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 | C | A | B | B | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 149 | 173 | 168 | 541 | 333 | 396 | 0 | 0 | 0 |
N.S. |
1 | 1.16 | 1.13 | 3.63 | 2.23 | 2.66 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.672 | 0.043 | 0.986 | 0.264 | 0.261 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | A | B | B | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 99 | 115 | 118 | 306 | 195 | 242 | 0 | 0 | 0 |
N.S. |
1 | 1.16 | 1.19 | 3.09 | 1.97 | 2.44 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.455 | 0.042 | 0.945 | 0.282 | 0.265 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | B | A | F | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 50 | 61 | 116 | 89 | 0 | 119 | 0 | 0 | 0 |
N.S. |
1 | 1.22 | 2.32 | 1.78 | 0.00 | 2.38 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.277 | 0.034 | 0.248 | 0.000 | 0.268 | 0.000 | 0.000 | 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 | 10 | 10 | 12 | 10 | 12 | 12 | 8 | 12 | 14 |
N.S. |
1 | 1.00 | 1.20 | 1.00 | 1.20 | 1.20 | 0.80 | 1.20 | 1.40 |
time (sec) |
N/A | 0.189 | 20.670 | 0.047 | 0.432 | 0.260 | 0.296 | 0.320 | 2.141 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | A | B | B | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 103 | 146 | 185 | 473 | 320 | 1159 | 0 | 0 | 0 |
N.S. |
1 | 1.42 | 1.80 | 4.59 | 3.11 | 11.25 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.709 | 0.485 | 0.802 | 0.347 | 0.271 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | A | B | F | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 74 | 106 | 137 | 240 | 0 | 623 | 0 | 0 | 0 |
N.S. |
1 | 1.43 | 1.85 | 3.24 | 0.00 | 8.42 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.496 | 0.342 | 0.855 | 0.000 | 0.270 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | A | A | B | B | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 29 | 33 | 52 | 56 | 91 | 166 | 0 | 80 | 49 |
N.S. |
1 | 1.14 | 1.79 | 1.93 | 3.14 | 5.72 | 0.00 | 2.76 | 1.69 |
time (sec) |
N/A | 0.262 | 0.048 | 0.805 | 0.198 | 0.274 | 0.000 | 0.295 | 2.113 |
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 | 12 | 12 | 14 | 12 | 73 | 14 | 10 | 14 | 14 |
N.S. |
1 | 1.00 | 1.17 | 1.00 | 6.08 | 1.17 | 0.83 | 1.17 | 1.17 |
time (sec) |
N/A | 0.197 | 6.404 | 0.090 | 0.299 | 0.267 | 0.323 | 0.291 | 2.115 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | A | B | B | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 256 | 294 | 440 | 876 | 605 | 4008 | 0 | 0 | 0 |
N.S. |
1 | 1.15 | 1.72 | 3.42 | 2.36 | 15.66 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 1.203 | 2.102 | 1.012 | 0.369 | 0.310 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | B | B | B | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 154 | 184 | 420 | 444 | 393 | 2218 | 0 | 0 | 0 |
N.S. |
1 | 1.19 | 2.73 | 2.88 | 2.55 | 14.40 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.740 | 6.237 | 0.940 | 0.342 | 0.289 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | B | B | F | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 92 | 108 | 271 | 197 | 0 | 1026 | 0 | 0 | 0 |
N.S. |
1 | 1.17 | 2.95 | 2.14 | 0.00 | 11.15 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.416 | 0.080 | 0.767 | 0.000 | 0.260 | 0.000 | 0.000 | 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 | 12 | 12 | 14 | 12 | 157 | 14 | 10 | 14 | 14 |
N.S. |
1 | 1.00 | 1.17 | 1.00 | 13.08 | 1.17 | 0.83 | 1.17 | 1.17 |
time (sec) |
N/A | 0.202 | 42.663 | 0.097 | 0.332 | 0.244 | 0.326 | 0.643 | 2.138 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | F(-2) | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 24 | 24 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.71 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.240 | 0.035 | 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(-2) | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 24 | 24 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.71 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.241 | 0.051 | 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(-2) | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 47 | 47 | 45 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.96 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.272 | 0.072 | 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(-2) | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 76 | 76 | 67 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.88 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.343 | 0.188 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | B | F | F | B | F | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD |
size | 448 | 452 | 1792 | 0 | 0 | 1976 | 0 | 0 | 0 |
N.S. |
1 | 1.01 | 4.00 | 0.00 | 0.00 | 4.41 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 2.445 | 20.202 | 0.000 | 0.000 | 0.303 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | B | F | F | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 330 | 334 | 937 | 0 | 0 | 1265 | 0 | 0 | 0 |
N.S. |
1 | 1.01 | 2.84 | 0.00 | 0.00 | 3.83 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 1.740 | 11.570 | 0.000 | 0.000 | 0.290 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | F | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 212 | 209 | 266 | 483 | 0 | 692 | 0 | 0 | 0 |
N.S. |
1 | 0.99 | 1.25 | 2.28 | 0.00 | 3.26 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 1.038 | 0.392 | 1.170 | 0.000 | 0.289 | 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 | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 34 | 30 | 30 | 49 | 83 | 132 | 0 | 60 | 31 |
N.S. |
1 | 0.88 | 0.88 | 1.44 | 2.44 | 3.88 | 0.00 | 1.76 | 0.91 |
time (sec) |
N/A | 0.327 | 0.014 | 0.665 | 0.198 | 0.263 | 0.000 | 0.301 | 2.116 |
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 | 26 | 26 | 28 | 26 | 164 | 34 | 22 | 28 | 30 |
N.S. |
1 | 1.00 | 1.08 | 1.00 | 6.31 | 1.31 | 0.85 | 1.08 | 1.15 |
time (sec) |
N/A | 0.336 | 45.950 | 0.201 | 0.393 | 0.267 | 6.271 | 0.405 | 2.318 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | B | F | F | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 696 | 641 | 2218 | 0 | 0 | 3847 | 0 | 0 | 0 |
N.S. |
1 | 0.92 | 3.19 | 0.00 | 0.00 | 5.53 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 3.555 | 9.937 | 0.000 | 0.000 | 0.348 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | B | F | F | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 510 | 484 | 1216 | 0 | 0 | 2410 | 0 | 0 | 0 |
N.S. |
1 | 0.95 | 2.38 | 0.00 | 0.00 | 4.73 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 2.829 | 4.935 | 0.000 | 0.000 | 0.314 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | A | B | F | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 327 | 324 | 610 | 1012 | 0 | 1284 | 0 | 0 | 0 |
N.S. |
1 | 0.99 | 1.87 | 3.09 | 0.00 | 3.93 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 1.592 | 1.968 | 4.514 | 0.000 | 0.302 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | A | A | A | B | F | A | B |
verified |
N/A | No | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 95 | 107 | 109 | 167 | 168 | 446 | 0 | 155 | 212 |
N.S. |
1 | 1.13 | 1.15 | 1.76 | 1.77 | 4.69 | 0.00 | 1.63 | 2.23 |
time (sec) |
N/A | 0.611 | 0.343 | 4.001 | 0.283 | 0.279 | 0.000 | 0.304 | 2.440 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | F | B | F | F | B | F(-1) | F | F(-1) |
verified |
N/A | N/A | No | N/A | TBD | TBD | TBD | TBD | TBD |
size | 864 | 0 | 7404 | 0 | 0 | 7980 | 0 | 0 | 0 |
N.S. |
1 | 0.00 | 8.57 | 0.00 | 0.00 | 9.24 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.000 | 32.297 | 0.000 | 0.000 | 0.377 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | F | B | F | F | B | F | F | F(-1) |
verified |
N/A | N/A | No | N/A | TBD | TBD | TBD | TBD | TBD |
size | 636 | 0 | 2129 | 0 | 0 | 4887 | 0 | 0 | 0 |
N.S. |
1 | 0.00 | 3.35 | 0.00 | 0.00 | 7.68 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.000 | 16.795 | 0.000 | 0.000 | 0.336 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | F | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 400 | 360 | 696 | 1102 | 0 | 2465 | 0 | 0 | 0 |
N.S. |
1 | 0.90 | 1.74 | 2.76 | 0.00 | 6.16 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 2.296 | 1.179 | 17.620 | 0.000 | 0.307 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | B | B | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 85 | 75 | 75 | 244 | 183 | 652 | 0 | 145 | 180 |
N.S. |
1 | 0.88 | 0.88 | 2.87 | 2.15 | 7.67 | 0.00 | 1.71 | 2.12 |
time (sec) |
N/A | 0.394 | 0.101 | 17.864 | 0.220 | 0.268 | 0.000 | 0.317 | 2.472 |