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 | Mupad | |
grade |
N/A | A | A | B | F(-2) | A | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 57 | 57 | 56 | 96 | 0 | 227 | 688 | 361 | 1537 |
N.S. |
1 | 1.00 | 0.98 | 1.68 | 0.00 | 3.98 | 12.07 | 6.33 | 26.96 |
time (sec) |
N/A | 0.320 | 0.112 | 0.354 | 0.000 | 0.265 | 10.690 | 0.317 | 6.172 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 19 | 19 | 19 | 19 | 19 | 28 | 17 | 18 | 18 |
N.S. |
1 | 1.00 | 1.00 | 1.00 | 1.00 | 1.47 | 0.89 | 0.95 | 0.95 |
time (sec) |
N/A | 0.250 | 0.032 | 0.198 | 0.225 | 0.246 | 0.146 | 0.299 | 3.113 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 23 | 23 | 20 | 31 | 19 | 25 | 27 | 39 | 30 |
N.S. |
1 | 1.00 | 0.87 | 1.35 | 0.83 | 1.09 | 1.17 | 1.70 | 1.30 |
time (sec) |
N/A | 0.264 | 0.166 | 0.205 | 0.200 | 0.230 | 0.235 | 0.314 | 2.718 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 58 | 58 | 57 | 97 | 0 | 231 | 804 | 407 | 2219 |
N.S. |
1 | 1.00 | 0.98 | 1.67 | 0.00 | 3.98 | 13.86 | 7.02 | 38.26 |
time (sec) |
N/A | 0.311 | 0.088 | 0.312 | 0.000 | 0.264 | 11.396 | 0.316 | 13.516 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 58 | 58 | 55 | 85 | 0 | 235 | 748 | 103 | 2213 |
N.S. |
1 | 1.00 | 0.95 | 1.47 | 0.00 | 4.05 | 12.90 | 1.78 | 38.16 |
time (sec) |
N/A | 0.326 | 0.140 | 0.319 | 0.000 | 0.262 | 11.271 | 0.305 | 13.900 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 65 | 65 | 61 | 108 | 0 | 263 | 0 | 121 | 1540 |
N.S. |
1 | 1.00 | 0.94 | 1.66 | 0.00 | 4.05 | 0.00 | 1.86 | 23.69 |
time (sec) |
N/A | 0.342 | 0.253 | 0.512 | 0.000 | 0.300 | 0.000 | 0.293 | 6.282 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 100 | 100 | 134 | 96 | 0 | 266 | 0 | 116 | 419 |
N.S. |
1 | 1.00 | 1.34 | 0.96 | 0.00 | 2.66 | 0.00 | 1.16 | 4.19 |
time (sec) |
N/A | 0.372 | 0.334 | 0.529 | 0.000 | 2.767 | 0.000 | 0.318 | 4.195 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 99 | 99 | 116 | 95 | 0 | 265 | 0 | 115 | 417 |
N.S. |
1 | 1.00 | 1.17 | 0.96 | 0.00 | 2.68 | 0.00 | 1.16 | 4.21 |
time (sec) |
N/A | 0.498 | 0.280 | 0.395 | 0.000 | 3.585 | 0.000 | 0.311 | 3.626 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | B | F(-2) | B | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 247 | 247 | 526 | 484 | 0 | 1049 | 0 | 607 | 9992 |
N.S. |
1 | 1.00 | 2.13 | 1.96 | 0.00 | 4.25 | 0.00 | 2.46 | 40.45 |
time (sec) |
N/A | 0.685 | 5.444 | 2.135 | 0.000 | 94.181 | 0.000 | 0.393 | 13.036 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | B | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 170 | 170 | 335 | 291 | 0 | 747 | 0 | 340 | 6735 |
N.S. |
1 | 1.00 | 1.97 | 1.71 | 0.00 | 4.39 | 0.00 | 2.00 | 39.62 |
time (sec) |
N/A | 0.580 | 2.468 | 1.322 | 0.000 | 21.241 | 0.000 | 0.376 | 10.762 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | B | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 103 | 103 | 135 | 165 | 0 | 505 | 0 | 196 | 3577 |
N.S. |
1 | 1.00 | 1.31 | 1.60 | 0.00 | 4.90 | 0.00 | 1.90 | 34.73 |
time (sec) |
N/A | 0.509 | 2.023 | 0.842 | 0.000 | 2.940 | 0.000 | 0.363 | 7.959 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 76 | 76 | 112 | 92 | 0 | 296 | 0 | 128 | 345 |
N.S. |
1 | 1.00 | 1.47 | 1.21 | 0.00 | 3.89 | 0.00 | 1.68 | 4.54 |
time (sec) |
N/A | 0.442 | 0.492 | 0.559 | 0.000 | 0.516 | 0.000 | 0.319 | 5.078 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 121 | 121 | 106 | 108 | 0 | 1022 | 0 | 507 | 2665 |
N.S. |
1 | 1.00 | 0.88 | 0.89 | 0.00 | 8.45 | 0.00 | 4.19 | 22.02 |
time (sec) |
N/A | 0.509 | 0.653 | 1.043 | 0.000 | 1.619 | 0.000 | 0.408 | 5.248 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | B | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 187 | 221 | 205 | 210 | 0 | 2835 | 0 | 333 | 20827 |
N.S. |
1 | 1.18 | 1.10 | 1.12 | 0.00 | 15.16 | 0.00 | 1.78 | 111.37 |
time (sec) |
N/A | 0.960 | 1.459 | 2.897 | 0.000 | 81.938 | 0.000 | 0.382 | 17.788 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | F(-1) | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 458 | 458 | 319 | 412 | 0 | 0 | 0 | 748 | 52103 |
N.S. |
1 | 1.00 | 0.70 | 0.90 | 0.00 | 0.00 | 0.00 | 1.63 | 113.76 |
time (sec) |
N/A | 1.185 | 2.953 | 7.633 | 0.000 | 0.000 | 0.000 | 0.539 | 23.607 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 213 | 213 | 184 | 318 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.86 | 1.49 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.820 | 8.944 | 6.349 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | F | F(-1) | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 102 | 102 | 187 | 204 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.83 | 2.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.444 | 9.666 | 2.838 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 87 | 87 | 82 | 142 | 0 | 326 | 672 | 454 | 886 |
N.S. |
1 | 1.00 | 0.94 | 1.63 | 0.00 | 3.75 | 7.72 | 5.22 | 10.18 |
time (sec) |
N/A | 0.476 | 0.509 | 0.551 | 0.000 | 0.301 | 14.147 | 0.338 | 5.870 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | F(-1) | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 120 | 131 | 115 | 141 | 0 | 444 | 0 | 167 | 126 |
N.S. |
1 | 1.09 | 0.96 | 1.18 | 0.00 | 3.70 | 0.00 | 1.39 | 1.05 |
time (sec) |
N/A | 0.536 | 0.632 | 0.545 | 0.000 | 0.305 | 0.000 | 0.331 | 2.923 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | B | F(-1) | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 187 | 213 | 175 | 270 | 0 | 830 | 0 | 480 | 283 |
N.S. |
1 | 1.14 | 0.94 | 1.44 | 0.00 | 4.44 | 0.00 | 2.57 | 1.51 |
time (sec) |
N/A | 0.762 | 0.955 | 0.875 | 0.000 | 0.314 | 0.000 | 0.370 | 6.576 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | B | F(-1) | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 260 | 302 | 302 | 456 | 0 | 1334 | 0 | 918 | 502 |
N.S. |
1 | 1.16 | 1.16 | 1.75 | 0.00 | 5.13 | 0.00 | 3.53 | 1.93 |
time (sec) |
N/A | 1.061 | 1.316 | 1.342 | 0.000 | 0.348 | 0.000 | 0.385 | 5.715 |