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 | N/A | N/A | N/A | N/A | F(-2) | N/A | F(-2) | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 16 | 16 | 18 | 14 | 16 | 0 | 15 | 0 | 16 |
N.S. |
1 | 1.00 | 1.12 | 0.88 | 1.00 | 0.00 | 0.94 | 0.00 | 1.00 |
time (sec) |
N/A | 0.029 | 0.891 | 0.073 | 0.717 | 0.000 | 8.407 | 0.000 | 0.096 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | F(-2) | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 16 | 16 | 18 | 14 | 16 | 0 | 17 | 16 | 16 |
N.S. |
1 | 1.00 | 1.12 | 0.88 | 1.00 | 0.00 | 1.06 | 1.00 | 1.00 |
time (sec) |
N/A | 0.026 | 4.786 | 0.102 | 0.703 | 0.000 | 16.239 | 1.566 | 0.095 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | A | F | C | A | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 120 | 120 | 100 | 144 | 0 | 99 | 85 | 0 | 0 |
N.S. |
1 | 1.00 | 0.83 | 1.20 | 0.00 | 0.82 | 0.71 | 0.00 | 0.00 |
time (sec) |
N/A | 0.048 | 0.028 | 1.267 | 0.000 | 0.099 | 68.116 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | A | F | C | A | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 124 | 124 | 66 | 138 | 0 | 85 | 85 | 0 | 0 |
N.S. |
1 | 1.00 | 0.53 | 1.11 | 0.00 | 0.69 | 0.69 | 0.00 | 0.00 |
time (sec) |
N/A | 0.060 | 0.023 | 0.636 | 0.000 | 0.094 | 11.194 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | A | F | C | A | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 88 | 88 | 66 | 119 | 0 | 68 | 85 | 0 | 0 |
N.S. |
1 | 1.00 | 0.75 | 1.35 | 0.00 | 0.77 | 0.97 | 0.00 | 0.00 |
time (sec) |
N/A | 0.026 | 0.017 | 0.408 | 0.000 | 0.092 | 3.264 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | A | F | C | F(-2) | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 89 | 89 | 45 | 98 | 0 | 53 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.51 | 1.10 | 0.00 | 0.60 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.047 | 0.016 | 0.274 | 0.000 | 0.094 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | A | F | C | F(-2) | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 55 | 55 | 40 | 85 | 0 | 49 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.73 | 1.55 | 0.00 | 0.89 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.021 | 0.014 | 0.162 | 0.000 | 0.085 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | A | F | C | F(-2) | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 125 | 125 | 42 | 129 | 0 | 70 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.34 | 1.03 | 0.00 | 0.56 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.060 | 0.017 | 0.296 | 0.000 | 0.087 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | F | F(-1) | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 109 | 109 | 90 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.83 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.085 | 0.049 | 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 | F | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 109 | 109 | 90 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.83 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.087 | 0.043 | 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 | F | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 109 | 109 | 90 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.83 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.085 | 0.046 | 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 | F(-2) | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 107 | 107 | 90 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.84 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.078 | 0.039 | 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 | F(-2) | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 105 | 105 | 87 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.83 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.081 | 0.037 | 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 | F(-2) | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 109 | 109 | 87 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.80 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.086 | 0.042 | 0.000 | 0.000 | 0.000 | 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 | F(-2) | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 18 | 18 | 20 | 16 | 421 | 53 | 17 | 0 | 18 |
N.S. |
1 | 1.00 | 1.11 | 0.89 | 23.39 | 2.94 | 0.94 | 0.00 | 1.00 |
time (sec) |
N/A | 0.098 | 58.945 | 0.098 | 3.550 | 0.260 | 78.696 | 0.000 | 0.129 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | F(-2) | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 18 | 18 | 20 | 16 | 398 | 44 | 17 | 0 | 18 |
N.S. |
1 | 1.00 | 1.11 | 0.89 | 22.11 | 2.44 | 0.94 | 0.00 | 1.00 |
time (sec) |
N/A | 0.095 | 142.451 | 0.068 | 3.593 | 0.254 | 8.465 | 0.000 | 0.142 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | F(-2) | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 18 | 18 | 20 | 16 | 438 | 50 | 0 | 18 | 18 |
N.S. |
1 | 1.00 | 1.11 | 0.89 | 24.33 | 2.78 | 0.00 | 1.00 | 1.00 |
time (sec) |
N/A | 0.087 | 71.710 | 0.021 | 3.579 | 0.243 | 0.000 | 0.571 | 0.140 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | F(-2) | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 18 | 18 | 20 | 16 | 469 | 50 | 0 | 18 | 18 |
N.S. |
1 | 1.00 | 1.11 | 0.89 | 26.06 | 2.78 | 0.00 | 1.00 | 1.00 |
time (sec) |
N/A | 0.091 | 59.244 | 0.110 | 3.519 | 0.251 | 0.000 | 0.602 | 0.130 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | F(-2) | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 18 | 18 | 20 | 16 | 471 | 50 | 0 | 18 | 18 |
N.S. |
1 | 1.00 | 1.11 | 0.89 | 26.17 | 2.78 | 0.00 | 1.00 | 1.00 |
time (sec) |
N/A | 0.100 | 41.968 | 0.131 | 3.587 | 0.255 | 0.000 | 0.631 | 0.133 |
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 | 18 | 18 | 20 | 16 | 18 | 20 | 15 | 18 | 18 |
N.S. |
1 | 1.00 | 1.11 | 0.89 | 1.00 | 1.11 | 0.83 | 1.00 | 1.00 |
time (sec) |
N/A | 0.018 | 1.852 | 0.079 | 0.422 | 0.242 | 4.158 | 0.306 | 0.092 |
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 | 18 | 18 | 20 | 16 | 18 | 18 | 15 | 18 | 18 |
N.S. |
1 | 1.00 | 1.11 | 0.89 | 1.00 | 1.00 | 0.83 | 1.00 | 1.00 |
time (sec) |
N/A | 0.016 | 1.604 | 0.077 | 0.445 | 0.229 | 0.486 | 0.305 | 0.093 |
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 | 18 | 18 | 20 | 16 | 18 | 23 | 17 | 18 | 18 |
N.S. |
1 | 1.00 | 1.11 | 0.89 | 1.00 | 1.28 | 0.94 | 1.00 | 1.00 |
time (sec) |
N/A | 0.016 | 0.960 | 0.080 | 0.469 | 0.238 | 1.425 | 0.290 | 0.094 |
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 | 18 | 18 | 20 | 16 | 18 | 31 | 17 | 18 | 18 |
N.S. |
1 | 1.00 | 1.11 | 0.89 | 1.00 | 1.72 | 0.94 | 1.00 | 1.00 |
time (sec) |
N/A | 0.019 | 0.794 | 0.085 | 0.506 | 0.234 | 3.485 | 0.289 | 0.093 |
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 | 18 | 18 | 20 | 16 | 182 | 34 | 17 | 18 | 18 |
N.S. |
1 | 1.00 | 1.11 | 0.89 | 10.11 | 1.89 | 0.94 | 1.00 | 1.00 |
time (sec) |
N/A | 0.019 | 8.687 | 0.081 | 1.910 | 0.240 | 10.744 | 0.310 | 0.114 |
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 | 18 | 18 | 20 | 16 | 180 | 32 | 17 | 18 | 18 |
N.S. |
1 | 1.00 | 1.11 | 0.89 | 10.00 | 1.78 | 0.94 | 1.00 | 1.00 |
time (sec) |
N/A | 0.015 | 8.426 | 0.083 | 1.901 | 0.229 | 1.992 | 0.311 | 0.119 |
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 | 18 | 18 | 20 | 16 | 195 | 39 | 19 | 18 | 18 |
N.S. |
1 | 1.00 | 1.11 | 0.89 | 10.83 | 2.17 | 1.06 | 1.00 | 1.00 |
time (sec) |
N/A | 0.015 | 24.106 | 0.096 | 1.690 | 0.246 | 4.072 | 0.318 | 0.121 |
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 | 18 | 18 | 20 | 16 | 219 | 51 | 19 | 18 | 18 |
N.S. |
1 | 1.00 | 1.11 | 0.89 | 12.17 | 2.83 | 1.06 | 1.00 | 1.00 |
time (sec) |
N/A | 0.018 | 16.641 | 0.089 | 1.971 | 0.231 | 11.473 | 0.295 | 0.114 |