Detailed conclusion table per each integral is given by 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 just an F.
In this table,the column N.S. in the table below, which stands for normalized size is defined as \(\frac {\text {antiderivative leaf size}}{\text {optimal antiderivative leaf size}}\). To help make the table fit, Mathematica was abbreviated to MMA.
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | C | A | A | A | B | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 22 | 22 | 77 | 21 | 20 | 25 | 61 | 18 | 16 |
N.S. | 1 | 1.00 | 3.50 | 0.95 | 0.91 | 1.14 | 2.77 | 0.82 | 0.73 |
time (sec) |
N/A | 0.010 | 0.047 | 0.090 | 1.373 | 0.714 | 4.152 | 0.453 | 0.141 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | F | A | F | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 32 | 32 | 41 | 50 | 0 | 44 | 0 | 147 | 49 |
N.S. | 1 | 1.00 | 1.28 | 1.56 | 0.00 | 1.38 | 0.00 | 4.59 | 1.53 |
time (sec) |
N/A | 0.021 | 0.096 | 0.010 | 0.000 | 0.736 | 0.000 | 0.470 | 0.622 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 25 | 25 | 16 | 16 | 23 | 20 | 14 | 15 | 14 |
N.S. | 1 | 1.00 | 0.64 | 0.64 | 0.92 | 0.80 | 0.56 | 0.60 | 0.56 |
time (sec) |
N/A | 0.011 | 0.009 | 0.017 | 1.234 | 0.604 | 0.156 | 0.445 | 0.163 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | C | A | C | A | F | C | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 58 | 58 | 68 | 42 | 112 | 46 | 0 | 43 | -1 |
N.S. | 1 | 1.00 | 1.17 | 0.72 | 1.93 | 0.79 | 0.00 | 0.74 | -0.02 |
time (sec) |
N/A | 0.054 | 0.014 | 0.059 | 0.752 | 0.711 | 0.000 | 0.462 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | B | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 50 | 50 | 26 | 42 | 56 | 67 | 838 | 32 | 88 |
N.S. | 1 | 1.00 | 0.52 | 0.84 | 1.12 | 1.34 | 16.76 | 0.64 | 1.76 |
time (sec) |
N/A | 0.017 | 0.027 | 0.121 | 0.743 | 0.770 | 3.456 | 0.496 | 0.269 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | C | C | F | B | A | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 334 | 334 | 30 | 36 | 0 | 1058 | 26 | 248 | 313 |
N.S. | 1 | 1.00 | 0.09 | 0.11 | 0.00 | 3.17 | 0.08 | 0.74 | 0.94 |
time (sec) |
N/A | 0.246 | 0.004 | 0.018 | 0.000 | 0.769 | 1.145 | 0.475 | 0.262 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | F | A | C | A | F(-2) | F | F | F |
verified |
N/A | N/A | NO | TBD | TBD | TBD | TBD | TBD | TBD |
size | 291 | 0 | 430 | 172 | 366 | 0 | 0 | 0 | -1 |
N.S. |
1 | 0.00 | 1.48 | 0.59 | 1.26 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A | 0.033 | 0.765 | 0.148 | 0.915 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | F | B | C | A | F(-2) | F | F | F |
verified |
N/A | N/A | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 308 | 0 | 654 | 199 | 378 | 0 | 0 | 0 | -1 |
N.S. |
1 | 0.00 | 2.12 | 0.65 | 1.23 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A | 0.030 | 0.457 | 0.011 | 2.147 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | F | F | A | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 84 | 84 | 105 | 0 | 0 | 66 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.25 | 0.00 | 0.00 | 0.79 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.039 | 0.129 | 0.010 | 0.000 | 1.051 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | B | A | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 41 | 41 | 41 | 34 | 51 | 101 | 112 | 51 | -1 |
N.S. | 1 | 1.00 | 1.00 | 0.83 | 1.24 | 2.46 | 2.73 | 1.24 | -0.02 |
time (sec) |
N/A | 0.135 | 0.060 | 0.056 | 2.465 | 1.014 | 8.009 | 0.475 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | A | B | F | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 73 | 73 | 69 | 175 | 63 | 112 | 0 | 67 | -1 |
N.S. | 1 | 1.00 | 0.95 | 2.40 | 0.86 | 1.53 | 0.00 | 0.92 | -0.01 |
time (sec) |
N/A | 0.076 | 0.066 | 0.096 | 1.778 | 0.992 | 0.000 | 0.618 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 73 | 73 | 52 | 60 | 77 | 110 | 0 | 79 | -1 |
N.S. | 1 | 1.00 | 0.71 | 0.82 | 1.05 | 1.51 | 0.00 | 1.08 | -0.01 |
time (sec) |
N/A | 0.188 | 0.051 | 0.018 | 2.080 | 0.749 | 0.000 | 0.545 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | C | C | F | B | F | F(-2) | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 365 | 365 | 217 | 109 | 0 | 5235 | 0 | 0 | -1 |
N.S. |
1 | 1.00 | 0.59 | 0.30 | 0.00 | 14.34 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A | 0.583 | 0.172 | 0.059 | 0.000 | 9.864 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | C | C | F | B | F | F(-2) | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 337 | 337 | 212 | 105 | 0 | 4535 | 0 | 0 | -1 |
N.S. |
1 | 1.00 | 0.63 | 0.31 | 0.00 | 13.46 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A | 0.408 | 0.151 | 0.059 | 0.000 | 5.823 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | F | F | A | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 77 | 77 | 74 | 0 | 0 | 56 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 0.96 | 0.00 | 0.00 | 0.73 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.048 | 10.033 | 0.007 | 0.000 | 1.427 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | F | F | A | F | F(-2) | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 118 | 118 | 112 | 0 | 0 | 73 | 0 | 0 | -1 |
N.S. |
1 | 1.00 | 0.95 | 0.00 | 0.00 | 0.62 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.128 | 10.065 | 0.012 | 0.000 | 2.163 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | F | A | F | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 83 | 83 | 77 | 298 | 0 | 81 | 0 | 188 | -1 |
N.S. | 1 | 1.00 | 0.93 | 3.59 | 0.00 | 0.98 | 0.00 | 2.27 | -0.01 |
time (sec) |
N/A | 0.070 | 0.158 | 0.018 | 0.000 | 3.669 | 0.000 | 0.617 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | F | F | A | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 96 | 96 | 89 | 0 | 0 | 122 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 0.93 | 0.00 | 0.00 | 1.27 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.058 | 0.389 | 0.014 | 0.000 | 3.603 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | B | B | A | A | A | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 25 | 25 | 65 | 49 | 36 | 34 | 61 | 55 | -1 |
N.S. | 1 | 1.00 | 2.60 | 1.96 | 1.44 | 1.36 | 2.44 | 2.20 | -0.04 |
time (sec) |
N/A | 0.053 | 0.111 | 0.036 | 0.917 | 0.890 | 2.011 | 0.549 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | B | A | A | B | F | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 25 | 25 | 126 | 33 | 35 | 55 | 0 | 54 | -1 |
N.S. | 1 | 1.00 | 5.04 | 1.32 | 1.40 | 2.20 | 0.00 | 2.16 | -0.04 |
time (sec) |
N/A | 0.035 | 0.086 | 0.069 | 1.639 | 1.061 | 0.000 | 0.562 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | B | C | A | B | B | F | A | B |
verified |
N/A | Yes | NO | TBD | TBD | TBD | TBD | TBD | TBD |
size | 108 | 786 | 478 | 95 | 12209 | 219 | 0 | 104 | 307 |
N.S. | 1 | 7.28 | 4.43 | 0.88 | 113.05 | 2.03 | 0.00 | 0.96 | 2.84 |
time (sec) |
N/A | 0.754 | 6.254 | 0.129 | 6.059 | 1.380 | 0.000 | 0.559 | 1.247 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | B | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 29 | 29 | 56 | 27 | 40 | 46 | 66 | 30 | 26 |
N.S. | 1 | 1.00 | 1.93 | 0.93 | 1.38 | 1.59 | 2.28 | 1.03 | 0.90 |
time (sec) |
N/A | 0.013 | 0.025 | 0.086 | 2.188 | 0.904 | 0.347 | 0.504 | 0.278 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | B | B | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 26 | 26 | 26 | 20 | 43 | 68 | 0 | 21 | 19 |
N.S. | 1 | 1.00 | 1.00 | 0.77 | 1.65 | 2.62 | 0.00 | 0.81 | 0.73 |
time (sec) |
N/A | 0.011 | 0.014 | 0.056 | 2.818 | 0.804 | 0.000 | 0.537 | 0.162 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | F | B | F | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 110 | 147 | 121 | 364 | 0 | 859 | 0 | 379 | -1 |
N.S. | 1 | 1.34 | 1.10 | 3.31 | 0.00 | 7.81 | 0.00 | 3.45 | -0.01 |
time (sec) |
N/A | 0.401 | 0.092 | 0.097 | 0.000 | 0.618 | 0.000 | 0.625 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | C | A | F | C | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 40 | 40 | 54 | 75 | 0 | 23 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.35 | 1.88 | 0.00 | 0.58 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A | 0.055 | 21.193 | 0.151 | 0.000 | 0.201 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | C | B | F | B | F | B | B |
verified |
N/A | Yes | NO | TBD | TBD | TBD | TBD | TBD | TBD |
size | 185 | 349 | 910 | 392 | 0 | 452 | 0 | 301 | 608 |
N.S. | 1 | 1.89 | 4.92 | 2.12 | 0.00 | 2.44 | 0.00 | 1.63 | 3.29 |
time (sec) |
N/A | 0.664 | 0.387 | 0.101 | 0.000 | 0.799 | 0.000 | 0.544 | 2.183 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 102 | 102 | 102 | 83 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.00 | 0.81 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.101 | 0.015 | 0.023 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | F | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 159 | 159 | 122 | 0 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 0.77 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.132 | 0.073 | 0.099 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | F | F | F | F(-1) | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 395 | 395 | 347 | 0 | 0 | 0 | 0 | 0 | -1 |
N.S. |
1 | 1.00 | 0.88 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A | 0.373 | 0.168 | 0.075 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 981 | 981 | 868 | 722 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 0.88 | 0.74 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A | 0.876 | 0.348 | 0.051 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | F | F | F | F | F | F |
verified |
N/A | Yes | NO | TBD | TBD | TBD | TBD | TBD | TBD |
size | 555 | 555 | 1076 | 0 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.94 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A | 0.487 | 0.987 | 0.007 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 313 | 313 | 303 | 252 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 0.97 | 0.81 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A | 0.264 | 0.062 | 0.014 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | B | A | A | B | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 80 | 80 | 262 | 113 | 84 | 220 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 3.28 | 1.41 | 1.05 | 2.75 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.055 | 0.176 | 0.253 | 2.741 | 0.788 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 57 | 57 | 44 | 71 | 31 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 0.77 | 1.25 | 0.54 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A | 0.053 | 0.044 | 0.493 | 8.552 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | NO | TBD | TBD | TBD | TBD | TBD | TBD |
size | 121 | 121 | 131 | 171 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.08 | 1.41 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.076 | 0.152 | 0.331 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |