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 | A | A | A | A | C | C | B | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 81 | 89 | 81 | 89 | 98 | 197 | 39 | 228 | 16 | 61 |
N.S. |
1 | 1.10 | 1.00 | 1.10 | 1.21 | 2.43 | 0.48 | 2.81 | 0.20 | 0.75 |
time (sec) |
N/A | 0.204 | 0.081 | 0.692 | 0.108 | 0.090 | 0.803 | 0.127 | 0.222 | 0.487 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | C | C | B | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 108 | 120 | 95 | 113 | 141 | 223 | 42 | 252 | 16 | 86 |
N.S. |
1 | 1.11 | 0.88 | 1.05 | 1.31 | 2.06 | 0.39 | 2.33 | 0.15 | 0.80 |
time (sec) |
N/A | 0.223 | 0.144 | 0.648 | 0.110 | 0.094 | 1.662 | 0.121 | 0.220 | 0.582 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 133 | 151 | 92 | 0 | 0 | 0 | 29 | 0 | 16 | 0 |
N.S. |
1 | 1.14 | 0.69 | 0.00 | 0.00 | 0.00 | 0.22 | 0.00 | 0.12 | 0.00 |
time (sec) |
N/A | 0.241 | 7.416 | 0.000 | 0.000 | 0.000 | 0.768 | 0.000 | 0.227 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 108 | 120 | 79 | 0 | 0 | 0 | 29 | 0 | 16 | 0 |
N.S. |
1 | 1.11 | 0.73 | 0.00 | 0.00 | 0.00 | 0.27 | 0.00 | 0.15 | 0.00 |
time (sec) |
N/A | 0.225 | 7.057 | 0.000 | 0.000 | 0.000 | 0.574 | 0.000 | 0.223 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 85 | 89 | 66 | 0 | 0 | 0 | 29 | 0 | 16 | 0 |
N.S. |
1 | 1.05 | 0.78 | 0.00 | 0.00 | 0.00 | 0.34 | 0.00 | 0.19 | 0.00 |
time (sec) |
N/A | 0.195 | 6.710 | 0.000 | 0.000 | 0.000 | 0.485 | 0.000 | 0.200 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 59 | 59 | 52 | 0 | 0 | 0 | 29 | 0 | 14 | 0 |
N.S. |
1 | 1.00 | 0.88 | 0.00 | 0.00 | 0.00 | 0.49 | 0.00 | 0.24 | 0.00 |
time (sec) |
N/A | 0.171 | 6.228 | 0.000 | 0.000 | 0.000 | 0.446 | 0.000 | 0.215 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 85 | 84 | 52 | 0 | 0 | 0 | 32 | 0 | 16 | 0 |
N.S. |
1 | 0.99 | 0.61 | 0.00 | 0.00 | 0.00 | 0.38 | 0.00 | 0.19 | 0.00 |
time (sec) |
N/A | 0.189 | 10.012 | 0.000 | 0.000 | 0.000 | 0.545 | 0.000 | 0.255 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 108 | 115 | 52 | 0 | 0 | 0 | 34 | 0 | 16 | 0 |
N.S. |
1 | 1.06 | 0.48 | 0.00 | 0.00 | 0.00 | 0.31 | 0.00 | 0.15 | 0.00 |
time (sec) |
N/A | 0.213 | 10.013 | 0.000 | 0.000 | 0.000 | 0.704 | 0.000 | 0.269 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 133 | 146 | 52 | 0 | 0 | 0 | 34 | 0 | 16 | 0 |
N.S. |
1 | 1.10 | 0.39 | 0.00 | 0.00 | 0.00 | 0.26 | 0.00 | 0.12 | 0.00 |
time (sec) |
N/A | 0.237 | 10.014 | 0.000 | 0.000 | 0.000 | 0.845 | 0.000 | 0.229 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | C | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 202 | 286 | 161 | 202 | 274 | 240 | 39 | 0 | 16 | 0 |
N.S. |
1 | 1.42 | 0.80 | 1.00 | 1.36 | 1.19 | 0.19 | 0.00 | 0.08 | 0.00 |
time (sec) |
N/A | 0.436 | 0.571 | 1.287 | 0.118 | 0.101 | 1.641 | 0.000 | 0.203 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | C | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 173 | 255 | 148 | 175 | 222 | 214 | 39 | 0 | 16 | 0 |
N.S. |
1 | 1.47 | 0.86 | 1.01 | 1.28 | 1.24 | 0.23 | 0.00 | 0.09 | 0.00 |
time (sec) |
N/A | 0.396 | 0.514 | 0.766 | 0.149 | 0.090 | 0.718 | 0.000 | 0.227 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | C | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 148 | 224 | 116 | 138 | 176 | 146 | 39 | 0 | 16 | 0 |
N.S. |
1 | 1.51 | 0.78 | 0.93 | 1.19 | 0.99 | 0.26 | 0.00 | 0.11 | 0.00 |
time (sec) |
N/A | 0.345 | 0.361 | 0.678 | 0.124 | 0.091 | 0.521 | 0.000 | 0.237 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | C | F | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 20 | 20 | 20 | 19 | 18 | 18 | 80 | 0 | 16 | 18 |
N.S. |
1 | 1.00 | 1.00 | 0.95 | 0.90 | 0.90 | 4.00 | 0.00 | 0.80 | 0.90 |
time (sec) |
N/A | 0.148 | 0.215 | 0.585 | 0.050 | 0.086 | 0.414 | 0.000 | 0.232 | 0.277 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | C | F | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 46 | 46 | 32 | 27 | 36 | 26 | 311 | 0 | 16 | 26 |
N.S. |
1 | 1.00 | 0.70 | 0.59 | 0.78 | 0.57 | 6.76 | 0.00 | 0.35 | 0.57 |
time (sec) |
N/A | 0.176 | 0.249 | 0.604 | 0.032 | 0.079 | 0.587 | 0.000 | 0.264 | 0.361 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | C | F | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 71 | 77 | 43 | 40 | 55 | 39 | 1110 | 0 | 16 | 39 |
N.S. |
1 | 1.08 | 0.61 | 0.56 | 0.77 | 0.55 | 15.63 | 0.00 | 0.23 | 0.55 |
time (sec) |
N/A | 0.201 | 0.322 | 0.623 | 0.031 | 0.081 | 0.963 | 0.000 | 0.245 | 0.463 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | C | F | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 96 | 108 | 54 | 51 | 73 | 50 | 1928 | 0 | 16 | 80 |
N.S. |
1 | 1.12 | 0.56 | 0.53 | 0.76 | 0.52 | 20.08 | 0.00 | 0.17 | 0.83 |
time (sec) |
N/A | 0.229 | 0.391 | 0.642 | 0.031 | 0.075 | 1.298 | 0.000 | 0.209 | 0.564 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 134 | 148 | 91 | 0 | 0 | 0 | 39 | 0 | 16 | 0 |
N.S. |
1 | 1.10 | 0.68 | 0.00 | 0.00 | 0.00 | 0.29 | 0.00 | 0.12 | 0.00 |
time (sec) |
N/A | 0.300 | 7.019 | 0.000 | 0.000 | 0.000 | 0.733 | 0.000 | 0.203 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 109 | 117 | 80 | 0 | 0 | 0 | 39 | 0 | 16 | 0 |
N.S. |
1 | 1.07 | 0.73 | 0.00 | 0.00 | 0.00 | 0.36 | 0.00 | 0.15 | 0.00 |
time (sec) |
N/A | 0.274 | 6.735 | 0.000 | 0.000 | 0.000 | 0.530 | 0.000 | 0.220 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 86 | 86 | 64 | 0 | 0 | 0 | 39 | 0 | 16 | 0 |
N.S. |
1 | 1.00 | 0.74 | 0.00 | 0.00 | 0.00 | 0.45 | 0.00 | 0.19 | 0.00 |
time (sec) |
N/A | 0.234 | 6.323 | 0.000 | 0.000 | 0.000 | 0.460 | 0.000 | 0.239 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | C | F | F | B |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 63 | 63 | 47 | 0 | 0 | 0 | 37 | 0 | 12 | 38 |
N.S. |
1 | 1.00 | 0.75 | 0.00 | 0.00 | 0.00 | 0.59 | 0.00 | 0.19 | 0.60 |
time (sec) |
N/A | 0.216 | 0.004 | 0.000 | 0.000 | 0.000 | 0.441 | 0.000 | 0.275 | 0.291 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 88 | 88 | 52 | 0 | 0 | 0 | 34 | 0 | 16 | 0 |
N.S. |
1 | 1.00 | 0.59 | 0.00 | 0.00 | 0.00 | 0.39 | 0.00 | 0.18 | 0.00 |
time (sec) |
N/A | 0.243 | 10.013 | 0.000 | 0.000 | 0.000 | 0.572 | 0.000 | 0.202 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 111 | 119 | 52 | 0 | 0 | 0 | 31 | 0 | 16 | 0 |
N.S. |
1 | 1.07 | 0.47 | 0.00 | 0.00 | 0.00 | 0.28 | 0.00 | 0.14 | 0.00 |
time (sec) |
N/A | 0.266 | 10.011 | 0.000 | 0.000 | 0.000 | 0.684 | 0.000 | 0.208 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 136 | 150 | 52 | 0 | 0 | 0 | 34 | 0 | 16 | 0 |
N.S. |
1 | 1.10 | 0.38 | 0.00 | 0.00 | 0.00 | 0.25 | 0.00 | 0.12 | 0.00 |
time (sec) |
N/A | 0.298 | 10.013 | 0.000 | 0.000 | 0.000 | 0.955 | 0.000 | 0.212 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 81 | 81 | 55 | 0 | 0 | 0 | 39 | 0 | 37 | 0 |
N.S. |
1 | 1.00 | 0.68 | 0.00 | 0.00 | 0.00 | 0.48 | 0.00 | 0.46 | 0.00 |
time (sec) |
N/A | 0.244 | 6.937 | 0.000 | 0.000 | 0.000 | 0.472 | 0.000 | 0.196 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 48 | 47 | 40 | 42 | 47 | 58 | 418 | 51 | 52 | 68 |
N.S. |
1 | 0.98 | 0.83 | 0.88 | 0.98 | 1.21 | 8.71 | 1.06 | 1.08 | 1.42 |
time (sec) |
N/A | 0.195 | 0.051 | 0.595 | 0.039 | 0.090 | 1.766 | 0.126 | 0.209 | 0.324 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 23 | 23 | 23 | 22 | 21 | 25 | 104 | 21 | 26 | 21 |
N.S. |
1 | 1.00 | 1.00 | 0.96 | 0.91 | 1.09 | 4.52 | 0.91 | 1.13 | 0.91 |
time (sec) |
N/A | 0.153 | 0.029 | 0.540 | 0.031 | 0.091 | 0.570 | 0.120 | 0.241 | 0.298 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F | C | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 41 | 41 | 41 | 0 | 0 | 0 | 39 | 0 | 42 | 0 |
N.S. |
1 | 1.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.95 | 0.00 | 1.02 | 0.00 |
time (sec) |
N/A | 0.171 | 0.043 | 0.000 | 0.000 | 0.000 | 2.604 | 0.000 | 0.235 | 0.000 |