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 | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 114 | 112 | 149 | 129 | 706 | 929 | 0 | 334 | 1088 |
N.S. |
1 | 0.98 | 1.31 | 1.13 | 6.19 | 8.15 | 0.00 | 2.93 | 9.54 |
time (sec) |
N/A | 0.302 | 3.450 | 2.008 | 0.198 | 0.258 | 0.000 | 0.305 | 2.317 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | B | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 74 | 75 | 113 | 83 | 334 | 461 | 0 | 182 | 508 |
N.S. |
1 | 1.01 | 1.53 | 1.12 | 4.51 | 6.23 | 0.00 | 2.46 | 6.86 |
time (sec) |
N/A | 0.268 | 2.722 | 1.549 | 0.189 | 0.246 | 0.000 | 0.269 | 2.261 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | B | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 43 | 47 | 84 | 45 | 121 | 180 | 0 | 81 | 166 |
N.S. |
1 | 1.09 | 1.95 | 1.05 | 2.81 | 4.19 | 0.00 | 1.88 | 3.86 |
time (sec) |
N/A | 0.235 | 0.584 | 1.210 | 0.181 | 0.276 | 0.000 | 0.268 | 2.350 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 16 | 16 | 16 | 17 | 23 | 36 | 0 | 23 | 23 |
N.S. |
1 | 1.00 | 1.00 | 1.06 | 1.44 | 2.25 | 0.00 | 1.44 | 1.44 |
time (sec) |
N/A | 0.160 | 0.007 | 0.837 | 0.184 | 0.251 | 0.000 | 0.250 | 2.316 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | F(-2) | A | F | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 50 | 50 | 52 | 124 | 0 | 457 | 0 | 0 | 471 |
N.S. |
1 | 1.00 | 1.04 | 2.48 | 0.00 | 9.14 | 0.00 | 0.00 | 9.42 |
time (sec) |
N/A | 0.290 | 2.173 | 0.423 | 0.000 | 0.276 | 0.000 | 0.000 | 2.792 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | F(-2) | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 100 | 122 | 199 | 315 | 0 | 1747 | 0 | 0 | 0 |
N.S. |
1 | 1.22 | 1.99 | 3.15 | 0.00 | 17.47 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.300 | 6.505 | 0.546 | 0.000 | 0.297 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | F(-2) | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 156 | 192 | 210 | 442 | 0 | 6569 | 0 | 0 | 0 |
N.S. |
1 | 1.23 | 1.35 | 2.83 | 0.00 | 42.11 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.393 | 7.020 | 0.999 | 0.000 | 0.339 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | F(-2) | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 220 | 268 | 273 | 672 | 0 | 17376 | 0 | 0 | 0 |
N.S. |
1 | 1.22 | 1.24 | 3.05 | 0.00 | 78.98 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.492 | 8.437 | 2.313 | 0.000 | 0.439 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | B | F | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 174 | 175 | 231 | 0 | 0 | 12590 | 0 | 0 | 0 |
N.S. |
1 | 1.01 | 1.33 | 0.00 | 0.00 | 72.36 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.401 | 3.390 | 0.000 | 0.000 | 0.728 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | B | F | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 126 | 125 | 193 | 0 | 0 | 6645 | 0 | 0 | 0 |
N.S. |
1 | 0.99 | 1.53 | 0.00 | 0.00 | 52.74 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.298 | 0.649 | 0.000 | 0.000 | 0.450 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | B | F | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 84 | 82 | 143 | 0 | 0 | 4389 | 0 | 0 | 0 |
N.S. |
1 | 0.98 | 1.70 | 0.00 | 0.00 | 52.25 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.260 | 0.216 | 0.000 | 0.000 | 0.365 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | F | F | B | F | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 38 | 41 | 97 | 0 | 0 | 1645 | 0 | 0 | 0 |
N.S. |
1 | 1.08 | 2.55 | 0.00 | 0.00 | 43.29 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.219 | 0.201 | 0.000 | 0.000 | 0.289 | 0.000 | 0.000 | 0.000 |
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 | 82 | 80 | 148 | 0 | 0 | 3009 | 0 | 0 | 0 |
N.S. |
1 | 0.98 | 1.80 | 0.00 | 0.00 | 36.70 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.253 | 0.348 | 0.000 | 0.000 | 0.347 | 0.000 | 0.000 | 0.000 |
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 | 135 | 148 | 174 | 0 | 0 | 7831 | 0 | 0 | 0 |
N.S. |
1 | 1.10 | 1.29 | 0.00 | 0.00 | 58.01 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.318 | 1.258 | 0.000 | 0.000 | 0.607 | 0.000 | 0.000 | 0.000 |
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 | 193 | 220 | 234 | 0 | 0 | 18681 | 0 | 0 | 0 |
N.S. |
1 | 1.14 | 1.21 | 0.00 | 0.00 | 96.79 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.392 | 1.093 | 0.000 | 0.000 | 1.802 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | A | C | A | B | F | B | F(-1) |
verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD |
size | 29 | 32 | 28 | 26 | 44 | 190 | 0 | 73 | 0 |
N.S. |
1 | 1.10 | 0.97 | 0.90 | 1.52 | 6.55 | 0.00 | 2.52 | 0.00 |
time (sec) |
N/A | 0.337 | 0.065 | 0.473 | 0.267 | 0.260 | 0.000 | 0.282 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | C | A | A | F | B | F(-1) |
verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD |
size | 14 | 14 | 18 | 17 | 22 | 18 | 0 | 27 | 0 |
N.S. |
1 | 1.00 | 1.29 | 1.21 | 1.57 | 1.29 | 0.00 | 1.93 | 0.00 |
time (sec) |
N/A | 0.287 | 0.012 | 0.397 | 0.271 | 0.254 | 0.000 | 0.282 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | A | A | F | B | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 14 | 14 | 14 | 79 | 13 | 18 | 0 | 30 | 0 |
N.S. |
1 | 1.00 | 1.00 | 5.64 | 0.93 | 1.29 | 0.00 | 2.14 | 0.00 |
time (sec) |
N/A | 0.291 | 0.010 | 0.227 | 0.269 | 0.253 | 0.000 | 0.257 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | B | F | B | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 46 | 46 | 92 | 0 | 0 | 528 | 0 | 253 | 0 |
N.S. |
1 | 1.00 | 2.00 | 0.00 | 0.00 | 11.48 | 0.00 | 5.50 | 0.00 |
time (sec) |
N/A | 0.242 | 0.198 | 0.000 | 0.000 | 0.258 | 0.000 | 0.321 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | F | F | B | F | B | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 26 | 26 | 65 | 0 | 0 | 221 | 0 | 110 | 0 |
N.S. |
1 | 1.00 | 2.50 | 0.00 | 0.00 | 8.50 | 0.00 | 4.23 | 0.00 |
time (sec) |
N/A | 0.214 | 0.067 | 0.000 | 0.000 | 0.267 | 0.000 | 0.277 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | F | F | B | F | B | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 16 | 16 | 45 | 0 | 0 | 163 | 0 | 85 | 0 |
N.S. |
1 | 1.00 | 2.81 | 0.00 | 0.00 | 10.19 | 0.00 | 5.31 | 0.00 |
time (sec) |
N/A | 0.203 | 0.063 | 0.000 | 0.000 | 0.267 | 0.000 | 0.291 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | B | F | B | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 47 | 47 | 90 | 0 | 0 | 666 | 0 | 412 | 0 |
N.S. |
1 | 1.00 | 1.91 | 0.00 | 0.00 | 14.17 | 0.00 | 8.77 | 0.00 |
time (sec) |
N/A | 0.232 | 0.097 | 0.000 | 0.000 | 0.258 | 0.000 | 0.391 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | F | F | B | F | B | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 33 | 33 | 68 | 0 | 0 | 358 | 0 | 157 | 0 |
N.S. |
1 | 1.00 | 2.06 | 0.00 | 0.00 | 10.85 | 0.00 | 4.76 | 0.00 |
time (sec) |
N/A | 0.216 | 0.049 | 0.000 | 0.000 | 0.266 | 0.000 | 0.302 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | F | F | B | F | B | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 14 | 14 | 48 | 0 | 0 | 213 | 0 | 69 | 0 |
N.S. |
1 | 1.00 | 3.43 | 0.00 | 0.00 | 15.21 | 0.00 | 4.93 | 0.00 |
time (sec) |
N/A | 0.196 | 0.043 | 0.000 | 0.000 | 0.254 | 0.000 | 0.282 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | A | B | C | C | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 34 | 34 | 30 | 95 | 44 | 59 | 0 | 84 | 0 |
N.S. |
1 | 1.00 | 0.88 | 2.79 | 1.29 | 1.74 | 0.00 | 2.47 | 0.00 |
time (sec) |
N/A | 0.335 | 0.034 | 0.257 | 0.271 | 0.251 | 0.000 | 0.279 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | C | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 16 | 16 | 20 | 26 | 22 | 13 | 0 | 32 | 0 |
N.S. |
1 | 1.00 | 1.25 | 1.62 | 1.38 | 0.81 | 0.00 | 2.00 | 0.00 |
time (sec) |
N/A | 0.275 | 0.012 | 0.364 | 0.278 | 0.266 | 0.000 | 0.261 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | C | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 16 | 16 | 16 | 29 | 13 | 13 | 0 | 35 | 0 |
N.S. |
1 | 1.00 | 1.00 | 1.81 | 0.81 | 0.81 | 0.00 | 2.19 | 0.00 |
time (sec) |
N/A | 0.271 | 0.008 | 0.430 | 0.263 | 0.258 | 0.000 | 0.272 | 0.000 |