Detailed conclusion table per each integral is given by the 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 as F.
In this table, the column N.S.
means normalized size and is defined as \(\frac {\text {antiderivative leaf size}}{\text {optimal antiderivative leaf size}}\). To make the table
fit the page, the name Mathematica
was abbreviated to MMA
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | B | F(-2) | A | B | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 57 | 57 | 56 | 96 | 0 | 227 | 688 | 361 | 132 | 1537 |
N.S. |
1 | 1.00 | 0.98 | 1.68 | 0.00 | 3.98 | 12.07 | 6.33 | 2.32 | 26.96 |
time (sec) |
N/A | 0.329 | 0.132 | 0.178 | 0.000 | 0.126 | 11.756 | 0.130 | 0.149 | 2.988 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 19 | 19 | 19 | 17 | 19 | 28 | 17 | 18 | 18 | 18 |
N.S. |
1 | 1.00 | 1.00 | 0.89 | 1.00 | 1.47 | 0.89 | 0.95 | 0.95 | 0.95 |
time (sec) |
N/A | 0.260 | 0.042 | 0.090 | 0.028 | 0.120 | 0.158 | 0.111 | 0.149 | 1.006 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 23 | 23 | 20 | 28 | 19 | 25 | 27 | 39 | 39 | 30 |
N.S. |
1 | 1.00 | 0.87 | 1.22 | 0.83 | 1.09 | 1.17 | 1.70 | 1.70 | 1.30 |
time (sec) |
N/A | 0.275 | 0.229 | 0.103 | 0.034 | 0.107 | 0.336 | 0.116 | 0.164 | 1.034 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | B | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 58 | 58 | 57 | 97 | 0 | 231 | 804 | 407 | 179 | 2219 |
N.S. |
1 | 1.00 | 0.98 | 1.67 | 0.00 | 3.98 | 13.86 | 7.02 | 3.09 | 38.26 |
time (sec) |
N/A | 0.327 | 0.129 | 0.183 | 0.000 | 0.156 | 12.188 | 0.133 | 0.235 | 6.932 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | B | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 58 | 58 | 55 | 85 | 0 | 235 | 748 | 103 | 179 | 2213 |
N.S. |
1 | 1.00 | 0.95 | 1.47 | 0.00 | 4.05 | 12.90 | 1.78 | 3.09 | 38.16 |
time (sec) |
N/A | 0.339 | 0.200 | 0.192 | 0.000 | 0.150 | 15.220 | 0.123 | 0.239 | 6.908 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | F | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 65 | 65 | 61 | 108 | 0 | 263 | 0 | 121 | 159 | 1540 |
N.S. |
1 | 1.00 | 0.94 | 1.66 | 0.00 | 4.05 | 0.00 | 1.86 | 2.45 | 23.69 |
time (sec) |
N/A | 0.359 | 0.352 | 0.288 | 0.000 | 0.155 | 0.000 | 0.125 | 0.250 | 3.006 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | F | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 100 | 100 | 134 | 96 | 0 | 266 | 0 | 116 | 97 | 419 |
N.S. |
1 | 1.00 | 1.34 | 0.96 | 0.00 | 2.66 | 0.00 | 1.16 | 0.97 | 4.19 |
time (sec) |
N/A | 0.394 | 0.478 | 0.319 | 0.000 | 2.835 | 0.000 | 0.129 | 0.162 | 1.703 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | F | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 99 | 99 | 116 | 95 | 0 | 265 | 0 | 115 | 97 | 417 |
N.S. |
1 | 1.00 | 1.17 | 0.96 | 0.00 | 2.68 | 0.00 | 1.16 | 0.98 | 4.21 |
time (sec) |
N/A | 0.523 | 0.375 | 0.229 | 0.000 | 3.774 | 0.000 | 0.127 | 0.154 | 1.678 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | B | B | F(-2) | B | F | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 247 | 247 | 526 | 484 | 0 | 1049 | 0 | 607 | 2335 | 9992 |
N.S. |
1 | 1.00 | 2.13 | 1.96 | 0.00 | 4.25 | 0.00 | 2.46 | 9.45 | 40.45 |
time (sec) |
N/A | 0.716 | 7.770 | 1.230 | 0.000 | 96.820 | 0.000 | 0.207 | 0.211 | 6.705 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-2) | B | F | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 170 | 170 | 335 | 291 | 0 | 747 | 0 | 340 | 1631 | 6735 |
N.S. |
1 | 1.00 | 1.97 | 1.71 | 0.00 | 4.39 | 0.00 | 2.00 | 9.59 | 39.62 |
time (sec) |
N/A | 0.601 | 3.542 | 0.757 | 0.000 | 21.484 | 0.000 | 0.193 | 0.181 | 5.703 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-2) | B | F | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 103 | 103 | 135 | 165 | 0 | 505 | 0 | 196 | 445 | 3577 |
N.S. |
1 | 1.00 | 1.31 | 1.60 | 0.00 | 4.90 | 0.00 | 1.90 | 4.32 | 34.73 |
time (sec) |
N/A | 0.531 | 2.916 | 0.501 | 0.000 | 2.907 | 0.000 | 0.170 | 0.165 | 4.283 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | F | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 76 | 76 | 112 | 92 | 0 | 296 | 0 | 128 | 193 | 345 |
N.S. |
1 | 1.00 | 1.47 | 1.21 | 0.00 | 3.89 | 0.00 | 1.68 | 2.54 | 4.54 |
time (sec) |
N/A | 0.462 | 0.615 | 0.310 | 0.000 | 0.381 | 0.000 | 0.133 | 0.176 | 1.781 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | F | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 121 | 121 | 106 | 108 | 0 | 1022 | 0 | 507 | 291 | 2665 |
N.S. |
1 | 1.00 | 0.88 | 0.89 | 0.00 | 8.45 | 0.00 | 4.19 | 2.40 | 22.02 |
time (sec) |
N/A | 0.521 | 0.954 | 0.571 | 0.000 | 1.647 | 0.000 | 0.190 | 0.173 | 2.565 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-2) | B | F | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 187 | 221 | 205 | 210 | 0 | 2835 | 0 | 333 | 1678 | 20827 |
N.S. |
1 | 1.18 | 1.10 | 1.12 | 0.00 | 15.16 | 0.00 | 1.78 | 8.97 | 111.37 |
time (sec) |
N/A | 0.994 | 2.053 | 1.652 | 0.000 | 87.078 | 0.000 | 0.170 | 0.215 | 9.106 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-2) | F(-1) | F | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 458 | 458 | 319 | 412 | 0 | 0 | 0 | 748 | 4916 | 52103 |
N.S. |
1 | 1.00 | 0.70 | 0.90 | 0.00 | 0.00 | 0.00 | 1.63 | 10.73 | 113.76 |
time (sec) |
N/A | 1.220 | 4.573 | 4.630 | 0.000 | 0.000 | 0.000 | 0.298 | 0.225 | 11.950 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 213 | 213 | 184 | 318 | 0 | 0 | 0 | 0 | 26 | 0 |
N.S. |
1 | 1.00 | 0.86 | 1.49 | 0.00 | 0.00 | 0.00 | 0.00 | 0.12 | 0.00 |
time (sec) |
N/A | 0.886 | 10.692 | 7.799 | 0.000 | 0.000 | 0.000 | 0.000 | 0.163 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | B | F | F(-1) | F | F | F | F(-1) |
verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 102 | 102 | 187 | 206 | 0 | 0 | 0 | 0 | 53 | 0 |
N.S. |
1 | 1.00 | 1.83 | 2.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.52 | 0.00 |
time (sec) |
N/A | 0.451 | 12.048 | 4.852 | 0.000 | 0.000 | 0.000 | 0.000 | 0.167 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | B | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 87 | 87 | 82 | 142 | 0 | 326 | 672 | 454 | 26 | 886 |
N.S. |
1 | 1.00 | 0.94 | 1.63 | 0.00 | 3.75 | 7.72 | 5.22 | 0.30 | 10.18 |
time (sec) |
N/A | 0.495 | 0.473 | 0.339 | 0.000 | 0.118 | 13.950 | 0.145 | 0.151 | 2.188 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | F(-1) | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 120 | 131 | 115 | 141 | 0 | 444 | 0 | 167 | 173 | 126 |
N.S. |
1 | 1.09 | 0.96 | 1.18 | 0.00 | 3.70 | 0.00 | 1.39 | 1.44 | 1.05 |
time (sec) |
N/A | 0.561 | 0.598 | 0.324 | 0.000 | 0.119 | 0.000 | 0.135 | 0.161 | 1.257 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-2) | B | F(-1) | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 187 | 213 | 175 | 270 | 0 | 830 | 0 | 480 | 504 | 283 |
N.S. |
1 | 1.14 | 0.94 | 1.44 | 0.00 | 4.44 | 0.00 | 2.57 | 2.70 | 1.51 |
time (sec) |
N/A | 0.793 | 0.908 | 0.527 | 0.000 | 0.187 | 0.000 | 0.179 | 0.159 | 3.112 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-2) | B | F(-1) | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 260 | 302 | 302 | 456 | 0 | 1334 | 0 | 918 | 1239 | 502 |
N.S. |
1 | 1.16 | 1.16 | 1.75 | 0.00 | 5.13 | 0.00 | 3.53 | 4.77 | 1.93 |
time (sec) |
N/A | 1.101 | 1.266 | 0.917 | 0.000 | 0.225 | 0.000 | 0.192 | 0.162 | 2.876 |