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, 52
}
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 3, 5, 7, 9, 11, 13, 20, 23, 24, 25, 26, 27, 28, 30, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 49
}
B grade: { }
C grade: { 2, 4, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19, 21, 22, 29, 31, 32, 39, 46, 47, 48, 50, 51, 52
}
F grade: { }
2.1.3 Maple
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 45 }
B grade: { }
C grade: { 43, 44 }
F grade: { 23, 24, 37, 38, 39, 40, 41, 42, 46, 47, 48, 49, 50, 51, 52 }
2.1.4 Maxima
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 43, 44, 45 }
B grade: { }
C grade: { }
F grade: { 23, 24, 37, 38, 39, 40, 41, 42, 46, 47, 48, 49, 50, 51, 52 }
2.1.5 FriCAS
A grade: { 1, 3, 25, 33, 43, 44, 45 }
B grade: { 2, 4, 5, 6, 7, 8, 19, 20, 26, 27, 28, 34, 35, 36 }
C grade: { }
F grade: { 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24, 29, 30, 31, 32, 37, 38, 39, 40, 41, 42, 46, 47,
48, 49, 50, 51, 52 }
2.1.6 Sympy
A grade: { 1, 2, 3, 4, 5, 6, 7, 8 }
B grade: { }
C grade: { }
F grade: { 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, 52 }
2.1.7 Giac
A grade: { 1, 25, 26, 27, 33, 34, 35, 36 }
B grade: { 2, 3, 4, 5, 6, 7, 8, 28 }
C grade: { }
F grade: { 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47, 48, 49, 50, 51, 52 }
2.1.8 Mupad
A grade: { }
B grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27, 43, 44, 45
}
C grade: { }
F grade: { 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 46, 47, 48, 49, 50, 51, 52
}
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 | 11 | 11 | 19 | 17 | 11 | 19 | 29 | 12 | 24 |
normalized size | 1 | 1.00 | 1.73 | 1.55 | 1.00 | 1.73 | 2.64 | 1.09 | 2.18 |
time (sec) |
N/A |
0.005 |
0.015 |
0.024 |
1.229 |
0.854 |
0.154 |
0.396 |
0.787 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 15 | 15 | 29 | 26 | 18 | 39 | 17 | 35 | 15 |
normalized size | 1 | 1.00 | 1.93 | 1.73 | 1.20 | 2.60 | 1.13 | 2.33 | 1.00 |
time (sec) |
N/A |
0.009 |
0.016 |
0.025 |
0.416 |
0.435 |
0.127 |
0.410 |
0.096 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 28 | 34 | 31 | 23 | 47 | 58 | 118 | 78 |
normalized size | 1 | 1.00 | 1.21 | 1.11 | 0.82 | 1.68 | 2.07 | 4.21 | 2.79 |
time (sec) |
N/A |
0.013 |
0.092 |
0.031 |
0.357 |
0.542 |
0.434 |
0.342 |
3.028 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 33 | 32 | 34 | 84 | 27 | 62 | 23 |
normalized size | 1 | 1.00 | 1.22 | 1.19 | 1.26 | 3.11 | 1.00 | 2.30 | 0.85 |
time (sec) |
N/A |
0.017 |
0.013 |
0.033 |
0.416 |
0.443 |
0.192 |
0.510 |
0.125 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 42 | 42 | 46 | 44 | 38 | 83 | 71 | 164 | 182 |
normalized size | 1 | 1.00 | 1.10 | 1.05 | 0.90 | 1.98 | 1.69 | 3.90 | 4.33 |
time (sec) |
N/A |
0.028 |
0.109 |
0.027 |
0.327 |
0.457 |
0.771 |
0.442 |
4.808 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 45 | 45 | 33 | 46 | 44 | 123 | 39 | 91 | 36 |
normalized size | 1 | 1.00 | 0.73 | 1.02 | 0.98 | 2.73 | 0.87 | 2.02 | 0.80 |
time (sec) |
N/A |
0.024 |
0.030 |
0.024 |
0.626 |
0.614 |
0.407 |
1.143 |
0.086 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 56 | 57 | 48 | 126 | 85 | 208 | 340 |
normalized size | 1 | 1.00 | 0.97 | 0.98 | 0.83 | 2.17 | 1.47 | 3.59 | 5.86 |
time (sec) |
N/A |
0.030 |
0.305 |
0.026 |
0.398 |
0.497 |
1.222 |
0.440 |
7.463 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 57 | 57 | 33 | 52 | 54 | 168 | 51 | 116 | 43 |
normalized size | 1 | 1.00 | 0.58 | 0.91 | 0.95 | 2.95 | 0.89 | 2.04 | 0.75 |
time (sec) |
N/A |
0.034 |
0.010 |
0.025 |
0.543 |
0.528 |
0.777 |
0.341 |
0.112 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
F(-2) |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 232 | 232 | 175 | 200 | 197 | 0 | 0 | 0 | 91 |
normalized size | 1 | 1.00 | 0.75 | 0.86 | 0.85 | 0.00 | 0.00 | 0.00 | 0.39 |
time (sec) |
N/A |
0.192 |
0.521 |
0.434 |
0.422 |
0.000 |
0.000 |
0.000 |
0.842 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F(-2) |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 212 | 212 | 40 | 182 | 185 | 0 | 0 | 0 | 74 |
normalized size | 1 | 1.00 | 0.19 | 0.86 | 0.87 | 0.00 | 0.00 | 0.00 | 0.35 |
time (sec) |
N/A |
0.143 |
0.077 |
0.384 |
0.552 |
0.000 |
0.000 |
0.000 |
0.425 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
F(-2) |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 210 | 210 | 159 | 176 | 179 | 0 | 0 | 0 | 75 |
normalized size | 1 | 1.00 | 0.76 | 0.84 | 0.85 | 0.00 | 0.00 | 0.00 | 0.36 |
time (sec) |
N/A |
0.140 |
0.221 |
0.382 |
0.497 |
0.000 |
0.000 |
0.000 |
0.369 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F(-2) |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 192 | 192 | 40 | 160 | 165 | 0 | 0 | 0 | 50 |
normalized size | 1 | 1.00 | 0.21 | 0.83 | 0.86 | 0.00 | 0.00 | 0.00 | 0.26 |
time (sec) |
N/A |
0.115 |
0.040 |
0.417 |
0.416 |
0.000 |
0.000 |
0.000 |
0.246 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
F(-2) |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 192 | 192 | 131 | 166 | 165 | 0 | 0 | 0 | 57 |
normalized size | 1 | 1.00 | 0.68 | 0.86 | 0.86 | 0.00 | 0.00 | 0.00 | 0.30 |
time (sec) |
N/A |
0.112 |
0.096 |
0.346 |
0.431 |
0.000 |
0.000 |
0.000 |
0.335 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F(-2) |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 212 | 212 | 38 | 184 | 187 | 0 | 0 | 0 | 76 |
normalized size | 1 | 1.00 | 0.18 | 0.87 | 0.88 | 0.00 | 0.00 | 0.00 | 0.36 |
time (sec) |
N/A |
0.140 |
0.068 |
0.311 |
0.514 |
0.000 |
0.000 |
0.000 |
0.364 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F(-2) |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 214 | 214 | 40 | 184 | 188 | 0 | 0 | 0 | 77 |
normalized size | 1 | 1.00 | 0.19 | 0.86 | 0.88 | 0.00 | 0.00 | 0.00 | 0.36 |
time (sec) |
N/A |
0.143 |
0.081 |
0.280 |
0.529 |
0.000 |
0.000 |
0.000 |
0.623 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F(-2) |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 234 | 234 | 40 | 202 | 205 | 0 | 0 | 0 | 91 |
normalized size | 1 | 1.00 | 0.17 | 0.86 | 0.88 | 0.00 | 0.00 | 0.00 | 0.39 |
time (sec) |
N/A |
0.171 |
0.114 |
0.446 |
0.421 |
0.000 |
0.000 |
0.000 |
0.680 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F(-2) |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 242 | 242 | 38 | 214 | 196 | 0 | 0 | 0 | 246 |
normalized size | 1 | 1.00 | 0.16 | 0.88 | 0.81 | 0.00 | 0.00 | 0.00 | 1.02 |
time (sec) |
N/A |
0.470 |
0.032 |
0.534 |
0.501 |
0.000 |
0.000 |
0.000 |
0.628 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F(-2) |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 225 | 225 | 40 | 203 | 182 | 0 | 0 | 0 | 260 |
normalized size | 1 | 1.00 | 0.18 | 0.90 | 0.81 | 0.00 | 0.00 | 0.00 | 1.16 |
time (sec) |
N/A |
0.380 |
0.047 |
0.447 |
0.543 |
0.000 |
0.000 |
0.000 |
0.816 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
B |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 131 | 131 | 40 | 114 | 102 | 211 | 0 | 0 | 134 |
normalized size | 1 | 1.00 | 0.31 | 0.87 | 0.78 | 1.61 | 0.00 | 0.00 | 1.02 |
time (sec) |
N/A |
0.102 |
0.040 |
0.361 |
0.485 |
0.652 |
0.000 |
0.000 |
0.520 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 131 | 131 | 98 | 114 | 103 | 639 | 0 | 0 | 128 |
normalized size | 1 | 1.00 | 0.75 | 0.87 | 0.79 | 4.88 | 0.00 | 0.00 | 0.98 |
time (sec) |
N/A |
0.096 |
0.155 |
0.271 |
0.418 |
0.766 |
0.000 |
0.000 |
0.579 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F(-2) |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 225 | 225 | 38 | 209 | 182 | 0 | 0 | 0 | 231 |
normalized size | 1 | 1.00 | 0.17 | 0.93 | 0.81 | 0.00 | 0.00 | 0.00 | 1.03 |
time (sec) |
N/A |
0.310 |
0.028 |
0.358 |
0.421 |
0.000 |
0.000 |
0.000 |
0.546 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F(-2) |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 244 | 244 | 38 | 229 | 204 | 0 | 0 | 0 | 277 |
normalized size | 1 | 1.00 | 0.16 | 0.94 | 0.84 | 0.00 | 0.00 | 0.00 | 1.14 |
time (sec) |
N/A |
0.425 |
0.059 |
0.408 |
0.486 |
0.000 |
0.000 |
0.000 |
0.434 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
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 | 46 | 46 | 48 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.04 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.028 |
0.045 |
1.347 |
0.000 |
1.028 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
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 | 51 | 51 | 54 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.06 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.032 |
0.063 |
1.184 |
0.000 |
2.994 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 36 | 36 | 27 | 29 | 30 | 52 | 0 | 31 | -1 |
normalized size | 1 | 1.00 | 0.75 | 0.81 | 0.83 | 1.44 | 0.00 | 0.86 | -0.03 |
time (sec) |
N/A |
0.018 |
0.021 |
0.149 |
0.494 |
1.317 |
0.000 |
0.602 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 16 | 16 | 16 | 22 | 20 | 43 | 0 | 20 | -1 |
normalized size | 1 | 1.00 | 1.00 | 1.38 | 1.25 | 2.69 | 0.00 | 1.25 | -0.06 |
time (sec) |
N/A |
0.021 |
0.007 |
0.170 |
0.454 |
0.753 |
0.000 |
0.362 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 17 | 17 | 17 | 28 | 12 | 45 | 0 | 19 | 25 |
normalized size | 1 | 1.00 | 1.00 | 1.65 | 0.71 | 2.65 | 0.00 | 1.12 | 1.47 |
time (sec) |
N/A |
0.012 |
0.008 |
0.188 |
0.495 |
0.705 |
0.000 |
0.449 |
0.290 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 39 | 39 | 30 | 37 | 22 | 74 | 0 | 139 | -1 |
normalized size | 1 | 1.00 | 0.77 | 0.95 | 0.56 | 1.90 | 0.00 | 3.56 | -0.03 |
time (sec) |
N/A |
0.018 |
0.031 |
0.122 |
0.568 |
0.586 |
0.000 |
0.911 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 200 | 200 | 39 | 189 | 113 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.20 | 0.94 | 0.56 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.097 |
0.055 |
0.165 |
0.531 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 176 | 176 | 122 | 165 | 94 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.69 | 0.94 | 0.53 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.087 |
0.111 |
0.173 |
0.458 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 176 | 176 | 28 | 164 | 94 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.16 | 0.93 | 0.53 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.090 |
0.012 |
0.204 |
0.533 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 212 | 212 | 30 | 185 | 109 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.14 | 0.87 | 0.51 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.096 |
0.015 |
0.115 |
0.460 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 70 | 70 | 39 | 40 | 37 | 110 | 0 | 57 | -1 |
normalized size | 1 | 1.00 | 0.56 | 0.57 | 0.53 | 1.57 | 0.00 | 0.81 | -0.01 |
time (sec) |
N/A |
0.027 |
0.142 |
0.170 |
0.475 |
1.392 |
0.000 |
2.413 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 32 | 32 | 20 | 27 | 16 | 61 | 0 | 21 | -1 |
normalized size | 1 | 1.00 | 0.62 | 0.84 | 0.50 | 1.91 | 0.00 | 0.66 | -0.03 |
time (sec) |
N/A |
0.015 |
0.016 |
0.204 |
0.479 |
1.415 |
0.000 |
0.392 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 21 | 26 | 13 | 80 | 0 | 13 | -1 |
normalized size | 1 | 1.00 | 0.68 | 0.84 | 0.42 | 2.58 | 0.00 | 0.42 | -0.03 |
time (sec) |
N/A |
0.016 |
0.023 |
0.221 |
0.456 |
1.348 |
0.000 |
0.313 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 42 | 42 | 29 | 142 | 0 | 43 | -1 |
normalized size | 1 | 1.00 | 0.55 | 0.55 | 0.38 | 1.84 | 0.00 | 0.56 | -0.01 |
time (sec) |
N/A |
0.026 |
0.111 |
0.115 |
0.453 |
1.369 |
0.000 |
0.419 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F(-1) |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 58 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.97 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.040 |
0.055 |
180.000 |
0.000 |
1.204 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F(-1) |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 62 | 62 | 60 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.97 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.045 |
0.048 |
180.000 |
0.000 |
1.174 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 87 | 289 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 3.32 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.097 |
1.802 |
2.955 |
0.000 |
1.316 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
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 | 84 | 84 | 83 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.99 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.101 |
0.546 |
2.637 |
0.000 |
0.557 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 64 | 64 | 67 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.05 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.037 |
0.092 |
1.619 |
0.000 |
1.194 |
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 |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 90 | 90 | 83 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.92 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.153 |
0.474 |
2.900 |
0.000 |
2.774 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
F(-1) |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 76 | 76 | 73 | 21900 | 84 | 145 | 0 | 0 | 123 |
normalized size | 1 | 1.00 | 0.96 | 288.16 | 1.11 | 1.91 | 0.00 | 0.00 | 1.62 |
time (sec) |
N/A |
0.071 |
0.252 |
4.521 |
0.346 |
0.945 |
0.000 |
0.000 |
1.576 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
A |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 45 | 10907 | 56 | 87 | 0 | 0 | 84 |
normalized size | 1 | 1.00 | 0.88 | 213.86 | 1.10 | 1.71 | 0.00 | 0.00 | 1.65 |
time (sec) |
N/A |
0.053 |
0.131 |
1.915 |
0.374 |
0.926 |
0.000 |
0.000 |
0.670 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 25 | 26 | 26 | 25 | 41 | 0 | 0 | 73 |
normalized size | 1 | 1.00 | 1.04 | 1.04 | 1.00 | 1.64 | 0.00 | 0.00 | 2.92 |
time (sec) |
N/A |
0.042 |
0.019 |
0.257 |
0.321 |
1.319 |
0.000 |
0.000 |
0.341 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 509 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 9.98 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.049 |
3.084 |
4.186 |
0.000 |
1.419 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 1099 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 21.55 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.048 |
7.658 |
4.953 |
0.000 |
1.065 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 784 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 9.92 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.041 |
14.077 |
1.635 |
0.000 |
0.836 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
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 | 77 | 77 | 69 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.90 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.029 |
0.154 |
1.474 |
0.000 |
1.099 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 73 | 73 | 264 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 3.62 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.041 |
1.100 |
2.256 |
0.000 |
0.776 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 477 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 6.04 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.042 |
2.427 |
3.886 |
0.000 |
0.618 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 83 | 83 | 306 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 3.69 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.046 |
1.890 |
2.952 |
0.000 |
0.621 |
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
[1.000]
|
|
|
|
|
|
|
| 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 |
1 |
1 |
1.00 |
6 |
0.167 |
|
|
|
|
|
|
|
2 |
A |
2 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
3 |
A |
2 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
4 |
A |
3 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
5 |
A |
3 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
6 |
A |
4 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
7 |
A |
4 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
8 |
A |
5 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
9 |
A |
13 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
10 |
A |
12 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
11 |
A |
12 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
12 |
A |
11 |
8 |
1.00 |
12 |
0.667 |
|
|
|
|
|
|
|
13 |
A |
11 |
8 |
1.00 |
12 |
0.667 |
|
|
|
|
|
|
|
14 |
A |
12 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
15 |
A |
12 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
16 |
A |
13 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
17 |
A |
13 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
18 |
A |
12 |
8 |
1.00 |
12 |
0.667 |
|
|
|
|
|
|
|
19 |
A |
9 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
20 |
A |
9 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
21 |
A |
12 |
8 |
1.00 |
12 |
0.667 |
|
|
|
|
|
|
|
22 |
A |
13 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
23 |
A |
2 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
24 |
A |
2 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
25 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
26 |
A |
2 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
27 |
A |
2 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
28 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
29 |
A |
14 |
10 |
1.00 |
10 |
1.000 |
|
|
|
|
|
|
|
30 |
A |
13 |
10 |
1.00 |
10 |
1.000 |
|
|
|
|
|
|
|
31 |
A |
13 |
10 |
1.00 |
10 |
1.000 |
|
|
|
|
|
|
|
32 |
A |
14 |
10 |
1.00 |
10 |
1.000 |
|
|
|
|
|
|
|
33 |
A |
5 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
34 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
35 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
36 |
A |
5 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
37 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
38 |
A |
3 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
39 |
A |
2 |
2 |
1.00 |
21 |
0.095 |
|
|
|
|
|
|
|
40 |
A |
2 |
2 |
1.00 |
21 |
0.095 |
|
|
|
|
|
|
|
41 |
A |
3 |
3 |
1.00 |
21 |
0.143 |
|
|
|
|
|
|
|
42 |
A |
3 |
3 |
1.00 |
21 |
0.143 |
|
|
|
|
|
|
|
43 |
A |
3 |
2 |
1.00 |
19 |
0.105 |
|
|
|
|
|
|
|
44 |
A |
3 |
2 |
1.00 |
19 |
0.105 |
|
|
|
|
|
|
|
45 |
A |
2 |
2 |
1.00 |
19 |
0.105 |
|
|
|
|
|
|
|
46 |
A |
2 |
2 |
1.00 |
19 |
0.105 |
|
|
|
|
|
|
|
47 |
A |
2 |
2 |
1.00 |
19 |
0.105 |
|
|
|
|
|
|
|
48 |
A |
1 |
1 |
1.00 |
19 |
0.053 |
|
|
|
|
|
|
|
49 |
A |
1 |
1 |
1.00 |
17 |
0.059 |
|
|
|
|
|
|
|
50 |
A |
1 |
1 |
1.00 |
17 |
0.059 |
|
|
|
|
|
|
|
51 |
A |
1 |
1 |
1.00 |
19 |
0.053 |
|
|
|
|
|
|
|
52 |
A |
1 |
1 |
1.00 |
21 |
0.048 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|