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Chapter 2
detailed summary tables of results

 2.1 List of integrals sorted by grade for each CAS
  2.1.1 Rubi
  2.1.2 Mathematica
  2.1.3 Maple
  2.1.4 Maxima
  2.1.5 FriCAS
  2.1.6 Sympy
  2.1.7 Giac
  2.1.8 Mupad
 2.2 Detailed conclusion table per each integral for all CAS systems
 2.3 Detailed conclusion table specific for Rubi results

2.1 List of integrals sorted by grade for each CAS

2.1.1 Rubi

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111 }

B grade: { }

C grade: { }

F grade: { }

2.1.2 Mathematica

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102 }

B grade: { }

C grade: { 15, 16, 17, 18, 19, 31, 32, 33, 34, 35, 47, 48, 49, 50, 51, 105 }

F grade: { 103, 104, 106, 107, 108, 109, 110, 111 }

2.1.3 Maple

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 20, 26, 27, 28, 31, 32, 33, 34, 42, 43, 44, 47, 48, 49, 50, 51, 61, 62, 63, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 105, 108, 109, 110, 111 }

B grade: { 11, 12, 13, 14, 18, 19, 21, 22, 23, 24, 25, 29, 30, 35, 36, 37, 38, 39, 40, 41, 45, 46, 52, 53, 54, 55, 56, 57, 58, 59, 60, 64, 65, 66, 103, 104, 106, 107 }

C grade: { }

F grade: { }

2.1.4 Maxima

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 60, 61, 62, 63, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 108, 109, 110, 111 }

B grade: { }

C grade: { }

F grade: { 20, 21, 22, 23, 24, 25, 36, 37, 38, 39, 40, 41, 52, 53, 54, 55, 56, 57, 58, 59, 64, 65, 66, 103, 104, 105, 106, 107 }

2.1.5 FriCAS

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 31, 32, 33, 34, 35, 47, 48, 49, 60, 61, 62, 63, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 108, 109, 110, 111 }

B grade: { 26, 27, 28, 29, 30, 42, 43, 44, 45, 46, 50, 51, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102 }

C grade: { }

F grade: { 20, 21, 22, 23, 24, 25, 36, 37, 38, 39, 40, 41, 52, 53, 54, 55, 56, 57, 58, 59, 64, 65, 66, 103, 104, 105, 106, 107 }

2.1.6 Sympy

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 60, 61, 62, 63, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 85, 91, 97 }

B grade: { 10, 11, 26, 27, 42, 43, 80, 81, 82, 86, 92, 98 }

C grade: { 15, 16, 31, 32, 47, 48 }

F grade: { 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 64, 65, 66, 83, 84, 87, 88, 89, 90, 93, 94, 95, 96, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111 }

2.1.7 Giac

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 60, 61, 62, 63, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102 }

B grade: { 20, 21, 22, 23, 24, 25, 36, 37, 38, 39, 52, 53, 54, 55, 64, 65, 66, 108, 109, 110, 111 }

C grade: { }

F grade: { 40, 41, 56, 57, 58, 59, 103, 104, 105, 106, 107 }

2.1.8 Mupad

A grade: { }

B grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 108, 109, 110, 111 }

C grade: { }

F grade: { 103, 104, 105, 106, 107 }

2.2 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 Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 50 50 50 41 40 40 46 43 40
normalized size 1 1.00 1.00 0.82 0.80 0.80 0.92 0.86 0.80
time (sec) N/A 0.041 0.002 0.000 1.249 0.509 0.064 0.320 0.027
Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 69 69 69 58 57 61 65 64 59
normalized size 1 1.00 1.00 0.84 0.83 0.88 0.94 0.93 0.86
time (sec) N/A 0.045 0.021 0.002 1.207 0.789 0.069 0.244 0.033
Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 88 88 88 75 74 82 83 85 78
normalized size 1 1.00 1.00 0.85 0.84 0.93 0.94 0.97 0.89
time (sec) N/A 0.073 0.019 0.002 1.559 0.502 0.074 0.218 0.662
Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 105 105 105 90 89 103 102 106 95
normalized size 1 1.00 1.00 0.86 0.85 0.98 0.97 1.01 0.90
time (sec) N/A 0.095 0.034 0.002 1.643 0.695 0.079 0.360 0.659
Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 122 122 122 105 104 124 121 127 112
normalized size 1 1.00 1.00 0.86 0.85 1.02 0.99 1.04 0.92
time (sec) N/A 0.111 0.038 0.001 1.174 0.901 0.083 0.319 0.058
Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 112 112 97 95 94 100 116 106 94
normalized size 1 1.00 0.87 0.85 0.84 0.89 1.04 0.95 0.84
time (sec) N/A 0.126 0.049 0.001 1.060 1.094 0.084 0.389 0.056
Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 154 154 154 139 138 151 165 157 138
normalized size 1 1.00 1.00 0.90 0.90 0.98 1.07 1.02 0.90
time (sec) N/A 0.130 0.045 0.000 1.027 0.447 0.093 0.256 0.697
Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 196 196 196 183 182 202 209 208 182
normalized size 1 1.00 1.00 0.93 0.93 1.03 1.07 1.06 0.93
time (sec) N/A 0.168 0.057 0.000 1.400 0.783 0.102 0.299 0.716
Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 234 234 234 219 218 253 258 259 220
normalized size 1 1.00 1.00 0.94 0.93 1.08 1.10 1.11 0.94
time (sec) N/A 0.238 0.083 0.000 1.138 0.831 0.110 0.258 0.114
Problem 10 Optimal Rubi Mathematica 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 45 45 50 58 43 43 515 51 51
normalized size 1 1.00 1.11 1.29 0.96 0.96 11.44 1.13 1.13
time (sec) N/A 0.032 0.018 0.012 1.125 1.600 3.147 0.251 0.709
Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B A A B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 51 51 58 86 51 51 2195 59 63
normalized size 1 1.00 1.14 1.69 1.00 1.00 43.04 1.16 1.24
time (sec) N/A 0.057 0.026 0.009 1.122 1.018 110.122 0.306 0.715
Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B A A F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 57 57 68 114 61 61 0 69 75
normalized size 1 1.00 1.19 2.00 1.07 1.07 0.00 1.21 1.32
time (sec) N/A 0.072 0.032 0.008 1.355 1.481 0.000 0.310 0.742
Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B A A F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 64 64 81 145 72 72 0 80 90
normalized size 1 1.00 1.27 2.27 1.12 1.12 0.00 1.25 1.41
time (sec) N/A 0.147 0.045 0.010 1.238 4.816 0.000 0.434 0.815
Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B A A F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 76 76 98 179 88 88 0 96 108
normalized size 1 1.00 1.29 2.36 1.16 1.16 0.00 1.26 1.42
time (sec) N/A 0.192 0.064 0.009 1.245 18.713 0.000 0.261 1.190
Problem 15 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C A A A C A B
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 92 92 98 92 65 65 923 67 118
normalized size 1 1.00 1.07 1.00 0.71 0.71 10.03 0.73 1.28
time (sec) N/A 0.077 0.178 0.007 2.209 0.933 2.887 0.378 0.242
Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C A A A C A B
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 104 104 121 148 75 75 3589 77 159
normalized size 1 1.00 1.16 1.42 0.72 0.72 34.51 0.74 1.53
time (sec) N/A 0.085 0.138 0.003 2.579 0.997 98.602 0.228 0.951
Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C A A A F(-1) A B
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 127 127 150 204 83 83 0 85 199
normalized size 1 1.00 1.18 1.61 0.65 0.65 0.00 0.67 1.57
time (sec) N/A 0.101 0.481 0.003 2.385 1.299 0.000 0.291 1.127
Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C B A A F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 136 136 165 241 92 92 0 94 1209
normalized size 1 1.00 1.21 1.77 0.68 0.68 0.00 0.69 8.89
time (sec) N/A 0.140 0.603 0.006 2.618 4.555 0.000 0.301 6.108
Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C B A A F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 151 151 187 303 106 106 0 108 1509
normalized size 1 1.00 1.24 2.01 0.70 0.70 0.00 0.72 9.99
time (sec) N/A 0.176 0.582 0.006 2.367 17.273 0.000 0.307 7.805
Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 189 189 194 231 0 0 0 1248 1308
normalized size 1 1.00 1.03 1.22 0.00 0.00 0.00 6.60 6.92
time (sec) N/A 0.211 0.251 0.034 0.000 0.000 0.000 4.590 1.316
Problem 21 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 211 211 234 616 0 0 0 1618 3942
normalized size 1 1.00 1.11 2.92 0.00 0.00 0.00 7.67 18.68
time (sec) N/A 0.240 0.219 0.031 0.000 0.000 0.000 3.540 2.139
Problem 22 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 245 245 280 866 0 0 0 3272 15179
normalized size 1 1.00 1.14 3.53 0.00 0.00 0.00 13.36 61.96
time (sec) N/A 0.159 0.289 0.033 0.000 0.000 0.000 2.827 2.539
Problem 23 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 290 290 383 1132 0 0 0 5201 5981
normalized size 1 1.00 1.32 3.90 0.00 0.00 0.00 17.93 20.62
time (sec) N/A 0.725 0.500 0.042 0.000 0.000 0.000 4.908 1.749
Problem 24 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 321 321 441 1435 0 0 0 6096 11383
normalized size 1 1.00 1.37 4.47 0.00 0.00 0.00 18.99 35.46
time (sec) N/A 0.534 0.647 0.043 0.000 0.000 0.000 3.720 2.030
Problem 25 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 545 545 816 3835 0 0 0 11831 49150
normalized size 1 1.00 1.50 7.04 0.00 0.00 0.00 21.71 90.18
time (sec) N/A 4.213 1.293 0.080 0.000 0.000 0.000 7.208 4.306
Problem 26 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 94 94 90 122 83 169 604 93 84
normalized size 1 1.00 0.96 1.30 0.88 1.80 6.43 0.99 0.89
time (sec) N/A 0.052 0.054 0.019 1.677 1.317 3.565 0.230 0.088
Problem 27 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 115 115 112 182 106 217 2689 115 107
normalized size 1 1.00 0.97 1.58 0.92 1.89 23.38 1.00 0.93
time (sec) N/A 0.140 0.080 0.018 1.068 1.464 118.426 0.252 0.104
Problem 28 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 138 138 134 242 127 262 0 136 128
normalized size 1 1.00 0.97 1.75 0.92 1.90 0.00 0.99 0.93
time (sec) N/A 0.154 0.054 0.023 0.971 1.855 0.000 0.253 0.136
Problem 29 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B A B F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 150 150 159 302 145 304 0 158 146
normalized size 1 1.00 1.06 2.01 0.97 2.03 0.00 1.05 0.97
time (sec) N/A 0.214 0.072 0.017 1.183 5.398 0.000 0.298 0.870
Problem 30 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B A B F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 162 162 185 362 163 346 0 179 164
normalized size 1 1.00 1.14 2.23 1.01 2.14 0.00 1.10 1.01
time (sec) N/A 0.232 0.086 0.020 1.349 25.374 0.000 0.316 0.583
Problem 31 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C A A A C A B
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 140 140 146 146 96 154 952 100 149
normalized size 1 1.00 1.04 1.04 0.69 1.10 6.80 0.71 1.06
time (sec) N/A 0.098 0.489 0.014 2.417 1.089 3.495 0.239 0.252
Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C A A A C A B
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 165 165 186 214 120 212 4106 128 201
normalized size 1 1.00 1.13 1.30 0.73 1.28 24.88 0.78 1.22
time (sec) N/A 0.129 0.419 0.013 2.391 0.928 108.823 0.234 0.315
Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C A A A F(-1) A B
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 179 179 200 260 135 239 0 142 237
normalized size 1 1.00 1.12 1.45 0.75 1.34 0.00 0.79 1.32
time (sec) N/A 0.141 0.434 0.016 2.581 1.475 0.000 0.308 1.154
Problem 34 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C A A A F(-1) A B
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 187 187 234 328 143 255 0 155 1547
normalized size 1 1.00 1.25 1.75 0.76 1.36 0.00 0.83 8.27
time (sec) N/A 0.167 0.606 0.015 2.951 4.343 0.000 0.318 5.349
Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C B A A F(-1) A B
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 194 194 243 374 155 279 0 169 1894
normalized size 1 1.00 1.25 1.93 0.80 1.44 0.00 0.87 9.76
time (sec) N/A 0.197 0.659 0.015 2.628 19.792 0.000 0.305 8.177
Problem 36 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 330 330 341 1237 0 0 0 3434 2382
normalized size 1 1.00 1.03 3.75 0.00 0.00 0.00 10.41 7.22
time (sec) N/A 0.745 0.756 0.145 0.000 0.000 0.000 5.020 1.504
Problem 37 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 368 368 398 1813 0 0 0 5164 4707
normalized size 1 1.00 1.08 4.93 0.00 0.00 0.00 14.03 12.79
time (sec) N/A 0.870 1.170 0.179 0.000 0.000 0.000 6.669 1.709
Problem 38 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 386 386 421 2310 0 0 0 5579 7373
normalized size 1 1.00 1.09 5.98 0.00 0.00 0.00 14.45 19.10
time (sec) N/A 0.490 1.296 0.175 0.000 0.000 0.000 6.114 1.771
Problem 39 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 439 439 489 1801 0 0 0 7502 13024
normalized size 1 1.00 1.11 4.10 0.00 0.00 0.00 17.09 29.67
time (sec) N/A 1.894 1.882 0.070 0.000 0.000 0.000 8.028 2.306
Problem 40 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 468 468 524 1917 0 0 0 0 18449
normalized size 1 1.00 1.12 4.10 0.00 0.00 0.00 0.00 39.42
time (sec) N/A 1.118 2.112 0.049 0.000 0.000 0.000 0.000 3.116
Problem 41 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 770 770 935 4570 0 0 0 0 82785
normalized size 1 1.00 1.21 5.94 0.00 0.00 0.00 0.00 107.51
time (sec) N/A 7.835 5.697 0.104 0.000 0.000 0.000 0.000 13.909
Problem 42 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 143 143 128 186 121 307 668 123 118
normalized size 1 1.00 0.90 1.30 0.85 2.15 4.67 0.86 0.83
time (sec) N/A 0.076 0.098 0.020 1.062 1.495 3.687 0.334 0.092
Problem 43 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 175 175 161 278 155 389 2822 157 151
normalized size 1 1.00 0.92 1.59 0.89 2.22 16.13 0.90 0.86
time (sec) N/A 0.224 0.126 0.023 1.098 1.917 124.287 0.353 0.113
Problem 44 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 204 204 193 370 188 470 0 190 182
normalized size 1 1.00 0.95 1.81 0.92 2.30 0.00 0.93 0.89
time (sec) N/A 0.252 0.084 0.022 1.082 2.865 0.000 0.394 0.847
Problem 45 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B A B F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 224 224 231 462 214 544 0 224 209
normalized size 1 1.00 1.03 2.06 0.96 2.43 0.00 1.00 0.93
time (sec) N/A 0.307 0.118 0.022 1.065 6.125 0.000 0.333 0.248
Problem 46 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B A B F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 239 239 261 554 238 616 0 257 233
normalized size 1 1.00 1.09 2.32 1.00 2.58 0.00 1.08 0.97
time (sec) N/A 0.345 0.129 0.021 1.117 27.719 0.000 0.370 0.616
Problem 47 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C A A A C A B
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 185 185 186 180 137 278 1103 131 185
normalized size 1 1.00 1.01 0.97 0.74 1.50 5.96 0.71 1.00
time (sec) N/A 0.117 0.746 0.017 2.554 0.734 3.615 0.362 0.260
Problem 48 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C A A A C A B
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 223 223 235 264 173 384 4496 171 249
normalized size 1 1.00 1.05 1.18 0.78 1.72 20.16 0.77 1.12
time (sec) N/A 0.215 0.592 0.017 2.568 0.893 117.113 0.365 1.008
Problem 49 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C A A A F(-1) A B
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 243 243 259 322 200 435 0 198 295
normalized size 1 1.00 1.07 1.33 0.82 1.79 0.00 0.81 1.21
time (sec) N/A 0.227 0.658 0.019 2.606 2.172 0.000 0.379 1.170
Problem 50 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C A A B F(-1) A B
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 263 263 303 396 217 485 0 228 1611
normalized size 1 1.00 1.15 1.51 0.83 1.84 0.00 0.87 6.13
time (sec) N/A 0.263 0.904 0.023 3.147 5.214 0.000 0.388 5.453
Problem 51 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C A A B F(-1) A B
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 269 269 325 454 229 521 0 255 1963
normalized size 1 1.00 1.21 1.69 0.85 1.94 0.00 0.95 7.30
time (sec) N/A 0.286 0.977 0.019 2.120 23.869 0.000 0.375 8.217
Problem 52 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 474 474 488 3725 0 0 0 3397 4225
normalized size 1 1.00 1.03 7.86 0.00 0.00 0.00 7.17 8.91
time (sec) N/A 2.193 1.911 0.359 0.000 0.000 0.000 13.320 2.344
Problem 53 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 621 621 625 7858 0 0 0 5288 8689
normalized size 1 1.00 1.01 12.65 0.00 0.00 0.00 8.52 13.99
time (sec) N/A 4.512 3.609 0.616 0.000 0.000 0.000 10.792 3.265
Problem 54 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 646 646 661 10222 0 0 0 5439 13431
normalized size 1 1.00 1.02 15.82 0.00 0.00 0.00 8.42 20.79
time (sec) N/A 3.299 4.293 0.448 0.000 0.000 0.000 10.391 4.558
Problem 55 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 679 679 845 3492 0 0 0 6861 23811
normalized size 1 1.00 1.24 5.14 0.00 0.00 0.00 10.10 35.07
time (sec) N/A 4.182 6.548 0.096 0.000 0.000 0.000 13.218 5.347
Problem 56 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 728 728 980 3824 0 0 0 0 36653
normalized size 1 1.00 1.35 5.25 0.00 0.00 0.00 0.00 50.35
time (sec) N/A 2.733 6.674 0.063 0.000 0.000 0.000 0.000 7.160
Problem 57 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 1150 1144 1590 6026 0 0 0 0 114377
normalized size 1 0.99 1.38 5.24 0.00 0.00 0.00 0.00 99.46
time (sec) N/A 8.164 7.480 0.125 0.000 0.000 0.000 0.000 20.572
Problem 58 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 645 645 775 3107 0 0 0 0 53538
normalized size 1 1.00 1.20 4.82 0.00 0.00 0.00 0.00 83.00
time (sec) N/A 3.367 4.405 0.089 0.000 0.000 0.000 0.000 8.852
Problem 59 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 1177 1179 1649 6130 0 0 0 0 97905
normalized size 1 1.00 1.40 5.21 0.00 0.00 0.00 0.00 83.18
time (sec) N/A 7.926 7.346 0.129 0.000 0.000 0.000 0.000 17.175
Problem 60 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 416 416 416 829 418 463 503 478 398
normalized size 1 1.00 1.00 1.99 1.00 1.11 1.21 1.15 0.96
time (sec) N/A 0.629 0.122 0.003 0.517 0.809 0.161 0.428 0.383
Problem 61 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 259 259 259 354 251 285 309 295 246
normalized size 1 1.00 1.00 1.37 0.97 1.10 1.19 1.14 0.95
time (sec) N/A 0.332 0.046 0.002 0.698 0.696 0.125 0.306 0.948
Problem 62 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 154 154 154 161 138 151 165 157 138
normalized size 1 1.00 1.00 1.05 0.90 0.98 1.07 1.02 0.90
time (sec) N/A 0.152 0.032 0.001 0.588 0.725 0.096 0.284 0.089
Problem 63 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 20 20 20 17 16 16 15 17 16
normalized size 1 1.00 1.00 0.85 0.80 0.80 0.75 0.85 0.80
time (sec) N/A 0.033 0.002 0.001 0.617 0.896 0.090 1.774 0.027
Problem 64 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 211 211 234 616 0 0 0 1620 3942
normalized size 1 1.00 1.11 2.92 0.00 0.00 0.00 7.68 18.68
time (sec) N/A 0.318 0.059 0.024 0.000 0.000 0.000 4.239 1.174
Problem 65 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 368 368 398 1813 0 0 0 5164 4707
normalized size 1 1.00 1.08 4.93 0.00 0.00 0.00 14.03 12.79
time (sec) N/A 0.923 1.197 0.140 0.000 0.000 0.000 11.929 1.523
Problem 66 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B F F(-1) F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 621 621 625 7858 0 0 0 5288 8689
normalized size 1 1.00 1.01 12.65 0.00 0.00 0.00 8.52 13.99
time (sec) N/A 4.594 3.584 0.381 0.000 0.000 0.000 6.426 3.161
Problem 67 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 4 4 4 5 4 4 3 5 4
normalized size 1 1.00 1.00 1.25 1.00 1.00 0.75 1.25 1.00
time (sec) N/A 0.011 0.001 0.002 0.430 0.774 0.069 0.307 0.018
Problem 68 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 14 14 16 18 14 14 12 17 14
normalized size 1 1.00 1.14 1.29 1.00 1.00 0.86 1.21 1.00
time (sec) N/A 0.024 0.004 0.003 0.436 1.301 0.121 0.331 0.728
Problem 69 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 31 31 30 35 27 27 26 30 27
normalized size 1 1.00 0.97 1.13 0.87 0.87 0.84 0.97 0.87
time (sec) N/A 0.052 0.012 0.003 0.455 1.187 0.146 0.278 0.037
Problem 70 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 51 51 45 58 43 43 41 49 44
normalized size 1 1.00 0.88 1.14 0.84 0.84 0.80 0.96 0.86
time (sec) N/A 0.085 0.025 0.003 0.448 1.239 0.176 0.248 0.038
Problem 71 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 68 68 68 87 62 62 63 74 64
normalized size 1 1.00 1.00 1.28 0.91 0.91 0.93 1.09 0.94
time (sec) N/A 0.117 0.018 0.002 0.436 1.209 0.209 0.230 0.034
Problem 72 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 92 92 92 122 84 84 88 105 87
normalized size 1 1.00 1.00 1.33 0.91 0.91 0.96 1.14 0.95
time (sec) N/A 0.149 0.032 0.003 0.463 0.896 0.247 0.267 0.038
Problem 73 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 11 11 11 12 11 11 8 13 8
normalized size 1 1.00 1.00 1.09 1.00 1.00 0.73 1.18 0.73
time (sec) N/A 0.010 0.003 0.004 0.429 0.926 0.107 0.279 0.083
Problem 74 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 22 22 23 29 22 22 29 26 22
normalized size 1 1.00 1.05 1.32 1.00 1.00 1.32 1.18 1.00
time (sec) N/A 0.021 0.007 0.004 0.437 0.829 0.283 0.285 0.799
Problem 75 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 29 29 30 45 29 29 44 33 29
normalized size 1 1.00 1.03 1.55 1.00 1.00 1.52 1.14 1.00
time (sec) N/A 0.050 0.013 0.006 0.434 0.951 0.510 0.250 0.071
Problem 76 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 47 47 44 69 45 45 66 49 45
normalized size 1 1.00 0.94 1.47 0.96 0.96 1.40 1.04 0.96
time (sec) N/A 0.068 0.019 0.006 0.448 0.837 0.858 0.228 0.764
Problem 77 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 66 66 67 98 62 62 94 69 63
normalized size 1 1.00 1.02 1.48 0.94 0.94 1.42 1.05 0.95
time (sec) N/A 0.085 0.023 0.008 0.444 0.825 1.531 0.293 0.074
Problem 78 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 90 90 91 134 84 84 122 97 86
normalized size 1 1.00 1.01 1.49 0.93 0.93 1.36 1.08 0.96
time (sec) N/A 0.107 0.036 0.007 0.441 0.816 2.591 0.388 0.084
Problem 79 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 29 29 29 20 19 19 19 22 19
normalized size 1 1.00 1.00 0.69 0.66 0.66 0.66 0.76 0.66
time (sec) N/A 0.021 0.007 0.008 0.439 0.899 0.141 0.237 0.078
Problem 80 Optimal Rubi Mathematica 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 42 42 39 44 32 32 304 38 38
normalized size 1 1.00 0.93 1.05 0.76 0.76 7.24 0.90 0.90
time (sec) N/A 0.052 0.019 0.006 0.444 0.751 1.760 0.287 0.843
Problem 81 Optimal Rubi Mathematica 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 47 47 44 65 37 37 716 43 47
normalized size 1 1.00 0.94 1.38 0.79 0.79 15.23 0.91 1.00
time (sec) N/A 0.064 0.021 0.006 0.436 0.928 12.723 0.367 0.111
Problem 82 Optimal Rubi Mathematica 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 57 57 55 89 47 47 1389 53 59
normalized size 1 1.00 0.96 1.56 0.82 0.82 24.37 0.93 1.04
time (sec) N/A 0.079 0.025 0.007 0.438 0.926 91.466 0.374 0.820
Problem 83 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 74 74 71 120 62 62 0 68 78
normalized size 1 1.00 0.96 1.62 0.84 0.84 0.00 0.92 1.05
time (sec) N/A 0.107 0.033 0.007 0.450 1.032 0.000 0.328 0.880
Problem 84 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 96 96 91 156 82 82 0 90 99
normalized size 1 1.00 0.95 1.62 0.85 0.85 0.00 0.94 1.03
time (sec) N/A 0.137 0.046 0.009 0.449 1.300 0.000 0.243 0.883
Problem 85 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 46 46 42 33 32 45 34 36 32
normalized size 1 1.00 0.91 0.72 0.70 0.98 0.74 0.78 0.70
time (sec) N/A 0.051 0.022 0.010 0.436 0.946 0.260 0.252 0.049
Problem 86 Optimal Rubi Mathematica 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 71 71 66 74 57 93 1188 66 64
normalized size 1 1.00 0.93 1.04 0.80 1.31 16.73 0.93 0.90
time (sec) N/A 0.174 0.047 0.010 0.439 0.971 10.543 0.256 0.807
Problem 87 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 82 82 77 110 68 116 0 77 79
normalized size 1 1.00 0.94 1.34 0.83 1.41 0.00 0.94 0.96
time (sec) N/A 0.199 0.061 0.010 0.461 1.100 0.000 0.251 0.842
Problem 88 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 95 95 90 146 81 141 0 90 94
normalized size 1 1.00 0.95 1.54 0.85 1.48 0.00 0.95 0.99
time (sec) N/A 0.221 0.049 0.013 0.444 2.419 0.000 0.329 0.876
Problem 89 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 106 106 102 182 92 164 0 101 108
normalized size 1 1.00 0.96 1.72 0.87 1.55 0.00 0.95 1.02
time (sec) N/A 0.266 0.057 0.010 0.442 11.237 0.000 0.289 1.364
Problem 90 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 122 122 118 221 108 200 0 117 127
normalized size 1 1.00 0.97 1.81 0.89 1.64 0.00 0.96 1.04
time (sec) N/A 0.315 0.063 0.013 0.450 66.478 0.000 0.367 1.673
Problem 91 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 56 56 48 40 42 72 46 46 42
normalized size 1 1.00 0.86 0.71 0.75 1.29 0.82 0.82 0.75
time (sec) N/A 0.057 0.024 0.010 0.433 0.740 0.291 0.351 0.046
Problem 92 Optimal Rubi Mathematica 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 89 89 80 90 75 153 1255 85 79
normalized size 1 1.00 0.90 1.01 0.84 1.72 14.10 0.96 0.89
time (sec) N/A 0.260 0.051 0.013 0.454 0.723 10.508 0.381 0.101
Problem 93 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 105 105 97 134 91 191 0 101 97
normalized size 1 1.00 0.92 1.28 0.87 1.82 0.00 0.96 0.92
time (sec) N/A 0.320 0.074 0.013 0.442 0.880 0.000 0.322 0.828
Problem 94 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 117 117 114 178 107 229 0 117 115
normalized size 1 1.00 0.97 1.52 0.91 1.96 0.00 1.00 0.98
time (sec) N/A 0.246 0.055 0.015 0.440 2.548 0.000 0.380 0.905
Problem 95 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 131 131 136 222 123 267 0 133 133
normalized size 1 1.00 1.04 1.69 0.94 2.04 0.00 1.02 1.02
time (sec) N/A 0.280 0.062 0.013 0.454 11.825 0.000 0.328 1.332
Problem 96 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 147 147 153 266 139 305 0 149 151
normalized size 1 1.00 1.04 1.81 0.95 2.07 0.00 1.01 1.03
time (sec) N/A 0.330 0.083 0.014 0.452 70.058 0.000 0.394 1.680
Problem 97 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 68 68 60 47 52 103 53 56 52
normalized size 1 1.00 0.88 0.69 0.76 1.51 0.78 0.82 0.76
time (sec) N/A 0.058 0.031 0.013 0.442 0.655 0.305 0.400 0.046
Problem 98 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 105 105 97 106 88 211 1034 98 90
normalized size 1 1.00 0.92 1.01 0.84 2.01 9.85 0.93 0.86
time (sec) N/A 0.196 0.089 0.014 0.441 0.938 8.787 0.311 0.093
Problem 99 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 122 122 121 158 108 267 0 118 113
normalized size 1 1.00 0.99 1.30 0.89 2.19 0.00 0.97 0.93
time (sec) N/A 0.222 0.051 0.015 0.442 1.241 0.000 0.326 0.126
Problem 100 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 141 141 144 210 126 321 0 136 131
normalized size 1 1.00 1.02 1.49 0.89 2.28 0.00 0.96 0.93
time (sec) N/A 0.253 0.074 0.017 0.447 2.534 0.000 0.320 0.881
Problem 101 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 158 158 169 262 145 376 0 155 152
normalized size 1 1.00 1.07 1.66 0.92 2.38 0.00 0.98 0.96
time (sec) N/A 0.289 0.090 0.016 0.450 12.105 0.000 0.365 1.392
Problem 102 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B F(-1) A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 177 177 195 314 163 430 0 173 170
normalized size 1 1.00 1.10 1.77 0.92 2.43 0.00 0.98 0.96
time (sec) N/A 0.343 0.108 0.017 0.462 70.992 0.000 0.430 1.755
Problem 103 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A F(-1) B F F F F F
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 717 717 0 3038 0 0 0 0 -1
normalized size 1 1.00 0.00 4.24 0.00 0.00 0.00 0.00 -0.00
time (sec) N/A 0.596 0.000 0.019 0.000 1.554 0.000 0.000 0.000
Problem 104 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A F(-1) B F F F F F
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 505 505 0 1585 0 0 0 0 -1
normalized size 1 1.00 0.00 3.14 0.00 0.00 0.00 0.00 -0.00
time (sec) N/A 0.279 0.000 0.014 0.000 1.514 0.000 0.000 0.000
Problem 105 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C A F F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 359 359 526 453 0 0 0 0 -1
normalized size 1 1.00 1.47 1.26 0.00 0.00 0.00 0.00 -0.00
time (sec) N/A 0.159 1.382 0.011 0.000 0.984 0.000 0.000 0.000
Problem 106 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A F(-1) B F F F F F
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 447 447 0 1005 0 0 0 0 -1
normalized size 1 1.00 0.00 2.25 0.00 0.00 0.00 0.00 -0.00
time (sec) N/A 0.273 0.000 0.026 0.000 0.607 0.000 0.000 0.000
Problem 107 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A F(-1) B F F F(-1) F F
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 680 680 0 1395 0 0 0 0 -1
normalized size 1 1.00 0.00 2.05 0.00 0.00 0.00 0.00 -0.00
time (sec) N/A 0.515 0.000 0.057 0.000 0.962 0.000 0.000 0.000
Problem 108 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A F(-1) A A A F B B
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 19 19 0 18 17 17 0 60 17
normalized size 1 1.00 0.00 0.95 0.89 0.89 0.00 3.16 0.89
time (sec) N/A 0.018 0.000 0.005 0.633 0.715 0.000 1.911 0.987
Problem 109 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A F(-1) A A A F B B
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 57 57 0 52 51 82 0 142 51
normalized size 1 1.00 0.00 0.91 0.89 1.44 0.00 2.49 0.89
time (sec) N/A 0.067 0.000 0.005 0.638 0.825 0.000 2.012 0.928
Problem 110 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A F(-1) A A A F B B
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 57 57 0 53 49 80 0 136 51
normalized size 1 1.00 0.00 0.93 0.86 1.40 0.00 2.39 0.89
time (sec) N/A 0.080 0.000 0.005 0.630 0.608 0.000 1.946 0.957
Problem 111 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A F(-1) A A A F B B
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 69 69 0 63 94 92 0 166 62
normalized size 1 1.00 0.00 0.91 1.36 1.33 0.00 2.41 0.90
time (sec) N/A 0.091 0.000 0.005 0.678 0.754 0.000 2.099 0.977

2.3 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 [47] had the largest ratio of [.6875]

Table 2.1: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 2 1 1.00 18 0.056
2 A 2 1 1.00 23 0.043
3 A 2 1 1.00 28 0.036
4 A 2 1 1.00 33 0.030
5 A 2 1 1.00 38 0.026
6 A 2 1 1.00 20 0.050
7 A 2 1 1.00 25 0.040
8 A 2 1 1.00 30 0.033
9 A 2 1 1.00 35 0.029
10 A 10 7 1.00 18 0.389
11 A 9 7 1.00 23 0.304
12 A 8 6 1.00 28 0.214
13 A 10 7 1.00 33 0.212
14 A 12 8 1.00 38 0.210
15 A 15 8 1.00 16 0.500
16 A 14 8 1.00 21 0.381
17 A 15 7 1.00 26 0.269
18 A 17 8 1.00 31 0.258
19 A 19 9 1.00 36 0.250
20 A 9 7 1.00 20 0.350
21 A 8 7 1.00 25 0.280
22 A 9 8 1.00 30 0.267
23 A 11 9 1.00 35 0.257
24 A 13 10 1.00 40 0.250
25 A 13 10 1.00 55 0.182
26 A 12 9 1.00 18 0.500
27 A 11 9 1.00 23 0.391
28 A 10 8 1.00 28 0.286
29 A 10 8 1.00 33 0.242
30 A 11 9 1.00 38 0.237
31 A 17 10 1.00 16 0.625
32 A 16 10 1.00 21 0.476
33 A 15 9 1.00 26 0.346
34 A 15 9 1.00 31 0.290
35 A 16 10 1.00 36 0.278
36 A 11 9 1.00 20 0.450
37 A 10 9 1.00 25 0.360
38 A 9 8 1.00 30 0.267
39 A 9 8 1.00 35 0.229
40 A 10 9 1.00 40 0.225
41 A 13 11 1.00 55 0.200
42 A 14 10 1.00 18 0.556
43 A 13 9 1.00 23 0.391
44 A 12 9 1.00 28 0.321
45 A 12 10 1.00 33 0.303
46 A 13 11 1.00 38 0.290
47 A 19 11 1.00 16 0.688
48 A 18 10 1.00 21 0.476
49 A 17 10 1.00 26 0.385
50 A 17 11 1.00 31 0.355
51 A 18 12 1.00 36 0.333
52 A 13 10 1.00 20 0.500
53 A 12 9 1.00 25 0.360
54 A 11 9 1.00 30 0.300
55 A 11 10 1.00 35 0.286
56 A 12 11 1.00 40 0.275
57 A 11 9 0.99 55 0.164
58 A 11 10 1.00 50 0.200
59 A 13 10 1.00 50 0.200
60 A 2 1 1.00 63 0.016
61 A 2 1 1.00 63 0.016
62 A 2 1 1.00 61 0.016
63 A 2 1 1.00 63 0.016
64 A 9 8 1.00 63 0.127
65 A 11 10 1.00 63 0.159
66 A 13 10 1.00 63 0.159
67 A 2 2 1.00 26 0.077
68 A 3 2 1.00 31 0.065
69 A 3 2 1.00 36 0.056
70 A 3 2 1.00 41 0.049
71 A 3 2 1.00 46 0.043
72 A 3 2 1.00 51 0.039
73 A 4 3 1.00 21 0.143
74 A 4 3 1.00 26 0.115
75 A 6 4 1.00 31 0.129
76 A 6 4 1.00 36 0.111
77 A 6 4 1.00 41 0.098
78 A 6 4 1.00 46 0.087
79 A 3 2 1.00 16 0.125
80 A 3 2 1.00 21 0.095
81 A 3 2 1.00 26 0.077
82 A 3 2 1.00 31 0.065
83 A 3 2 1.00 36 0.056
84 A 3 2 1.00 41 0.049
85 A 3 2 1.00 26 0.077
86 A 3 2 1.00 31 0.065
87 A 3 2 1.00 36 0.056
88 A 3 2 1.00 41 0.049
89 A 3 2 1.00 46 0.043
90 A 3 2 1.00 51 0.039
91 A 9 5 1.00 21 0.238
92 A 9 5 1.00 26 0.192
93 A 9 5 1.00 31 0.161
94 A 3 2 1.00 36 0.056
95 A 3 2 1.00 41 0.049
96 A 3 2 1.00 46 0.043
97 A 3 2 1.00 16 0.125
98 A 3 2 1.00 21 0.095
99 A 3 2 1.00 26 0.077
100 A 3 2 1.00 31 0.065
101 A 3 2 1.00 36 0.056
102 A 3 2 1.00 41 0.049
103 A 12 10 1.00 32 0.312
104 A 10 10 1.00 32 0.312
105 A 8 8 1.00 32 0.250
106 A 7 7 1.00 32 0.219
107 A 9 8 1.00 32 0.250
108 A 1 1 1.00 28 0.036
109 A 5 5 1.00 31 0.161
110 A 5 5 1.00 33 0.152
111 A 4 4 1.00 36 0.111

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