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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 F(-2) A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 305 305 334 329 0 4134 165 397 309
normalized size 1 1. 1.1 1.08 0. 13.55 0.54 1.3 1.01
time (sec) N/A 0.559 0.169 0.111 0. 0.359 7.335 0.297 114.599
Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A F(-2) A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 323 323 337 380 0 4082 168 424 311
normalized size 1 1. 1.04 1.18 0. 12.64 0.52 1.31 0.96
time (sec) N/A 0.415 0.189 0.109 0. 0.377 7.569 0.281 75.131
Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F A A A F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 754 754 534 34 0 3762 199 811 0
normalized size 1 1. 0.71 0.05 0. 4.99 0.26 1.08 0.
time (sec) N/A 2.794 1.593 0.027 0. 0.486 37.057 0.312 0.
Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 329 329 425 39 0 3752 202 855 304
normalized size 1 1. 1.29 0.12 0. 11.4 0.61 2.6 0.92
time (sec) N/A 0.456 0.234 0.02 0. 0.493 36.872 0.311 77.931
Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A A F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 791 791 67 53 0 3082 136 0 0
normalized size 1 1. 0.08 0.07 0. 3.9 0.17 0. 0.
time (sec) N/A 1.872 0.062 0.059 0. 0.331 20.696 0. 0.
Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A A F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 791 791 67 53 0 3082 136 0 0
normalized size 1 1. 0.08 0.07 0. 3.9 0.17 0. 0.
time (sec) N/A 1.752 0.048 0.06 0. 0.314 18.967 0. 0.
Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 349 349 69 55 0 3079 136 0 333
normalized size 1 1. 0.2 0.16 0. 8.82 0.39 0. 0.95
time (sec) N/A 0.925 0.061 0.041 0. 0.329 20.679 0. 76.343
Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 751 751 69 55 0 3079 136 0 333
normalized size 1 1. 0.09 0.07 0. 4.1 0.18 0. 0.44
time (sec) N/A 1.946 0.051 0.043 0. 0.327 19.166 0. 74.154
Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 411 411 55 42 0 1458 75 0 304
normalized size 1 1. 0.13 0.1 0. 3.55 0.18 0. 0.74
time (sec) N/A 0.606 0.036 0.064 0. 0.293 7.435 0. 82.881
Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A A F A
verified N/A NO Yes TBD TBD TBD TBD TBD TBD
size 469 451 55 42 0 1681 24 0 590
normalized size 1 0.96 0.12 0.09 0. 3.58 0.05 0. 1.26
time (sec) N/A 0.782 0.023 0.01 0. 0.301 3.761 0. 88.896
Problem 11 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 85 85 64 58 97 131 73 97 73
normalized size 1 1. 0.75 0.68 1.14 1.54 0.86 1.14 0.86
time (sec) N/A 0.087 0.031 0.004 0.836 0.288 0.416 0.271 15.586
Problem 12 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 NO TBD TBD TBD TBD TBD TBD
size 140 140 135 109 0 212 190 0 128
normalized size 1 1. 0.96 0.78 0. 1.51 1.36 0. 0.91
time (sec) N/A 0.185 0.324 0.026 0. 0.292 2.878 0. 28.947
Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 347 347 258 27 0 1343 19 333 270
normalized size 1 1. 0.74 0.08 0. 3.87 0.05 0.96 0.78
time (sec) N/A 0.565 0.335 0.009 0. 0.281 4.448 0.299 39.272
Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 331 331 55 42 0 1048 20 331 270
normalized size 1 1. 0.17 0.13 0. 3.17 0.06 1. 0.82
time (sec) N/A 0.479 0.024 0.011 0. 0.293 4.851 0.283 43.016
Problem 15 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 27 27 31 42 36 58 26 39 17
normalized size 1 1. 1.15 1.56 1.33 2.15 0.96 1.44 0.63
time (sec) N/A 0.021 0.021 0.018 0.824 0.268 0.435 0.269 5.046
Problem 16 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 131 131 131 96 0 289 49 198 141
normalized size 1 1. 1. 0.73 0. 2.21 0.37 1.51 1.08
time (sec) N/A 0.181 0.124 0.045 0. 0.313 3.196 0.342 14.183
Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 157 157 53 40 0 331 24 0 148
normalized size 1 1. 0.34 0.25 0. 2.11 0.15 0. 0.94
time (sec) N/A 0.177 0.02 0.011 0. 0.288 0.538 0. 17.447
Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 171 171 55 42 0 701 24 0 168
normalized size 1 1. 0.32 0.25 0. 4.1 0.14 0. 0.98
time (sec) N/A 0.272 0.021 0.012 0. 0.289 0.563 0. 17.76
Problem 19 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 117 117 111 78 0 390 49 166 100
normalized size 1 1. 0.95 0.67 0. 3.33 0.42 1.42 0.85
time (sec) N/A 0.113 0.073 0.07 0. 0.286 3.171 0.346 10.398
Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 511 511 57 44 0 1458 76 0 359
normalized size 1 1. 0.11 0.09 0. 2.85 0.15 0. 0.7
time (sec) N/A 0.875 0.036 0.004 0. 0.292 7.471 0. 119.994
Problem 21 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A A F A
verified N/A NO Yes TBD TBD TBD TBD TBD TBD
size 429 411 57 44 0 1007 26 0 590
normalized size 1 0.96 0.13 0.1 0. 2.35 0.06 0. 1.38
time (sec) N/A 0.705 0.021 0.009 0. 0.295 3.801 0. 85.792
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 97 97 90 68 111 173 82 111 82
normalized size 1 1. 0.93 0.7 1.14 1.78 0.85 1.14 0.85
time (sec) N/A 0.108 0.111 0.011 0.818 0.266 0.512 0.285 17.103
Problem 23 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 NO TBD TBD TBD TBD TBD TBD
size 140 140 129 109 0 190 148 0 128
normalized size 1 1. 0.92 0.78 0. 1.36 1.06 0. 0.91
time (sec) N/A 0.204 0.3 0.019 0. 0.286 2.688 0. 40.927
Problem 24 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 347 347 257 29 0 1343 20 333 468
normalized size 1 1. 0.74 0.08 0. 3.87 0.06 0.96 1.35
time (sec) N/A 0.67 0.289 0.01 0. 0.284 4.305 0.314 68.009
Problem 25 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 355 355 57 44 0 1223 26 342 495
normalized size 1 1. 0.16 0.12 0. 3.45 0.07 0.96 1.39
time (sec) N/A 0.659 0.024 0.01 0. 0.289 4.897 0.286 72.816
Problem 26 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 13 13 25 10 23 23 17 26 8
normalized size 1 1. 1.92 0.77 1.77 1.77 1.31 2. 0.62
time (sec) N/A 0.011 0.006 0.002 0.817 0.268 0.354 0.267 3.973
Problem 27 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 129 129 129 110 0 401 51 198 141
normalized size 1 1. 1. 0.85 0. 3.11 0.4 1.53 1.09
time (sec) N/A 0.244 0.122 0.034 0. 0.28 3.187 0.344 13.309
Problem 28 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 165 165 55 42 0 581 26 0 168
normalized size 1 1. 0.33 0.25 0. 3.52 0.16 0. 1.02
time (sec) N/A 0.212 0.021 0.01 0. 0.3 0.584 0. 17.078
Problem 29 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 169 169 57 44 0 693 26 0 168
normalized size 1 1. 0.34 0.26 0. 4.1 0.15 0. 0.99
time (sec) N/A 0.293 0.021 0.01 0. 0.31 0.576 0. 18.865
Problem 30 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 125 125 114 90 0 347 51 182 121
normalized size 1 1. 0.91 0.72 0. 2.78 0.41 1.46 0.97
time (sec) N/A 0.147 0.085 0.034 0. 0.29 3.154 0.347 10.295
Problem 31 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F F(-2) A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 135 135 71 47 0 0 136 144 202
normalized size 1 1. 0.53 0.35 0. 0. 1.01 1.07 1.5
time (sec) N/A 0.25 0.052 0.076 0. 0. 3.848 0.291 51.762
Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F F(-2) F(-2) A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 164 164 72 62 0 0 0 166 165
normalized size 1 1. 0.44 0.38 0. 0. 0. 1.01 1.01
time (sec) N/A 0.211 0.053 0.054 0. 0. 0. 0.298 44.585
Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F F(-2) F(-2) A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 180 180 89 62 0 0 0 177 216
normalized size 1 1. 0.49 0.34 0. 0. 0. 0.98 1.2
time (sec) N/A 0.276 0.069 0.013 0. 0. 0. 0.29 54.45
Problem 34 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A F(-2) A A A F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 49 49 49 43 0 1 112 58 0
normalized size 1 1. 1. 0.88 0. 0.02 2.29 1.18 0.
time (sec) N/A 0.084 0.039 0.006 0. 0.299 1.681 0.268 0.
Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A F(-2) A A A F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 86 86 86 161 0 1 423 115 0
normalized size 1 1. 1. 1.87 0. 0.01 4.92 1.34 0.
time (sec) N/A 0.184 0.146 0.005 0. 0.268 4.567 0.267 0.
Problem 36 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A F(-2) A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 253 253 293 266 0 1018 109 347 235
normalized size 1 1. 1.16 1.05 0. 4.02 0.43 1.37 0.93
time (sec) N/A 0.42 0.17 0.007 0. 0.271 3.119 0.273 72.46
Problem 37 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 208 208 251 560 0 3429 428 1 214
normalized size 1 1. 1.21 2.69 0. 16.49 2.06 0. 1.03
time (sec) N/A 1.104 0.325 0.031 0. 0.333 34.703 1.288 76.268
Problem 38 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A F(-2) A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 311 311 346 334 0 4070 167 406 314
normalized size 1 1. 1.11 1.07 0. 13.09 0.54 1.31 1.01
time (sec) N/A 0.639 0.211 0.07 0. 0.35 8.909 0.325 124.579
Problem 39 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F F(-1) F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 716 716 88 67 0 0 0 0 0
normalized size 1 1. 0.12 0.09 0. 0. 0. 0. 0.
time (sec) N/A 3.498 0.089 0.034 0. 0. 0. 0. 0.
Problem 40 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F A A A F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 753 753 551 45 0 3742 204 873 0
normalized size 1 1. 0.73 0.06 0. 4.97 0.27 1.16 0.
time (sec) N/A 3.01 2.933 0.007 0. 0.515 47.512 0.311 0.
Problem 41 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A C C F A F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 433 433 88 67 0 17781 0 0 432
normalized size 1 1. 0.2 0.15 0. 41.06 0. 0. 1.
time (sec) N/A 2.038 0.105 0.008 0. 3.285 0. 0. 171.85
Problem 42 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 141 141 128 0 0 0 0 0 151
normalized size 1 1. 0.91 0. 0. 0. 0. 0. 1.07
time (sec) N/A 0.289 0.188 0.105 0. 0. 0. 0. 27.745
Problem 43 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 107 107 107 0 0 0 0 0 104
normalized size 1 1. 1. 0. 0. 0. 0. 0. 0.97
time (sec) N/A 0.197 0.122 0.088 0. 0. 0. 0. 19.29
Problem 44 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 83 83 82 0 0 0 0 0 60
normalized size 1 1. 0.99 0. 0. 0. 0. 0. 0.72
time (sec) N/A 0.066 0.044 0.059 0. 0. 0. 0. 9.966
Problem 45 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 152 152 131 0 0 0 0 0 0
normalized size 1 1. 0.86 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.224 0.153 0.102 0. 0. 0. 0. 0.
Problem 46 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 205 205 188 0 0 0 0 0 0
normalized size 1 1. 0.92 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.355 1.13 0.231 0. 0. 0. 0. 0.
Problem 47 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 81 81 80 0 0 0 0 0 56
normalized size 1 1. 0.99 0. 0. 0. 0. 0. 0.69
time (sec) N/A 0.07 0.069 0.063 0. 0. 0. 0. 10.371
Problem 48 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 288 288 165 0 0 0 0 0 158
normalized size 1 1. 0.57 0. 0. 0. 0. 0. 0.55
time (sec) N/A 0.522 0.493 0.102 0. 0. 0. 0. 27.384
Problem 49 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 203 203 142 0 0 0 0 0 109
normalized size 1 1. 0.7 0. 0. 0. 0. 0. 0.54
time (sec) N/A 0.345 0.464 0.112 0. 0. 0. 0. 18.857
Problem 50 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 134 134 137 0 0 0 0 0 100
normalized size 1 1. 1.02 0. 0. 0. 0. 0. 0.75
time (sec) N/A 0.126 0.153 0.097 0. 0. 0. 0. 19.185
Problem 51 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 333 333 245 0 0 0 0 0 0
normalized size 1 1. 0.74 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.481 0.585 0.187 0. 0. 0. 0. 0.
Problem 52 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 410 410 495 0 0 0 0 0 0
normalized size 1 1. 1.21 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.787 1.328 0.21 0. 0. 0. 0. 0.
Problem 53 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 424 424 252 0 0 0 0 0 158
normalized size 1 1. 0.59 0. 0. 0. 0. 0. 0.37
time (sec) N/A 0.829 1.226 0.124 0. 0. 0. 0. 27.846
Problem 54 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 272 272 212 0 0 0 0 0 109
normalized size 1 1. 0.78 0. 0. 0. 0. 0. 0.4
time (sec) N/A 0.516 0.825 0.13 0. 0. 0. 0. 19.391
Problem 55 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 184 184 164 0 0 0 0 0 151
normalized size 1 1. 0.89 0. 0. 0. 0. 0. 0.82
time (sec) N/A 0.216 0.437 0.117 0. 0. 0. 0. 31.452
Problem 56 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 582 582 1031 0 0 0 0 0 0
normalized size 1 1. 1.77 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.941 2.252 0.242 0. 0. 0. 0. 0.
Problem 57 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 701 701 1241 0 0 0 0 0 0
normalized size 1 1. 1.77 0. 0. 0. 0. 0. 0.
time (sec) N/A 1.492 4.606 0.274 0. 0. 0. 0. 0.
Problem 58 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F F A
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 171 171 0 0 0 0 0 0 138
normalized size 1 1. 0. 0. 0. 0. 0. 0. 0.81
time (sec) N/A 0.425 0.121 0.081 0. 0. 0. 0. 73.619
Problem 59 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A F(-1) A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 24 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.021 0.189 0.226 0. 0. 0. 0. 0.
Problem 60 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F(-2) A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 299 299 213 0 0 0 0 0 240
normalized size 1 1. 0.71 0. 0. 0. 0. 0. 0.8
time (sec) N/A 0.319 0.396 0.169 0. 0. 0. 0. 41.21
Problem 61 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F(-2) A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 217 217 171 0 0 0 0 0 170
normalized size 1 1. 0.79 0. 0. 0. 0. 0. 0.78
time (sec) N/A 0.217 0.207 0.154 0. 0. 0. 0. 29.317
Problem 62 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 135 135 110 0 0 0 0 0 104
normalized size 1 1. 0.81 0. 0. 0. 0. 0. 0.77
time (sec) N/A 0.129 0.093 0.138 0. 0. 0. 0. 17.075
Problem 63 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 167 167 0 0 0 0 0 0 128
normalized size 1 1. 0. 0. 0. 0. 0. 0. 0.77
time (sec) N/A 0.332 0.07 0.13 0. 0. 0. 0. 70.947
Problem 64 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 261 261 0 0 0 0 0 0 206
normalized size 1 1. 0. 0. 0. 0. 0. 0. 0.79
time (sec) N/A 0.558 0.117 0.104 0. 0. 0. 0. 137.215
Problem 65 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F(-1)
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 357 357 0 0 0 0 0 0 0
normalized size 1 1. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.795 0.411 0.122 0. 0. 0. 0. 0.
Problem 66 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A F(-2) A A A F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 62 62 57 66 0 185 656 308 0
normalized size 1 1. 0.92 1.06 0. 2.98 10.58 4.97 0.
time (sec) N/A 0.083 0.199 0.018 0. 0.272 3.841 0.27 0.
Problem 67 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A F(-2) A A A F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 132 132 123 138 0 668 3128 1206 0
normalized size 1 1. 0.93 1.05 0. 5.06 23.7 9.14 0.
time (sec) N/A 0.203 0.364 0.022 0. 0.29 42.678 0.29 0.
Problem 68 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A F(-2) A F(-1) A F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 218 218 205 226 0 1632 0 1 0
normalized size 1 1. 0.94 1.04 0. 7.49 0. 0. 0.
time (sec) N/A 0.378 0.778 0.027 0. 0.279 0. 0.295 0.
Problem 69 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 308 308 455 0 0 0 0 0 566
normalized size 1 1. 1.48 0. 0. 0. 0. 0. 1.84
time (sec) N/A 1.507 3.904 0.07 0. 0. 0. 0. 177.191
Problem 70 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 224 224 348 0 0 0 0 0 410
normalized size 1 1. 1.55 0. 0. 0. 0. 0. 1.83
time (sec) N/A 0.931 1.476 0.058 0. 0. 0. 0. 120.378
Problem 71 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 154 154 279 0 0 0 0 0 148
normalized size 1 1. 1.81 0. 0. 0. 0. 0. 0.96
time (sec) N/A 0.276 0.827 0.037 0. 0. 0. 0. 32.276
Problem 72 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 243 243 379 0 0 0 0 0 0
normalized size 1 1. 1.56 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.829 2.408 0.084 0. 0. 0. 0. 0.
Problem 73 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 368 368 2302 0 0 0 0 0 0
normalized size 1 1. 6.26 0. 0. 0. 0. 0. 0.
time (sec) N/A 1.365 6.291 0.176 0. 0. 0. 0. 0.
Problem 74 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 552 552 4111 0 0 0 0 0 0
normalized size 1 1. 7.45 0. 0. 0. 0. 0. 0.
time (sec) N/A 2.355 6.474 0.234 0. 0. 0. 0. 0.
Problem 75 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 750 750 5537 0 0 0 0 0 0
normalized size 1 1. 7.38 0. 0. 0. 0. 0. 0.
time (sec) N/A 5.684 6.488 0.097 0. 0. 0. 0. 0.
Problem 76 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 543 543 4177 0 0 0 0 0 0
normalized size 1 1. 7.69 0. 0. 0. 0. 0. 0.
time (sec) N/A 5.109 6.41 0.097 0. 0. 0. 0. 0.
Problem 77 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 362 328 3152 0 0 0 0 0 0
normalized size 1 0.91 8.71 0. 0. 0. 0. 0. 0.
time (sec) N/A 1.385 6.262 0.09 0. 0. 0. 0. 0.
Problem 78 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 726 726 11767 0 0 0 0 0 0
normalized size 1 1. 16.21 0. 0. 0. 0. 0. 0.
time (sec) N/A 4.916 7.106 0.262 0. 0. 0. 0. 0.
Problem 79 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 1129 1129 16855 0 0 0 0 0 0
normalized size 1 1. 14.93 0. 0. 0. 0. 0. 0.
time (sec) N/A 9.802 7.577 0.302 0. 0. 0. 0. 0.
Problem 80 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 1707 1707 13018 0 0 0 0 0 0
normalized size 1 1. 7.63 0. 0. 0. 0. 0. 0.
time (sec) N/A 13.928 7.797 0.188 0. 0. 0. 0. 0.
Problem 81 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 1191 1191 10910 0 0 0 0 0 0
normalized size 1 1. 9.16 0. 0. 0. 0. 0. 0.
time (sec) N/A 8.987 6.821 0.156 0. 0. 0. 0. 0.
Problem 82 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 713 713 8593 0 0 0 0 0 0
normalized size 1 1. 12.05 0. 0. 0. 0. 0. 0.
time (sec) N/A 4.999 6.678 0.148 0. 0. 0. 0. 0.
Problem 83 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 1708 1708 43535 0 0 0 0 0 0
normalized size 1 1. 25.49 0. 0. 0. 0. 0. 0.
time (sec) N/A 14.907 8.956 0.435 0. 0. 0. 0. 0.
Problem 84 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 2446 2446 56566 0 0 0 0 0 0
normalized size 1 1. 23.13 0. 0. 0. 0. 0. 0.
time (sec) N/A 24.827 9.886 0.608 0. 0. 0. 0. 0.
Problem 85 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F(-2) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 292 292 3778 0 0 0 0 0 262
normalized size 1 1. 12.94 0. 0. 0. 0. 0. 0.9
time (sec) N/A 0.842 6.259 0.087 0. 0. 0. 0. 80.164
Problem 86 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F(-2) F(-1) F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 294 294 10587 0 0 0 0 0 265
normalized size 1 1. 36.01 0. 0. 0. 0. 0. 0.9
time (sec) N/A 0.829 6.915 0.092 0. 0. 0. 0. 84.961
Problem 87 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F(-2) F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 292 292 688 0 0 0 0 0 258
normalized size 1 1. 2.36 0. 0. 0. 0. 0. 0.88
time (sec) N/A 0.836 0.861 0.031 0. 0. 0. 0. 89.888
Problem 88 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F(-2) F(-1) F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 298 298 3012 0 0 0 0 0 262
normalized size 1 1. 10.11 0. 0. 0. 0. 0. 0.88
time (sec) N/A 0.856 6.261 0.02 0. 0. 0. 0. 87.669
Problem 89 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F(-2) F(-1) F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 298 298 8781 0 0 0 0 0 262
normalized size 1 1. 29.47 0. 0. 0. 0. 0. 0.88
time (sec) N/A 0.843 6.605 0.019 0. 0. 0. 0. 103.334
Problem 90 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A F(-1) A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 29 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.026 0.321 0.192 0. 0. 0. 0. 0.
Problem 91 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F(-2) A
verified N/A NO NO TBD TBD TBD TBD TBD TBD
size 606 606 2025 0 0 0 0 0 520
normalized size 1 1. 3.34 0. 0. 0. 0. 0. 0.86
time (sec) N/A 1.393 24.419 0.161 0. 0. 0. 0. 165.131
Problem 92 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F(-2) A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 447 447 1522 0 0 0 0 0 382
normalized size 1 1. 3.4 0. 0. 0. 0. 0. 0.85
time (sec) N/A 0.993 6.261 0.134 0. 0. 0. 0. 123.616
Problem 93 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 288 288 902 0 0 0 0 0 245
normalized size 1 1. 3.13 0. 0. 0. 0. 0. 0.85
time (sec) N/A 0.62 1.361 0.134 0. 0. 0. 0. 80.888
Problem 94 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A F(-1) A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 29 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.028 0.099 0.122 0. 0. 0. 0. 0.
Problem 95 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A F(-1) A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 29 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.028 0.165 0.084 0. 0. 0. 0. 0.
Problem 96 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A F(-1) A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 29 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.028 1.015 0.11 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 [4] had the largest ratio of [ 0.5556 ]

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 12 7 1. 17 0.412
2 A 13 7 1. 18 0.389
3 A 19 6 1. 17 0.353
4 A 13 10 1. 18 0.556
5 A 19 6 1. 26 0.231
6 A 19 6 1. 26 0.231
7 A 7 4 1. 27 0.148
8 A 19 6 1. 27 0.222
9 A 19 6 1. 18 0.333
10 A 19 7 0.96 18 0.389
11 A 10 7 1. 18 0.389
12 A 19 6 1. 16 0.375
13 A 19 6 1. 13 0.462
14 A 19 6 1. 18 0.333
15 A 5 5 1. 18 0.278
16 A 7 4 1. 18 0.222
17 A 7 4 1. 18 0.222
18 A 7 4 1. 18 0.222
19 A 7 4 1. 18 0.222
20 A 19 6 1. 20 0.3
21 A 19 7 0.96 20 0.35
22 A 11 8 1. 20 0.4
23 A 19 6 1. 18 0.333
24 A 19 6 1. 15 0.4
25 A 19 6 1. 20 0.3
26 A 5 5 1. 20 0.25
27 A 7 4 1. 20 0.2
28 A 7 4 1. 20 0.2
29 A 7 4 1. 20 0.2
30 A 7 4 1. 20 0.2
31 A 9 6 1. 25 0.24
32 A 9 6 1. 26 0.231
33 A 9 6 1. 33 0.182
34 A 5 5 1. 17 0.294
35 A 6 6 1. 22 0.273
36 A 11 8 1. 17 0.471
37 A 5 4 1. 22 0.182
38 A 14 9 1. 17 0.529
39 A 15 9 1. 22 0.409
40 A 21 8 1. 17 0.471
41 A 9 6 1. 22 0.273
42 A 5 4 1. 21 0.19
43 A 5 4 1. 21 0.19
44 A 3 3 1. 19 0.158
45 A 6 4 1. 21 0.19
46 A 7 4 1. 21 0.19
47 A 3 3 1. 20 0.15
48 A 9 5 1. 21 0.238
49 A 7 5 1. 21 0.238
50 A 4 4 1. 19 0.21
51 A 10 5 1. 21 0.238
52 A 11 5 1. 21 0.238
53 A 11 5 1. 21 0.238
54 A 8 5 1. 21 0.238
55 A 5 4 1. 19 0.21
56 A 15 5 1. 21 0.238
57 A 16 5 1. 21 0.238
58 A 6 5 1. 23 0.217
59 A 0 0 0. 0 0.
60 A 10 5 1. 21 0.238
61 A 8 5 1. 21 0.238
62 A 6 5 1. 19 0.263
63 A 6 5 1. 21 0.238
64 A 8 5 1. 21 0.238
65 A 10 5 1. 21 0.238
66 A 2 1 1. 22 0.045
67 A 2 1 1. 24 0.042
68 A 2 1 1. 24 0.042
69 A 5 3 1. 26 0.115
70 A 5 3 1. 26 0.115
71 A 3 2 1. 24 0.083
72 A 6 3 1. 26 0.115
73 A 7 3 1. 26 0.115
74 A 8 3 1. 26 0.115
75 A 9 4 1. 26 0.154
76 A 9 5 1. 26 0.192
77 A 4 3 0.91 24 0.125
78 A 10 4 1. 26 0.154
79 A 11 4 1. 26 0.154
80 A 11 4 1. 26 0.154
81 A 11 5 1. 26 0.192
82 A 5 3 1. 24 0.125
83 A 15 4 1. 26 0.154
84 A 16 4 1. 26 0.154
85 A 6 5 1. 26 0.192
86 A 6 5 1. 26 0.192
87 A 6 5 1. 26 0.192
88 A 6 5 1. 26 0.192
89 A 6 5 1. 26 0.192
90 A 0 0 0. 0 0.
91 A 10 5 1. 26 0.192
92 A 8 5 1. 26 0.192
93 A 6 5 1. 24 0.208
94 A 0 0 0. 0 0.
95 A 0 0 0. 0 0.
96 A 0 0 0. 0 0.

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