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 | A | B | A | F | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 42 | 172 | 30 | 80 | 1701 | 50 | 0 | 0 | 42 | 85 |
N.S. |
1 | 4.10 | 0.71 | 1.90 | 40.50 | 1.19 | 0.00 | 0.00 | 1.00 | 2.02 |
time (sec) |
N/A | 0.330 | 0.460 | 14.750 | 0.271 | 0.072 | 0.000 | 0.000 | 0.155 | 23.727 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | C | A | A | B | C | F(-1) | C | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 110 | 152 | 79 | 116 | 974 | 83 | 0 | 839 | 23 | 171 |
N.S. |
1 | 1.38 | 0.72 | 1.05 | 8.85 | 0.75 | 0.00 | 7.63 | 0.21 | 1.55 |
time (sec) |
N/A | 0.434 | 1.385 | 170.404 | 0.134 | 0.080 | 0.000 | 11.187 | 0.156 | 26.315 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | B | C | B | A | F | F | B | B |
verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 49 | 49 | 127 | 211 | 139 | 56 | 0 | 0 | 43 | 45 |
N.S. |
1 | 1.00 | 2.59 | 4.31 | 2.84 | 1.14 | 0.00 | 0.00 | 0.88 | 0.92 |
time (sec) |
N/A | 0.249 | 0.137 | 0.164 | 0.048 | 0.063 | 0.000 | 0.000 | 0.156 | 22.736 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | B | A | B | A | F | A | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 58 | 51 | 137 | 67 | 153 | 57 | 0 | 74 | 17 | 55 |
N.S. |
1 | 0.88 | 2.36 | 1.16 | 2.64 | 0.98 | 0.00 | 1.28 | 0.29 | 0.95 |
time (sec) |
N/A | 0.238 | 0.108 | 92.029 | 0.084 | 0.064 | 0.000 | 1.024 | 0.150 | 22.724 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | B | A | B | B | F | B | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 51 | 51 | 137 | 67 | 162 | 57 | 0 | 83 | 19 | 38 |
N.S. |
1 | 1.00 | 2.69 | 1.31 | 3.18 | 1.12 | 0.00 | 1.63 | 0.37 | 0.75 |
time (sec) |
N/A | 0.239 | 0.108 | 109.371 | 0.079 | 0.065 | 0.000 | 1.004 | 0.160 | 24.416 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | A | F | F | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 96 | 103 | 155 | 0 | 0 | 150 | 0 | 0 | 32 | 0 |
N.S. |
1 | 1.07 | 1.61 | 0.00 | 0.00 | 1.56 | 0.00 | 0.00 | 0.33 | 0.00 |
time (sec) |
N/A | 0.330 | 1.226 | 0.000 | 0.000 | 0.074 | 0.000 | 0.000 | 0.176 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | B | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 71 | 112 | 128 | 0 | 0 | 150 | 0 | 0 | 37 | 0 |
N.S. |
1 | 1.58 | 1.80 | 0.00 | 0.00 | 2.11 | 0.00 | 0.00 | 0.52 | 0.00 |
time (sec) |
N/A | 0.338 | 1.957 | 0.000 | 0.000 | 0.074 | 0.000 | 0.000 | 0.188 | 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 | 109 | 109 | 115 | 0 | 0 | 0 | 0 | 0 | 14 | 0 |
N.S. |
1 | 1.00 | 1.06 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.13 | 0.00 |
time (sec) |
N/A | 0.304 | 0.313 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.153 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | C | F | F | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 60 | 59 | 58 | 102 | 0 | 78 | 0 | 0 | 18 | 89 |
N.S. |
1 | 0.98 | 0.97 | 1.70 | 0.00 | 1.30 | 0.00 | 0.00 | 0.30 | 1.48 |
time (sec) |
N/A | 0.299 | 0.101 | 0.658 | 0.000 | 0.079 | 0.000 | 0.000 | 0.151 | 22.146 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | B | F | F | F(-2) | F | F(-1) | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 109 | 109 | 411 | 0 | 0 | 0 | 0 | 0 | 26 | 0 |
N.S. |
1 | 1.00 | 3.77 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.24 | 0.00 |
time (sec) |
N/A | 0.308 | 4.544 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.155 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | B | F | C | F | F(-1) | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 95 | 92 | 72 | 190 | 0 | 111 | 0 | 0 | 29 | 0 |
N.S. |
1 | 0.97 | 0.76 | 2.00 | 0.00 | 1.17 | 0.00 | 0.00 | 0.31 | 0.00 |
time (sec) |
N/A | 0.369 | 0.120 | 0.677 | 0.000 | 0.083 | 0.000 | 0.000 | 0.160 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F(-2) | F(-1) | F(-1) | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 109 | 109 | 174 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
N.S. |
1 | 1.00 | 1.60 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.26 | 0.00 |
time (sec) |
N/A | 0.299 | 1.124 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.150 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | C | F(-1) | F(-1) | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 99 | 96 | 73 | 131 | 0 | 145 | 0 | 0 | 31 | 0 |
N.S. |
1 | 0.97 | 0.74 | 1.32 | 0.00 | 1.46 | 0.00 | 0.00 | 0.31 | 0.00 |
time (sec) |
N/A | 0.364 | 0.160 | 0.779 | 0.000 | 0.082 | 0.000 | 0.000 | 0.162 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | B | F | F | F(-2) | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 110 | 110 | 377 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
N.S. |
1 | 1.00 | 3.43 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.00 |
time (sec) |
N/A | 0.303 | 3.068 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.158 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | B | F | C | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 60 | 59 | 58 | 129 | 0 | 82 | 0 | 0 | 31 | 0 |
N.S. |
1 | 0.98 | 0.97 | 2.15 | 0.00 | 1.37 | 0.00 | 0.00 | 0.52 | 0.00 |
time (sec) |
N/A | 0.291 | 0.095 | 0.687 | 0.000 | 0.081 | 0.000 | 0.000 | 0.161 | 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 | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 109 | 109 | 186 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
N.S. |
1 | 1.00 | 1.71 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.26 | 0.00 |
time (sec) |
N/A | 0.290 | 1.494 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.150 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | C | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 99 | 96 | 76 | 131 | 0 | 107 | 0 | 0 | 31 | 0 |
N.S. |
1 | 0.97 | 0.77 | 1.32 | 0.00 | 1.08 | 0.00 | 0.00 | 0.31 | 0.00 |
time (sec) |
N/A | 0.364 | 0.145 | 0.686 | 0.000 | 0.083 | 0.000 | 0.000 | 0.162 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | B | F | F | F(-2) | F(-1) | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 110 | 110 | 579 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
N.S. |
1 | 1.00 | 5.26 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.00 |
time (sec) |
N/A | 0.297 | 6.907 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.155 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | B | F | C | F(-1) | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 99 | 96 | 88 | 205 | 0 | 130 | 0 | 0 | 31 | 0 |
N.S. |
1 | 0.97 | 0.89 | 2.07 | 0.00 | 1.31 | 0.00 | 0.00 | 0.31 | 0.00 |
time (sec) |
N/A | 0.361 | 0.155 | 0.681 | 0.000 | 0.084 | 0.000 | 0.000 | 0.164 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | B | F | F | F | F(-1) | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 122 | 138 | 367 | 0 | 0 | 0 | 0 | 0 | 26 | 0 |
N.S. |
1 | 1.13 | 3.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.21 | 0.00 |
time (sec) |
N/A | 0.345 | 1.687 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.163 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 119 | 136 | 225 | 0 | 0 | 0 | 0 | 0 | 26 | 0 |
N.S. |
1 | 1.14 | 1.89 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.22 | 0.00 |
time (sec) |
N/A | 0.338 | 13.017 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.156 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 123 | 134 | 181 | 0 | 0 | 0 | 0 | 0 | 24 | 0 |
N.S. |
1 | 1.09 | 1.47 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.20 | 0.00 |
time (sec) |
N/A | 0.329 | 0.295 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.155 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F(-2) | F(-1) | F(-1) | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 130 | 126 | 165 | 0 | 0 | 0 | 0 | 0 | 31 | 0 |
N.S. |
1 | 0.97 | 1.27 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.24 | 0.00 |
time (sec) |
N/A | 0.341 | 2.179 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.160 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | B | F | F | F(-2) | F(-1) | F(-1) | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 130 | 126 | 466 | 0 | 0 | 0 | 0 | 0 | 29 | 0 |
N.S. |
1 | 0.97 | 3.58 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.22 | 0.00 |
time (sec) |
N/A | 0.332 | 7.177 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.156 | 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 | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 130 | 130 | 138 | 0 | 0 | 0 | 0 | 0 | 18 | 0 |
N.S. |
1 | 1.00 | 1.06 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.14 | 0.00 |
time (sec) |
N/A | 0.324 | 0.675 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.151 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | B | F | F | F(-2) | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 129 | 126 | 441 | 0 | 0 | 0 | 0 | 0 | 31 | 0 |
N.S. |
1 | 0.98 | 3.42 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.24 | 0.00 |
time (sec) |
N/A | 0.335 | 5.467 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.159 | 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 | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 130 | 126 | 218 | 0 | 0 | 0 | 0 | 0 | 31 | 0 |
N.S. |
1 | 0.97 | 1.68 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.24 | 0.00 |
time (sec) |
N/A | 0.333 | 1.598 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.157 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 139 | 170 | 169 | 0 | 0 | 0 | 0 | 0 | 26 | 0 |
N.S. |
1 | 1.22 | 1.22 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.19 | 0.00 |
time (sec) |
N/A | 0.371 | 1.180 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.164 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 106 | 106 | 142 | 0 | 0 | 0 | 0 | 0 | 17 | 0 |
N.S. |
1 | 1.00 | 1.34 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.16 | 0.00 |
time (sec) |
N/A | 0.326 | 0.918 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.170 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 107 | 111 | 142 | 0 | 0 | 0 | 0 | 0 | 15 | 0 |
N.S. |
1 | 1.04 | 1.33 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.14 | 0.00 |
time (sec) |
N/A | 0.306 | 0.741 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.160 | 0.000 |