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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 A A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 110 110 110 191 127 162 0 0 92
normalized size 1 1. 1. 1.74 1.15 1.47 0. 0. 0.84
time (sec) N/A 0.308 0.02 0.02 0.799 0.249 0. 0. 21.588
Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B A A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 84 84 84 166 97 135 0 0 66
normalized size 1 1. 1. 1.98 1.15 1.61 0. 0. 0.79
time (sec) N/A 0.263 0.016 0.007 0.812 0.266 0. 0. 18.233
Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B A A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 58 58 58 133 65 97 0 0 39
normalized size 1 1. 1. 2.29 1.12 1.67 0. 0. 0.67
time (sec) N/A 0.154 0.014 0.006 0.871 0.266 0. 0. 9.748
Problem 4 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 26 26 26 35 43 32 17 45 56
normalized size 1 1. 1. 1.35 1.65 1.23 0.65 1.73 2.15
time (sec) N/A 0.036 0.007 0.001 0.781 0.248 0.235 0.265 8.032
Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A A A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 20 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.074 0.104 0.017 0. 0. 0. 0. 0.
Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A A A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 20 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.071 0.236 0.019 0. 0. 0. 0. 0.
Problem 7 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 26 26 21 37 46 31 17 47 53
normalized size 1 1. 0.81 1.42 1.77 1.19 0.65 1.81 2.04
time (sec) N/A 0.041 0.013 0.004 0.786 0.259 0.249 0.245 9.025
Problem 8 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 28 28 19 41 46 34 19 47 60
normalized size 1 1. 0.68 1.46 1.64 1.21 0.68 1.68 2.14
time (sec) N/A 0.044 0.013 0.004 0.757 0.258 0.254 0.286 8.758
Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A F F F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 217 217 158 382 263 446 0 0 0
normalized size 1 1. 0.73 1.76 1.21 2.06 0. 0. 0.
time (sec) N/A 0.79 0.278 0.072 0.797 0.261 0. 0. 0.
Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B A A F F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 165 165 113 324 201 355 0 0 0
normalized size 1 1. 0.68 1.96 1.22 2.15 0. 0. 0.
time (sec) N/A 0.633 0.327 0.062 0.803 0.255 0. 0. 0.
Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C A A F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 107 107 85 231 128 238 0 0 0
normalized size 1 1. 0.79 2.16 1.2 2.22 0. 0. 0.
time (sec) N/A 0.324 0.154 0.041 0.792 0.255 0. 0. 0.
Problem 12 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 46 46 41 54 69 81 39 70 88
normalized size 1 1. 0.89 1.17 1.5 1.76 0.85 1.52 1.91
time (sec) N/A 0.065 0.058 0.003 0.862 0.255 0.311 0.269 12.947
Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A A A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 20 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.07 0.859 0.041 0. 0. 0. 0. 0.
Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A A A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 20 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.066 0.749 0.051 0. 0. 0. 0. 0.
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 48 48 42 58 74 88 39 76 83
normalized size 1 1. 0.88 1.21 1.54 1.83 0.81 1.58 1.73
time (sec) N/A 0.066 0.052 0.003 0.789 0.275 0.318 0.255 14.041
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 52 52 35 64 77 99 42 78 94
normalized size 1 1. 0.67 1.23 1.48 1.9 0.81 1.5 1.81
time (sec) N/A 0.07 0.051 0.003 0.862 0.26 0.335 0.256 13.867
Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A F F F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 333 333 241 548 409 948 0 0 0
normalized size 1 1. 0.72 1.65 1.23 2.85 0. 0. 0.
time (sec) N/A 1.694 0.305 0.089 0.866 0.257 0. 0. 0.
Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A F F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 243 243 203 385 316 703 0 0 0
normalized size 1 1. 0.84 1.58 1.3 2.89 0. 0. 0.
time (sec) N/A 1.193 0.23 0.078 0.981 0.27 0. 0. 0.
Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C A A F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 159 159 120 393 201 456 0 0 0
normalized size 1 1. 0.75 2.47 1.26 2.87 0. 0. 0.
time (sec) N/A 0.534 0.183 0.036 0.924 0.255 0. 0. 0.
Problem 20 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 69 69 69 74 113 171 76 93 122
normalized size 1 1. 1. 1.07 1.64 2.48 1.1 1.35 1.77
time (sec) N/A 0.085 0.117 0.001 0.799 0.246 0.392 0.25 16.624
Problem 21 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A A A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 20 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.071 0.984 0.055 0. 0. 0. 0. 0.
Problem 22 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A A A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 20 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.069 0.841 0.085 0. 0. 0. 0. 0.
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 72 72 62 79 119 178 78 99 116
normalized size 1 1. 0.86 1.1 1.65 2.47 1.08 1.38 1.61
time (sec) N/A 0.085 0.081 0.003 0.791 0.259 0.424 0.262 17.749
Problem 24 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 78 78 54 87 124 189 85 104 129
normalized size 1 1. 0.69 1.12 1.59 2.42 1.09 1.33 1.65
time (sec) N/A 0.092 0.08 0.003 0.769 0.262 0.422 0.249 17.488
Problem 25 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F(-2) A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 153 153 130 0 0 360 332 1 144
normalized size 1 1. 0.85 0. 0. 2.35 2.17 0.01 0.94
time (sec) N/A 0.412 0.209 0.039 0. 0.262 0.656 0.318 45.59
Problem 26 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 115 115 91 199 0 224 196 1 105
normalized size 1 1. 0.79 1.73 0. 1.95 1.7 0.01 0.91
time (sec) N/A 0.269 0.145 0.046 0. 0.262 0.514 0.277 28.909
Problem 27 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 77 77 73 105 0 117 94 1 65
normalized size 1 1. 0.95 1.36 0. 1.52 1.22 0.01 0.84
time (sec) N/A 0.136 0.168 0.03 0. 0.28 0.398 0.275 13.937
Problem 28 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 30 30 30 31 0 50 32 43 0
normalized size 1 1. 1. 1.03 0. 1.67 1.07 1.43 0.
time (sec) N/A 0.035 0.033 0.004 0. 0.26 0.232 0.265 0.
Problem 29 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 68 68 56 0 0 68 0 0 65
normalized size 1 1. 0.82 0. 0. 1. 0. 0. 0.96
time (sec) N/A 0.226 0.099 0.07 0. 0.283 0. 0. 15.413
Problem 30 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F A F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 100 100 78 0 0 117 0 0 92
normalized size 1 1. 0.78 0. 0. 1.17 0. 0. 0.92
time (sec) N/A 0.287 0.303 0.068 0. 0.278 0. 0. 21.44
Problem 31 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F A F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 147 147 111 0 0 216 0 0 138
normalized size 1 1. 0.76 0. 0. 1.47 0. 0. 0.94
time (sec) N/A 0.397 0.299 0.032 0. 0.279 0. 0. 33.072
Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F(-2) A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 322 322 239 0 0 651 709 1 309
normalized size 1 1. 0.74 0. 0. 2.02 2.2 0. 0.96
time (sec) N/A 0.831 0.377 0.027 0. 0.267 1.158 0.446 109.006
Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F(-2) A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 239 239 171 0 0 387 439 1 221
normalized size 1 1. 0.72 0. 0. 1.62 1.84 0. 0.92
time (sec) N/A 0.559 0.317 0.024 0. 0.292 0.91 0.405 70.021
Problem 34 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 156 156 117 220 0 188 233 1 138
normalized size 1 1. 0.75 1.41 0. 1.21 1.49 0.01 0.88
time (sec) N/A 0.286 0.263 0.05 0. 0.284 0.682 0.355 36.249
Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A A A F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 67 67 52 90 101 82 94 914 0
normalized size 1 1. 0.78 1.34 1.51 1.22 1.4 13.64 0.
time (sec) N/A 0.082 0.1 0.023 0.838 0.28 0.379 0.283 0.
Problem 36 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 134 134 108 0 0 128 0 0 138
normalized size 1 1. 0.81 0. 0. 0.96 0. 0. 1.03
time (sec) N/A 0.438 0.269 0.079 0. 0.255 0. 0. 33.932
Problem 37 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 202 202 136 0 0 231 0 0 202
normalized size 1 1. 0.67 0. 0. 1.14 0. 0. 1.
time (sec) N/A 0.588 0.813 0.032 0. 0.28 0. 0. 48.122
Problem 38 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F A F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 286 286 217 0 0 424 0 0 284
normalized size 1 1. 0.76 0. 0. 1.48 0. 0. 0.99
time (sec) N/A 0.819 0.601 0.032 0. 0.282 0. 0. 78.454
Problem 39 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F(-2) A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 496 496 341 0 0 956 1074 1 478
normalized size 1 1. 0.69 0. 0. 1.93 2.17 0. 0.96
time (sec) N/A 1.238 0.582 0.032 0. 0.305 1.573 0.568 174.116
Problem 40 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F(-2) A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 366 366 248 0 0 559 653 1 347
normalized size 1 1. 0.68 0. 0. 1.53 1.78 0. 0.95
time (sec) N/A 0.812 0.486 0.03 0. 0.289 1.202 0.495 110.148
Problem 41 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F(-2) A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 236 236 161 0 0 259 350 1 212
normalized size 1 1. 0.68 0. 0. 1.1 1.48 0. 0.9
time (sec) N/A 0.408 0.382 0.018 0. 0.271 0.887 0.45 56.599
Problem 42 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A A A F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 103 103 74 124 155 113 153 1393 0
normalized size 1 1. 0.72 1.2 1.5 1.1 1.49 13.52 0.
time (sec) N/A 0.112 0.136 0.005 0.818 0.299 0.487 0.305 0.
Problem 43 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F A F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 200 200 160 0 0 189 0 0 211
normalized size 1 1. 0.8 0. 0. 0.94 0. 0. 1.05
time (sec) N/A 0.592 0.485 0.034 0. 0.266 0. 0. 50.486
Problem 44 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F A F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 305 305 250 0 0 350 0 0 313
normalized size 1 1. 0.82 0. 0. 1.15 0. 0. 1.03
time (sec) N/A 0.849 1.88 0.031 0. 0.284 0. 0. 73.323
Problem 45 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F A F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 447 447 325 0 0 641 0 0 457
normalized size 1 1. 0.73 0. 0. 1.43 0. 0. 1.02
time (sec) N/A 1.194 0.983 0.033 0. 0.281 0. 0. 124.056
Problem 46 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A F(-1) F A
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 192 192 0 2495 0 558 0 0 160
normalized size 1 1. 0. 12.99 0. 2.91 0. 0. 0.83
time (sec) N/A 0.548 2.844 0.083 0. 0.266 0. 0. 94.025
Problem 47 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A F F A
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 145 145 0 1341 0 366 0 0 114
normalized size 1 1. 0. 9.25 0. 2.52 0. 0. 0.79
time (sec) N/A 0.464 2.661 0.089 0. 0.265 0. 0. 52.758
Problem 48 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A F F A
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 98 98 0 526 0 198 0 0 66
normalized size 1 1. 0. 5.37 0. 2.02 0. 0. 0.67
time (sec) N/A 0.257 89.733 0.06 0. 0.266 0. 0. 26.609
Problem 49 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B A A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 40 40 100 65 89 59 27 0 46
normalized size 1 1. 2.5 1.62 2.22 1.48 0.68 0. 1.15
time (sec) N/A 0.064 0.017 0.004 0.88 0.264 0.348 0. 13.08
Problem 50 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A F(-1) A F(-1)
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 0.205 0.118 0.116 0. 0. 0. 0. 0.
Problem 51 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A F(-1) A F(-1)
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 0.185 0.461 0.135 0. 0. 0. 0. 0.
Problem 52 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A F F F(-1)
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 388 388 0 2553 0 1877 0 0 0
normalized size 1 1. 0. 6.58 0. 4.84 0. 0. 0.
time (sec) N/A 1.421 3.575 0.072 0. 0.292 0. 0. 0.
Problem 53 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A F F F(-1)
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 294 294 0 1251 0 1126 0 0 0
normalized size 1 1. 0. 4.26 0. 3.83 0. 0. 0.
time (sec) N/A 1.118 3.27 0.05 0. 0.283 0. 0. 0.
Problem 54 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A F F F
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 191 191 0 437 0 540 0 0 0
normalized size 1 1. 0. 2.29 0. 2.83 0. 0. 0.
time (sec) N/A 0.569 90.721 0.085 0. 0.267 0. 0. 0.
Problem 55 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 74 74 68 99 135 135 66 0 75
normalized size 1 1. 0.92 1.34 1.82 1.82 0.89 0. 1.01
time (sec) N/A 0.108 0.192 0.003 0.773 0.276 0.428 0. 20.243
Problem 56 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 0.2 1.462 0.415 0. 0. 0. 0. 0.
Problem 57 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 0.185 1.51 0.219 0. 0. 0. 0. 0.
Problem 58 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A F F F(-1)
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 594 594 0 3116 0 3650 0 0 0
normalized size 1 1. 0. 5.25 0. 6.14 0. 0. 0.
time (sec) N/A 3.152 4.084 0.079 0. 0.285 0. 0. 0.
Problem 59 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A F F F(-1)
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 439 439 0 1457 0 2049 0 0 0
normalized size 1 1. 0. 3.32 0. 4.67 0. 0. 0.
time (sec) N/A 2.185 3.536 0.053 0. 0.27 0. 0. 0.
Problem 60 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A F A F F F(-1)
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 276 276 0 514 0 940 0 0 0
normalized size 1 1. 0. 1.86 0. 3.41 0. 0. 0.
time (sec) N/A 0.954 90.36 0.039 0. 0.252 0. 0. 0.
Problem 61 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 111 111 97 134 196 257 116 0 105
normalized size 1 1. 0.87 1.21 1.77 2.32 1.05 0. 0.95
time (sec) N/A 0.142 0.34 0.004 0.891 0.281 0.56 0. 24.563
Problem 62 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 0.198 2.743 0.227 0. 0. 0. 0. 0.
Problem 63 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 0.182 2.183 0.405 0. 0. 0. 0. 0.
Problem 64 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 71 71 106 77 111 101 76 93 61
normalized size 1 1. 1.49 1.08 1.56 1.42 1.07 1.31 0.86
time (sec) N/A 0.151 0.594 0.008 0.807 0.294 4.425 0.321 12.45
Problem 65 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 145 145 197 144 216 182 167 182 133
normalized size 1 1. 1.36 0.99 1.49 1.26 1.15 1.26 0.92
time (sec) N/A 0.332 0.766 0.006 0.853 0.255 6.622 0.264 25.878
Problem 66 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 224 224 285 211 321 267 265 271 212
normalized size 1 1. 1.27 0.94 1.43 1.19 1.18 1.21 0.95
time (sec) N/A 0.449 1.086 0.007 0.878 0.257 9.355 0.24 37.185
Problem 67 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A A A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 22 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.063 0.352 0.038 0. 0. 0. 0. 0.
Problem 68 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A A A F(-1)
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 22 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.06 0.785 0.062 0. 0. 0. 0. 0.
Problem 69 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A F(-1) A F(-1)
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 22 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.059 1.061 0.075 0. 0. 0. 0. 0.
Problem 70 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F(-2) A F(-1) F A
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 340 340 0 0 0 365 0 0 333
normalized size 1 1. 0. 0. 0. 1.07 0. 0. 0.98
time (sec) N/A 0.833 0.258 0.064 0. 0.28 0. 0. 77.94
Problem 71 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F(-2) A F(-1) F A
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 228 228 0 0 0 258 0 0 219
normalized size 1 1. 0. 0. 0. 1.13 0. 0. 0.96
time (sec) N/A 0.499 0.2 0.051 0. 0.276 0. 0. 50.253
Problem 72 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F(-2) A F(-1) F A
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 116 116 0 0 0 150 0 0 105
normalized size 1 1. 0. 0. 0. 1.29 0. 0. 0.91
time (sec) N/A 0.239 0.145 0.052 0. 0.271 0. 0. 22.981
Problem 73 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 0.181 0.11 0.08 0. 0. 0. 0. 0.
Problem 74 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 0.173 0.148 0.333 0. 0. 0. 0. 0.
Problem 75 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 0.172 0.354 0.246 0. 0. 0. 0. 0.
Problem 76 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 115 115 103 217 180 181 0 0 104
normalized size 1 1. 0.9 1.89 1.57 1.57 0. 0. 0.9
time (sec) N/A 0.217 0.051 0.04 0.816 0.301 0. 0. 32.533
Problem 77 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B A A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 85 85 76 187 144 146 0 0 75
normalized size 1 1. 0.89 2.2 1.69 1.72 0. 0. 0.88
time (sec) N/A 0.18 0.027 0.026 0.827 0.287 0. 0. 27.057
Problem 78 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B A A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 54 54 47 148 107 101 0 0 44
normalized size 1 1. 0.87 2.74 1.98 1.87 0. 0. 0.81
time (sec) N/A 0.105 0.023 0.023 0.797 0.272 0. 0. 18.6
Problem 79 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 23 23 23 24 31 31 17 32 17
normalized size 1 1. 1. 1.04 1.35 1.35 0.74 1.39 0.74
time (sec) N/A 0.058 0.005 0.003 0.789 0.251 0.285 0.252 13.455
Problem 80 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A A A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 27 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.103 0.053 0.037 0. 0. 0. 0. 0.
Problem 81 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A A A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 27 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.1 0.055 0.036 0. 0. 0. 0. 0.
Problem 82 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 140 140 137 267 181 332 0 0 110
normalized size 1 1. 0.98 1.91 1.29 2.37 0. 0. 0.79
time (sec) N/A 0.396 0.168 0.032 0.811 0.24 0. 0. 39.027
Problem 83 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B A A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 107 107 103 225 143 251 0 0 78
normalized size 1 1. 0.96 2.1 1.34 2.35 0. 0. 0.73
time (sec) N/A 0.286 0.115 0.028 0.803 0.255 0. 0. 29.054
Problem 84 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 69 69 54 67 97 100 58 0 92
normalized size 1 1. 0.78 0.97 1.41 1.45 0.84 0. 1.33
time (sec) N/A 0.127 0.079 0.019 0.851 0.246 0.403 0. 22.489
Problem 85 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 26 34 34 26 34 19
normalized size 1 1. 1. 1.04 1.36 1.36 1.04 1.36 0.76
time (sec) N/A 0.057 0.02 0.003 0.777 0.256 0.268 0.227 7.625
Problem 86 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A A A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 61 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.183 0.144 0.036 0. 0. 0. 0. 0.
Problem 87 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A A A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 61 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.181 0.153 0.048 0. 0. 0. 0. 0.
Problem 88 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A F F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 261 261 220 488 355 779 0 0 0
normalized size 1 1. 0.84 1.87 1.36 2.98 0. 0. 0.
time (sec) N/A 0.804 0.458 0.053 0.813 0.273 0. 0. 0.
Problem 89 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 182 182 177 295 289 512 0 0 0
normalized size 1 1. 0.97 1.62 1.59 2.81 0. 0. 0.
time (sec) N/A 0.486 0.215 0.04 0.799 0.267 0. 0. 0.
Problem 90 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 106 106 98 127 203 200 122 0 141
normalized size 1 1. 0.92 1.2 1.92 1.89 1.15 0. 1.33
time (sec) N/A 0.162 0.097 0.026 0.813 0.265 0.517 0. 30.51
Problem 91 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 27 26 34 62 53 34 22
normalized size 1 1. 1. 0.96 1.26 2.3 1.96 1.26 0.81
time (sec) N/A 0.058 0.017 0.003 0.771 0.255 0.299 0.24 8.529
Problem 92 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A A A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 65 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.183 0.832 0.091 0. 0. 0. 0. 0.
Problem 93 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade N/A A A A A A A A A
verified N/A N/A N/A TBD TBD TBD TBD TBD TBD
size 63 0 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.179 0.908 0.119 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 [19] had the largest ratio of [ 0.7333 ]

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 6 6 1. 17 0.353
2 A 5 5 1. 17 0.294
3 A 4 4 1. 15 0.267
4 A 4 4 1. 13 0.308
5 A 0 0 0. 0 0.
6 A 0 0 0. 0 0.
7 A 4 4 1. 14 0.286
8 A 4 4 1. 16 0.25
9 A 13 8 1. 17 0.471
10 A 11 9 1. 17 0.529
11 A 10 10 1. 15 0.667
12 A 3 2 1. 13 0.154
13 A 0 0 0. 0 0.
14 A 0 0 0. 0 0.
15 A 3 2 1. 14 0.143
16 A 3 2 1. 16 0.125
17 A 26 10 1. 17 0.588
18 A 23 12 1. 17 0.706
19 A 15 11 1. 15 0.733
20 A 3 2 1. 13 0.154
21 A 0 0 0. 0 0.
22 A 0 0 0. 0 0.
23 A 3 2 1. 14 0.143
24 A 3 2 1. 16 0.125
25 A 6 3 1. 23 0.13
26 A 5 3 1. 23 0.13
27 A 4 3 1. 21 0.143
28 A 2 1 1. 15 0.067
29 A 4 3 1. 23 0.13
30 A 5 4 1. 23 0.174
31 A 6 4 1. 23 0.174
32 A 10 3 1. 25 0.12
33 A 8 3 1. 25 0.12
34 A 6 3 1. 23 0.13
35 A 4 3 1. 17 0.176
36 A 6 3 1. 25 0.12
37 A 8 4 1. 25 0.16
38 A 10 4 1. 25 0.16
39 A 14 3 1. 25 0.12
40 A 11 3 1. 25 0.12
41 A 8 3 1. 23 0.13
42 A 4 3 1. 17 0.176
43 A 8 3 1. 25 0.12
44 A 11 4 1. 25 0.16
45 A 14 4 1. 25 0.16
46 A 6 6 1. 25 0.24
47 A 5 5 1. 25 0.2
48 A 4 4 1. 23 0.174
49 A 5 5 1. 17 0.294
50 A 0 0 0. 0 0.
51 A 0 0 0. 0 0.
52 A 13 8 1. 25 0.32
53 A 11 9 1. 25 0.36
54 A 11 11 1. 23 0.478
55 A 4 3 1. 17 0.176
56 A 0 0 0. 0 0.
57 A 0 0 0. 0 0.
58 A 26 10 1. 25 0.4
59 A 24 13 1. 25 0.52
60 A 17 12 1. 23 0.522
61 A 4 3 1. 17 0.176
62 A 0 0 0. 0 0.
63 A 0 0 0. 0 0.
64 A 5 4 1. 17 0.235
65 A 8 4 1. 19 0.21
66 A 11 4 1. 19 0.21
67 A 0 0 0. 0 0.
68 A 0 0 0. 0 0.
69 A 0 0 0. 0 0.
70 A 8 3 1. 25 0.12
71 A 6 3 1. 25 0.12
72 A 4 3 1. 23 0.13
73 A 0 0 0. 0 0.
74 A 0 0 0. 0 0.
75 A 0 0 0. 0 0.
76 A 5 5 1. 24 0.208
77 A 4 4 1. 24 0.167
78 A 3 3 1. 22 0.136
79 A 2 2 1. 21 0.095
80 A 0 0 0. 0 0.
81 A 0 0 0. 0 0.
82 A 6 6 1. 24 0.25
83 A 5 5 1. 24 0.208
84 A 5 5 1. 22 0.227
85 A 2 2 1. 21 0.095
86 A 0 0 0. 0 0.
87 A 0 0 0. 0 0.
88 A 12 9 1. 24 0.375
89 A 11 10 1. 24 0.417
90 A 4 3 1. 22 0.136
91 A 2 2 1. 21 0.095
92 A 0 0 0. 0 0.
93 A 0 0 0. 0 0.

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