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 | C | B | F | C | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 342 | 371 | 466 | 3535 | 0 | 579 | 0 | 0 | 0 |
| N.S. |
1 | 1.08 | 1.36 | 10.34 | 0.00 | 1.69 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.987 | 19.913 | 11.508 | 0.000 | 0.147 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F | C | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 266 | 290 | 369 | 2625 | 0 | 510 | 0 | 0 | 0 |
| N.S. |
1 | 1.09 | 1.39 | 9.87 | 0.00 | 1.92 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.302 | 17.169 | 10.789 | 0.000 | 0.129 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | C | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 276 | 298 | 106414 | 2153 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.08 | 385.56 | 7.80 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 3.118 | 31.817 | 12.664 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | C | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 272 | 301 | 106467 | 2147 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.11 | 391.42 | 7.89 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 3.023 | 31.373 | 13.788 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | C | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 339 | 367 | 80411 | 2504 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.08 | 237.20 | 7.39 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 3.820 | 42.348 | 15.225 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | F | C | C | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | N/A | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 421 | 0 | 151903 | 3577 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 0.00 | 360.81 | 8.50 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.000 | 34.777 | 16.107 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | F(-1) | F | C | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 519 | 553 | 3647 | 0 | 0 | 743 | 0 | 0 | 0 |
| N.S. |
1 | 1.07 | 7.03 | 0.00 | 0.00 | 1.43 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 4.565 | 30.394 | 180.000 | 0.000 | 0.169 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | F(-1) | F | C | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 425 | 454 | 542 | 0 | 0 | 656 | 0 | 0 | 0 |
| N.S. |
1 | 1.07 | 1.28 | 0.00 | 0.00 | 1.54 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 3.934 | 23.763 | 180.000 | 0.000 | 0.151 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | F(-1) | F | C | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 340 | 369 | 470 | 0 | 0 | 579 | 0 | 0 | 0 |
| N.S. |
1 | 1.09 | 1.38 | 0.00 | 0.00 | 1.70 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 3.066 | 21.064 | 180.000 | 0.000 | 0.139 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | F(-1) | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 342 | 369 | 152077 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.08 | 444.67 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 3.928 | 32.039 | 180.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | C | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 349 | 375 | 151943 | 3088 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.07 | 435.37 | 8.85 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 3.987 | 32.534 | 9.081 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | C | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 359 | 381 | 152213 | 3320 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.06 | 423.99 | 9.25 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 3.919 | 33.089 | 3.916 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | F | C | C | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | N/A | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 422 | 0 | 152026 | 3746 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 0.00 | 360.25 | 8.88 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.000 | 38.607 | 3.865 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | F | C | C | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | N/A | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 513 | 0 | 197608 | 4915 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 0.00 | 385.20 | 9.58 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.000 | 36.451 | 3.925 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F | C | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 280 | 313 | 363 | 2461 | 0 | 510 | 0 | 0 | 0 |
| N.S. |
1 | 1.12 | 1.30 | 8.79 | 0.00 | 1.82 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.414 | 20.278 | 4.530 | 0.000 | 0.142 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F | C | F(-1) | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 212 | 237 | 311 | 1484 | 0 | 452 | 0 | 0 | 0 |
| N.S. |
1 | 1.12 | 1.47 | 7.00 | 0.00 | 2.13 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.757 | 13.075 | 1.417 | 0.000 | 0.113 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F | C | F | F | F(-1) |
| verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD |
| size | 150 | 171 | 260 | 1097 | 0 | 371 | 0 | 0 | 0 |
| N.S. |
1 | 1.14 | 1.73 | 7.31 | 0.00 | 2.47 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.280 | 11.600 | 1.858 | 0.000 | 0.102 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | C | F | F(-1) | F | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 138 | 159 | 16611 | 245 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.15 | 120.37 | 1.78 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.346 | 33.645 | 1.462 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | C | F | F(-1) | F | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 256 | 290 | 51757 | 1196 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.13 | 202.18 | 4.67 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.555 | 37.005 | 1.398 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | C | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 344 | 364 | 78884 | 2335 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.06 | 229.31 | 6.79 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 3.565 | 36.662 | 1.459 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F(-1) | C | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 423 | 448 | 533 | 2914 | 0 | 928 | 0 | 0 | 0 |
| N.S. |
1 | 1.06 | 1.26 | 6.89 | 0.00 | 2.19 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 3.610 | 27.467 | 3.957 | 0.000 | 0.188 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F(-1) | C | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 326 | 354 | 469 | 2044 | 0 | 804 | 0 | 0 | 0 |
| N.S. |
1 | 1.09 | 1.44 | 6.27 | 0.00 | 2.47 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.663 | 22.533 | 1.447 | 0.000 | 0.159 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F(-1) | C | F | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 235 | 270 | 365 | 1239 | 0 | 699 | 0 | 0 | 0 |
| N.S. |
1 | 1.15 | 1.55 | 5.27 | 0.00 | 2.97 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.961 | 14.166 | 1.408 | 0.000 | 0.144 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F | C | F | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 215 | 247 | 328 | 777 | 0 | 626 | 0 | 0 | 0 |
| N.S. |
1 | 1.15 | 1.53 | 3.61 | 0.00 | 2.91 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.791 | 13.519 | 0.338 | 0.000 | 0.119 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | C | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 220 | 250 | 37663 | 1305 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.14 | 171.20 | 5.93 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.011 | 36.459 | 1.394 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | C | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 371 | 407 | 96750 | 2242 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.10 | 260.78 | 6.04 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 4.199 | 36.480 | 1.428 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | F | C | C | F(-1) | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | N/A | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 487 | 0 | 141550 | 3532 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 0.00 | 290.66 | 7.25 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.000 | 37.155 | 1.485 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F | C | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 588 | 615 | 4179 | 7501 | 0 | 1553 | 0 | 0 | 0 |
| N.S. |
1 | 1.05 | 7.11 | 12.76 | 0.00 | 2.64 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 5.124 | 31.647 | 4.120 | 0.000 | 0.327 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F | C | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 472 | 502 | 626 | 5950 | 0 | 1365 | 0 | 0 | 0 |
| N.S. |
1 | 1.06 | 1.33 | 12.61 | 0.00 | 2.89 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 3.834 | 21.181 | 1.527 | 0.000 | 0.248 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F | C | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 368 | 405 | 621 | 4415 | 0 | 1210 | 0 | 0 | 0 |
| N.S. |
1 | 1.10 | 1.69 | 12.00 | 0.00 | 3.29 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.910 | 22.844 | 1.536 | 0.000 | 0.200 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F | C | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 346 | 380 | 463 | 3233 | 0 | 1093 | 0 | 0 | 0 |
| N.S. |
1 | 1.10 | 1.34 | 9.34 | 0.00 | 3.16 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.771 | 21.142 | 1.491 | 0.000 | 0.172 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F | C | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 329 | 359 | 487 | 2658 | 0 | 973 | 0 | 0 | 0 |
| N.S. |
1 | 1.09 | 1.48 | 8.08 | 0.00 | 2.96 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.673 | 21.774 | 1.441 | 0.000 | 0.155 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | C | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 399 | 439 | 98645 | 4386 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.10 | 247.23 | 10.99 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 4.351 | 36.291 | 1.518 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | F | C | C | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | N/A | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 526 | 0 | 243381 | 7297 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 0.00 | 462.70 | 13.87 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.000 | 34.834 | 1.570 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |