[next] [prev] [prev-tail] [tail] [up]

2 detailed summary tables of results

 2.1 Detailed conclusion table per each integral for all CAS systems
 2.2 Detailed conclusion table specific for Rubi results

2.1 Detailed conclusion table per each integral for all CAS systems

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 normalized size is defined as \(\frac{\text{antiderivative leaf size}}{\text{optimal antiderivative leaf size}}\)

Problem 1 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 15 15 15 14 18 18 12 18 12
normalized size 1 1. 1. 0.93 1.2 1.2 0.8 1.2 0.8
time (sec) N/A 0.009 0.005 0.004 1.365 0.206 0.038 0.199 0.669
Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A C C C A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 15 52 37 34 55 1 42 55 42
normalized size 1 3.47 2.47 2.27 3.67 0.07 2.8 3.67 2.8
time (sec) N/A 0.097 0.02 0.005 1.375 0.21 52.66 0.212 8.535
Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A F A F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 82 82 119 370 0 363 0 239 0
normalized size 1 1. 1.45 4.51 0. 4.43 0. 2.91 0.
time (sec) N/A 0.144 0.159 0.062 0. 0.22 0. 0.206 0.
Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A F A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 43 43 43 172 0 201 0 127 34
normalized size 1 1. 1. 4. 0. 4.67 0. 2.95 0.79
time (sec) N/A 0.038 0.042 0.069 0. 0.216 0. 0.233 4.029
Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A F F(-2) F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 74 74 65 115 0 274 0 0 0
normalized size 1 1. 0.88 1.55 0. 3.7 0. 0. 0.
time (sec) N/A 0.146 0.146 0.023 0. 0.225 0. 0. 0.
Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B B A A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 64 64 165 125 72 230 0 0 53
normalized size 1 1. 2.58 1.95 1.12 3.59 0. 0. 0.83
time (sec) N/A 0.076 0.179 0.038 1.546 0.231 0. 0. 5.039
Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A F A F(-2) A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 48 48 90 45 0 216 0 136 41
normalized size 1 1. 1.88 0.94 0. 4.5 0. 2.83 0.85
time (sec) N/A 0.039 0.071 0.026 0. 0.222 0. 0.203 3.715
Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A F A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 30 30 29 21 0 24 53 27 26
normalized size 1 1. 0.97 0.7 0. 0.8 1.77 0.9 0.87
time (sec) N/A 0.128 0.023 0.004 0. 0.208 1.502 0.206 4.971
Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A F F F
verified N/A NO N/A TBD TBD TBD TBD TBD TBD
size 220 365 0 902 0 1185 0 0 0
normalized size 1 1.66 0. 4.1 0. 5.39 0. 0. 0.
time (sec) N/A 0.997 4.931 0.177 0. 0.253 0. 0. 0.
Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A B A B F A F(-1) F F(-1)
verified N/A NO N/A TBD TBD TBD TBD TBD TBD
size 220 541 0 1542 0 1185 0 0 0
normalized size 1 2.46 0. 7.01 0. 5.39 0. 0. 0.
time (sec) N/A 1.561 1.66 0.033 0. 0.255 0. 0. 0.
Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B B F(-2) A F(-2) F(-2) A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 138 138 403 278 0 680 0 0 184
normalized size 1 1. 2.92 2.01 0. 4.93 0. 0. 1.33
time (sec) N/A 0.162 0.413 0.032 0. 0.287 0. 0. 18.456
Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F(-1) F F A
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 125 125 0 0 0 0 0 0 202
normalized size 1 1. 0. 0. 0. 0. 0. 0. 1.62
time (sec) N/A 0.328 0.154 0.036 0. 0. 0. 0. 21.574
Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F(-1) F F A
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 81 81 0 0 0 0 0 0 155
normalized size 1 1. 0. 0. 0. 0. 0. 0. 1.91
time (sec) N/A 0.278 0.119 0.037 0. 0. 0. 0. 17.113
Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 31 31 145 0 0 81 15 0 29
normalized size 1 1. 4.68 0. 0. 2.61 0.48 0. 0.94
time (sec) N/A 0.089 2.357 0.024 0. 0.39 2.212 0. 3.712
Problem 15 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B C F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 33 33 162 22 0 39 15 0 29
normalized size 1 1. 4.91 0.67 0. 1.18 0.45 0. 0.88
time (sec) N/A 0.091 1.435 0.055 0. 0.4 1.164 0. 3.951
Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 19 19 19 16 20 31 56 20 17
normalized size 1 1. 1. 0.84 1.05 1.63 2.95 1.05 0.89
time (sec) N/A 0.434 0.025 0.004 1.341 0.203 12.491 0.214 11.273
Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 52 52 36 120 0 43 2147 0 0
normalized size 1 1. 0.69 2.31 0. 0.83 41.29 0. 0.
time (sec) N/A 0.047 0.04 0.045 0. 0.229 4.98 0. 0.
Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F A F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 56 56 39 0 0 45 0 0 0
normalized size 1 1. 0.7 0. 0. 0.8 0. 0. 0.
time (sec) N/A 0.046 0.04 0.045 0. 0.23 0. 0. 0.
Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 17 17 17 0 20 20 311 0 12
normalized size 1 1. 1. 0. 1.18 1.18 18.29 0. 0.71
time (sec) N/A 0.087 0.027 0.029 1.396 0.23 5.359 0. 4.054
Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 20 20 20 0 24 24 36 0 14
normalized size 1 1. 1. 0. 1.2 1.2 1.8 0. 0.7
time (sec) N/A 0.092 0.028 0.045 1.397 0.232 3.559 0. 4.364
Problem 21 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 42 42 37 48 54 70 36 55 39
normalized size 1 1. 0.88 1.14 1.29 1.67 0.86 1.31 0.93
time (sec) N/A 0.063 0.063 0.017 1.352 0.224 0.176 0.198 5.391
Problem 22 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 22 22 22 21 27 26 22 26 15
normalized size 1 1. 1. 0.95 1.23 1.18 1. 1.18 0.68
time (sec) N/A 0.041 0.021 0.004 1.342 0.216 0.116 0.201 4.678
Problem 23 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 62 62 49 51 69 78 49 100 54
normalized size 1 1. 0.79 0.82 1.11 1.26 0.79 1.61 0.87
time (sec) N/A 0.154 0.052 0.019 1.368 0.231 0.216 0.199 10.102
Problem 24 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C B F A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 86 86 961 455 0 720 0 0 87
normalized size 1 1. 11.17 5.29 0. 8.37 0. 0. 1.01
time (sec) N/A 0.226 5.47 0.057 0. 0.252 0. 0. 18.81
Problem 25 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 19 19 19 0 0 20 15 0 15
normalized size 1 1. 1. 0. 0. 1.05 0.79 0. 0.79
time (sec) N/A 0.106 0.026 0.039 0. 0.218 0.282 0. 3.644
Problem 26 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 26 26 26 0 0 30 27 0 20
normalized size 1 1. 1. 0. 0. 1.15 1.04 0. 0.77
time (sec) N/A 0.161 0.041 0.05 0. 0.223 1.344 0. 4.681
Problem 27 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 63 63 85 0 0 1 46 0 58
normalized size 1 1. 1.35 0. 0. 0.02 0.73 0. 0.92
time (sec) N/A 0.365 0.251 0.028 0. 0.243 2.87 0. 9.559
Problem 28 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 82 82 161 25 0 1 51 0 73
normalized size 1 1. 1.96 0.3 0. 0.01 0.62 0. 0.89
time (sec) N/A 0.14 0.081 0.008 0. 0.241 2.482 0. 8.607
Problem 29 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F A F F(-1) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 679 679 412 0 699 0 0 0 0
normalized size 1 1. 0.61 0. 1.03 0. 0. 0. 0.
time (sec) N/A 7.496 0.267 0.063 1.399 0. 0. 0. 0.
Problem 30 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 17 17 17 0 20 20 311 0 12
normalized size 1 1. 1. 0. 1.18 1.18 18.29 0. 0.71
time (sec) N/A 0.088 0.04 0.037 1.402 0.241 7.301 0. 3.919
Problem 31 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 52 52 36 120 0 43 2147 0 0
normalized size 1 1. 0.69 2.31 0. 0.83 41.29 0. 0.
time (sec) N/A 0.046 0.023 0.033 0. 0.244 6.697 0. 0.
Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 34 34 33 44 65 65 0 0 27
normalized size 1 1. 0.97 1.29 1.91 1.91 0. 0. 0.79
time (sec) N/A 0.06 0.056 0.066 1.766 0.224 0. 0. 9.298
Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 58 58 41 65 84 84 36 86 49
normalized size 1 1. 0.71 1.12 1.45 1.45 0.62 1.48 0.84
time (sec) N/A 0.078 0.021 0.056 1.497 0.214 1.607 0.207 2.456
Problem 34 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 58 58 41 48 84 84 37 86 49
normalized size 1 1. 0.71 0.83 1.45 1.45 0.64 1.48 0.84
time (sec) N/A 0.069 0.02 0.063 1.5 0.216 1.72 0.204 2.453
Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 49 49 86 12 105 122 29 0 71
normalized size 1 1. 1.76 0.24 2.14 2.49 0.59 0. 1.45
time (sec) N/A 0.016 0.071 0.034 1.502 0.216 1.61 0. 3.587
Problem 36 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 55 55 39 65 84 96 32 85 48
normalized size 1 1. 0.71 1.18 1.53 1.75 0.58 1.55 0.87
time (sec) N/A 0.07 0.019 0.057 1.474 0.215 1.605 0.209 2.299
Problem 37 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F(-1) F F F
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 121 97 0 0 0 0 0 0 0
normalized size 1 0.8 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.083 0.073 0.049 0. 0. 0. 0. 0.
Problem 38 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F(-2) F F F
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 145 145 0 0 0 0 0 0 0
normalized size 1 1. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.194 0.089 0.059 0. 0. 0. 0. 0.
Problem 39 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C F F F(-1) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 110 176 109 0 0 0 0 0 155
normalized size 1 1.6 0.99 0. 0. 0. 0. 0. 1.41
time (sec) N/A 0.099 0.19 0.098 0. 0. 0. 0. 4.678
Problem 40 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C F F F(-1) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 81 131 85 0 0 0 0 0 204
normalized size 1 1.62 1.05 0. 0. 0. 0. 0. 2.52
time (sec) N/A 0.139 0.022 0.019 0. 0. 0. 0. 7.393
Problem 41 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C F A F(-1) F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 66 117 49 0 104 0 0 0 155
normalized size 1 1.77 0.74 0. 1.58 0. 0. 0. 2.35
time (sec) N/A 0.113 0.035 0.064 1.524 0. 0. 0. 5.457
Problem 42 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C F F F(-1) F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 79 145 61 0 0 0 0 0 216
normalized size 1 1.84 0.77 0. 0. 0. 0. 0. 2.73
time (sec) N/A 0.188 0.058 0.02 0. 0. 0. 0. 9.593
Problem 43 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A F C F F F(-1) F(-1) F F
verified N/A N/A Yes TBD TBD TBD TBD TBD TBD
size 118 0 72 0 0 0 0 0 0
normalized size 1 0. 0.61 0. 0. 0. 0. 0. 0.
time (sec) N/A 27.68 0.228 0.028 0. 0. 0. 0. 0.
Problem 44 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A F A F F F(-1) F(-1) F F(-1)
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 111 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 1.338 1.235 0.062 0. 0. 0. 0. 0.
Problem 45 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A F A F F F(-1) F(-1) F F(-1)
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 176 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.82 0.415 0.073 0. 0. 0. 0. 0.
Problem 46 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A C A F F F(-2) F F F(-1)
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 326 576 0 0 0 0 0 0 0
normalized size 1 1.77 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 1.525 0.165 0.183 0. 0. 0. 0. 0.
Problem 47 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C A F A F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 407 407 206 584 0 1827 0 0 478
normalized size 1 1. 0.51 1.43 0. 4.49 0. 0. 1.17
time (sec) N/A 1.477 2.259 0.132 0. 0.281 0. 0. 48.171
Problem 48 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C A F A F(-1) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 648 648 662 719 0 2839 0 0 0
normalized size 1 1. 1.02 1.11 0. 4.38 0. 0. 0.
time (sec) N/A 2.686 6.077 0.072 0. 0.34 0. 0. 0.
Problem 49 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C A F F(-1) F(-1) A F
verified N/A NO Yes TBD TBD TBD TBD TBD TBD
size 1058 1058 1242 989 0 0 0 1 0
normalized size 1 1. 1.17 0.93 0. 0. 0. 0. 0.
time (sec) N/A 5.375 6.165 0.138 0. 0. 0. 0.227 0.
Problem 50 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C B F A F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 378 378 1236 21028 0 6251 0 0 379
normalized size 1 1. 3.27 55.63 0. 16.54 0. 0. 1.
time (sec) N/A 1.586 6.467 0.765 0. 0.363 0. 0. 79.056
Problem 51 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C B F F(-1) F(-1) A F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 638 638 1431 86793 0 0 0 1 0
normalized size 1 1. 2.24 136.04 0. 0. 0. 0. 0.
time (sec) N/A 2.835 6.717 4.895 0. 0. 0. 0.351 0.
Problem 52 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A C C C F A F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 66 204 213 1275 0 1 0 0 0
normalized size 1 3.09 3.23 19.32 0. 0.02 0. 0. 0.
time (sec) N/A 2.372 0.563 0.168 0. 0.617 0. 0. 0.
Problem 53 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C F F F(-2) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 198 198 205 0 0 0 0 0 231
normalized size 1 1. 1.04 0. 0. 0. 0. 0. 1.17
time (sec) N/A 0.241 0.468 0.052 0. 0. 0. 0. 13.653
Problem 54 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C F F F(-2) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 198 198 220 0 0 0 0 0 282
normalized size 1 1. 1.11 0. 0. 0. 0. 0. 1.42
time (sec) N/A 0.212 0.442 0.073 0. 0. 0. 0. 13.12
Problem 55 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C F F F(-1) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 88 97 273 0 0 0 0 0 94
normalized size 1 1.1 3.1 0. 0. 0. 0. 0. 1.07
time (sec) N/A 0.047 0.81 0.199 0. 0. 0. 0. 2.42
Problem 56 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 107 107 34 69 142 143 32 0 85
normalized size 1 1. 0.32 0.64 1.33 1.34 0.3 0. 0.79
time (sec) N/A 0.102 0.018 0.069 1.666 0.21 1.783 0. 4.775
Problem 57 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 67 67 48 49 96 115 37 97 56
normalized size 1 1. 0.72 0.73 1.43 1.72 0.55 1.45 0.84
time (sec) N/A 0.084 0.025 0.073 1.534 0.208 1.678 0.24 2.768
Problem 58 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A F A F F F(-2) F F F
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 1 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.099 0.372 0.072 0. 0. 0. 0. 0.
Problem 59 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A F A F F F(-1) F F F
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 280 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.385 0.042 0.112 0. 0. 0. 0. 0.
Problem 60 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 25 25 25 24 31 31 22 31 22
normalized size 1 1. 1. 0.96 1.24 1.24 0.88 1.24 0.88
time (sec) N/A 0.012 0.012 0.001 1.39 0.194 0.107 0.245 3.601
Problem 61 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C A F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 59 59 86 52 0 51 73 0 56
normalized size 1 1. 1.46 0.88 0. 0.86 1.24 0. 0.95
time (sec) N/A 0.06 0.027 0.043 0. 0.195 0.213 0. 7.633
Problem 62 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C A F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 78 78 99 75 0 89 100 0 80
normalized size 1 1. 1.27 0.96 0. 1.14 1.28 0. 1.03
time (sec) N/A 0.124 0.029 0.033 0. 0.198 0.218 0. 14.771
Problem 63 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C B F A F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 49 49 110 100 0 105 0 0 36
normalized size 1 1. 2.24 2.04 0. 2.14 0. 0. 0.73
time (sec) N/A 0.026 0.144 0.022 0. 0.257 0. 0. 2.408
Problem 64 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 53 53 108 365 0 81 0 0 46
normalized size 1 1. 2.04 6.89 0. 1.53 0. 0. 0.87
time (sec) N/A 0.037 0.134 0.04 0. 0.254 0. 0. 2.585
Problem 65 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F A F F A
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 75 75 0 1421 0 1 0 0 63
normalized size 1 1. 0. 18.95 0. 0.01 0. 0. 0.84
time (sec) N/A 0.161 0.329 0.098 0. 0.27 0. 0. 9.109
Problem 66 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C B F A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 171 171 322 456 0 2419 0 0 144
normalized size 1 1. 1.88 2.67 0. 14.15 0. 0. 0.84
time (sec) N/A 0.142 0.427 0.101 0. 0.339 0. 0. 6.94
Problem 67 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C F F F(-1) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 80 80 203 0 0 0 0 0 178
normalized size 1 1. 2.54 0. 0. 0. 0. 0. 2.22
time (sec) N/A 0.118 0.56 0.088 0. 0. 0. 0. 28.634
Problem 68 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C F F F(-1) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 88 88 221 0 0 0 0 0 199
normalized size 1 1. 2.51 0. 0. 0. 0. 0. 2.26
time (sec) N/A 0.131 0.555 0.043 0. 0. 0. 0. 29.508
Problem 69 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C F F F(-1) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 149 149 205 0 0 0 0 0 170
normalized size 1 1. 1.38 0. 0. 0. 0. 0. 1.14
time (sec) N/A 0.136 0.446 0.07 0. 0. 0. 0. 39.319
Problem 70 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C F F F(-1) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 135 135 219 0 0 0 0 0 173
normalized size 1 1. 1.62 0. 0. 0. 0. 0. 1.28
time (sec) N/A 0.109 0.425 0.045 0. 0. 0. 0. 36.093
Problem 71 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 127 127 120 164 0 1451 0 0 155
normalized size 1 1. 0.94 1.29 0. 11.43 0. 0. 1.22
time (sec) N/A 0.068 0.161 0.289 0. 0.358 0. 0. 3.059
Problem 72 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 157 157 135 240 0 2641 0 0 223
normalized size 1 1. 0.86 1.53 0. 16.82 0. 0. 1.42
time (sec) N/A 0.101 0.249 0.153 0. 0.461 0. 0. 4.538
Problem 73 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 74 74 118 286 0 950 0 0 37
normalized size 1 1. 1.59 3.86 0. 12.84 0. 0. 0.5
time (sec) N/A 0.288 0.191 0.169 0. 0.342 0. 0. 4.325
Problem 74 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 103 103 139 383 0 914 0 0 22
normalized size 1 1. 1.35 3.72 0. 8.87 0. 0. 0.21
time (sec) N/A 0.545 0.181 0.197 0. 0.328 0. 0. 3.314
Problem 75 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C F F F(-1) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 81 81 126 0 0 0 0 0 17
normalized size 1 1. 1.56 0. 0. 0. 0. 0. 0.21
time (sec) N/A 0.04 0.152 0.07 0. 0. 0. 0. 2.603
Problem 76 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C F F F(-1) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 81 81 126 0 0 0 0 0 20
normalized size 1 1. 1.56 0. 0. 0. 0. 0. 0.25
time (sec) N/A 0.037 0.14 0.04 0. 0. 0. 0. 3.072
Problem 77 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C F F F(-1) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 113 113 118 0 0 0 0 0 144
normalized size 1 1. 1.04 0. 0. 0. 0. 0. 1.27
time (sec) N/A 0.046 0.131 0.05 0. 0. 0. 0. 5.869
Problem 78 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C F F F(-1) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 109 109 124 0 0 0 0 0 192
normalized size 1 1. 1.14 0. 0. 0. 0. 0. 1.76
time (sec) N/A 0.044 0.102 0.056 0. 0. 0. 0. 5.238
Problem 79 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A F F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 87 87 159 206 0 72 0 0 0
normalized size 1 1. 1.83 2.37 0. 0.83 0. 0. 0.
time (sec) N/A 1.479 0.553 0.053 0. 0.266 0. 0. 0.
Problem 80 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A C C C F A F F F(-1)
verified N/A NO Yes TBD TBD TBD TBD TBD TBD
size 1 579 100 317 0 95 0 0 0
normalized size 1 579. 100. 317. 0. 95. 0. 0. 0.
time (sec) N/A 3.072 0.464 0.063 0. 0.264 0. 0. 0.
Problem 81 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A C C C F A F(-2) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 46 180 133 536 0 85 0 0 0
normalized size 1 3.91 2.89 11.65 0. 1.85 0. 0. 0.
time (sec) N/A 2.933 0.59 0.079 0. 0.259 0. 0. 0.
Problem 82 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F F(-2) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 32 32 323 258 0 0 0 0 469
normalized size 1 1. 10.09 8.06 0. 0. 0. 0. 14.66
time (sec) N/A 0.145 0.516 0.093 0. 0. 0. 0. 63.878
Problem 83 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 23 23 262 240 0 59 0 0 371
normalized size 1 1. 11.39 10.43 0. 2.57 0. 0. 16.13
time (sec) N/A 0.091 0.283 0.031 0. 0.258 0. 0. 41.593
Problem 84 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F F(-1) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 218 218 206 353 0 0 0 0 34
normalized size 1 1. 0.94 1.62 0. 0. 0. 0. 0.16
time (sec) N/A 0.116 0.43 0.283 0. 0. 0. 0. 3.449
Problem 85 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F F(-1) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 210 210 207 350 0 0 0 0 34
normalized size 1 1. 0.99 1.67 0. 0. 0. 0. 0.16
time (sec) N/A 0.105 0.656 0.266 0. 0. 0. 0. 7.693
Problem 86 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F F(-1) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 222 222 196 349 0 0 0 0 53
normalized size 1 1. 0.88 1.57 0. 0. 0. 0. 0.24
time (sec) N/A 0.1 0.623 0.257 0. 0. 0. 0. 11.817
Problem 87 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F F(-1) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 214 214 198 350 0 0 0 0 51
normalized size 1 1. 0.93 1.64 0. 0. 0. 0. 0.24
time (sec) N/A 0.096 0.539 0.256 0. 0. 0. 0. 6.47
Problem 88 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F F(-1) F F F(-1)
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 65 65 685 327 0 0 0 0 0
normalized size 1 1. 10.54 5.03 0. 0. 0. 0. 0.
time (sec) N/A 0.211 4.031 0.24 0. 0. 0. 0. 0.
Problem 89 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F F(-1) F F F(-1)
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 63 63 1137 311 0 0 0 0 0
normalized size 1 1. 18.05 4.94 0. 0. 0. 0. 0.
time (sec) N/A 0.213 10.714 0.235 0. 0. 0. 0. 0.
Problem 90 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F(-2) F F F
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 53 53 0 0 0 0 0 0 0
normalized size 1 1. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.096 0.144 0.045 0. 0. 0. 0. 0.
Problem 91 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F(-1) F F F
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 108 108 0 0 0 0 0 0 0
normalized size 1 1. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.16 0.047 0.039 0. 0. 0. 0. 0.

2.2 Detailed conclusion table specific for Rubi results

The following table is specific to Rubi. It gives additional statistics for each integral. the column steps is the number of steps used by Rubi to obtain the antiderivative. The rules column is the number of unique rules used. The integrand size column is the leaf size of the integrand. Finally the ratio \(\frac{\text{number of rules}}{\text{integrand size}}\) is given. The larger this ratio is, the harder the integral was to solve. In this test, problem number [29] had the largest ratio of [ 2.143 ]

Table 1: Rubi specific breakdown of results for each integral
# grade
number of
steps
used
number of
unique
rules
normalized
antiderivative
leaf size
integrand
leaf size
\(\frac{\text{number of rules}}{\text{integrand leaf size}}\)
1 A 1 1 1. 10 0.1
2 C 5 2 3.47 37 0.054
3 A 9 6 1. 15 0.4
4 A 3 3 1. 19 0.158
5 A 8 7 1. 17 0.412
6 A 6 6 1. 17 0.353
7 A 3 3 1. 17 0.176
8 A 4 2 1. 23 0.087
9 A 18 12 1.66 27 0.444
10 B 25 13 2.46 39 0.333
11 A 7 3 1. 45 0.067
12 A 7 4 1. 32 0.125
13 A 5 3 1. 32 0.094
14 A 2 2 1. 27 0.074
15 A 2 2 1. 29 0.069
16 A 2 1 1. 30 0.033
17 A 3 2 1. 13 0.154
18 A 3 2 1. 15 0.133
19 A 2 2 1. 23 0.087
20 A 2 2 1. 25 0.08
21 A 3 2 1. 11 0.182
22 A 2 2 1. 18 0.111
23 A 6 6 1. 20 0.3
24 A 6 5 1. 31 0.161
25 A 2 2 1. 29 0.069
26 A 2 2 1. 35 0.057
27 A 5 5 1. 32 0.156
28 A 6 6 1. 21 0.286
29 A 359 30 1. 14 2.143
30 A 2 2 1. 23 0.087
31 A 3 2 1. 13 0.154
32 A 2 2 1. 33 0.061
33 A 5 5 1. 15 0.333
34 A 5 5 1. 15 0.333
35 A 1 1 1. 11 0.091
36 A 5 5 1. 15 0.333
37 A 1 1 0.8 17 0.059
38 A 3 3 1. 18 0.167
39 A 2 2 1.6 16 0.125
40 A 5 5 1.62 17 0.294
41 A 5 5 1.77 13 0.385
42 A 5 5 1.84 16 0.312
43 F 0 0 N/A 0 N/A
44 F 0 0 N/A 0 N/A
45 F 0 0 N/A 0 N/A
46 C 7 3 1.77 32 0.094
47 A 19 9 1. 20 0.45
48 A 29 9 1. 20 0.45
49 A 49 9 1. 20 0.45
50 A 14 6 1. 23 0.261
51 A 24 6 1. 23 0.261
52 C 9 8 3.09 48 0.167
53 A 7 7 1. 24 0.292
54 A 7 7 1. 24 0.292
55 A 1 1 1.1 18 0.056
56 A 8 8 1. 13 0.615
57 A 6 6 1. 15 0.4
58 F 0 0 N/A 0 N/A
59 F 0 0 N/A 0 N/A
60 A 1 1 1. 38 0.026
61 A 1 1 1. 33 0.03
62 A 2 2 1. 39 0.051
63 A 1 1 1. 19 0.053
64 A 4 4 1. 19 0.21
65 A 4 4 1. 24 0.167
66 A 1 1 1. 24 0.042
67 A 7 7 1. 24 0.292
68 A 7 7 1. 24 0.292
69 A 3 3 1. 26 0.115
70 A 3 3 1. 22 0.136
71 A 1 1 1. 22 0.045
72 A 1 1 1. 23 0.043
73 A 8 8 1. 18 0.444
74 A 8 8 1. 23 0.348
75 A 1 1 1. 21 0.048
76 A 1 1 1. 19 0.053
77 A 1 1 1. 19 0.053
78 A 1 1 1. 19 0.053
79 A 4 4 1. 34 0.118
80 C 5 5 579. 40 0.125
81 C 7 7 3.91 51 0.137
82 A 2 2 1. 29 0.069
83 A 2 2 1. 18 0.111
84 A 1 1 1. 25 0.04
85 A 1 1 1. 25 0.04
86 A 1 1 1. 25 0.04
87 A 1 1 1. 25 0.04
88 A 2 2 1. 40 0.05
89 A 2 2 1. 40 0.05
90 A 1 1 1. 18 0.056
91 A 3 3 1. 15 0.2

[next] [prev] [prev-tail] [front] [up]