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 | F | B | B | F(-1) | F | B |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 384 | 312 | 899 | 0 | 892 | 777 | 0 | 0 | 1110 |
| N.S. |
1 | 0.81 | 2.34 | 0.00 | 2.32 | 2.02 | 0.00 | 0.00 | 2.89 |
| time (sec) |
N/A | 1.471 | 13.448 | 0.000 | 0.394 | 0.321 | 0.000 | 0.000 | 22.874 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | B | A | F(-1) | F | B |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 285 | 255 | 264 | 0 | 648 | 461 | 0 | 0 | 714 |
| N.S. |
1 | 0.89 | 0.93 | 0.00 | 2.27 | 1.62 | 0.00 | 0.00 | 2.51 |
| time (sec) |
N/A | 1.188 | 5.871 | 0.000 | 0.361 | 0.305 | 0.000 | 0.000 | 20.948 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | B | A | F | F | B |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 180 | 183 | 160 | 0 | 441 | 261 | 0 | 0 | 185 |
| N.S. |
1 | 1.02 | 0.89 | 0.00 | 2.45 | 1.45 | 0.00 | 0.00 | 1.03 |
| time (sec) |
N/A | 0.949 | 1.992 | 0.000 | 0.370 | 0.294 | 0.000 | 0.000 | 17.566 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | F | F | F(-1) | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 123 | 128 | 103 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.04 | 0.84 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.640 | 32.981 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | F | F | F(-2) | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 202 | 214 | 110 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.06 | 0.54 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.109 | 52.516 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | F | F(-1) | F(-2) | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 207 | 223 | 131 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.08 | 0.63 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.144 | 36.826 | 0.000 | 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 | A | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 167 | 181 | 190 | 541 | 0 | 0 | 0 | 197 | 0 |
| N.S. |
1 | 1.08 | 1.14 | 3.24 | 0.00 | 0.00 | 0.00 | 1.18 | 0.00 |
| time (sec) |
N/A | 1.123 | 2.958 | 3.852 | 0.000 | 0.000 | 0.000 | 0.451 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 257 | 306 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.19 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.184 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 366 | 388 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.06 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.101 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 392 | 423 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.08 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.288 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 385 | 406 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.05 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.162 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 375 | 396 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.06 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.125 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 365 | 395 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.08 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.158 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 413 | 431 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.04 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.269 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 424 | 451 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.06 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.288 | 0.000 | 0.000 | 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 | A | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 174 | 188 | 196 | 747 | 0 | 0 | 0 | 220 | 0 |
| N.S. |
1 | 1.08 | 1.13 | 4.29 | 0.00 | 0.00 | 0.00 | 1.26 | 0.00 |
| time (sec) |
N/A | 1.136 | 3.129 | 3.905 | 0.000 | 0.000 | 0.000 | 0.467 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 269 | 318 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.18 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.205 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | C | F | B | B | F(-1) | F | B |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 435 | 336 | 1029 | 0 | 1324 | 937 | 0 | 0 | 1253 |
| N.S. |
1 | 0.77 | 2.37 | 0.00 | 3.04 | 2.15 | 0.00 | 0.00 | 2.88 |
| time (sec) |
N/A | 1.486 | 13.210 | 0.000 | 0.423 | 0.377 | 0.000 | 0.000 | 23.241 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | B | A | F(-1) | F | B |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 322 | 279 | 306 | 0 | 950 | 563 | 0 | 0 | 790 |
| N.S. |
1 | 0.87 | 0.95 | 0.00 | 2.95 | 1.75 | 0.00 | 0.00 | 2.45 |
| time (sec) |
N/A | 1.220 | 6.435 | 0.000 | 0.380 | 0.340 | 0.000 | 0.000 | 24.214 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | B | A | F | F | B |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 197 | 200 | 177 | 0 | 644 | 310 | 0 | 0 | 510 |
| N.S. |
1 | 1.02 | 0.90 | 0.00 | 3.27 | 1.57 | 0.00 | 0.00 | 2.59 |
| time (sec) |
N/A | 0.996 | 2.066 | 0.000 | 0.375 | 0.293 | 0.000 | 0.000 | 19.684 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | F | F | F(-1) | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 170 | 175 | 159 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.03 | 0.94 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.931 | 31.181 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | F | F | F(-2) | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 216 | 228 | 162 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.06 | 0.75 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.168 | 34.092 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | F | F(-1) | F(-2) | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 230 | 246 | 228 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.07 | 0.99 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.247 | 29.424 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | F | F | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 232 | 299 | 196 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.29 | 0.84 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.278 | 1.161 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 383 | 410 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.07 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.172 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 410 | 441 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.08 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.331 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 406 | 425 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.05 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.216 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 396 | 415 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.05 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.167 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 389 | 415 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.07 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.215 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 433 | 451 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.04 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.344 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 451 | 478 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.06 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.410 | 0.000 | 0.000 | 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 | A | A | B | A | B |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 81 | 118 | 92 | 80 | 102 | 71 | 189 | 76 | 93 |
| N.S. |
1 | 1.46 | 1.14 | 0.99 | 1.26 | 0.88 | 2.33 | 0.94 | 1.15 |
| time (sec) |
N/A | 0.349 | 0.278 | 1.126 | 0.224 | 0.270 | 0.135 | 0.340 | 14.098 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | B | F | C | F | F | B |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 117 | 120 | 97 | 396 | 0 | 232 | 0 | 0 | 169 |
| N.S. |
1 | 1.03 | 0.83 | 3.38 | 0.00 | 1.98 | 0.00 | 0.00 | 1.44 |
| time (sec) |
N/A | 0.655 | 1.567 | 2.800 | 0.000 | 0.108 | 0.000 | 0.000 | 15.719 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | N/A | N/A | N/A | F(-1) | N/A | F(-1) | F(-1) | N/A |
| verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 45 | 45 | 47 | 45 | 0 | 54 | 0 | 0 | 47 |
| N.S. |
1 | 1.00 | 1.04 | 1.00 | 0.00 | 1.20 | 0.00 | 0.00 | 1.04 |
| time (sec) |
N/A | 0.292 | 24.925 | 2.037 | 0.000 | 1.429 | 0.000 | 0.000 | 73.796 |