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 | A | C | A | C | F(-1) | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 172 | 180 | 221 | 108 | 224 | 334837 | 0 | 339 | 570 | 1393 |
N.S. |
1 | 1.05 | 1.28 | 0.63 | 1.30 | 1946.73 | 0.00 | 1.97 | 3.31 | 8.10 |
time (sec) |
N/A | 0.664 | 0.458 | 0.111 | 0.107 | 21.129 | 0.000 | 0.128 | 0.167 | 6.753 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | A | C | F(-1) | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 236 | 235 | 263 | 147 | 284 | 343626 | 0 | 387 | 912 | 1002 |
N.S. |
1 | 1.00 | 1.11 | 0.62 | 1.20 | 1456.04 | 0.00 | 1.64 | 3.86 | 4.25 |
time (sec) |
N/A | 0.969 | 0.399 | 0.113 | 0.117 | 31.067 | 0.000 | 0.141 | 0.162 | 6.532 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | A | C | F(-1) | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 281 | 285 | 313 | 188 | 345 | 343822 | 0 | 435 | 1255 | 1056 |
N.S. |
1 | 1.01 | 1.11 | 0.67 | 1.23 | 1223.57 | 0.00 | 1.55 | 4.47 | 3.76 |
time (sec) |
N/A | 1.139 | 0.224 | 0.122 | 0.111 | 44.661 | 0.000 | 0.133 | 0.174 | 6.512 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 179 | 179 | 152 | 186 | 184 | 184 | 216 | 188 | 189 | 180 |
N.S. |
1 | 1.00 | 0.85 | 1.04 | 1.03 | 1.03 | 1.21 | 1.05 | 1.06 | 1.01 |
time (sec) |
N/A | 0.709 | 0.063 | 0.154 | 0.031 | 0.095 | 0.037 | 0.119 | 0.168 | 0.265 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 131 | 131 | 111 | 127 | 126 | 126 | 146 | 128 | 129 | 126 |
N.S. |
1 | 1.00 | 0.85 | 0.97 | 0.96 | 0.96 | 1.11 | 0.98 | 0.98 | 0.96 |
time (sec) |
N/A | 0.580 | 0.024 | 0.155 | 0.033 | 0.092 | 0.039 | 0.118 | 0.173 | 5.651 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 82 | 82 | 77 | 67 | 66 | 66 | 76 | 68 | 69 | 67 |
N.S. |
1 | 1.00 | 0.94 | 0.82 | 0.80 | 0.80 | 0.93 | 0.83 | 0.84 | 0.82 |
time (sec) |
N/A | 0.403 | 0.016 | 0.132 | 0.033 | 0.080 | 0.023 | 0.122 | 0.166 | 0.041 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | A | C | F(-1) | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 247 | 319 | 311 | 58 | 328 | 622377 | 0 | 338 | 550 | 5042 |
N.S. |
1 | 1.29 | 1.26 | 0.23 | 1.33 | 2519.74 | 0.00 | 1.37 | 2.23 | 20.41 |
time (sec) |
N/A | 0.901 | 0.221 | 0.098 | 0.112 | 34.261 | 0.000 | 0.127 | 0.169 | 6.605 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | A | C | F(-1) | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 269 | 349 | 319 | 105 | 350 | 352423 | 0 | 360 | 1082 | 1383 |
N.S. |
1 | 1.30 | 1.19 | 0.39 | 1.30 | 1310.12 | 0.00 | 1.34 | 4.02 | 5.14 |
time (sec) |
N/A | 0.900 | 0.128 | 0.109 | 0.120 | 24.428 | 0.000 | 0.132 | 0.172 | 6.725 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | A | C | F(-1) | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 334 | 408 | 366 | 144 | 412 | 358509 | 0 | 410 | 1681 | 1001 |
N.S. |
1 | 1.22 | 1.10 | 0.43 | 1.23 | 1073.38 | 0.00 | 1.23 | 5.03 | 3.00 |
time (sec) |
N/A | 1.179 | 0.196 | 0.119 | 0.120 | 40.344 | 0.000 | 0.133 | 0.173 | 0.721 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | A | C | F(-1) | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 377 | 456 | 411 | 185 | 472 | 358702 | 0 | 459 | 2279 | 1053 |
N.S. |
1 | 1.21 | 1.09 | 0.49 | 1.25 | 951.46 | 0.00 | 1.22 | 6.05 | 2.79 |
time (sec) |
N/A | 1.355 | 0.240 | 0.112 | 0.131 | 72.053 | 0.000 | 0.137 | 0.169 | 6.520 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | A | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 121 | 121 | 79 | 96 | 0 | 72 | 61 | 0 | 72 | 0 |
N.S. |
1 | 1.00 | 0.65 | 0.79 | 0.00 | 0.60 | 0.50 | 0.00 | 0.60 | 0.00 |
time (sec) |
N/A | 0.442 | 10.042 | 0.449 | 0.000 | 0.125 | 1.261 | 0.000 | 0.169 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | A | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 127 | 127 | 85 | 101 | 0 | 82 | 66 | 0 | 57 | 0 |
N.S. |
1 | 1.00 | 0.67 | 0.80 | 0.00 | 0.65 | 0.52 | 0.00 | 0.45 | 0.00 |
time (sec) |
N/A | 0.443 | 10.043 | 0.458 | 0.000 | 0.118 | 1.420 | 0.000 | 0.179 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | A | F | A | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 87 | 87 | 81 | 90 | 0 | 79 | 95 | 0 | 88 | 0 |
N.S. |
1 | 1.00 | 0.93 | 1.03 | 0.00 | 0.91 | 1.09 | 0.00 | 1.01 | 0.00 |
time (sec) |
N/A | 0.406 | 10.040 | 0.455 | 0.000 | 0.114 | 1.451 | 0.000 | 0.239 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | A | F | A | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 89 | 89 | 83 | 95 | 0 | 73 | 90 | 0 | 79 | 0 |
N.S. |
1 | 1.00 | 0.93 | 1.07 | 0.00 | 0.82 | 1.01 | 0.00 | 0.89 | 0.00 |
time (sec) |
N/A | 0.411 | 10.038 | 0.466 | 0.000 | 0.114 | 1.365 | 0.000 | 0.182 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | A | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 258 | 257 | 131 | 193 | 0 | 128 | 102 | 0 | 99 | 0 |
N.S. |
1 | 1.00 | 0.51 | 0.75 | 0.00 | 0.50 | 0.40 | 0.00 | 0.38 | 0.00 |
time (sec) |
N/A | 0.615 | 10.097 | 0.691 | 0.000 | 0.112 | 1.561 | 0.000 | 0.188 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | C | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 14 | 14 | 14 | 13 | 12 | 12 | 80 | 12 | 20 | 12 |
N.S. |
1 | 1.00 | 1.00 | 0.93 | 0.86 | 0.86 | 5.71 | 0.86 | 1.43 | 0.86 |
time (sec) |
N/A | 0.237 | 1.174 | 0.646 | 0.061 | 0.087 | 3.421 | 0.148 | 0.174 | 5.970 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | C | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 27 | 29 | 27 | 24 | 25 | 34 | 104 | 22 | 31 | 23 |
N.S. |
1 | 1.07 | 1.00 | 0.89 | 0.93 | 1.26 | 3.85 | 0.81 | 1.15 | 0.85 |
time (sec) |
N/A | 0.307 | 10.031 | 0.900 | 0.062 | 0.098 | 4.965 | 0.137 | 0.171 | 5.714 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 34 | 25 | 27 | 24 | 23 | 33 | 109 | 22 | 31 | 20 |
N.S. |
1 | 0.74 | 0.79 | 0.71 | 0.68 | 0.97 | 3.21 | 0.65 | 0.91 | 0.59 |
time (sec) |
N/A | 0.307 | 10.033 | 0.875 | 0.062 | 0.091 | 5.654 | 0.128 | 0.175 | 5.830 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 47 | 38 | 38 | 35 | 44 | 44 | 133 | 30 | 42 | 29 |
N.S. |
1 | 0.81 | 0.81 | 0.74 | 0.94 | 0.94 | 2.83 | 0.64 | 0.89 | 0.62 |
time (sec) |
N/A | 0.324 | 10.047 | 1.017 | 0.076 | 0.094 | 7.068 | 0.127 | 0.169 | 5.472 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | A | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 388 | 385 | 281 | 287 | 0 | 204 | 260 | 0 | 259 | 0 |
N.S. |
1 | 0.99 | 0.72 | 0.74 | 0.00 | 0.53 | 0.67 | 0.00 | 0.67 | 0.00 |
time (sec) |
N/A | 1.232 | 10.162 | 2.921 | 0.000 | 0.237 | 3.833 | 0.000 | 0.216 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | C | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 21 | 22 | 21 | 20 | 21 | 21 | 165 | 21 | 27 | 19 |
N.S. |
1 | 1.05 | 1.00 | 0.95 | 1.00 | 1.00 | 7.86 | 1.00 | 1.29 | 0.90 |
time (sec) |
N/A | 0.320 | 10.047 | 1.584 | 0.062 | 0.089 | 5.989 | 0.136 | 0.178 | 5.381 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 38 | 27 | 27 | 24 | 25 | 25 | 196 | 27 | 33 | 23 |
N.S. |
1 | 0.71 | 0.71 | 0.63 | 0.66 | 0.66 | 5.16 | 0.71 | 0.87 | 0.61 |
time (sec) |
N/A | 0.333 | 10.046 | 1.638 | 0.063 | 0.086 | 8.400 | 0.132 | 0.172 | 5.437 |