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 | Reduce | Mupad | |
| grade |
N/A | A | C | F | B | B | F(-1) | F | F | B |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 384 | 312 | 899 | 0 | 892 | 777 | 0 | 0 | 215 | 1110 |
| N.S. |
1 | 0.81 | 2.34 | 0.00 | 2.32 | 2.02 | 0.00 | 0.00 | 0.56 | 2.89 |
| time (sec) |
N/A | 1.475 | 15.587 | 0.000 | 0.182 | 0.146 | 0.000 | 0.000 | 0.322 | 54.808 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | B | A | F(-1) | F | F | B |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 285 | 255 | 264 | 0 | 648 | 461 | 0 | 0 | 141 | 714 |
| N.S. |
1 | 0.89 | 0.93 | 0.00 | 2.27 | 1.62 | 0.00 | 0.00 | 0.49 | 2.51 |
| time (sec) |
N/A | 1.198 | 9.399 | 0.000 | 0.173 | 0.127 | 0.000 | 0.000 | 0.237 | 47.979 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | B | A | F | F | F | B |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 180 | 183 | 160 | 0 | 441 | 261 | 0 | 0 | 68 | 185 |
| N.S. |
1 | 1.02 | 0.89 | 0.00 | 2.45 | 1.45 | 0.00 | 0.00 | 0.38 | 1.03 |
| time (sec) |
N/A | 0.931 | 2.819 | 0.000 | 0.156 | 0.100 | 0.000 | 0.000 | 0.204 | 42.354 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 123 | 128 | 103 | 0 | 0 | 0 | 0 | 0 | 92 | 0 |
| N.S. |
1 | 1.04 | 0.84 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.75 | 0.00 |
| time (sec) |
N/A | 0.648 | 36.804 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.183 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | F | F | F | F(-2) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 202 | 214 | 110 | 0 | 0 | 0 | 0 | 0 | 111 | 0 |
| N.S. |
1 | 1.06 | 0.54 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.55 | 0.00 |
| time (sec) |
N/A | 1.099 | 64.316 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.191 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | F | F | F(-1) | F(-2) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 207 | 223 | 131 | 0 | 0 | 0 | 0 | 0 | 132 | 0 |
| N.S. |
1 | 1.08 | 0.63 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.64 | 0.00 |
| time (sec) |
N/A | 1.157 | 53.966 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.186 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | B | F | F | F | F(-2) | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 167 | 181 | 190 | 588 | 0 | 0 | 0 | 0 | 130 | 0 |
| N.S. |
1 | 1.08 | 1.14 | 3.52 | 0.00 | 0.00 | 0.00 | 0.00 | 0.78 | 0.00 |
| time (sec) |
N/A | 1.120 | 4.783 | 6.955 | 0.000 | 0.000 | 0.000 | 0.000 | 0.189 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 256 | 306 | 0 | 0 | 0 | 0 | 0 | 0 | 69 | 0 |
| N.S. |
1 | 1.20 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.27 | 0.00 |
| time (sec) |
N/A | 1.193 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.233 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 358 | 388 | 0 | 0 | 0 | 0 | 0 | 0 | 67 | 0 |
| N.S. |
1 | 1.08 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.19 | 0.00 |
| time (sec) |
N/A | 1.097 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.219 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 382 | 423 | 0 | 0 | 0 | 0 | 0 | 0 | 173 | 0 |
| N.S. |
1 | 1.11 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.45 | 0.00 |
| time (sec) |
N/A | 1.348 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.226 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 372 | 406 | 0 | 0 | 0 | 0 | 0 | 0 | 140 | 0 |
| N.S. |
1 | 1.09 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.38 | 0.00 |
| time (sec) |
N/A | 1.219 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.265 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 366 | 396 | 0 | 0 | 0 | 0 | 0 | 0 | 65 | 0 |
| N.S. |
1 | 1.08 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.18 | 0.00 |
| time (sec) |
N/A | 1.180 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.213 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 364 | 395 | 0 | 0 | 0 | 0 | 0 | 0 | 89 | 0 |
| N.S. |
1 | 1.09 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.24 | 0.00 |
| time (sec) |
N/A | 1.182 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.195 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 405 | 431 | 0 | 0 | 0 | 0 | 0 | 0 | 121 | 0 |
| N.S. |
1 | 1.06 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.30 | 0.00 |
| time (sec) |
N/A | 1.295 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.204 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | B | F | F | F | F(-2) | F | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 174 | 188 | 196 | 897 | 0 | 0 | 0 | 0 | 191 | 0 |
| N.S. |
1 | 1.08 | 1.13 | 5.16 | 0.00 | 0.00 | 0.00 | 0.00 | 1.10 | 0.00 |
| time (sec) |
N/A | 1.151 | 4.956 | 13.925 | 0.000 | 0.000 | 0.000 | 0.000 | 0.187 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 268 | 318 | 0 | 0 | 0 | 0 | 0 | 0 | 105 | 0 |
| N.S. |
1 | 1.19 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.39 | 0.00 |
| time (sec) |
N/A | 1.224 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.335 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | C | F | B | B | F(-1) | F | F | B |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 435 | 336 | 1029 | 0 | 1324 | 937 | 0 | 0 | 322 | 1253 |
| N.S. |
1 | 0.77 | 2.37 | 0.00 | 3.04 | 2.15 | 0.00 | 0.00 | 0.74 | 2.88 |
| time (sec) |
N/A | 1.508 | 15.927 | 0.000 | 0.213 | 0.169 | 0.000 | 0.000 | 0.360 | 47.127 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | B | A | F(-1) | F | F | B |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 322 | 279 | 306 | 0 | 950 | 563 | 0 | 0 | 212 | 790 |
| N.S. |
1 | 0.87 | 0.95 | 0.00 | 2.95 | 1.75 | 0.00 | 0.00 | 0.66 | 2.45 |
| time (sec) |
N/A | 1.241 | 11.047 | 0.000 | 0.192 | 0.135 | 0.000 | 0.000 | 0.252 | 51.634 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | B | A | F | F | F | B |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 197 | 200 | 177 | 0 | 644 | 310 | 0 | 0 | 102 | 510 |
| N.S. |
1 | 1.02 | 0.90 | 0.00 | 3.27 | 1.57 | 0.00 | 0.00 | 0.52 | 2.59 |
| time (sec) |
N/A | 0.992 | 3.076 | 0.000 | 0.178 | 0.104 | 0.000 | 0.000 | 0.221 | 39.554 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 170 | 175 | 159 | 0 | 0 | 0 | 0 | 0 | 136 | 0 |
| N.S. |
1 | 1.03 | 0.94 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.80 | 0.00 |
| time (sec) |
N/A | 0.929 | 40.443 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.192 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | F | F | F | F(-2) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 216 | 228 | 162 | 0 | 0 | 0 | 0 | 0 | 165 | 0 |
| N.S. |
1 | 1.06 | 0.75 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.76 | 0.00 |
| time (sec) |
N/A | 1.189 | 54.181 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.203 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | F | F | F(-1) | F(-2) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 230 | 246 | 228 | 0 | 0 | 0 | 0 | 0 | 196 | 0 |
| N.S. |
1 | 1.07 | 0.99 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.85 | 0.00 |
| time (sec) |
N/A | 1.246 | 53.123 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.192 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | F | F | F | F | F | F(-1) |
| verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 236 | 299 | 196 | 0 | 0 | 0 | 0 | 0 | 247 | 0 |
| N.S. |
1 | 1.27 | 0.83 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.05 | 0.00 |
| time (sec) |
N/A | 1.296 | 2.016 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.197 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 373 | 410 | 0 | 0 | 0 | 0 | 0 | 0 | 102 | 0 |
| N.S. |
1 | 1.10 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.27 | 0.00 |
| time (sec) |
N/A | 1.174 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.239 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 400 | 441 | 0 | 0 | 0 | 0 | 0 | 0 | 261 | 0 |
| N.S. |
1 | 1.10 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.65 | 0.00 |
| time (sec) |
N/A | 1.379 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.231 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 392 | 425 | 0 | 0 | 0 | 0 | 0 | 0 | 212 | 0 |
| N.S. |
1 | 1.08 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.54 | 0.00 |
| time (sec) |
N/A | 1.265 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.299 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 386 | 415 | 0 | 0 | 0 | 0 | 0 | 0 | 99 | 0 |
| N.S. |
1 | 1.08 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.26 | 0.00 |
| time (sec) |
N/A | 1.275 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.233 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 386 | 415 | 0 | 0 | 0 | 0 | 0 | 0 | 135 | 0 |
| N.S. |
1 | 1.08 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.35 | 0.00 |
| time (sec) |
N/A | 1.273 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.199 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 421 | 451 | 0 | 0 | 0 | 0 | 0 | 0 | 183 | 0 |
| N.S. |
1 | 1.07 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.43 | 0.00 |
| time (sec) |
N/A | 1.387 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.209 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | A | A | A | B | A | B | B |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 81 | 118 | 92 | 80 | 102 | 71 | 189 | 76 | 105 | 93 |
| N.S. |
1 | 1.46 | 1.14 | 0.99 | 1.26 | 0.88 | 2.33 | 0.94 | 1.30 | 1.15 |
| time (sec) |
N/A | 0.364 | 0.476 | 2.224 | 0.033 | 0.081 | 0.132 | 0.432 | 0.179 | 38.757 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | B | F | C | F | F | F | B |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 117 | 120 | 97 | 396 | 0 | 220 | 0 | 0 | 89 | 169 |
| N.S. |
1 | 1.03 | 0.83 | 3.38 | 0.00 | 1.88 | 0.00 | 0.00 | 0.76 | 1.44 |
| time (sec) |
N/A | 0.673 | 2.912 | 0.846 | 0.000 | 0.105 | 0.000 | 0.000 | 0.185 | 39.255 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | N/A | N/A | N/A | F(-1) | N/A | F(-2) | F(-1) | N/A | N/A |
| verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 45 | 45 | 47 | 45 | 0 | 54 | 0 | 0 | 102 | 47 |
| N.S. |
1 | 1.00 | 1.04 | 1.00 | 0.00 | 1.20 | 0.00 | 0.00 | 2.27 | 1.04 |
| time (sec) |
N/A | 0.307 | 40.356 | 0.925 | 0.000 | 1.423 | 0.000 | 0.000 | 0.297 | 173.106 |