
1.00 |
8 |
0.500 |
|
|
|
|
|
|
|
11 |
A |
3 |
3 |
1.00 |
6 |
0.500 |
|
|
|
|
|
|
|
12 |
A |
6 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
13 |
A |
5 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
14 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
15 |
A |
6 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
16 |
A |
7 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
17 |
A |
9 |
8 |
1.00 |
10 |
0.800 |
|
|
|
|
|
|
|
18 |
A |
8 |
7 |
1.00 |
10 |
0.700 |
|
|
|
|
|
|
|
19 |
A |
7 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
20 |
A |
6 |
6 |
1.00 |
8 |
0.750 |
|
|
|
|
|
|
|
21 |
A |
5 |
5 |
1.00 |
6 |
0.833 |
|
|
|
|
|
|
|
22 |
A |
14 |
8 |
1.00 |
10 |
0.800 |
|
|
|
|
|
|
|
23 |
A |
6 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
24 |
A |
8 |
8 |
1.00 |
10 |
0.800 |
|
|
|
|
|
|
|
25 |
A |
9 |
9 |
1.00 |
10 |
0.900 |
|
|
|
|
|
|
|
26 |
A |
10 |
9 |
1.00 |
10 |
0.900 |
|
|
|
|
|
|
|
27 |
A |
20 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
28 |
A |
17 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
29 |
A |
11 |
8 |
1.00 |
10 |
0.800 |
|
|
|
|
|
|
|
30 |
A |
8 |
6 |
1.00 |
8 |
0.750 |
|
|
|
|
|
|
|
31 |
A |
17 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
32 |
A |
12 |
8 |
1.00 |
12 |
0.667 |
|
|
|
|
|
|
|
33 |
A |
25 |
14 |
1.00 |
12 |
1.167 |
|
|
|
|
|
|
|
34 |
A |
16 |
12 |
1.00 |
10 |
1.200 |
|
|
|
|
|
|
|
35 |
A |
10 |
7 |
1.00 |
8 |
0.875 |
|
|
|
|
|
|
|
36 |
A |
20 |
10 |
1.00 |
12 |
0.833 |
|
|
|
|
|
|
|
37 |
A |
14 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
38 |
A |
6 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
39 |
A |
6 |
6 |
1.00 |
14 |
0.429 |
|
|
|
|
|
|
|
40 |
A |
7 |
7 |
1.00 |
10 |
0.700 |
|
|
|
|
|
|
|
41 |
A |
6 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
42 |
A |
6 |
4 |
1.00 |
8 |
0.500 |
|
|
|
|
|
|
|
43 |
A |
5 |
3 |
1.00 |
6 |
0.500 |
|
|
|
|
|
|
|
44 |
A |
6 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
45 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
46 |
A |
5 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
47 |
A |
6 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
48 |
A |
6 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
49 |
A |
8 |
8 |
1.00 |
19 |
0.421 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|