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 N.S. in the table below, which stands for normalized size is defined as \(\frac {\text {antiderivative leaf size}}{\text {optimal antiderivative leaf size}}\). To help make the table fit, Mathematica was abbreviated to MMA.
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | B | A | B | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 55 | 55 | 111 | 84 | 194 | 58 | 0 | 80 | 83 |
N.S. | 1 | 1.00 | 2.02 | 1.53 | 3.53 | 1.05 | 0.00 | 1.45 | 1.51 |
time (sec) |
N/A | 0.070 | 0.163 | 0.112 | 0.472 | 4.127 | 0.000 | 0.428 | 0.479 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 68 | 68 | 50 | 44 | 58 | 71 | 0 | 52 | 134 |
N.S. | 1 | 1.00 | 0.74 | 0.65 | 0.85 | 1.04 | 0.00 | 0.76 | 1.97 |
time (sec) |
N/A | 0.052 | 0.152 | 0.109 | 0.263 | 3.172 | 0.000 | 0.405 | 0.435 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | B | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 39 | 39 | 62 | 54 | 94 | 38 | 0 | 49 | 51 |
N.S. | 1 | 1.00 | 1.59 | 1.38 | 2.41 | 0.97 | 0.00 | 1.26 | 1.31 |
time (sec) |
N/A | 0.058 | 0.109 | 0.080 | 0.467 | 2.780 | 0.000 | 0.420 | 0.291 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 40 | 40 | 30 | 28 | 31 | 35 | 0 | 34 | 62 |
N.S. | 1 | 1.00 | 0.75 | 0.70 | 0.78 | 0.88 | 0.00 | 0.85 | 1.55 |
time (sec) |
N/A | 0.037 | 0.062 | 0.095 | 0.262 | 3.124 | 0.000 | 0.426 | 0.339 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 9 | 9 | 9 | 17 | 9 | 9 | 0 | 9 | 25 |
N.S. | 1 | 1.00 | 1.00 | 1.89 | 1.00 | 1.00 | 0.00 | 1.00 | 2.78 |
time (sec) |
N/A | 0.018 | 0.012 | 0.060 | 0.262 | 3.769 | 0.000 | 0.412 | 0.294 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 15 | 15 | 30 | 18 | 30 | 25 | 0 | 17 | 45 |
N.S. | 1 | 1.00 | 2.00 | 1.20 | 2.00 | 1.67 | 0.00 | 1.13 | 3.00 |
time (sec) |
N/A | 0.029 | 0.038 | 0.073 | 0.467 | 2.975 | 0.000 | 0.411 | 0.282 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 16 | 16 | 11 | 13 | 18 | 19 | 0 | 14 | 36 |
N.S. | 1 | 1.00 | 0.69 | 0.81 | 1.12 | 1.19 | 0.00 | 0.88 | 2.25 |
time (sec) |
N/A | 0.030 | 0.024 | 0.069 | 0.261 | 3.074 | 0.000 | 0.413 | 0.304 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | B | A | B | B | F | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 31 | 31 | 90 | 51 | 86 | 64 | 0 | 65 | 85 |
N.S. | 1 | 1.00 | 2.90 | 1.65 | 2.77 | 2.06 | 0.00 | 2.10 | 2.74 |
time (sec) |
N/A | 0.044 | 0.043 | 0.095 | 0.466 | 4.479 | 0.000 | 0.422 | 0.298 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 36 | 36 | 26 | 25 | 29 | 45 | 0 | 29 | 80 |
N.S. | 1 | 1.00 | 0.72 | 0.69 | 0.81 | 1.25 | 0.00 | 0.81 | 2.22 |
time (sec) |
N/A | 0.036 | 0.034 | 0.112 | 0.267 | 3.564 | 0.000 | 0.415 | 0.297 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | B | A | B | B | F | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 49 | 49 | 163 | 83 | 134 | 104 | 0 | 109 | 123 |
N.S. | 1 | 1.00 | 3.33 | 1.69 | 2.73 | 2.12 | 0.00 | 2.22 | 2.51 |
time (sec) |
N/A | 0.061 | 0.050 | 0.097 | 0.465 | 3.360 | 0.000 | 0.424 | 0.333 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 58 | 58 | 39 | 41 | 42 | 70 | 0 | 41 | 113 |
N.S. | 1 | 1.00 | 0.67 | 0.71 | 0.72 | 1.21 | 0.00 | 0.71 | 1.95 |
time (sec) |
N/A | 0.043 | 0.058 | 0.113 | 0.271 | 2.621 | 0.000 | 0.413 | 0.431 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 178 | 178 | 172 | 160 | 251 | 260 | 0 | 252 | 445 |
N.S. | 1 | 1.00 | 0.97 | 0.90 | 1.41 | 1.46 | 0.00 | 1.42 | 2.50 |
time (sec) |
N/A | 0.206 | 0.867 | 0.272 | 0.272 | 3.755 | 0.000 | 0.401 | 2.300 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 113 | 113 | 115 | 101 | 120 | 144 | 0 | 139 | 172 |
N.S. | 1 | 1.00 | 1.02 | 0.89 | 1.06 | 1.27 | 0.00 | 1.23 | 1.52 |
time (sec) |
N/A | 0.118 | 0.527 | 0.162 | 0.268 | 2.879 | 0.000 | 0.403 | 1.137 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 66 | 66 | 56 | 60 | 50 | 58 | 0 | 53 | 80 |
N.S. | 1 | 1.00 | 0.85 | 0.91 | 0.76 | 0.88 | 0.00 | 0.80 | 1.21 |
time (sec) |
N/A | 0.064 | 0.066 | 0.107 | 0.260 | 3.195 | 0.000 | 0.403 | 0.442 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 19 | 19 | 11 | 21 | 11 | 11 | 0 | 12 | 55 |
N.S. | 1 | 1.00 | 0.58 | 1.11 | 0.58 | 0.58 | 0.00 | 0.63 | 2.89 |
time (sec) |
N/A | 0.017 | 0.011 | 0.055 | 0.263 | 4.124 | 0.000 | 0.430 | 0.418 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 38 | 38 | 39 | 43 | 42 | 47 | 0 | 44 | 75 |
N.S. | 1 | 1.00 | 1.03 | 1.13 | 1.11 | 1.24 | 0.00 | 1.16 | 1.97 |
time (sec) |
N/A | 0.045 | 0.054 | 0.117 | 0.265 | 3.052 | 0.000 | 0.427 | 0.459 |
Optimal | Rubi | MMA | 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 | 72 | 72 | 85 | 86 | 84 | 151 | 0 | 90 | 157 |
N.S. | 1 | 1.00 | 1.18 | 1.19 | 1.17 | 2.10 | 0.00 | 1.25 | 2.18 |
time (sec) |
N/A | 0.061 | 0.150 | 0.164 | 0.255 | 2.503 | 0.000 | 0.412 | 0.567 |
Optimal | Rubi | MMA | 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 | 122 | 122 | 132 | 149 | 149 | 327 | 0 | 155 | 284 |
N.S. | 1 | 1.00 | 1.08 | 1.22 | 1.22 | 2.68 | 0.00 | 1.27 | 2.33 |
time (sec) |
N/A | 0.090 | 0.312 | 0.206 | 0.262 | 2.904 | 0.000 | 0.406 | 0.870 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 159 | 205 | 194 | 190 | 0 | 483 | 0 | 216 | 2500 |
N.S. |
1 | 1.29 | 1.22 | 1.19 | 0.00 | 3.04 | 0.00 | 1.36 | 15.72 |
time (sec) |
N/A | 0.216 | 1.161 | 0.204 | 0.000 | 3.505 | 0.000 | 0.424 | 7.377 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 84 | 112 | 87 | 106 | 0 | 296 | 0 | 103 | 2341 |
N.S. |
1 | 1.33 | 1.04 | 1.26 | 0.00 | 3.52 | 0.00 | 1.23 | 27.87 |
time (sec) |
N/A | 0.123 | 0.276 | 0.138 | 0.000 | 3.433 | 0.000 | 0.419 | 2.258 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 61 | 61 | 71 | 75 | 0 | 206 | 0 | 80 | 697 |
N.S. |
1 | 1.00 | 1.16 | 1.23 | 0.00 | 3.38 | 0.00 | 1.31 | 11.43 |
time (sec) |
N/A | 0.121 | 0.075 | 0.123 | 0.000 | 4.278 | 0.000 | 0.435 | 0.526 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | B | F | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 91 | 91 | 158 | 141 | 0 | 453 | 0 | 163 | 2163 |
N.S. |
1 | 1.00 | 1.74 | 1.55 | 0.00 | 4.98 | 0.00 | 1.79 | 23.77 |
time (sec) |
N/A | 0.190 | 0.446 | 0.197 | 0.000 | 4.760 | 0.000 | 0.405 | 1.190 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | B | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 186 | 186 | 312 | 251 | 0 | 852 | 0 | 299 | 2500 |
N.S. |
1 | 1.00 | 1.68 | 1.35 | 0.00 | 4.58 | 0.00 | 1.61 | 13.44 |
time (sec) |
N/A | 0.184 | 0.968 | 0.256 | 0.000 | 4.157 | 0.000 | 0.443 | 1.586 |