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 | A | A | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 225 | 244 | 149 | 252 | 755 | 430 | 0 | 163 | 0 |
N.S. |
1 | 1.08 | 0.66 | 1.12 | 3.36 | 1.91 | 0.00 | 0.72 | 0.00 |
time (sec) |
N/A | 0.829 | 0.231 | 0.405 | 0.422 | 0.257 | 0.000 | 0.294 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 111 | 118 | 130 | 124 | 611 | 343 | 0 | 121 | 0 |
N.S. |
1 | 1.06 | 1.17 | 1.12 | 5.50 | 3.09 | 0.00 | 1.09 | 0.00 |
time (sec) |
N/A | 0.362 | 0.135 | 0.359 | 0.388 | 0.260 | 0.000 | 0.298 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | A | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 91 | 91 | 105 | 83 | 464 | 111 | 0 | 79 | 0 |
N.S. |
1 | 1.00 | 1.15 | 0.91 | 5.10 | 1.22 | 0.00 | 0.87 | 0.00 |
time (sec) |
N/A | 0.283 | 0.068 | 0.336 | 0.365 | 0.239 | 0.000 | 0.283 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 15 | 15 | 17 | 15 | 17 | 17 | 14 | 17 | 17 |
N.S. |
1 | 1.00 | 1.13 | 1.00 | 1.13 | 1.13 | 0.93 | 1.13 | 1.13 |
time (sec) |
N/A | 0.175 | 5.777 | 0.259 | 0.736 | 0.234 | 0.592 | 0.293 | 1.078 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 107 | 107 | 132 | 0 | 0 | 331 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.23 | 0.00 | 0.00 | 3.09 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.264 | 0.445 | 0.000 | 0.000 | 0.255 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 227 | 246 | 152 | 244 | 834 | 467 | 0 | 167 | 0 |
N.S. |
1 | 1.08 | 0.67 | 1.07 | 3.67 | 2.06 | 0.00 | 0.74 | 0.00 |
time (sec) |
N/A | 0.822 | 0.230 | 0.382 | 0.439 | 0.254 | 0.000 | 0.285 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 112 | 119 | 134 | 120 | 674 | 385 | 0 | 123 | 0 |
N.S. |
1 | 1.06 | 1.20 | 1.07 | 6.02 | 3.44 | 0.00 | 1.10 | 0.00 |
time (sec) |
N/A | 0.363 | 0.132 | 0.422 | 0.402 | 0.261 | 0.000 | 0.282 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | A | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 91 | 91 | 109 | 79 | 512 | 115 | 0 | 81 | 0 |
N.S. |
1 | 1.00 | 1.20 | 0.87 | 5.63 | 1.26 | 0.00 | 0.89 | 0.00 |
time (sec) |
N/A | 0.277 | 0.058 | 0.377 | 0.361 | 0.254 | 0.000 | 0.271 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 16 | 16 | 18 | 16 | 22 | 21 | 14 | 18 | 18 |
N.S. |
1 | 1.00 | 1.12 | 1.00 | 1.38 | 1.31 | 0.88 | 1.12 | 1.12 |
time (sec) |
N/A | 0.176 | 6.641 | 0.287 | 0.717 | 0.249 | 0.603 | 0.280 | 1.117 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 108 | 108 | 136 | 0 | 0 | 371 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.26 | 0.00 | 0.00 | 3.44 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.260 | 0.499 | 0.000 | 0.000 | 0.250 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | C | B | B | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 66 | 119 | 72 | 75 | 183 | 127 | 0 | 53 | 0 |
N.S. |
1 | 1.80 | 1.09 | 1.14 | 2.77 | 1.92 | 0.00 | 0.80 | 0.00 |
time (sec) |
N/A | 0.556 | 0.112 | 0.342 | 0.273 | 0.266 | 0.000 | 0.286 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | C | B | B | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 52 | 57 | 75 | 49 | 123 | 108 | 0 | 43 | 0 |
N.S. |
1 | 1.10 | 1.44 | 0.94 | 2.37 | 2.08 | 0.00 | 0.83 | 0.00 |
time (sec) |
N/A | 0.291 | 0.053 | 0.391 | 0.268 | 0.255 | 0.000 | 0.278 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | C | B | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 39 | 39 | 24 | 25 | 94 | 16 | 0 | 21 | 0 |
N.S. |
1 | 1.00 | 0.62 | 0.64 | 2.41 | 0.41 | 0.00 | 0.54 | 0.00 |
time (sec) |
N/A | 0.231 | 0.019 | 0.391 | 0.258 | 0.254 | 0.000 | 0.275 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 13 | 13 | 15 | 11 | 13 | 13 | 12 | 13 | 13 |
N.S. |
1 | 1.00 | 1.15 | 0.85 | 1.00 | 1.00 | 0.92 | 1.00 | 1.00 |
time (sec) |
N/A | 0.165 | 6.331 | 0.243 | 0.648 | 0.239 | 0.585 | 0.268 | 1.086 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 13 | 13 | 15 | 11 | 13 | 13 | 14 | 13 | 13 |
N.S. |
1 | 1.00 | 1.15 | 0.85 | 1.00 | 1.00 | 1.08 | 1.00 | 1.00 |
time (sec) |
N/A | 0.397 | 8.128 | 0.262 | 0.476 | 0.264 | 0.418 | 0.285 | 1.106 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 268 | 268 | 176 | 281 | 294 | 766 | 0 | 182 | 0 |
N.S. |
1 | 1.00 | 0.66 | 1.05 | 1.10 | 2.86 | 0.00 | 0.68 | 0.00 |
time (sec) |
N/A | 0.468 | 0.540 | 0.847 | 0.301 | 0.261 | 0.000 | 0.287 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 136 | 136 | 155 | 141 | 200 | 648 | 0 | 142 | 0 |
N.S. |
1 | 1.00 | 1.14 | 1.04 | 1.47 | 4.76 | 0.00 | 1.04 | 0.00 |
time (sec) |
N/A | 0.290 | 0.271 | 1.058 | 0.278 | 0.257 | 0.000 | 0.270 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 110 | 110 | 140 | 94 | 96 | 128 | 0 | 94 | 0 |
N.S. |
1 | 1.00 | 1.27 | 0.85 | 0.87 | 1.16 | 0.00 | 0.85 | 0.00 |
time (sec) |
N/A | 0.249 | 0.100 | 0.697 | 0.265 | 0.249 | 0.000 | 0.284 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 17 | 17 | 19 | 17 | 51 | 19 | 15 | 19 | 19 |
N.S. |
1 | 1.00 | 1.12 | 1.00 | 3.00 | 1.12 | 0.88 | 1.12 | 1.12 |
time (sec) |
N/A | 0.213 | 12.589 | 0.254 | 0.352 | 0.253 | 0.764 | 0.303 | 1.105 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 268 | 268 | 181 | 273 | 322 | 843 | 0 | 186 | 0 |
N.S. |
1 | 1.00 | 0.68 | 1.02 | 1.20 | 3.15 | 0.00 | 0.69 | 0.00 |
time (sec) |
N/A | 0.455 | 0.518 | 0.725 | 0.301 | 0.249 | 0.000 | 0.286 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 136 | 136 | 159 | 137 | 216 | 730 | 0 | 144 | 0 |
N.S. |
1 | 1.00 | 1.17 | 1.01 | 1.59 | 5.37 | 0.00 | 1.06 | 0.00 |
time (sec) |
N/A | 0.285 | 0.292 | 0.696 | 0.285 | 0.254 | 0.000 | 0.287 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 110 | 110 | 144 | 90 | 96 | 132 | 0 | 96 | 0 |
N.S. |
1 | 1.00 | 1.31 | 0.82 | 0.87 | 1.20 | 0.00 | 0.87 | 0.00 |
time (sec) |
N/A | 0.260 | 0.123 | 0.665 | 0.271 | 0.257 | 0.000 | 0.280 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 18 | 18 | 20 | 18 | 51 | 22 | 15 | 20 | 20 |
N.S. |
1 | 1.00 | 1.11 | 1.00 | 2.83 | 1.22 | 0.83 | 1.11 | 1.11 |
time (sec) |
N/A | 0.209 | 14.141 | 0.245 | 0.352 | 0.243 | 0.755 | 0.304 | 1.090 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 68 | 68 | 99 | 77 | 113 | 268 | 0 | 61 | 0 |
N.S. |
1 | 1.00 | 1.46 | 1.13 | 1.66 | 3.94 | 0.00 | 0.90 | 0.00 |
time (sec) |
N/A | 0.268 | 0.158 | 0.799 | 0.304 | 0.261 | 0.000 | 0.276 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | C | C | B | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 75 | 75 | 88 | 75 | 121 | 307 | 0 | 70 | 0 |
N.S. |
1 | 1.00 | 1.17 | 1.00 | 1.61 | 4.09 | 0.00 | 0.93 | 0.00 |
time (sec) |
N/A | 0.238 | 0.162 | 0.786 | 0.316 | 0.253 | 0.000 | 0.275 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | C | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 56 | 56 | 48 | 49 | 45 | 41 | 0 | 42 | 0 |
N.S. |
1 | 1.00 | 0.86 | 0.88 | 0.80 | 0.73 | 0.00 | 0.75 | 0.00 |
time (sec) |
N/A | 0.214 | 0.057 | 0.700 | 0.269 | 0.249 | 0.000 | 0.281 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 15 | 15 | 17 | 13 | 43 | 15 | 14 | 15 | 15 |
N.S. |
1 | 1.00 | 1.13 | 0.87 | 2.87 | 1.00 | 0.93 | 1.00 | 1.00 |
time (sec) |
N/A | 0.204 | 13.851 | 0.320 | 0.325 | 0.245 | 0.681 | 0.289 | 1.092 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 261 | 269 | 194 | 460 | 536 | 634 | 0 | 227 | 0 |
N.S. |
1 | 1.03 | 0.74 | 1.76 | 2.05 | 2.43 | 0.00 | 0.87 | 0.00 |
time (sec) |
N/A | 0.862 | 0.400 | 0.635 | 0.369 | 0.260 | 0.000 | 0.290 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 128 | 127 | 146 | 200 | 254 | 413 | 0 | 139 | 0 |
N.S. |
1 | 0.99 | 1.14 | 1.56 | 1.98 | 3.23 | 0.00 | 1.09 | 0.00 |
time (sec) |
N/A | 0.382 | 0.187 | 0.336 | 0.281 | 0.253 | 0.000 | 0.282 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 19 | 19 | 21 | 19 | 21 | 21 | 17 | 21 | 21 |
N.S. |
1 | 1.00 | 1.11 | 1.00 | 1.11 | 1.11 | 0.89 | 1.11 | 1.11 |
time (sec) |
N/A | 0.180 | 3.104 | 0.418 | 0.670 | 0.237 | 0.509 | 0.288 | 1.138 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 311 | 311 | 240 | 528 | 601 | 1142 | 0 | 263 | 0 |
N.S. |
1 | 1.00 | 0.77 | 1.70 | 1.93 | 3.67 | 0.00 | 0.85 | 0.00 |
time (sec) |
N/A | 0.588 | 0.949 | 0.903 | 0.459 | 0.271 | 0.000 | 0.294 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 160 | 160 | 177 | 231 | 299 | 777 | 0 | 165 | 0 |
N.S. |
1 | 1.00 | 1.11 | 1.44 | 1.87 | 4.86 | 0.00 | 1.03 | 0.00 |
time (sec) |
N/A | 0.346 | 0.467 | 0.725 | 0.365 | 0.273 | 0.000 | 0.291 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 21 | 21 | 23 | 21 | 72 | 23 | 19 | 23 | 23 |
N.S. |
1 | 1.00 | 1.10 | 1.00 | 3.43 | 1.10 | 0.90 | 1.10 | 1.10 |
time (sec) |
N/A | 0.230 | 6.414 | 0.477 | 0.355 | 0.247 | 40.258 | 0.328 | 1.122 |