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 | A | F | C | F | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 700 | 699 | 806 | 866 | 0 | 1236 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 1.15 | 1.24 | 0.00 | 1.77 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.651 | 28.405 | 2.793 | 0.000 | 0.150 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | A | F | C | F | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 405 | 404 | 450 | 625 | 0 | 842 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 1.11 | 1.54 | 0.00 | 2.08 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.567 | 24.188 | 2.610 | 0.000 | 0.131 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | A | F | C | F | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 284 | 284 | 319 | 498 | 0 | 671 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 1.12 | 1.75 | 0.00 | 2.36 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.124 | 19.292 | 2.578 | 0.000 | 0.123 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | A | F | F(-1) | F | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 313 | 313 | 245 | 478 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 0.78 | 1.53 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.503 | 22.947 | 2.962 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | A | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 678 | 678 | 3412 | 1208 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 5.03 | 1.78 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.000 | 34.128 | 3.956 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | B | F | F(-1) | F | F | F(-1) |
| verified |
N/A | No | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 981 | 976 | 21961 | 1814 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 0.99 | 22.39 | 1.85 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.301 | 36.590 | 5.178 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | B | F | F(-1) | F | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 736 | 735 | 8030 | 1544 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 10.91 | 2.10 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.499 | 42.322 | 5.146 | 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 | 442 | 442 | 586 | 848 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 1.33 | 1.92 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.316 | 24.657 | 6.528 | 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 | F | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 606 | 606 | 333 | 2250 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 0.55 | 3.71 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.727 | 26.023 | 7.694 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | B | F | F | F(-1) | F | F(-1) |
| verified |
N/A | No | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 1081 | 1080 | 10828 | 3389 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 10.02 | 3.14 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.252 | 39.612 | 10.176 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | B | F | F(-1) | F | F | F(-1) |
| verified |
N/A | No | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 898 | 897 | 15131 | 1809 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 16.85 | 2.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.717 | 35.437 | 5.174 | 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 | 472 | 472 | 443 | 1560 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 0.94 | 3.31 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.410 | 36.156 | 5.172 | 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 | 449 | 449 | 723 | 855 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 1.61 | 1.90 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.318 | 25.235 | 6.278 | 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 | F | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 625 | 625 | 341 | 2298 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 0.55 | 3.68 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.727 | 25.784 | 7.746 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | B | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 1090 | 1090 | 10790 | 3571 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 9.90 | 3.28 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.018 | 38.038 | 9.421 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | A | F | C | F | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 721 | 720 | 825 | 880 | 0 | 1267 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 1.14 | 1.22 | 0.00 | 1.76 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.203 | 28.445 | 2.498 | 0.000 | 0.159 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | A | F | C | F | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 410 | 410 | 442 | 637 | 0 | 859 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 1.08 | 1.55 | 0.00 | 2.10 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.397 | 25.003 | 2.336 | 0.000 | 0.123 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | A | F | C | F | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 291 | 291 | 326 | 506 | 0 | 682 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 1.12 | 1.74 | 0.00 | 2.34 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.178 | 21.099 | 2.342 | 0.000 | 0.125 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | A | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 309 | 309 | 249 | 475 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 0.81 | 1.54 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.519 | 22.056 | 3.028 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | A | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 680 | 680 | 3419 | 1211 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 5.03 | 1.78 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.128 | 34.657 | 3.928 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | B | F | F(-1) | F | F | F(-1) |
| verified |
N/A | No | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 980 | 976 | 21961 | 1834 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 22.41 | 1.87 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.893 | 36.913 | 5.277 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | B | F | F(-1) | F | F | F(-1) |
| verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 734 | 732 | 8107 | 1552 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 11.04 | 2.11 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.483 | 42.832 | 4.901 | 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 | 436 | 436 | 583 | 856 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 1.34 | 1.96 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.370 | 25.181 | 6.313 | 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 | F(-1) | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 616 | 616 | 340 | 2249 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 0.55 | 3.65 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.813 | 26.043 | 7.511 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | B | F | F | F(-1) | F | F(-1) |
| verified |
N/A | No | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 1128 | 1119 | 10836 | 3425 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 0.99 | 9.61 | 3.04 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.987 | 40.007 | 10.267 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | A | F | C | F | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 1097 | 1083 | 1291 | 1238 | 0 | 1665 | 0 | 0 | 0 |
| N.S. |
1 | 0.99 | 1.18 | 1.13 | 0.00 | 1.52 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.128 | 32.538 | 3.368 | 0.000 | 0.174 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | A | F | C | F | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 611 | 608 | 686 | 824 | 0 | 1068 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 1.12 | 1.35 | 0.00 | 1.75 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.843 | 26.261 | 2.808 | 0.000 | 0.149 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | A | F | C | F | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 368 | 367 | 390 | 611 | 0 | 775 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 1.06 | 1.66 | 0.00 | 2.11 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.321 | 23.770 | 2.322 | 0.000 | 0.146 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | A | F | F(-1) | F | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 465 | 465 | 1036 | 750 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 2.23 | 1.61 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.505 | 26.549 | 3.018 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | A | F | F(-1) | F(-1) | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 738 | 738 | 3935 | 1269 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 5.33 | 1.72 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.225 | 35.647 | 3.939 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | A | F | F(-1) | F | F | F(-1) |
| verified |
N/A | No | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 1395 | 1376 | 39032 | 2228 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 0.99 | 27.98 | 1.60 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 3.978 | 40.077 | 6.740 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | B | F | F(-1) | F | F | F(-1) |
| verified |
N/A | No | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 937 | 936 | 16972 | 1794 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 18.11 | 1.91 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.556 | 36.207 | 5.232 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | A | F | F(-1) | F | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 757 | 757 | 6321 | 1065 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 8.35 | 1.41 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.741 | 34.840 | 6.366 | 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(-2) | F(-1) |
| verified |
N/A | No | No | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 867 | 867 | 721 | 2286 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 0.83 | 2.64 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.229 | 31.921 | 7.834 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | B | F | F | F(-1) | F | F(-1) |
| verified |
N/A | No | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 1070 | 1070 | 11363 | 3342 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 10.62 | 3.12 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.319 | 40.542 | 10.352 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |