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 | C | A | B | B | B | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 149 | 173 | 168 | 541 | 333 | 396 | 0 | 0 | 87 | 0 |
N.S. |
1 | 1.16 | 1.13 | 3.63 | 2.23 | 2.66 | 0.00 | 0.00 | 0.58 | 0.00 |
time (sec) |
N/A | 0.689 | 0.050 | 0.428 | 0.126 | 0.079 | 0.000 | 0.000 | 0.244 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | C | A | B | B | B | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 99 | 115 | 118 | 306 | 195 | 242 | 0 | 0 | 66 | 0 |
N.S. |
1 | 1.16 | 1.19 | 3.09 | 1.97 | 2.44 | 0.00 | 0.00 | 0.67 | 0.00 |
time (sec) |
N/A | 0.449 | 0.044 | 0.388 | 0.124 | 0.089 | 0.000 | 0.000 | 0.249 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | C | A | A | F | B | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 50 | 61 | 90 | 89 | 0 | 119 | 0 | 0 | 43 | 0 |
N.S. |
1 | 1.22 | 1.80 | 1.78 | 0.00 | 2.38 | 0.00 | 0.00 | 0.86 | 0.00 |
time (sec) |
N/A | 0.291 | 0.020 | 0.167 | 0.000 | 0.089 | 0.000 | 0.000 | 0.230 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | 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 | TBD |
size | 10 | 10 | 12 | 10 | 12 | 12 | 8 | 12 | 12 | 14 |
N.S. |
1 | 1.00 | 1.20 | 1.00 | 1.20 | 1.20 | 0.80 | 1.20 | 1.20 | 1.40 |
time (sec) |
N/A | 0.188 | 2.783 | 0.056 | 0.318 | 0.073 | 0.291 | 0.133 | 0.227 | 2.454 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | C | A | B | B | B | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 103 | 146 | 185 | 473 | 320 | 1159 | 0 | 0 | 657 | 0 |
N.S. |
1 | 1.42 | 1.80 | 4.59 | 3.11 | 11.25 | 0.00 | 0.00 | 6.38 | 0.00 |
time (sec) |
N/A | 0.712 | 0.448 | 0.310 | 0.207 | 0.092 | 0.000 | 0.000 | 0.237 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | C | A | B | F | B | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 74 | 106 | 137 | 240 | 0 | 623 | 0 | 0 | 327 | 0 |
N.S. |
1 | 1.43 | 1.85 | 3.24 | 0.00 | 8.42 | 0.00 | 0.00 | 4.42 | 0.00 |
time (sec) |
N/A | 0.496 | 0.312 | 0.264 | 0.000 | 0.091 | 0.000 | 0.000 | 0.257 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | C | A | A | B | B | F | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 29 | 33 | 52 | 56 | 91 | 166 | 0 | 80 | 118 | 49 |
N.S. |
1 | 1.14 | 1.79 | 1.93 | 3.14 | 5.72 | 0.00 | 2.76 | 4.07 | 1.69 |
time (sec) |
N/A | 0.257 | 0.045 | 0.280 | 0.044 | 0.089 | 0.000 | 0.115 | 0.216 | 2.521 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | 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 | TBD |
size | 12 | 12 | 14 | 12 | 73 | 14 | 10 | 14 | 14 | 14 |
N.S. |
1 | 1.00 | 1.17 | 1.00 | 6.08 | 1.17 | 0.83 | 1.17 | 1.17 | 1.17 |
time (sec) |
N/A | 0.208 | 5.663 | 0.061 | 0.144 | 0.078 | 0.305 | 0.106 | 0.248 | 2.462 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | C | A | B | B | B | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 256 | 294 | 440 | 876 | 605 | 4008 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.15 | 1.72 | 3.42 | 2.36 | 15.66 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 1.182 | 1.860 | 0.409 | 0.203 | 0.144 | 0.000 | 0.000 | 0.233 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | C | B | B | B | B | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 154 | 184 | 420 | 444 | 393 | 2218 | 0 | 0 | 1141 | 0 |
N.S. |
1 | 1.19 | 2.73 | 2.88 | 2.55 | 14.40 | 0.00 | 0.00 | 7.41 | 0.00 |
time (sec) |
N/A | 0.734 | 6.213 | 0.347 | 0.192 | 0.106 | 0.000 | 0.000 | 0.233 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | C | B | B | F | B | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 92 | 108 | 271 | 197 | 0 | 1026 | 0 | 0 | 516 | 0 |
N.S. |
1 | 1.17 | 2.95 | 2.14 | 0.00 | 11.15 | 0.00 | 0.00 | 5.61 | 0.00 |
time (sec) |
N/A | 0.497 | 0.077 | 0.266 | 0.000 | 0.104 | 0.000 | 0.000 | 0.234 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | 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 | TBD |
size | 12 | 12 | 14 | 12 | 157 | 14 | 10 | 14 | 14 | 14 |
N.S. |
1 | 1.00 | 1.17 | 1.00 | 13.08 | 1.17 | 0.83 | 1.17 | 1.17 | 1.17 |
time (sec) |
N/A | 0.225 | 25.404 | 0.068 | 0.182 | 0.078 | 0.296 | 0.341 | 0.229 | 2.389 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F(-2) | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 24 | 24 | 17 | 0 | 0 | 0 | 0 | 0 | 21 | 0 |
N.S. |
1 | 1.00 | 0.71 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.88 | 0.00 |
time (sec) |
N/A | 0.290 | 0.036 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.230 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F(-2) | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 24 | 24 | 17 | 0 | 0 | 0 | 0 | 0 | 25 | 0 |
N.S. |
1 | 1.00 | 0.71 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.04 | 0.00 |
time (sec) |
N/A | 0.271 | 0.049 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.220 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F(-2) | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 47 | 47 | 45 | 0 | 0 | 0 | 0 | 0 | 21 | 0 |
N.S. |
1 | 1.00 | 0.96 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.45 | 0.00 |
time (sec) |
N/A | 0.319 | 0.064 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.201 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F(-2) | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 76 | 76 | 67 | 0 | 0 | 0 | 0 | 0 | 25 | 0 |
N.S. |
1 | 1.00 | 0.88 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.33 | 0.00 |
time (sec) |
N/A | 0.382 | 0.178 | 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 | C | B | F | F | B | F | F | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 448 | 452 | 1792 | 0 | 0 | 1976 | 0 | 0 | 1077 | 0 |
N.S. |
1 | 1.01 | 4.00 | 0.00 | 0.00 | 4.41 | 0.00 | 0.00 | 2.40 | 0.00 |
time (sec) |
N/A | 2.683 | 19.333 | 0.000 | 0.000 | 0.128 | 0.000 | 0.000 | 0.293 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | C | B | F | F | B | F | F | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 330 | 334 | 937 | 0 | 0 | 1265 | 0 | 0 | 669 | 0 |
N.S. |
1 | 1.01 | 2.84 | 0.00 | 0.00 | 3.83 | 0.00 | 0.00 | 2.03 | 0.00 |
time (sec) |
N/A | 1.753 | 11.146 | 0.000 | 0.000 | 0.124 | 0.000 | 0.000 | 0.292 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | B | F | B | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 212 | 209 | 266 | 483 | 0 | 692 | 0 | 0 | 326 | 0 |
N.S. |
1 | 0.99 | 1.25 | 2.28 | 0.00 | 3.26 | 0.00 | 0.00 | 1.54 | 0.00 |
time (sec) |
N/A | 1.056 | 0.341 | 0.956 | 0.000 | 0.096 | 0.000 | 0.000 | 0.240 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | B | B | F | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 34 | 30 | 30 | 49 | 83 | 132 | 0 | 60 | 82 | 31 |
N.S. |
1 | 0.88 | 0.88 | 1.44 | 2.44 | 3.88 | 0.00 | 1.76 | 2.41 | 0.91 |
time (sec) |
N/A | 0.325 | 0.015 | 0.664 | 0.073 | 0.098 | 0.000 | 0.127 | 0.228 | 2.459 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | 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 | TBD |
size | 26 | 26 | 28 | 26 | 164 | 34 | 22 | 28 | 163 | 30 |
N.S. |
1 | 1.00 | 1.08 | 1.00 | 6.31 | 1.31 | 0.85 | 1.08 | 6.27 | 1.15 |
time (sec) |
N/A | 0.335 | 41.753 | 0.172 | 0.404 | 0.074 | 7.112 | 0.226 | 0.313 | 2.595 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | C | B | F | F | B | F | F | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 683 | 641 | 2218 | 0 | 0 | 3847 | 0 | 0 | 30 | 0 |
N.S. |
1 | 0.94 | 3.25 | 0.00 | 0.00 | 5.63 | 0.00 | 0.00 | 0.04 | 0.00 |
time (sec) |
N/A | 3.595 | 9.489 | 0.000 | 0.000 | 0.192 | 0.000 | 0.000 | 200.020 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | C | B | F | F | B | F | F | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 510 | 484 | 1216 | 0 | 0 | 2410 | 0 | 0 | 1482 | 0 |
N.S. |
1 | 0.95 | 2.38 | 0.00 | 0.00 | 4.73 | 0.00 | 0.00 | 2.91 | 0.00 |
time (sec) |
N/A | 2.815 | 4.029 | 0.000 | 0.000 | 0.128 | 0.000 | 0.000 | 1.453 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | C | A | B | F | B | F | F | F | F(-1) |
verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 321 | 324 | 610 | 1012 | 0 | 1284 | 0 | 0 | 28 | 0 |
N.S. |
1 | 1.01 | 1.90 | 3.15 | 0.00 | 4.00 | 0.00 | 0.00 | 0.09 | 0.00 |
time (sec) |
N/A | 1.867 | 1.687 | 4.509 | 0.000 | 0.134 | 0.000 | 0.000 | 200.022 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | C | A | A | A | B | F | A | B | B |
verified |
N/A | No | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 95 | 107 | 109 | 167 | 168 | 446 | 0 | 155 | 147 | 212 |
N.S. |
1 | 1.13 | 1.15 | 1.76 | 1.77 | 4.69 | 0.00 | 1.63 | 1.55 | 2.23 |
time (sec) |
N/A | 0.676 | 0.339 | 4.116 | 0.160 | 0.094 | 0.000 | 0.143 | 0.195 | 2.872 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | F | B | F | F | B | F(-1) | F | F | F(-1) |
verified |
N/A | N/A | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 864 | 0 | 3963 | 0 | 0 | 7980 | 0 | 0 | 30 | 0 |
N.S. |
1 | 0.00 | 4.59 | 0.00 | 0.00 | 9.24 | 0.00 | 0.00 | 0.03 | 0.00 |
time (sec) |
N/A | 0.000 | 31.034 | 0.000 | 0.000 | 0.204 | 0.000 | 0.000 | 200.019 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | F | B | F | F | B | F(-1) | F | F | F(-1) |
verified |
N/A | N/A | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 623 | 0 | 2129 | 0 | 0 | 4887 | 0 | 0 | 30 | 0 |
N.S. |
1 | 0.00 | 3.42 | 0.00 | 0.00 | 7.84 | 0.00 | 0.00 | 0.05 | 0.00 |
time (sec) |
N/A | 0.000 | 17.472 | 0.000 | 0.000 | 0.160 | 0.000 | 0.000 | 200.025 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | B | F | B | F | F | F | F(-1) |
verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 400 | 360 | 696 | 1102 | 0 | 2465 | 0 | 0 | 28 | 0 |
N.S. |
1 | 0.90 | 1.74 | 2.76 | 0.00 | 6.16 | 0.00 | 0.00 | 0.07 | 0.00 |
time (sec) |
N/A | 2.331 | 1.106 | 30.654 | 0.000 | 0.133 | 0.000 | 0.000 | 200.028 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | B | B | B | F | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 85 | 75 | 75 | 244 | 183 | 652 | 0 | 145 | 251 | 180 |
N.S. |
1 | 0.88 | 0.88 | 2.87 | 2.15 | 7.67 | 0.00 | 1.71 | 2.95 | 2.12 |
time (sec) |
N/A | 0.393 | 0.103 | 17.667 | 0.054 | 0.083 | 0.000 | 0.135 | 0.241 | 2.741 |