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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
 2.2 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, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145 }

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, 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, 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, 125, 126, 127, 128, 129, 130, 131, 137, 138, 139, 140, 141, 142, 143, 144, 145 }

B grade: { 135, 136 }

C grade: { 40, 41, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 132, 133, 134 }

F grade: { }

2.1.3 Maple

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 51, 64, 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, 131, 132, 133, 134, 143, 144, 145 }

B grade: { 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130 }

C grade: { 135, 136, 137, 138, 139, 140, 141, 142 }

F grade: { 30, 40, 41 }

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, 20, 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, 131, 132, 133, 134, 139, 140

B grade: { }

C grade: { }

F grade: { 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, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 135, 136, 137, 138, 141, 142, 143, 144, 145 }

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, 20, 47, 48, 49, 50, 51, 52, 53, 54, 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, 125, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145 }

B grade: { 37, 38, 39, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124 }

C grade: { }

F grade: { 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 40, 41, 42, 43, 44, 45, 46, 68, 73, 126, 127, 128, 129, 130 }

2.1.6 Sympy

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 39, 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, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124 }

B grade: { 47, 48, 49, 50, 57, 58, 63, 64, 125 }

C grade: { 134, 135, 136, 137, 139, 140, 141, 142 }

F grade: { 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 72, 73, 126, 127, 128, 129, 130, 131, 132, 133, 138, 143, 144, 145 }

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, 20, 47, 48, 49, 50, 51, 52, 53, 54, 61, 62, 63, 64, 65, 66, 67, 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, 125, 132, 133, 134, 139, 140 }

B grade: { 37, 38, 39, 131, 143, 144, 145 }

C grade: { 24, 25, 26, 27 }

F grade: { 21, 22, 23, 28, 29, 30, 31, 32, 33, 34, 35, 36, 40, 41, 42, 43, 44, 45, 46, 55, 56, 57, 58, 59, 60, 68, 69, 70, 71, 72, 73, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 126, 127, 128, 129, 130, 135, 136, 137, 138, 141, 142 }

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
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 74 74 74 61 81 169 68 86
normalized size 1 1. 1. 0.82 1.09 2.28 0.92 1.16
time (sec) N/A 0.082 0.015 0.001 0.932 1.052 0.087 1.093
Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 74 74 74 61 81 169 68 86
normalized size 1 1. 1. 0.82 1.09 2.28 0.92 1.16
time (sec) N/A 0.057 0.012 0.001 0.961 1.124 0.074 1.085
Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 69 69 69 58 77 161 65 82
normalized size 1 1. 1. 0.84 1.12 2.33 0.94 1.19
time (sec) N/A 0.036 0.013 0. 0.946 1.122 0.11 1.093
Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 65 65 65 60 74 143 63 81
normalized size 1 1. 1. 0.92 1.14 2.2 0.97 1.25
time (sec) N/A 0.04 0.015 0.003 0.941 1.271 0.45 1.095
Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 63 63 63 57 74 157 58 77
normalized size 1 1. 1. 0.9 1.17 2.49 0.92 1.22
time (sec) N/A 0.051 0.023 0.007 0.965 1.289 0.492 1.095
Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 63 63 58 58 74 154 60 78
normalized size 1 1. 0.92 0.92 1.17 2.44 0.95 1.24
time (sec) N/A 0.048 0.04 0.006 0.945 1.211 0.67 1.083
Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 63 63 60 57 76 149 61 76
normalized size 1 1. 0.95 0.9 1.21 2.37 0.97 1.21
time (sec) N/A 0.051 0.047 0.006 0.959 1.263 1.013 1.106
Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 63 63 62 58 76 154 61 77
normalized size 1 1. 0.98 0.92 1.21 2.44 0.97 1.22
time (sec) N/A 0.051 0.03 0.007 0.969 1.291 2.982 1.097
Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 63 63 63 60 76 159 63 77
normalized size 1 1. 1. 0.95 1.21 2.52 1. 1.22
time (sec) N/A 0.052 0.058 0.007 0.952 1.208 8.345 1.091
Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 68 68 68 63 80 159 66 81
normalized size 1 1. 1. 0.93 1.18 2.34 0.97 1.19
time (sec) N/A 0.048 0.048 0.006 0.968 1.272 16.138 1.096
Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 159 159 159 142 193 400 168 208
normalized size 1 1. 1. 0.89 1.21 2.52 1.06 1.31
time (sec) N/A 0.214 0.047 0.002 0.943 1.113 0.094 1.115
Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 159 159 159 142 193 397 163 208
normalized size 1 1. 1. 0.89 1.21 2.5 1.03 1.31
time (sec) N/A 0.143 0.036 0.001 0.949 1.13 0.095 1.1
Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 154 154 154 139 189 385 165 204
normalized size 1 1. 1. 0.9 1.23 2.5 1.07 1.32
time (sec) N/A 0.111 0.031 0.001 0.973 1.103 0.095 1.091
Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 150 150 150 149 186 335 156 201
normalized size 1 1. 1. 0.99 1.24 2.23 1.04 1.34
time (sec) N/A 0.107 0.04 0.003 0.955 1.247 0.5 1.095
Problem 15 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 145 145 145 147 185 365 156 198
normalized size 1 1. 1. 1.01 1.28 2.52 1.08 1.37
time (sec) N/A 0.121 0.097 0.007 0.931 1.251 0.503 1.088
Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 149 149 139 148 188 362 151 200
normalized size 1 1. 0.93 0.99 1.26 2.43 1.01 1.34
time (sec) N/A 0.123 0.109 0.008 0.944 1.3 0.641 1.136
Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 149 149 151 146 189 354 158 197
normalized size 1 1. 1.01 0.98 1.27 2.38 1.06 1.32
time (sec) N/A 0.137 0.082 0.006 0.956 1.288 0.886 1.088
Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 148 148 130 144 188 346 151 192
normalized size 1 1. 0.88 0.97 1.27 2.34 1.02 1.3
time (sec) N/A 0.142 0.083 0.008 0.984 1.237 2.622 1.114
Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 143 143 142 144 186 344 151 189
normalized size 1 1. 0.99 1.01 1.3 2.41 1.06 1.32
time (sec) N/A 0.147 0.082 0.007 0.97 1.204 8.278 1.121
Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 149 149 144 148 189 346 153 190
normalized size 1 1. 0.97 0.99 1.27 2.32 1.03 1.28
time (sec) N/A 0.143 0.101 0.009 0.953 1.217 29.718 1.129
Problem 21 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-1) F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 339 339 460 1622 0 0 0 0
normalized size 1 1. 1.36 4.78 0. 0. 0. 0.
time (sec) N/A 1.856 0.644 0.046 0. 0. 0. 0.
Problem 22 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 278 278 377 1171 0 0 0 0
normalized size 1 1. 1.36 4.21 0. 0. 0. 0.
time (sec) N/A 0.466 0.451 0.033 0. 0. 0. 0.
Problem 23 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-1) F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 270 270 360 1327 0 0 0 0
normalized size 1 1. 1.33 4.91 0. 0. 0. 0.
time (sec) N/A 0.835 0.396 0.036 0. 0. 0. 0.
Problem 24 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 223 223 240 728 0 0 0 7461
normalized size 1 1. 1.08 3.26 0. 0. 0. 33.46
time (sec) N/A 0.213 0.396 0.023 0. 0. 0. 3.237
Problem 25 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 211 211 234 616 0 0 0 9080
normalized size 1 1. 1.11 2.92 0. 0. 0. 43.03
time (sec) N/A 0.266 0.223 0.017 0. 0. 0. 2.946
Problem 26 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 229 229 285 488 0 0 0 4740
normalized size 1 1. 1.24 2.13 0. 0. 0. 20.7
time (sec) N/A 0.259 0.468 0.027 0. 0. 0. 2.447
Problem 27 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 260 260 315 811 0 0 0 7089
normalized size 1 1. 1.21 3.12 0. 0. 0. 27.27
time (sec) N/A 0.471 1.135 0.027 0. 0. 0. 3.037
Problem 28 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 288 288 377 1054 0 0 0 0
normalized size 1 1. 1.31 3.66 0. 0. 0. 0.
time (sec) N/A 0.474 0.984 0.037 0. 0. 0. 0.
Problem 29 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 412 412 444 1429 0 0 0 0
normalized size 1 1. 1.08 3.47 0. 0. 0. 0.
time (sec) N/A 1.334 1.561 0.042 0. 0. 0. 0.
Problem 30 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 347 347 358 0 0 0 0 0
normalized size 1 1. 1.03 0. 0. 0. 0. 0.
time (sec) N/A 0.618 0.995 180. 0. 0. 0. 0.
Problem 31 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 356 356 378 1119 0 0 0 0
normalized size 1 1. 1.06 3.14 0. 0. 0. 0.
time (sec) N/A 0.902 1.161 0.036 0. 0. 0. 0.
Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-1) F(-1) F(-1) F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 317 317 335 1344 0 0 0 0
normalized size 1 1. 1.06 4.24 0. 0. 0. 0.
time (sec) N/A 0.415 1.411 0.117 0. 0. 0. 0.
Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 368 368 393 1813 0 0 0 0
normalized size 1 1. 1.07 4.93 0. 0. 0. 0.
time (sec) N/A 0.867 1.456 0.105 0. 0. 0. 0.
Problem 34 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-1) F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 403 403 458 1603 0 0 0 0
normalized size 1 1. 1.14 3.98 0. 0. 0. 0.
time (sec) N/A 0.932 1.676 0.042 0. 0. 0. 0.
Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-1) F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 514 514 559 2398 0 0 0 0
normalized size 1 1. 1.09 4.67 0. 0. 0. 0.
time (sec) N/A 1.486 2.278 0.057 0. 0. 0. 0.
Problem 36 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-1) F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 534 534 655 2512 0 0 0 0
normalized size 1 1. 1.23 4.7 0. 0. 0. 0.
time (sec) N/A 1.992 2.761 0.062 0. 0. 0. 0.
Problem 37 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) B F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 399 399 296 5520 0 11187 0 10541
normalized size 1 1. 0.74 13.83 0. 28.04 0. 26.42
time (sec) N/A 0.425 1.271 0.016 0. 2.672 0. 1.338
Problem 38 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) B F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 260 260 185 2187 0 4469 0 4324
normalized size 1 1. 0.71 8.41 0. 17.19 0. 16.63
time (sec) N/A 0.223 0.383 0.01 0. 1.825 0. 1.195
Problem 39 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 137 137 90 585 0 1181 3735 1234
normalized size 1 1. 0.66 4.27 0. 8.62 27.26 9.01
time (sec) N/A 0.088 0.122 0.004 0. 1.577 2.79 1.123
Problem 40 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C F F F F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 368 368 168 0 0 0 0 0
normalized size 1 1. 0.46 0. 0. 0. 0. 0.
time (sec) N/A 0.622 0.22 0.043 0. 0. 0. 0.
Problem 41 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C F F F F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 685 670 242 0 0 0 0 0
normalized size 1 0.98 0.35 0. 0. 0. 0. 0.
time (sec) N/A 2.378 0.345 0.029 0. 0. 0. 0.
Problem 42 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 356 356 378 1119 0 0 0 0
normalized size 1 1. 1.06 3.14 0. 0. 0. 0.
time (sec) N/A 0.924 1.187 0. 0. 0. 0. 0.
Problem 43 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 356 356 378 1119 0 0 0 0
normalized size 1 1. 1.06 3.14 0. 0. 0. 0.
time (sec) N/A 0.371 0.793 0.025 0. 0. 0. 0.
Problem 44 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 356 356 378 1119 0 0 0 0
normalized size 1 1. 1.06 3.14 0. 0. 0. 0.
time (sec) N/A 0.37 0.171 0.022 0. 0. 0. 0.
Problem 45 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 356 356 378 1119 0 0 0 0
normalized size 1 1. 1.06 3.14 0. 0. 0. 0.
time (sec) N/A 0.356 0.165 0.021 0. 0. 0. 0.
Problem 46 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 356 356 378 1119 0 0 0 0
normalized size 1 1. 1.06 3.14 0. 0. 0. 0.
time (sec) N/A 0.359 0.16 0.024 0. 0. 0. 0.
Problem 47 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 273 273 260 622 0 1854 1392 413
normalized size 1 1. 0.95 2.28 0. 6.79 5.1 1.51
time (sec) N/A 0.854 0.208 0.007 0. 4.168 49.741 1.15
Problem 48 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 203 203 193 474 0 1404 1044 289
normalized size 1 1. 0.95 2.33 0. 6.92 5.14 1.42
time (sec) N/A 0.424 0.143 0.006 0. 2.823 35.705 1.162
Problem 49 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 144 144 136 321 0 994 721 190
normalized size 1 1. 0.94 2.23 0. 6.9 5.01 1.32
time (sec) N/A 0.272 0.107 0.005 0. 1.966 19.719 1.15
Problem 50 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 103 103 100 211 0 691 498 134
normalized size 1 1. 0.97 2.05 0. 6.71 4.83 1.3
time (sec) N/A 0.179 0.069 0.005 0. 1.747 11.093 1.147
Problem 51 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 97 97 178 165 0 683 0 131
normalized size 1 1. 1.84 1.7 0. 7.04 0. 1.35
time (sec) N/A 0.2 0.143 0.008 0. 2.633 0. 1.174
Problem 52 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 118 118 203 227 0 873 0 182
normalized size 1 1. 1.72 1.92 0. 7.4 0. 1.54
time (sec) N/A 0.285 0.162 0.009 0. 2.991 0. 1.129
Problem 53 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 174 174 314 356 0 1283 0 286
normalized size 1 1. 1.8 2.05 0. 7.37 0. 1.64
time (sec) N/A 0.407 0.353 0.011 0. 5.267 0. 1.137
Problem 54 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 244 244 416 523 0 1747 0 423
normalized size 1 1. 1.7 2.14 0. 7.16 0. 1.73
time (sec) N/A 0.573 0.352 0.014 0. 13.701 0. 1.146
Problem 55 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 369 369 456 1450 0 31190 0 0
normalized size 1 1. 1.24 3.93 0. 84.53 0. 0.
time (sec) N/A 4.577 0.546 0.036 0. 100.239 0. 0.
Problem 56 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-1) B F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 282 282 365 1035 0 18515 0 0
normalized size 1 1. 1.29 3.67 0. 65.66 0. 0.
time (sec) N/A 3.59 0.538 0.031 0. 22.088 0. 0.
Problem 57 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B B F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 219 219 258 676 0 11135 1151 0
normalized size 1 1. 1.18 3.09 0. 50.84 5.26 0.
time (sec) N/A 0.637 0.347 0.025 0. 11.18 90.156 0.
Problem 58 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B B F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 213 213 253 563 0 11429 1192 0
normalized size 1 1. 1.19 2.64 0. 53.66 5.6 0.
time (sec) N/A 0.839 0.328 0.025 0. 5.209 96.593 0.
Problem 59 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-1) B F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 267 267 284 727 0 19478 0 0
normalized size 1 1. 1.06 2.72 0. 72.95 0. 0.
time (sec) N/A 1.065 0.373 0.029 0. 27.674 0. 0.
Problem 60 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-1) B F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 329 329 394 1121 0 31905 0 0
normalized size 1 1. 1.2 3.41 0. 96.98 0. 0.
time (sec) N/A 1.942 0.604 0.036 0. 102.526 0. 0.
Problem 61 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) B F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 320 320 309 1167 0 4393 0 572
normalized size 1 1. 0.97 3.65 0. 13.73 0. 1.79
time (sec) N/A 1.233 0.551 0.018 0. 3.462 0. 19.825
Problem 62 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) B F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 236 236 236 832 0 3043 0 377
normalized size 1 1. 1. 3.53 0. 12.89 0. 1.6
time (sec) N/A 0.44 0.385 0.017 0. 2.305 0. 19.206
Problem 63 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) B B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 165 165 175 336 0 2033 1030 263
normalized size 1 1. 1.06 2.04 0. 12.32 6.24 1.59
time (sec) N/A 0.287 0.288 0.013 0. 1.7 145.522 19.602
Problem 64 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 123 123 130 205 0 1374 474 189
normalized size 1 1. 1.06 1.67 0. 11.17 3.85 1.54
time (sec) N/A 0.184 0.122 0.011 0. 1.446 40.153 19.251
Problem 65 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) B F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 166 166 268 462 0 2333 0 306
normalized size 1 1. 1.61 2.78 0. 14.05 0. 1.84
time (sec) N/A 0.394 0.469 0.018 0. 7.603 0. 19.512
Problem 66 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) B F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 234 234 403 722 0 3637 0 387
normalized size 1 1. 1.72 3.09 0. 15.54 0. 1.65
time (sec) N/A 0.725 0.692 0.023 0. 16.641 0. 21.974
Problem 67 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) B F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 329 329 592 1078 0 5341 0 722
normalized size 1 1. 1.8 3.28 0. 16.23 0. 2.19
time (sec) N/A 1.157 1.219 0.028 0. 39.111 0. 19.648
Problem 68 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-1) F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 550 550 648 2558 0 0 0 0
normalized size 1 1. 1.18 4.65 0. 0. 0. 0.
time (sec) N/A 13.227 2.313 0.053 0. 0. 0. 0.
Problem 69 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 436 436 511 1977 0 26999 0 0
normalized size 1 1. 1.17 4.53 0. 61.92 0. 0.
time (sec) N/A 5.541 1.74 0.042 0. 75.811 0. 0.
Problem 70 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 362 362 414 1300 0 18090 0 0
normalized size 1 1. 1.14 3.59 0. 49.97 0. 0.
time (sec) N/A 2.498 1.235 0.035 0. 37.949 0. 0.
Problem 71 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-1) B F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 346 346 382 1182 0 18375 0 0
normalized size 1 1. 1.1 3.42 0. 53.11 0. 0.
time (sec) N/A 1.896 1.196 0.033 0. 35.917 0. 0.
Problem 72 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 399 399 444 1575 0 28044 0 0
normalized size 1 1. 1.11 3.95 0. 70.29 0. 0.
time (sec) N/A 2.203 1.473 0.045 0. 79.06 0. 0.
Problem 73 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-1) F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 575 575 548 2180 0 0 0 0
normalized size 1 1. 0.95 3.79 0. 0. 0. 0.
time (sec) N/A 9.906 1.96 0.048 0. 0. 0. 0.
Problem 74 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 68 68 62 56 78 220 61 85
normalized size 1 1. 0.91 0.82 1.15 3.24 0.9 1.25
time (sec) N/A 0.126 0.035 0.015 0.972 1.962 0.163 1.119
Problem 75 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 61 61 61 51 72 208 54 78
normalized size 1 1. 1. 0.84 1.18 3.41 0.89 1.28
time (sec) N/A 0.118 0.027 0.016 1.016 1.949 0.156 1.111
Problem 76 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 54 54 54 46 65 186 48 72
normalized size 1 1. 1. 0.85 1.2 3.44 0.89 1.33
time (sec) N/A 0.108 0.024 0.014 0.996 1.964 0.155 1.131
Problem 77 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 49 49 49 41 58 169 42 61
normalized size 1 1. 1. 0.84 1.18 3.45 0.86 1.24
time (sec) N/A 0.086 0.022 0.013 1.027 1.975 0.156 1.093
Problem 78 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 42 42 42 36 51 144 36 54
normalized size 1 1. 1. 0.86 1.21 3.43 0.86 1.29
time (sec) N/A 0.049 0.017 0.015 0.965 2.003 0.15 1.121
Problem 79 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 44 44 44 38 59 185 39 63
normalized size 1 1. 1. 0.86 1.34 4.2 0.89 1.43
time (sec) N/A 0.077 0.022 0.017 1.019 1.749 0.164 1.084
Problem 80 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 55 55 50 45 72 219 51 72
normalized size 1 1. 0.91 0.82 1.31 3.98 0.93 1.31
time (sec) N/A 0.104 0.025 0.017 1.001 1.859 0.19 1.11
Problem 81 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 64 64 56 50 76 231 56 89
normalized size 1 1. 0.88 0.78 1.19 3.61 0.88 1.39
time (sec) N/A 0.111 0.029 0.018 0.953 1.727 0.204 1.094
Problem 82 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 70 70 71 56 78 247 66 78
normalized size 1 1. 1.01 0.8 1.11 3.53 0.94 1.11
time (sec) N/A 0.085 0.046 0.013 1.468 1.843 0.186 1.083
Problem 83 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 57 57 58 49 69 209 54 69
normalized size 1 1. 1.02 0.86 1.21 3.67 0.95 1.21
time (sec) N/A 0.082 0.048 0.013 1.486 1.756 0.185 1.084
Problem 84 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 56 56 57 46 65 198 53 65
normalized size 1 1. 1.02 0.82 1.16 3.54 0.95 1.16
time (sec) N/A 0.073 0.043 0.01 1.488 1.903 0.19 1.147
Problem 85 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 49 49 50 41 58 180 48 58
normalized size 1 1. 1.02 0.84 1.18 3.67 0.98 1.18
time (sec) N/A 0.066 0.039 0.01 1.487 2.14 0.186 1.101
Problem 86 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 48 48 46 38 54 170 44 54
normalized size 1 1. 0.96 0.79 1.12 3.54 0.92 1.12
time (sec) N/A 0.028 0.041 0.011 1.499 2.06 0.182 1.081
Problem 87 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 53 53 51 43 61 185 49 61
normalized size 1 1. 0.96 0.81 1.15 3.49 0.92 1.15
time (sec) N/A 0.073 0.049 0.012 1.458 2.18 0.201 1.123
Problem 88 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 62 62 56 48 70 213 56 70
normalized size 1 1. 0.9 0.77 1.13 3.44 0.9 1.13
time (sec) N/A 0.084 0.053 0.014 1.471 1.587 0.225 1.115
Problem 89 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 69 69 61 53 77 239 61 77
normalized size 1 1. 0.88 0.77 1.12 3.46 0.88 1.12
time (sec) N/A 0.09 0.06 0.016 1.466 1.53 0.243 1.142
Problem 90 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 76 76 77 58 84 271 66 84
normalized size 1 1. 1.01 0.76 1.11 3.57 0.87 1.11
time (sec) N/A 0.1 0.057 0.013 1.531 1.565 0.263 1.077
Problem 91 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 81 81 71 64 96 325 75 82
normalized size 1 1. 0.88 0.79 1.19 4.01 0.93 1.01
time (sec) N/A 0.112 0.06 0.013 1.494 1.585 0.241 1.096
Problem 92 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 80 80 66 62 92 313 75 78
normalized size 1 1. 0.82 0.78 1.15 3.91 0.94 0.98
time (sec) N/A 0.1 0.055 0.011 1.495 1.538 0.238 1.113
Problem 93 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 75 75 60 56 85 285 68 72
normalized size 1 1. 0.8 0.75 1.13 3.8 0.91 0.96
time (sec) N/A 0.091 0.061 0.011 1.496 1.626 0.241 1.085
Problem 94 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 72 72 55 54 81 274 65 68
normalized size 1 1. 0.76 0.75 1.12 3.81 0.9 0.94
time (sec) N/A 0.068 0.061 0.012 1.49 1.64 0.237 1.1
Problem 95 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 72 72 56 53 81 279 65 68
normalized size 1 1. 0.78 0.74 1.12 3.88 0.9 0.94
time (sec) N/A 0.066 0.063 0.012 1.497 1.521 0.236 1.105
Problem 96 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 72 72 56 53 81 281 65 68
normalized size 1 1. 0.78 0.74 1.12 3.9 0.9 0.94
time (sec) N/A 0.037 0.06 0.013 1.515 1.573 0.231 1.106
Problem 97 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 79 79 63 58 88 302 70 74
normalized size 1 1. 0.8 0.73 1.11 3.82 0.89 0.94
time (sec) N/A 0.103 0.068 0.014 1.486 1.567 0.26 1.127
Problem 98 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 86 86 78 64 97 336 76 84
normalized size 1 1. 0.91 0.74 1.13 3.91 0.88 0.98
time (sec) N/A 0.119 0.06 0.016 1.548 1.544 0.278 1.11
Problem 99 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 93 93 73 68 104 370 82 90
normalized size 1 1. 0.78 0.73 1.12 3.98 0.88 0.97
time (sec) N/A 0.134 0.079 0.015 1.474 1.569 0.301 1.121
Problem 100 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 86 86 78 74 96 270 85 103
normalized size 1 1. 0.91 0.86 1.12 3.14 0.99 1.2
time (sec) N/A 0.135 0.047 0.011 1.472 1.5 0.172 1.131
Problem 101 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 81 81 73 69 89 255 80 96
normalized size 1 1. 0.9 0.85 1.1 3.15 0.99 1.19
time (sec) N/A 0.127 0.031 0.01 1.499 1.545 0.168 1.121
Problem 102 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 74 74 66 64 80 235 73 89
normalized size 1 1. 0.89 0.86 1.08 3.18 0.99 1.2
time (sec) N/A 0.121 0.03 0.008 1.469 1.557 0.164 1.08
Problem 103 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 65 65 61 59 73 219 66 73
normalized size 1 1. 0.94 0.91 1.12 3.37 1.02 1.12
time (sec) N/A 0.105 0.027 0.008 1.471 1.503 0.168 1.112
Problem 104 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 58 58 58 54 66 194 60 66
normalized size 1 1. 1. 0.93 1.14 3.34 1.03 1.14
time (sec) N/A 0.067 0.022 0.01 1.47 1.527 0.162 1.109
Problem 105 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 66 66 93 58 74 238 65 84
normalized size 1 1. 1.41 0.88 1.12 3.61 0.98 1.27
time (sec) N/A 0.108 0.06 0.01 1.479 1.481 0.174 1.08
Problem 106 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 71 71 97 63 89 275 75 89
normalized size 1 1. 1.37 0.89 1.25 3.87 1.06 1.25
time (sec) N/A 0.134 0.051 0.014 1.458 1.615 0.201 1.098
Problem 107 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 80 80 105 68 96 289 80 107
normalized size 1 1. 1.31 0.85 1.2 3.61 1. 1.34
time (sec) N/A 0.137 0.061 0.012 1.475 1.49 0.214 1.083
Problem 108 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 87 87 110 73 103 317 85 113
normalized size 1 1. 1.26 0.84 1.18 3.64 0.98 1.3
time (sec) N/A 0.149 0.069 0.013 1.499 1.527 0.237 1.094
Problem 109 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 248 248 145 427 0 2484 71 0
normalized size 1 1. 0.58 1.72 0. 10.02 0.29 0.
time (sec) N/A 0.345 0.178 0.105 0. 1.747 0.537 0.
Problem 110 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 237 237 132 419 0 2055 63 0
normalized size 1 1. 0.56 1.77 0. 8.67 0.27 0.
time (sec) N/A 0.293 0.165 0.026 0. 1.665 0.541 0.
Problem 111 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 232 232 129 416 0 2102 58 0
normalized size 1 1. 0.56 1.79 0. 9.06 0.25 0.
time (sec) N/A 0.292 0.162 0.022 0. 1.775 0.533 0.
Problem 112 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 225 225 121 412 0 2072 51 0
normalized size 1 1. 0.54 1.83 0. 9.21 0.23 0.
time (sec) N/A 0.297 0.17 0.018 0. 1.756 0.535 0.
Problem 113 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 224 224 115 408 0 1993 48 0
normalized size 1 1. 0.51 1.82 0. 8.9 0.21 0.
time (sec) N/A 0.215 0.274 0.02 0. 1.724 0.522 0.
Problem 114 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 229 229 126 414 0 1871 53 0
normalized size 1 1. 0.55 1.81 0. 8.17 0.23 0.
time (sec) N/A 0.31 0.185 0.023 0. 1.706 0.551 0.
Problem 115 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 238 238 131 419 0 2205 60 0
normalized size 1 1. 0.55 1.76 0. 9.26 0.25 0.
time (sec) N/A 0.335 0.31 0.023 0. 1.723 0.575 0.
Problem 116 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 245 245 140 424 0 2392 65 0
normalized size 1 1. 0.57 1.73 0. 9.76 0.27 0.
time (sec) N/A 0.329 0.303 0.023 0. 1.741 0.59 0.
Problem 117 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 243 243 156 429 0 2853 82 0
normalized size 1 1. 0.64 1.77 0. 11.74 0.34 0.
time (sec) N/A 0.36 0.222 0.02 0. 1.794 0.591 0.
Problem 118 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 242 242 155 426 0 2421 80 0
normalized size 1 1. 0.64 1.76 0. 10. 0.33 0.
time (sec) N/A 0.31 0.211 0.022 0. 1.81 0.607 0.
Problem 119 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 235 235 138 422 0 2670 71 0
normalized size 1 1. 0.59 1.8 0. 11.36 0.3 0.
time (sec) N/A 0.3 0.33 0.022 0. 1.773 0.587 0.
Problem 120 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 238 238 129 418 0 2402 68 0
normalized size 1 1. 0.54 1.76 0. 10.09 0.29 0.
time (sec) N/A 0.29 0.304 0.019 0. 1.835 0.592 0.
Problem 121 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 246 246 133 418 0 2074 68 0
normalized size 1 1. 0.54 1.7 0. 8.43 0.28 0.
time (sec) N/A 0.284 0.299 0.022 0. 1.734 0.592 0.
Problem 122 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 248 248 129 418 0 2261 68 0
normalized size 1 1. 0.52 1.69 0. 9.12 0.27 0.
time (sec) N/A 0.254 0.302 0.022 0. 1.68 0.571 0.
Problem 123 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 253 253 140 424 0 2529 73 0
normalized size 1 1. 0.55 1.68 0. 10. 0.29 0.
time (sec) N/A 0.343 0.379 0.023 0. 1.733 0.607 0.
Problem 124 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F B A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 262 262 139 429 0 2952 80 0
normalized size 1 1. 0.53 1.64 0. 11.27 0.31 0.
time (sec) N/A 0.366 0.332 0.024 0. 1.75 0.627 0.
Problem 125 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 149 149 142 357 0 1021 789 197
normalized size 1 1. 0.95 2.4 0. 6.85 5.3 1.32
time (sec) N/A 0.295 0.127 0.005 0. 2.351 49.275 1.175
Problem 126 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 594 594 721 3028 0 0 0 0
normalized size 1 1. 1.21 5.1 0. 0. 0. 0.
time (sec) N/A 14.113 2.847 0.059 0. 0. 0. 0.
Problem 127 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-1) F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 471 471 575 2300 0 0 0 0
normalized size 1 1. 1.22 4.88 0. 0. 0. 0.
time (sec) N/A 6.662 2.116 0.05 0. 0. 0. 0.
Problem 128 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 449 449 512 1760 0 0 0 0
normalized size 1 1. 1.14 3.92 0. 0. 0. 0.
time (sec) N/A 2.867 1.833 0.042 0. 0. 0. 0.
Problem 129 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-1) F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 460 460 529 2045 0 0 0 0
normalized size 1 1. 1.15 4.45 0. 0. 0. 0.
time (sec) N/A 2.791 2.638 0.045 0. 0. 0. 0.
Problem 130 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F F(-1) F(-1) F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 542 542 612 2503 0 0 0 0
normalized size 1 1. 1.13 4.62 0. 0. 0. 0.
time (sec) N/A 7.265 2.336 0.048 0. 0. 0. 0.
Problem 131 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 20 20 20 21 42 63 0 78
normalized size 1 1. 1. 1.05 2.1 3.15 0. 3.9
time (sec) N/A 0.036 0.147 0.014 1.239 1.857 0. 1.197
Problem 132 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C A A A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 210 278 265 145 398 317 0 328
normalized size 1 1.32 1.26 0.69 1.9 1.51 0. 1.56
time (sec) N/A 0.315 1.437 0.006 1.647 1.774 0. 1.255
Problem 133 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C A A A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 159 213 232 109 293 238 0 239
normalized size 1 1.34 1.46 0.69 1.84 1.5 0. 1.5
time (sec) N/A 0.189 1.089 0.007 1.492 1.911 0. 1.148
Problem 134 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C A A A C A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 109 149 194 73 188 162 350 153
normalized size 1 1.37 1.78 0.67 1.72 1.49 3.21 1.4
time (sec) N/A 0.122 0.703 0.005 1.636 1.817 77.822 1.159
Problem 135 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B C F(-2) A C F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 93 151 217 143 0 178 304 0
normalized size 1 1.62 2.33 1.54 0. 1.91 3.27 0.
time (sec) N/A 0.165 0.891 0.044 0. 1.598 45.044 0.
Problem 136 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B C F(-2) A C F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 99 155 233 163 0 215 270 0
normalized size 1 1.57 2.35 1.65 0. 2.17 2.73 0.
time (sec) N/A 0.252 0.217 0.025 0. 1.297 64.196 0.
Problem 137 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A C F(-2) A C F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 126 182 134 222 0 224 253 0
normalized size 1 1.44 1.06 1.76 0. 1.78 2.01 0.
time (sec) N/A 0.277 0.167 0.025 0. 1.367 89.724 0.
Problem 138 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A C F(-2) A F(-2) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 212 248 173 306 0 298 0 0
normalized size 1 1.17 0.82 1.44 0. 1.41 0. 0.
time (sec) N/A 0.373 0.2 0.033 0. 1.449 0. 0.
Problem 139 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A C A A C A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 216 245 202 273 309 298 362 257
normalized size 1 1.13 0.94 1.26 1.43 1.38 1.68 1.19
time (sec) N/A 0.205 0.809 0.035 1.56 1.432 125.927 1.152
Problem 140 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A C A A C A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 128 179 157 191 201 227 325 170
normalized size 1 1.4 1.23 1.49 1.57 1.77 2.54 1.33
time (sec) N/A 0.091 0.582 0.017 1.506 1.58 37.972 1.138
Problem 141 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A C F(-2) A C F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 102 155 135 148 0 203 287 0
normalized size 1 1.52 1.32 1.45 0. 1.99 2.81 0.
time (sec) N/A 0.122 0.565 0.022 0. 1.524 54.216 0.
Problem 142 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A C F(-2) A C F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 157 157 81 146 0 203 257 0
normalized size 1 1. 0.52 0.93 0. 1.29 1.64 0.
time (sec) N/A 0.125 0.128 0.022 0. 1.382 77.442 0.
Problem 143 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 160 160 87 82 0 173 0 1489
normalized size 1 1. 0.54 0.51 0. 1.08 0. 9.31
time (sec) N/A 0.145 0.125 0.005 0. 1.418 0. 1.84
Problem 144 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 226 226 124 118 0 251 0 2048
normalized size 1 1. 0.55 0.52 0. 1.11 0. 9.06
time (sec) N/A 0.178 0.15 0.005 0. 1.54 0. 2.506
Problem 145 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 292 292 158 154 0 327 0 2607
normalized size 1 1. 0.54 0.53 0. 1.12 0. 8.93
time (sec) N/A 0.242 0.181 0.007 0. 1.861 0. 3.479

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 [35] had the largest ratio of [ 0.4643 ]

Table 2.1Rubi 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. 26 0.038
2 A 2 1 1. 24 0.042
3 A 2 1 1. 23 0.043
4 A 2 1 1. 26 0.038
5 A 2 1 1. 26 0.038
6 A 2 1 1. 26 0.038
7 A 2 1 1. 26 0.038
8 A 2 1 1. 26 0.038
9 A 2 1 1. 26 0.038
10 A 2 1 1. 26 0.038
11 A 2 1 1. 28 0.036
12 A 2 1 1. 26 0.038
13 A 2 1 1. 25 0.04
14 A 2 1 1. 28 0.036
15 A 2 1 1. 28 0.036
16 A 2 1 1. 28 0.036
17 A 2 1 1. 28 0.036
18 A 2 1 1. 28 0.036
19 A 2 1 1. 28 0.036
20 A 2 1 1. 28 0.036
21 A 13 11 1. 28 0.393
22 A 12 11 1. 28 0.393
23 A 11 10 1. 28 0.357
24 A 10 9 1. 26 0.346
25 A 8 7 1. 25 0.28
26 A 12 10 1. 28 0.357
27 A 13 12 1. 28 0.429
28 A 13 11 1. 28 0.393
29 A 11 10 1. 28 0.357
30 A 10 9 1. 28 0.321
31 A 10 9 1. 28 0.321
32 A 10 9 1. 26 0.346
33 A 10 9 1. 25 0.36
34 A 14 12 1. 28 0.429
35 A 15 13 1. 28 0.464
36 A 15 13 1. 28 0.464
37 A 2 1 1. 30 0.033
38 A 2 1 1. 30 0.033
39 A 2 1 1. 28 0.036
40 A 8 5 1. 30 0.167
41 A 10 6 0.98 30 0.2
42 A 10 9 1. 28 0.321
43 A 11 10 1. 30 0.333
44 A 11 10 1. 31 0.323
45 A 11 10 1. 34 0.294
46 A 11 10 1. 34 0.294
47 A 7 6 1. 30 0.2
48 A 7 6 1. 30 0.2
49 A 7 6 1. 30 0.2
50 A 7 6 1. 28 0.214
51 A 7 6 1. 30 0.2
52 A 7 6 1. 30 0.2
53 A 7 6 1. 30 0.2
54 A 7 6 1. 30 0.2
55 A 5 3 1. 30 0.1
56 A 5 3 1. 30 0.1
57 A 5 3 1. 27 0.111
58 A 5 3 1. 30 0.1
59 A 5 3 1. 30 0.1
60 A 5 3 1. 30 0.1
61 A 8 7 1. 30 0.233
62 A 7 7 1. 30 0.233
63 A 6 6 1. 30 0.2
64 A 5 5 1. 28 0.179
65 A 8 7 1. 30 0.233
66 A 8 7 1. 30 0.233
67 A 8 7 1. 30 0.233
68 A 6 4 1. 30 0.133
69 A 6 4 1. 30 0.133
70 A 4 3 1. 30 0.1
71 A 4 3 1. 27 0.111
72 A 6 4 1. 30 0.133
73 A 6 4 1. 30 0.133
74 A 7 5 1. 31 0.161
75 A 7 5 1. 31 0.161
76 A 7 5 1. 31 0.161
77 A 7 5 1. 31 0.161
78 A 5 4 1. 29 0.138
79 A 4 3 1. 31 0.097
80 A 4 3 1. 31 0.097
81 A 4 3 1. 31 0.097
82 A 6 4 1. 31 0.129
83 A 6 4 1. 31 0.129
84 A 6 4 1. 31 0.129
85 A 6 4 1. 31 0.129
86 A 4 3 1. 28 0.107
87 A 5 3 1. 31 0.097
88 A 5 3 1. 31 0.097
89 A 5 3 1. 31 0.097
90 A 5 3 1. 31 0.097
91 A 7 5 1. 31 0.161
92 A 7 5 1. 31 0.161
93 A 7 5 1. 31 0.161
94 A 5 4 1. 31 0.129
95 A 5 4 1. 31 0.129
96 A 5 4 1. 28 0.143
97 A 6 3 1. 31 0.097
98 A 6 3 1. 31 0.097
99 A 6 3 1. 31 0.097
100 A 8 7 1. 31 0.226
101 A 8 7 1. 31 0.226
102 A 8 7 1. 31 0.226
103 A 8 7 1. 31 0.226
104 A 6 6 1. 29 0.207
105 A 8 7 1. 31 0.226
106 A 8 7 1. 31 0.226
107 A 8 7 1. 31 0.226
108 A 8 7 1. 31 0.226
109 A 12 7 1. 31 0.226
110 A 12 7 1. 31 0.226
111 A 12 7 1. 31 0.226
112 A 12 7 1. 31 0.226
113 A 10 6 1. 28 0.214
114 A 12 7 1. 31 0.226
115 A 12 7 1. 31 0.226
116 A 12 7 1. 31 0.226
117 A 13 8 1. 31 0.258
118 A 13 8 1. 31 0.258
119 A 13 8 1. 31 0.258
120 A 11 7 1. 31 0.226
121 A 11 7 1. 31 0.226
122 A 11 7 1. 28 0.25
123 A 13 7 1. 31 0.226
124 A 13 7 1. 31 0.226
125 A 7 6 1. 33 0.182
126 A 6 4 1. 35 0.114
127 A 6 4 1. 35 0.114
128 A 4 3 1. 32 0.094
129 A 6 4 1. 35 0.114
130 A 6 4 1. 35 0.114
131 A 1 1 1. 42 0.024
132 A 5 4 1.32 35 0.114
133 A 4 3 1.34 35 0.086
134 A 4 3 1.37 33 0.091
135 A 6 5 1.62 35 0.143
136 A 6 6 1.57 35 0.171
137 A 6 6 1.44 35 0.171
138 A 7 7 1.17 35 0.2
139 A 6 6 1.13 35 0.171
140 A 5 5 1.4 32 0.156
141 A 5 5 1.52 35 0.143
142 A 5 5 1. 35 0.143
143 A 4 4 1. 35 0.114
144 A 5 5 1. 35 0.143
145 A 6 5 1. 35 0.143

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