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 | A | A | A | A | B | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 436 | 436 | 495 | 440 | 621 | 668 | 2288 | 854 | 713 |
N.S. | 1 | 1.00 | 1.14 | 1.01 | 1.42 | 1.53 | 5.25 | 1.96 | 1.64 |
time (sec) |
N/A | 0.271 | 0.570 | 0.118 | 0.316 | 1.067 | 114.327 | 0.751 | 3.903 |
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 | 324 | 324 | 323 | 319 | 387 | 410 | 1510 | 558 | 514 |
N.S. | 1 | 1.00 | 1.00 | 0.98 | 1.19 | 1.27 | 4.66 | 1.72 | 1.59 |
time (sec) |
N/A | 0.188 | 0.326 | 0.100 | 0.284 | 0.954 | 74.684 | 0.634 | 3.488 |
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 | 212 | 212 | 184 | 198 | 198 | 204 | 848 | 309 | 351 |
N.S. | 1 | 1.00 | 0.87 | 0.93 | 0.93 | 0.96 | 4.00 | 1.46 | 1.66 |
time (sec) |
N/A | 0.114 | 0.161 | 0.098 | 0.290 | 0.729 | 41.624 | 0.553 | 3.301 |
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 | 115 | 115 | 82 | 128 | 128 | 90 | 354 | 128 | 128 |
N.S. | 1 | 1.00 | 0.71 | 1.11 | 1.11 | 0.78 | 3.08 | 1.11 | 1.11 |
time (sec) |
N/A | 0.048 | 0.067 | 0.095 | 0.318 | 0.982 | 8.440 | 0.549 | 2.846 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | A | A | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 188 | 188 | 161 | 220 | 0 | 565 | 192 | 248 | -1 |
N.S. |
1 | 1.00 | 0.86 | 1.17 | 0.00 | 3.01 | 1.02 | 1.32 | -0.01 |
time (sec) |
N/A | 0.140 | 0.312 | 0.098 | 0.000 | 1.705 | 26.933 | 0.551 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | B | F(-1) | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 201 | 201 | 231 | 226 | 0 | 1004 | 0 | 271 | -1 |
N.S. |
1 | 1.00 | 1.15 | 1.12 | 0.00 | 5.00 | 0.00 | 1.35 | -0.00 |
time (sec) |
N/A | 0.340 | 0.754 | 0.099 | 0.000 | 1.457 | 0.000 | 0.586 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | B | F(-1) | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 279 | 279 | 298 | 347 | 0 | 1661 | 0 | 529 | -1 |
N.S. |
1 | 1.00 | 1.07 | 1.24 | 0.00 | 5.95 | 0.00 | 1.90 | -0.00 |
time (sec) |
N/A | 0.466 | 1.149 | 0.100 | 0.000 | 1.184 | 0.000 | 0.614 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | B | F(-1) | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 375 | 375 | 409 | 548 | 0 | 2446 | 0 | 976 | -1 |
N.S. |
1 | 1.00 | 1.09 | 1.46 | 0.00 | 6.52 | 0.00 | 2.60 | -0.00 |
time (sec) |
N/A | 0.567 | 1.663 | 0.101 | 0.000 | 1.662 | 0.000 | 0.566 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | B | F(-1) | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 495 | 495 | 602 | 786 | 0 | 3624 | 0 | 1512 | -1 |
N.S. |
1 | 1.00 | 1.22 | 1.59 | 0.00 | 7.32 | 0.00 | 3.05 | -0.00 |
time (sec) |
N/A | 0.682 | 4.321 | 0.102 | 0.000 | 1.735 | 0.000 | 0.555 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | A | A | A | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 434 | 434 | 500 | 1023 | 629 | 677 | 707 | 1067 | -1 |
N.S. | 1 | 1.00 | 1.15 | 2.36 | 1.45 | 1.56 | 1.63 | 2.46 | -0.00 |
time (sec) |
N/A | 0.225 | 0.599 | 0.101 | 0.299 | 1.176 | 92.939 | 0.678 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | A | A | A | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 322 | 322 | 326 | 616 | 395 | 419 | 435 | 651 | -1 |
N.S. | 1 | 1.00 | 1.01 | 1.91 | 1.23 | 1.30 | 1.35 | 2.02 | -0.00 |
time (sec) |
N/A | 0.171 | 0.388 | 0.099 | 0.322 | 0.882 | 46.664 | 0.585 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 210 | 210 | 188 | 294 | 206 | 213 | 230 | 323 | -1 |
N.S. | 1 | 1.00 | 0.90 | 1.40 | 0.98 | 1.01 | 1.10 | 1.54 | -0.00 |
time (sec) |
N/A | 0.112 | 0.195 | 0.099 | 0.334 | 0.685 | 21.146 | 0.572 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 113 | 113 | 82 | 110 | 102 | 100 | 114 | 127 | -1 |
N.S. | 1 | 1.00 | 0.73 | 0.97 | 0.90 | 0.88 | 1.01 | 1.12 | -0.01 |
time (sec) |
N/A | 0.045 | 0.064 | 0.097 | 0.287 | 1.357 | 8.189 | 0.599 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | B | A | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 193 | 193 | 191 | 179 | 0 | 866 | 172 | 200 | -1 |
N.S. |
1 | 1.00 | 0.99 | 0.93 | 0.00 | 4.49 | 0.89 | 1.04 | -0.01 |
time (sec) |
N/A | 0.161 | 0.462 | 0.100 | 0.000 | 1.538 | 32.511 | 0.655 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | B | F(-1) | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 253 | 253 | 259 | 228 | 0 | 1583 | 0 | 388 | -1 |
N.S. |
1 | 1.00 | 1.02 | 0.90 | 0.00 | 6.26 | 0.00 | 1.53 | -0.00 |
time (sec) |
N/A | 0.404 | 0.806 | 0.102 | 0.000 | 0.936 | 0.000 | 0.614 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | B | F(-1) | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 350 | 350 | 387 | 395 | 0 | 2594 | 0 | 617 | -1 |
N.S. |
1 | 1.00 | 1.11 | 1.13 | 0.00 | 7.41 | 0.00 | 1.76 | -0.00 |
time (sec) |
N/A | 0.538 | 1.805 | 0.101 | 0.000 | 1.032 | 0.000 | 0.554 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | B | F(-1) | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 463 | 463 | 561 | 681 | 0 | 3834 | 0 | 1085 | -1 |
N.S. |
1 | 1.00 | 1.21 | 1.47 | 0.00 | 8.28 | 0.00 | 2.34 | -0.00 |
time (sec) |
N/A | 0.738 | 2.979 | 0.101 | 0.000 | 1.570 | 0.000 | 0.549 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | A | A | A | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 434 | 434 | 485 | 927 | 627 | 689 | 583 | 1030 | -1 |
N.S. | 1 | 1.00 | 1.12 | 2.14 | 1.44 | 1.59 | 1.34 | 2.37 | -0.00 |
time (sec) |
N/A | 0.221 | 0.662 | 0.103 | 0.305 | 0.991 | 109.214 | 0.567 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 322 | 322 | 317 | 547 | 393 | 431 | 377 | 622 | -1 |
N.S. | 1 | 1.00 | 0.98 | 1.70 | 1.22 | 1.34 | 1.17 | 1.93 | -0.00 |
time (sec) |
N/A | 0.160 | 0.439 | 0.312 | 0.275 | 1.137 | 61.692 | 0.536 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 210 | 210 | 177 | 253 | 204 | 225 | 230 | 302 | -1 |
N.S. | 1 | 1.00 | 0.84 | 1.20 | 0.97 | 1.07 | 1.10 | 1.44 | -0.00 |
time (sec) |
N/A | 0.115 | 0.215 | 0.099 | 0.273 | 1.584 | 30.710 | 0.555 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | B | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 113 | 113 | 75 | 98 | 98 | 110 | 425 | 115 | -1 |
N.S. | 1 | 1.00 | 0.66 | 0.87 | 0.87 | 0.97 | 3.76 | 1.02 | -0.01 |
time (sec) |
N/A | 0.047 | 0.074 | 0.098 | 0.287 | 1.393 | 0.449 | 0.542 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | B | A | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 210 | 210 | 230 | 202 | 0 | 1287 | 214 | 281 | -1 |
N.S. |
1 | 1.00 | 1.10 | 0.96 | 0.00 | 6.13 | 1.02 | 1.34 | -0.00 |
time (sec) |
N/A | 0.202 | 0.549 | 0.100 | 0.000 | 1.318 | 55.169 | 0.837 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | B | F(-1) | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 336 | 336 | 354 | 275 | 0 | 2444 | 0 | 439 | -1 |
N.S. |
1 | 1.00 | 1.05 | 0.82 | 0.00 | 7.27 | 0.00 | 1.31 | -0.00 |
time (sec) |
N/A | 0.529 | 1.164 | 0.101 | 0.000 | 1.220 | 0.000 | 0.956 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | B | F(-1) | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 438 | 438 | 554 | 459 | 0 | 3889 | 0 | 767 | -1 |
N.S. |
1 | 1.00 | 1.26 | 1.05 | 0.00 | 8.88 | 0.00 | 1.75 | -0.00 |
time (sec) |
N/A | 0.809 | 1.931 | 0.101 | 0.000 | 1.452 | 0.000 | 0.662 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | B | B | B | B | B | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 455 | 455 | 977 | 3932 | 1802 | 4115 | 65321 | 9032 | -1 |
N.S. | 1 | 1.00 | 2.15 | 8.64 | 3.96 | 9.04 | 143.56 | 19.85 | -0.00 |
time (sec) |
N/A | 0.237 | 2.855 | 0.107 | 0.353 | 1.854 | 12.745 | 0.673 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | B | B | B | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 338 | 338 | 619 | 2175 | 1118 | 2258 | 32849 | 4972 | -1 |
N.S. | 1 | 1.00 | 1.83 | 6.43 | 3.31 | 6.68 | 97.19 | 14.71 | -0.00 |
time (sec) |
N/A | 0.169 | 1.218 | 0.102 | 0.326 | 1.500 | 7.197 | 1.357 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | B | B | B | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 226 | 226 | 341 | 964 | 596 | 988 | 13522 | 2224 | -1 |
N.S. | 1 | 1.00 | 1.51 | 4.27 | 2.64 | 4.37 | 59.83 | 9.84 | -0.00 |
time (sec) |
N/A | 0.112 | 0.627 | 0.099 | 0.317 | 1.242 | 2.800 | 1.002 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | A | B | B | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 126 | 126 | 148 | 308 | 234 | 394 | 3798 | 728 | -1 |
N.S. | 1 | 1.00 | 1.17 | 2.44 | 1.86 | 3.13 | 30.14 | 5.78 | -0.01 |
time (sec) |
N/A | 0.049 | 0.167 | 0.091 | 0.304 | 1.202 | 1.118 | 0.884 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | C | F | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 203 | 203 | 186 | 0 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 0.92 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A | 0.129 | 0.420 | 0.023 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | F | F | F | F | F(-2) | F | F |
verified |
N/A | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 220 | 220 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
N.S. |
1 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A | 0.350 | 0.776 | 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 | 329 | 329 | 215 | 0 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 0.65 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A | 0.394 | 2.025 | 0.030 | 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(-2) | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 141 | 141 | 117 | 0 | 0 | 0 | 0 | 0 | -1 |
N.S. |
1 | 1.00 | 0.83 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.061 | 0.116 | 0.020 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | C | F | F | F | F(-2) | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 268 | 266 | 189 | 0 | 0 | 0 | 0 | 0 | -1 |
N.S. |
1 | 0.99 | 0.71 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A | 0.208 | 0.370 | 0.017 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | C | F | F | F | F(-2) | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 610 | 605 | 242 | 0 | 0 | 0 | 0 | 0 | -1 |
N.S. |
1 | 0.99 | 0.40 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A | 0.703 | 0.582 | 0.016 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |