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 | B | F | F(-2) | F(-1) |
| verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD |
| size | 547 | 532 | 735 | 9581948 | 0 | 24241 | 0 | 0 | 0 |
| N.S. |
1 | 0.97 | 1.34 | 17517.27 | 0.00 | 44.32 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.159 | 6.770 | 3.548 | 0.000 | 6.095 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F | B | F | F(-2) | F(-1) |
| verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD |
| size | 384 | 376 | 397 | 9581108 | 0 | 23719 | 0 | 0 | 0 |
| N.S. |
1 | 0.98 | 1.03 | 24950.80 | 0.00 | 61.77 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.858 | 1.839 | 1.742 | 0.000 | 5.734 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F(-2) | B | F | F(-2) | F(-1) |
| verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD |
| size | 294 | 293 | 256 | 9339148 | 0 | 5853 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 0.87 | 31765.81 | 0.00 | 19.91 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.528 | 0.612 | 1.041 | 0.000 | 2.134 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | F(-1) | F | B | F | F(-2) | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 349 | 346 | 344 | 0 | 0 | 10333 | 0 | 0 | 0 |
| N.S. |
1 | 0.99 | 0.99 | 0.00 | 0.00 | 29.61 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.853 | 0.738 | 180.000 | 0.000 | 1.429 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | F(-1) | F | B | F | F(-2) | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 501 | 487 | 422 | 0 | 0 | 10495 | 0 | 0 | 0 |
| N.S. |
1 | 0.97 | 0.84 | 0.00 | 0.00 | 20.95 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.976 | 4.501 | 180.000 | 0.000 | 1.633 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | F | C | B | F | B | F | F | F(-1) |
| verified |
N/A | N/A | Yes | No | TBD | TBD | TBD | TBD | TBD |
| size | 976 | 0 | 1214 | 17768513 | 0 | 12721 | 0 | 0 | 0 |
| N.S. |
1 | 0.00 | 1.24 | 18205.44 | 0.00 | 13.03 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.000 | 6.709 | 6.857 | 0.000 | 3.017 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | F | C | B | F | B | F | F | F(-1) |
| verified |
N/A | N/A | Yes | No | TBD | TBD | TBD | TBD | TBD |
| size | 747 | 0 | 381 | 17768080 | 0 | 11897 | 0 | 0 | 0 |
| N.S. |
1 | 0.00 | 0.51 | 23785.92 | 0.00 | 15.93 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.000 | 6.297 | 3.965 | 0.000 | 2.413 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F | B | F | F | F(-1) |
| verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD |
| size | 602 | 666 | 324 | 17767874 | 0 | 11351 | 0 | 0 | 0 |
| N.S. |
1 | 1.11 | 0.54 | 29514.74 | 0.00 | 18.86 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 28.684 | 1.433 | 2.437 | 0.000 | 2.295 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | F | C | F(-1) | F(-2) | B | F | F | F(-1) |
| verified |
N/A | N/A | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 570 | 0 | 283 | 0 | 0 | 4847 | 0 | 0 | 0 |
| N.S. |
1 | 0.00 | 0.50 | 0.00 | 0.00 | 8.50 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.000 | 0.733 | 0.000 | 0.000 | 0.676 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | F | C | F(-1) | F | B | F | F | F(-1) |
| verified |
N/A | N/A | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 691 | 0 | 363 | 0 | 0 | 5018 | 0 | 0 | 0 |
| N.S. |
1 | 0.00 | 0.53 | 0.00 | 0.00 | 7.26 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.000 | 2.479 | 180.000 | 0.000 | 0.979 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F(-1) | B | F | F(-2) | F(-1) |
| verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD |
| size | 1189 | 1159 | 2097 | 13067599 | 0 | 89177 | 0 | 0 | 0 |
| N.S. |
1 | 0.97 | 1.76 | 10990.41 | 0.00 | 75.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 5.631 | 7.236 | 1.567 | 0.000 | 35.377 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F(-1) | B | F | F(-2) | F(-1) |
| verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD |
| size | 865 | 848 | 1674 | 13067692 | 0 | 84903 | 0 | 0 | 0 |
| N.S. |
1 | 0.98 | 1.94 | 15107.16 | 0.00 | 98.15 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 4.437 | 6.909 | 2.906 | 0.000 | 29.832 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F(-2) | B | F | F(-2) | F(-1) |
| verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD |
| size | 686 | 679 | 1419 | 13067316 | 0 | 21092 | 0 | 0 | 0 |
| N.S. |
1 | 0.99 | 2.07 | 19048.57 | 0.00 | 30.75 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 4.291 | 6.681 | 1.534 | 0.000 | 12.127 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | B | F(-2) | B | F | F(-2) | F(-1) |
| verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD |
| size | 635 | 716 | 460 | 13066366 | 0 | 21090 | 0 | 0 | 0 |
| N.S. |
1 | 1.13 | 0.72 | 20576.95 | 0.00 | 33.21 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.218 | 6.398 | 1.243 | 0.000 | 11.870 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | F(-1) | F(-1) | B | F | F(-2) | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 749 | 740 | 934 | 0 | 0 | 39885 | 0 | 0 | 0 |
| N.S. |
1 | 0.99 | 1.25 | 0.00 | 0.00 | 53.25 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 4.462 | 6.547 | 180.000 | 0.000 | 8.077 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | F(-1) | F(-1) | B | F | F(-2) | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 1008 | 985 | 1401 | 0 | 0 | 40633 | 0 | 0 | 0 |
| N.S. |
1 | 0.98 | 1.39 | 0.00 | 0.00 | 40.31 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 4.637 | 6.660 | 180.000 | 0.000 | 7.929 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | A | F(-1) | B | F | F(-1) | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 182 | 181 | 261 | 232 | 0 | 2100 | 0 | 0 | 0 |
| N.S. |
1 | 0.99 | 1.43 | 1.27 | 0.00 | 11.54 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.466 | 3.623 | 0.716 | 0.000 | 1.312 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | A | F | B | F | F(-1) | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 141 | 136 | 212 | 153 | 0 | 1695 | 0 | 0 | 0 |
| N.S. |
1 | 0.96 | 1.50 | 1.09 | 0.00 | 12.02 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.414 | 0.838 | 0.098 | 0.000 | 1.035 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | A | F | B | F | F(-1) | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 79 | 79 | 139 | 102 | 0 | 433 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 1.76 | 1.29 | 0.00 | 5.48 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.298 | 0.576 | 0.203 | 0.000 | 0.462 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | B | F | F(-1) | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 142 | 138 | 197 | 0 | 0 | 1141 | 0 | 0 | 0 |
| N.S. |
1 | 0.97 | 1.39 | 0.00 | 0.00 | 8.04 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.424 | 1.185 | 0.000 | 0.000 | 1.009 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F(-1) | A | F | F(-1) | F(-1) |
| verified |
N/A | No | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 249 | 235 | 258 | 0 | 0 | 1444 | 0 | 0 | 0 |
| N.S. |
1 | 0.94 | 1.04 | 0.00 | 0.00 | 5.80 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.505 | 1.379 | 0.000 | 0.000 | 1.377 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | A | F | B | F | F(-1) | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 270 | 279 | 1017 | 455 | 0 | 3019 | 0 | 0 | 0 |
| N.S. |
1 | 1.03 | 3.77 | 1.69 | 0.00 | 11.18 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.647 | 6.499 | 0.122 | 0.000 | 2.950 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | A | F | B | F | F(-1) | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 209 | 211 | 777 | 318 | 0 | 2344 | 0 | 0 | 0 |
| N.S. |
1 | 1.01 | 3.72 | 1.52 | 0.00 | 11.22 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.510 | 6.358 | 0.102 | 0.000 | 2.337 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | A | F | B | F | F(-1) | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 179 | 173 | 246 | 217 | 0 | 1932 | 0 | 0 | 0 |
| N.S. |
1 | 0.97 | 1.37 | 1.21 | 0.00 | 10.79 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.423 | 2.464 | 0.086 | 0.000 | 1.748 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | A | F | F(-1) | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 203 | 193 | 254 | 0 | 0 | 2522 | 0 | 0 | 0 |
| N.S. |
1 | 0.95 | 1.25 | 0.00 | 0.00 | 12.42 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.503 | 0.768 | 0.000 | 0.000 | 0.705 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | A | F | F(-1) | F(-1) |
| verified |
N/A | No | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 435 | 409 | 235 | 0 | 0 | 1282 | 0 | 0 | 0 |
| N.S. |
1 | 0.94 | 0.54 | 0.00 | 0.00 | 2.95 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.635 | 1.290 | 0.000 | 0.000 | 1.759 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | B | F(-2) | B | F | F(-1) | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 236 | 234 | 339 | 686 | 0 | 6011 | 0 | 0 | 0 |
| N.S. |
1 | 0.99 | 1.44 | 2.91 | 0.00 | 25.47 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.606 | 5.141 | 0.590 | 0.000 | 2.596 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | B | F | B | F | F(-1) | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 160 | 159 | 204 | 507 | 0 | 1743 | 0 | 0 | 0 |
| N.S. |
1 | 0.99 | 1.28 | 3.17 | 0.00 | 10.89 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.428 | 1.974 | 0.087 | 0.000 | 0.630 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | B | F | B | F | F(-1) | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 153 | 152 | 195 | 457 | 0 | 1717 | 0 | 0 | 0 |
| N.S. |
1 | 0.99 | 1.27 | 2.99 | 0.00 | 11.22 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.414 | 1.709 | 0.109 | 0.000 | 0.622 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | B | F | B | F | F(-1) | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 156 | 155 | 264 | 406 | 0 | 1771 | 0 | 0 | 0 |
| N.S. |
1 | 0.99 | 1.69 | 2.60 | 0.00 | 11.35 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.382 | 10.901 | 0.093 | 0.000 | 0.660 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F(-2) | B | F | F(-1) | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 280 | 271 | 351 | 0 | 0 | 4153 | 0 | 0 | 0 |
| N.S. |
1 | 0.97 | 1.25 | 0.00 | 0.00 | 14.83 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.549 | 13.755 | 0.000 | 0.000 | 2.669 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F(-1) | B | F | F(-1) | F(-1) |
| verified |
N/A | No | No | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 478 | 454 | 562 | 0 | 0 | 5274 | 0 | 0 | 0 |
| N.S. |
1 | 0.95 | 1.18 | 0.00 | 0.00 | 11.03 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.675 | 15.974 | 0.000 | 0.000 | 3.339 | 0.000 | 0.000 | 0.000 |