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 | B | A | B | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 55 | 55 | 111 | 84 | 194 | 58 | 0 | 80 | 83 |
N.S. |
1 | 1.00 | 2.02 | 1.53 | 3.53 | 1.05 | 0.00 | 1.45 | 1.51 |
time (sec) |
N/A | 0.135 | 0.298 | 0.554 | 0.329 | 0.240 | 0.000 | 0.262 | 18.989 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 68 | 68 | 50 | 44 | 58 | 71 | 0 | 52 | 134 |
N.S. |
1 | 1.00 | 0.74 | 0.65 | 0.85 | 1.04 | 0.00 | 0.76 | 1.97 |
time (sec) |
N/A | 0.085 | 0.163 | 0.439 | 0.236 | 0.250 | 0.000 | 0.265 | 19.047 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | C | B | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 39 | 39 | 62 | 50 | 94 | 38 | 0 | 49 | 51 |
N.S. |
1 | 1.00 | 1.59 | 1.28 | 2.41 | 0.97 | 0.00 | 1.26 | 1.31 |
time (sec) |
N/A | 0.112 | 0.233 | 0.341 | 0.308 | 0.243 | 0.000 | 0.279 | 19.839 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 40 | 40 | 30 | 28 | 31 | 35 | 0 | 34 | 62 |
N.S. |
1 | 1.00 | 0.75 | 0.70 | 0.78 | 0.88 | 0.00 | 0.85 | 1.55 |
time (sec) |
N/A | 0.063 | 0.067 | 0.351 | 0.207 | 0.248 | 0.000 | 0.264 | 18.909 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 9 | 9 | 9 | 17 | 9 | 9 | 0 | 9 | 25 |
N.S. |
1 | 1.00 | 1.00 | 1.89 | 1.00 | 1.00 | 0.00 | 1.00 | 2.78 |
time (sec) |
N/A | 0.025 | 0.013 | 0.300 | 0.245 | 0.243 | 0.000 | 0.266 | 18.824 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 15 | 15 | 30 | 18 | 30 | 25 | 0 | 17 | 45 |
N.S. |
1 | 1.00 | 2.00 | 1.20 | 2.00 | 1.67 | 0.00 | 1.13 | 3.00 |
time (sec) |
N/A | 0.049 | 0.170 | 0.410 | 0.341 | 0.264 | 0.000 | 0.285 | 18.869 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 16 | 16 | 11 | 13 | 18 | 19 | 0 | 14 | 36 |
N.S. |
1 | 1.00 | 0.69 | 0.81 | 1.12 | 1.19 | 0.00 | 0.88 | 2.25 |
time (sec) |
N/A | 0.056 | 0.030 | 0.611 | 0.243 | 0.248 | 0.000 | 0.262 | 18.846 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | A | B | B | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 31 | 31 | 90 | 51 | 86 | 64 | 0 | 65 | 85 |
N.S. |
1 | 1.00 | 2.90 | 1.65 | 2.77 | 2.06 | 0.00 | 2.10 | 2.74 |
time (sec) |
N/A | 0.084 | 0.161 | 0.843 | 0.347 | 0.263 | 0.000 | 0.271 | 18.975 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 36 | 36 | 26 | 25 | 29 | 45 | 0 | 29 | 80 |
N.S. |
1 | 1.00 | 0.72 | 0.69 | 0.81 | 1.25 | 0.00 | 0.81 | 2.22 |
time (sec) |
N/A | 0.062 | 0.035 | 1.260 | 0.260 | 0.261 | 0.000 | 0.266 | 18.693 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | A | B | B | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 49 | 49 | 163 | 83 | 134 | 104 | 0 | 109 | 123 |
N.S. |
1 | 1.00 | 3.33 | 1.69 | 2.73 | 2.12 | 0.00 | 2.22 | 2.51 |
time (sec) |
N/A | 0.102 | 0.166 | 1.701 | 0.334 | 0.263 | 0.000 | 0.274 | 19.086 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 58 | 58 | 39 | 41 | 42 | 70 | 0 | 41 | 113 |
N.S. |
1 | 1.00 | 0.67 | 0.71 | 0.72 | 1.21 | 0.00 | 0.71 | 1.95 |
time (sec) |
N/A | 0.081 | 0.062 | 2.518 | 0.222 | 0.243 | 0.000 | 0.270 | 19.058 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 178 | 178 | 172 | 160 | 251 | 260 | 0 | 252 | 445 |
N.S. |
1 | 1.00 | 0.97 | 0.90 | 1.41 | 1.46 | 0.00 | 1.42 | 2.50 |
time (sec) |
N/A | 0.356 | 0.872 | 1.192 | 0.232 | 0.373 | 0.000 | 0.278 | 22.462 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 113 | 113 | 115 | 101 | 120 | 144 | 0 | 139 | 172 |
N.S. |
1 | 1.00 | 1.02 | 0.89 | 1.06 | 1.27 | 0.00 | 1.23 | 1.52 |
time (sec) |
N/A | 0.210 | 0.546 | 0.619 | 0.242 | 0.307 | 0.000 | 0.268 | 19.838 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 66 | 66 | 56 | 60 | 50 | 58 | 0 | 53 | 80 |
N.S. |
1 | 1.00 | 0.85 | 0.91 | 0.76 | 0.88 | 0.00 | 0.80 | 1.21 |
time (sec) |
N/A | 0.109 | 0.072 | 0.430 | 0.251 | 0.270 | 0.000 | 0.273 | 19.249 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 19 | 19 | 11 | 21 | 11 | 11 | 0 | 12 | 55 |
N.S. |
1 | 1.00 | 0.58 | 1.11 | 0.58 | 0.58 | 0.00 | 0.63 | 2.89 |
time (sec) |
N/A | 0.030 | 0.016 | 0.350 | 0.260 | 0.247 | 0.000 | 0.264 | 18.764 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 38 | 38 | 39 | 43 | 42 | 47 | 0 | 44 | 75 |
N.S. |
1 | 1.00 | 1.03 | 1.13 | 1.11 | 1.24 | 0.00 | 1.16 | 1.97 |
time (sec) |
N/A | 0.080 | 0.058 | 0.921 | 0.252 | 0.275 | 0.000 | 0.268 | 19.080 |
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 | 72 | 72 | 85 | 86 | 84 | 151 | 0 | 90 | 157 |
N.S. |
1 | 1.00 | 1.18 | 1.19 | 1.17 | 2.10 | 0.00 | 1.25 | 2.18 |
time (sec) |
N/A | 0.105 | 0.152 | 2.214 | 0.243 | 0.270 | 0.000 | 0.259 | 19.145 |
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 | 122 | 122 | 132 | 149 | 149 | 327 | 0 | 155 | 284 |
N.S. |
1 | 1.00 | 1.08 | 1.22 | 1.22 | 2.68 | 0.00 | 1.27 | 2.33 |
time (sec) |
N/A | 0.147 | 0.315 | 4.687 | 0.226 | 0.303 | 0.000 | 0.275 | 19.374 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 159 | 205 | 194 | 190 | 0 | 483 | 0 | 216 | 4468 |
N.S. |
1 | 1.29 | 1.22 | 1.19 | 0.00 | 3.04 | 0.00 | 1.36 | 28.10 |
time (sec) |
N/A | 0.390 | 1.305 | 0.826 | 0.000 | 0.297 | 0.000 | 0.283 | 27.542 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 84 | 112 | 87 | 106 | 0 | 296 | 0 | 103 | 2341 |
N.S. |
1 | 1.33 | 1.04 | 1.26 | 0.00 | 3.52 | 0.00 | 1.23 | 27.87 |
time (sec) |
N/A | 0.206 | 0.384 | 0.470 | 0.000 | 0.290 | 0.000 | 0.281 | 20.923 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | A | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 61 | 61 | 71 | 75 | 0 | 206 | 0 | 80 | 697 |
N.S. |
1 | 1.00 | 1.16 | 1.23 | 0.00 | 3.38 | 0.00 | 1.31 | 11.43 |
time (sec) |
N/A | 0.205 | 0.102 | 0.681 | 0.000 | 0.284 | 0.000 | 0.290 | 19.064 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | B | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 91 | 91 | 158 | 141 | 0 | 453 | 0 | 163 | 2163 |
N.S. |
1 | 1.00 | 1.74 | 1.55 | 0.00 | 4.98 | 0.00 | 1.79 | 23.77 |
time (sec) |
N/A | 0.311 | 0.572 | 1.602 | 0.000 | 0.346 | 0.000 | 0.284 | 19.809 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | B | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 186 | 186 | 312 | 251 | 0 | 852 | 0 | 299 | 4075 |
N.S. |
1 | 1.00 | 1.68 | 1.35 | 0.00 | 4.58 | 0.00 | 1.61 | 21.91 |
time (sec) |
N/A | 0.299 | 1.096 | 3.311 | 0.000 | 0.565 | 0.000 | 0.292 | 20.277 |