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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, 29, 30, 31, 33, 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 }

B grade: { 34 }

C grade: { }

F grade: { 28, 32}

2.1.2 Mathematica

A grade: { 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 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, 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, 100, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 128, 129, 131, 134, 135, 136, 137, 138, 139, 140, 141, 142 }

B grade: { 7, 21, 22, 63, 99, 101, 102, 124, 130, 132, 133 }

C grade: { }

F grade: { }

2.1.3 Maple

A grade: { 2, 3, 4, 9, 11, 12, 13, 18, 29, 30, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 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, 139, 140 }

B grade: { 34, 35, 51, 67 }

C grade: { 31, 32, 33 }

F grade: { 1, 5, 6, 7, 8, 10, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 53, 54, 55, 56, 137, 138, 141, 142 }

2.1.4 Maxima

A grade: { 9, 18, 29, 30, 31, 32, 33, 38, 47, 48, 49, 50, 51, 52, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 76, 104, 107

B grade: { 2, 3, 4, 11, 12, 13, 34, 35, 39, 40, 41, 42, 43, 44, 45, 46, 63, 70, 74, 75, 105, 106, 135, 136, 139, 140 }

C grade: { }

F grade: { 1, 5, 6, 7, 8, 10, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 36, 37, 53, 54, 55, 56, 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, 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, 137, 138, 141, 142 }

2.1.5 FriCAS

A grade: { 2, 3, 4, 9, 11, 12, 13, 18, 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, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 76, 77, 78, 85, 86, 87, 88, 89, 94, 95, 96, 104, 107, 108, 109, 116, 117, 118, 119, 120, 125, 126, 127, 135, 136, 139, 140 }

B grade: { 63, 70, 74, 75, 79, 80, 81, 82, 83, 84, 90, 91, 92, 93, 97, 98, 99, 100, 101, 102, 103, 105, 106, 110, 111, 112, 113, 114, 115, 121, 122, 123, 124, 128, 129, 130, 131, 132, 133, 134 }

C grade: { }

F grade: { 1, 5, 6, 7, 8, 10, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 53, 54, 55, 56, 137, 138, 141, 142 }

2.1.6 Sympy

A grade: { 3, 4, 9, 12, 13, 18, 30, 34, 35, 38, 47, 48, 49, 50, 51, 52, 61, 62, 64, 65, 66, 67, 69, 70, 71, 72, 73, 104, 135, 136, 139, 140 }

B grade: { }

C grade: { }

F grade: { 1, 2, 5, 6, 7, 8, 10, 11, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 53, 54, 55, 56, 57, 58, 59, 60, 63, 68, 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, 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, 137, 138, 141, 142 }

2.1.7 Giac

A grade: { 9, 18, 29, 30, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 104, 105, 106 }

B grade: { 31, 32, 63, 70, 79, 80, 81, 82, 83, 84, 110, 111, 112, 113, 114, 115 }

C grade: { 2, 3, 4, 11, 12, 13, 34, 35, 135, 136, 139, 140 }

F grade: { 1, 5, 6, 7, 8, 10, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 33, 36, 37, 53, 54, 55, 56, 57, 58, 59, 60, 76, 77, 78, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 107, 108, 109, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 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 F F F F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 107 107 110 0 0 0 0 0
normalized size 1 1. 1.03 0. 0. 0. 0. 0.
time (sec) N/A 0.138 0.063 0.889 0. 0. 0. 0.
Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A F(-1) C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 199 199 154 336 1098 386 0 1770
normalized size 1 1. 0.77 1.69 5.52 1.94 0. 8.89
time (sec) N/A 0.07 0.685 0.138 1.338 0.508 0. 1.298
Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 128 128 86 153 481 207 627 1260
normalized size 1 1. 0.67 1.2 3.76 1.62 4.9 9.84
time (sec) N/A 0.052 0.212 0.049 1.201 0.492 159.735 1.255
Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 73 73 48 130 262 115 326 880
normalized size 1 1. 0.66 1.78 3.59 1.58 4.47 12.05
time (sec) N/A 0.016 0.116 0.015 1.08 0.481 70.008 1.226
Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 81 81 114 0 0 0 0 0
normalized size 1 1. 1.41 0. 0. 0. 0. 0.
time (sec) N/A 0.022 1.775 0.043 0. 0. 0. 0.
Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 78 78 101 0 0 0 0 0
normalized size 1 1. 1.29 0. 0. 0. 0. 0.
time (sec) N/A 0.029 1.501 0.083 0. 0. 0. 0.
Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B F F(-1) F F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 137 137 334 0 0 0 0 0
normalized size 1 1. 2.44 0. 0. 0. 0. 0.
time (sec) N/A 0.049 7.891 0.117 0. 0. 0. 0.
Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 141 141 173 0 0 0 0 0
normalized size 1 1. 1.23 0. 0. 0. 0. 0.
time (sec) N/A 0.055 2.987 0.134 0. 0. 0. 0.
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 54 54 33 34 50 115 70 47
normalized size 1 1. 0.61 0.63 0.93 2.13 1.3 0.87
time (sec) N/A 0.026 0.038 0.01 1.029 0.463 6.416 1.148
Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F F F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 107 107 110 0 0 0 0 0
normalized size 1 1. 1.03 0. 0. 0. 0. 0.
time (sec) N/A 0.112 0.055 0.498 0. 0. 0. 0.
Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A F(-1) C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 199 199 155 274 1098 331 0 1764
normalized size 1 1. 0.78 1.38 5.52 1.66 0. 8.86
time (sec) N/A 0.053 0.654 0.11 1.233 0.497 0. 1.32
Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 128 128 85 153 481 189 627 1260
normalized size 1 1. 0.66 1.2 3.76 1.48 4.9 9.84
time (sec) N/A 0.037 0.199 0.03 1.079 0.472 127.152 1.222
Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 72 72 47 133 259 115 352 876
normalized size 1 1. 0.65 1.85 3.6 1.6 4.89 12.17
time (sec) N/A 0.016 0.097 0.015 1.08 0.476 46.278 1.197
Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 84 84 84 0 0 0 0 0
normalized size 1 1. 1. 0. 0. 0. 0. 0.
time (sec) N/A 0.018 0.019 0.037 0. 0. 0. 0.
Problem 15 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 80 80 80 0 0 0 0 0
normalized size 1 1. 1. 0. 0. 0. 0. 0.
time (sec) N/A 0.024 0.016 0.109 0. 0. 0. 0.
Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 141 141 112 0 0 0 0 0
normalized size 1 1. 0.79 0. 0. 0. 0. 0.
time (sec) N/A 0.049 0.26 0.205 0. 0. 0. 0.
Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 143 143 111 0 0 0 0 0
normalized size 1 1. 0.78 0. 0. 0. 0. 0.
time (sec) N/A 0.053 0.199 0.235 0. 0. 0. 0.
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 54 54 33 34 50 113 70 47
normalized size 1 1. 0.61 0.63 0.93 2.09 1.3 0.87
time (sec) N/A 0.029 0.027 0.007 1.025 0.468 8.329 1.155
Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F F F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 194 194 212 0 0 0 0 0
normalized size 1 1. 1.09 0. 0. 0. 0. 0.
time (sec) N/A 0.199 2.126 0.102 0. 0. 0. 0.
Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 130 130 174 0 0 0 0 0
normalized size 1 1. 1.34 0. 0. 0. 0. 0.
time (sec) N/A 0.126 1.617 0.062 0. 0. 0. 0.
Problem 21 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B F F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 78 78 166 0 0 0 0 0
normalized size 1 1. 2.13 0. 0. 0. 0. 0.
time (sec) N/A 0.076 0.469 0.046 0. 0. 0. 0.
Problem 22 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B F F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 76 76 163 0 0 0 0 0
normalized size 1 1. 2.14 0. 0. 0. 0. 0.
time (sec) N/A 0.078 1.256 0.074 0. 0. 0. 0.
Problem 23 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 126 126 170 0 0 0 0 0
normalized size 1 1. 1.35 0. 0. 0. 0. 0.
time (sec) N/A 0.125 1.586 0.085 0. 0. 0. 0.
Problem 24 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F F F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 188 188 210 0 0 0 0 0
normalized size 1 1. 1.12 0. 0. 0. 0. 0.
time (sec) N/A 0.191 2.19 0.17 0. 0. 0. 0.
Problem 25 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 76 76 133 0 0 0 0 0
normalized size 1 1. 1.75 0. 0. 0. 0. 0.
time (sec) N/A 0.094 0.318 0.087 0. 0. 0. 0.
Problem 26 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 100 100 102 0 0 0 0 0
normalized size 1 1. 1.02 0. 0. 0. 0. 0.
time (sec) N/A 0.14 0.09 0.477 0. 0. 0. 0.
Problem 27 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 102 102 102 0 0 0 0 0
normalized size 1 1. 1. 0. 0. 0. 0. 0.
time (sec) N/A 0.157 0.103 0.554 0. 0. 0. 0.
Problem 28 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A F A F F A F(-1) F
verified N/A N/A Yes TBD TBD TBD TBD TBD
size 139 0 143 0 0 336 0 0
normalized size 1 0. 1.03 0. 0. 2.42 0. 0.
time (sec) N/A 0.491 0.578 0.366 0. 0.52 0. 0.
Problem 29 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade N/A A A A A A F(-1) A
verified N/A N/A N/A TBD TBD TBD TBD TBD
size 24 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.623 6.411 0.052 0. 0. 0. 0.
Problem 30 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade N/A A A A A A A A
verified N/A N/A N/A TBD TBD TBD TBD TBD
size 26 0 0 0 0 0 0 0
normalized size 1 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.951 10.633 0.071 0. 0. 0. 0.
Problem 31 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A C A A F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 24 24 26 213 43 65 0 8643
normalized size 1 1. 1.08 8.88 1.79 2.71 0. 360.12
time (sec) N/A 3.975 1.332 0.251 2.297 0.501 0. 2.324
Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A F A C A A F(-1) B
verified N/A N/A Yes TBD TBD TBD TBD TBD
size 23 0 23 201 41 57 0 6483
normalized size 1 0. 1. 8.74 1.78 2.48 0. 281.87
time (sec) N/A 2.34 0.898 0.161 2.132 0.495 0. 1.885
Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A C A A F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 22 22 22 199 36 51 0 0
normalized size 1 1. 1. 9.05 1.64 2.32 0. 0.
time (sec) N/A 2.563 0.864 0.15 2.146 0.494 0. 0.
Problem 34 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A B A B B A A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 17 327 17 682 1866 43 19 5310
normalized size 1 19.24 1. 40.12 109.76 2.53 1.12 312.35
time (sec) N/A 0.766 0.387 0.079 1.445 0.471 31.327 1.416
Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B B A A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 16 16 16 268 529 41 17 1710
normalized size 1 1. 1. 16.75 33.06 2.56 1.06 106.88
time (sec) N/A 0.029 0.027 0.015 1.149 0.473 3.959 1.284
Problem 36 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 20 20 19 40 0 43 0 0
normalized size 1 1. 0.95 2. 0. 2.15 0. 0.
time (sec) N/A 1.732 0.606 0.069 0. 0.47 0. 0.
Problem 37 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 20 20 19 40 0 46 0 0
normalized size 1 1. 0.95 2. 0. 2.3 0. 0.
time (sec) N/A 1.95 0.627 0.093 0. 0.471 0. 0.
Problem 38 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 44 60 59 130 342 74
normalized size 1 1. 0.7 0.95 0.94 2.06 5.43 1.17
time (sec) N/A 0.047 0.153 0.014 1.045 0.471 70.24 1.111
Problem 39 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 119 119 74 108 726 243 0 132
normalized size 1 1. 0.62 0.91 6.1 2.04 0. 1.11
time (sec) N/A 0.093 0.654 0.027 1.171 0.481 0. 1.114
Problem 40 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 129 129 82 118 743 302 0 150
normalized size 1 1. 0.64 0.91 5.76 2.34 0. 1.16
time (sec) N/A 0.088 0.931 0.017 1.144 0.487 0. 1.2
Problem 41 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 119 119 74 108 726 224 0 135
normalized size 1 1. 0.62 0.91 6.1 1.88 0. 1.13
time (sec) N/A 0.083 0.665 0.016 1.197 0.486 0. 1.153
Problem 42 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 79 79 57 71 319 208 0 89
normalized size 1 1. 0.72 0.9 4.04 2.63 0. 1.13
time (sec) N/A 0.076 0.385 0.018 1.046 0.481 0. 1.155
Problem 43 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 183 183 110 166 1550 454 0 209
normalized size 1 1. 0.6 0.91 8.47 2.48 0. 1.14
time (sec) N/A 0.126 0.913 0.021 1.292 0.512 0. 1.158
Problem 44 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 129 129 81 118 743 262 0 150
normalized size 1 1. 0.63 0.91 5.76 2.03 0. 1.16
time (sec) N/A 0.089 0.678 0.015 1.142 0.488 0. 1.119
Problem 45 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 183 183 110 166 1544 444 0 205
normalized size 1 1. 0.6 0.91 8.44 2.43 0. 1.12
time (sec) N/A 0.125 0.764 0.028 1.287 0.511 0. 1.12
Problem 46 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 129 129 111 118 743 356 0 150
normalized size 1 1. 0.86 0.91 5.76 2.76 0. 1.16
time (sec) N/A 0.101 0.949 0.022 1.153 0.494 0. 1.147
Problem 47 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 30 30 19 19 23 59 27 22
normalized size 1 1. 0.63 0.63 0.77 1.97 0.9 0.73
time (sec) N/A 0.039 0.038 0.007 1.025 0.456 0.889 1.106
Problem 48 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 50 50 25 27 35 81 48 34
normalized size 1 1. 0.5 0.54 0.7 1.62 0.96 0.68
time (sec) N/A 0.118 0.035 0.004 1.061 0.46 2.356 1.132
Problem 49 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 30 30 18 20 23 58 27 20
normalized size 1 1. 0.6 0.67 0.77 1.93 0.9 0.67
time (sec) N/A 0.04 0.027 0.006 1.02 0.455 0.907 1.126
Problem 50 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 51 51 23 28 35 80 48 32
normalized size 1 1. 0.45 0.55 0.69 1.57 0.94 0.63
time (sec) N/A 0.117 0.031 0.007 1.059 0.464 2.42 1.089
Problem 51 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 27 27 22 103 53 68 27 53
normalized size 1 1. 0.81 3.81 1.96 2.52 1. 1.96
time (sec) N/A 0.08 0.091 0.017 1.106 0.451 0.328 1.14
Problem 52 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 41 41 33 30 31 84 32 31
normalized size 1 1. 0.8 0.73 0.76 2.05 0.78 0.76
time (sec) N/A 0.025 0.065 0.007 1.035 0.463 0.6 1.159
Problem 53 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 82 82 64 0 0 0 0 0
normalized size 1 1. 0.78 0. 0. 0. 0. 0.
time (sec) N/A 0.136 2.569 0.187 0. 0. 0. 0.
Problem 54 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 82 82 64 0 0 0 0 0
normalized size 1 1. 0.78 0. 0. 0. 0. 0.
time (sec) N/A 0.128 2.572 0.175 0. 0. 0. 0.
Problem 55 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 80 80 68 0 0 0 0 0
normalized size 1 1. 0.85 0. 0. 0. 0. 0.
time (sec) N/A 0.116 0.666 0.064 0. 0. 0. 0.
Problem 56 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 78 78 66 0 0 0 0 0
normalized size 1 1. 0.85 0. 0. 0. 0. 0.
time (sec) N/A 0.118 0.627 0.093 0. 0. 0. 0.
Problem 57 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 69 69 43 42 50 88 0 0
normalized size 1 1. 0.62 0.61 0.72 1.28 0. 0.
time (sec) N/A 0.053 0.027 0.099 1.172 0.463 0. 0.
Problem 58 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 65 65 47 44 51 95 0 0
normalized size 1 1. 0.72 0.68 0.78 1.46 0. 0.
time (sec) N/A 0.047 0.02 0.039 1.026 0.474 0. 0.
Problem 59 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 81 81 81 52 69 122 0 0
normalized size 1 1. 1. 0.64 0.85 1.51 0. 0.
time (sec) N/A 0.07 0.08 0.041 1.056 0.474 0. 0.
Problem 60 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 77 77 82 54 70 127 0 0
normalized size 1 1. 1.06 0.7 0.91 1.65 0. 0.
time (sec) N/A 0.051 0.081 0.04 1.059 0.477 0. 0.
Problem 61 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 35 35 24 30 39 72 29 28
normalized size 1 1. 0.69 0.86 1.11 2.06 0.83 0.8
time (sec) N/A 0.077 0.037 0.011 1.007 0.461 13.765 1.141
Problem 62 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 6 6 6 6 7 15 5 7
normalized size 1 1. 1. 1. 1.17 2.5 0.83 1.17
time (sec) N/A 0.008 0.009 0.003 0.98 0.465 0.299 1.165
Problem 63 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B A B B F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 5 5 21 5 26 74 0 27
normalized size 1 1. 4.2 1. 5.2 14.8 0. 5.4
time (sec) N/A 0.022 0.018 0.022 1.073 0.476 0. 1.114
Problem 64 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 4 4 4 4 4 14 3 4
normalized size 1 1. 1. 1. 1. 3.5 0.75 1.
time (sec) N/A 0.008 0.009 0.007 0.984 0.458 0.496 1.133
Problem 65 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 10 10 10 8 9 24 7 9
normalized size 1 1. 1. 0.8 0.9 2.4 0.7 0.9
time (sec) N/A 0.011 0.01 0.005 1.001 0.457 0.502 1.117
Problem 66 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 10 10 10 8 9 27 10 9
normalized size 1 1. 1. 0.8 0.9 2.7 1. 0.9
time (sec) N/A 0.01 0.011 0.007 0.999 0.464 0.492 1.095
Problem 67 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 13 13 13 24 14 34 12 14
normalized size 1 1. 1. 1.85 1.08 2.62 0.92 1.08
time (sec) N/A 0.017 0.017 0.02 1.01 0.476 11.969 1.129
Problem 68 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 30 30 30 25 32 140 0 32
normalized size 1 1. 1. 0.83 1.07 4.67 0. 1.07
time (sec) N/A 0.036 0.061 0.051 0.987 0.483 0. 1.122
Problem 69 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 7 7 7 7 5 41 10 9
normalized size 1 1. 1. 1. 0.71 5.86 1.43 1.29
time (sec) N/A 0.01 0.008 0.002 0.995 0.474 0.643 1.202
Problem 70 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 5 5 5 9 11 65 10 39
normalized size 1 1. 1. 1.8 2.2 13. 2. 7.8
time (sec) N/A 0.01 0.005 0.002 1.006 0.482 3.565 1.185
Problem 71 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 4 4 4 4 7 16 3 7
normalized size 1 1. 1. 1. 1.75 4. 0.75 1.75
time (sec) N/A 0.02 0.01 0.008 1.012 0.465 1.06 1.186
Problem 72 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 6 6 6 6 9 27 5 9
normalized size 1 1. 1. 1. 1.5 4.5 0.83 1.5
time (sec) N/A 0.017 0.017 0.054 1.031 0.439 3.251 1.277
Problem 73 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 37 37 27 36 38 70 114 47
normalized size 1 1. 0.73 0.97 1.03 1.89 3.08 1.27
time (sec) N/A 0.013 0.058 0.003 0.988 0.463 1.492 1.281
Problem 74 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B B F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 115 115 108 88 377 549 0 171
normalized size 1 1. 0.94 0.77 3.28 4.77 0. 1.49
time (sec) N/A 0.123 0.157 0.124 1.929 0.493 0. 1.14
Problem 75 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B B F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 144 144 134 117 417 647 0 198
normalized size 1 1. 0.93 0.81 2.9 4.49 0. 1.38
time (sec) N/A 0.216 0.256 0.109 2.083 0.501 0. 1.169
Problem 76 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 81 81 81 52 69 122 0 0
normalized size 1 1. 1. 0.64 0.85 1.51 0. 0.
time (sec) N/A 0.055 0.023 0. 1.032 0.471 0. 0.
Problem 77 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 87 87 129 62 0 192 0 0
normalized size 1 1. 1.48 0.71 0. 2.21 0. 0.
time (sec) N/A 0.098 0.218 0.063 0. 0.483 0. 0.
Problem 78 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 155 155 165 127 0 427 0 0
normalized size 1 1. 1.06 0.82 0. 2.75 0. 0.
time (sec) N/A 0.202 0.562 0.208 0. 0.5 0. 0.
Problem 79 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 142 142 132 116 0 749 0 405
normalized size 1 1. 0.93 0.82 0. 5.27 0. 2.85
time (sec) N/A 0.21 0.229 0.267 0. 0.505 0. 1.288
Problem 80 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 157 157 156 139 0 770 0 703
normalized size 1 1. 0.99 0.89 0. 4.9 0. 4.48
time (sec) N/A 0.203 1.072 0.374 0. 0.517 0. 1.388
Problem 81 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 298 298 268 239 0 1519 0 803
normalized size 1 1. 0.9 0.8 0. 5.1 0. 2.69
time (sec) N/A 0.367 0.885 0.481 0. 0.546 0. 1.465
Problem 82 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 162 162 162 152 0 868 0 518
normalized size 1 1. 1. 0.94 0. 5.36 0. 3.2
time (sec) N/A 0.334 0.385 0.195 0. 0.521 0. 1.317
Problem 83 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 179 179 244 175 0 910 0 817
normalized size 1 1. 1.36 0.98 0. 5.08 0. 4.56
time (sec) N/A 0.352 1.123 0.34 0. 0.519 0. 1.354
Problem 84 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 340 340 323 311 0 1773 0 1030
normalized size 1 1. 0.95 0.91 0. 5.21 0. 3.03
time (sec) N/A 0.602 1.546 0.486 0. 0.556 0. 1.559
Problem 85 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 151 151 119 123 0 413 0 0
normalized size 1 1. 0.79 0.81 0. 2.74 0. 0.
time (sec) N/A 0.207 0.153 0.203 0. 0.497 0. 0.
Problem 86 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 171 171 132 145 0 452 0 0
normalized size 1 1. 0.77 0.85 0. 2.64 0. 0.
time (sec) N/A 0.218 0.25 0.2 0. 0.498 0. 0.
Problem 87 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 301 301 224 246 0 826 0 0
normalized size 1 1. 0.74 0.82 0. 2.74 0. 0.
time (sec) N/A 0.346 0.43 0.335 0. 0.516 0. 0.
Problem 88 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 107 107 170 84 0 301 0 0
normalized size 1 1. 1.59 0.79 0. 2.81 0. 0.
time (sec) N/A 0.198 0.463 0.129 0. 0.499 0. 0.
Problem 89 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 140 140 188 107 0 479 0 0
normalized size 1 1. 1.34 0.76 0. 3.42 0. 0.
time (sec) N/A 0.228 0.789 0.15 0. 0.505 0. 0.
Problem 90 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 213 213 386 166 0 871 0 0
normalized size 1 1. 1.81 0.78 0. 4.09 0. 0.
time (sec) N/A 0.332 2.288 0.271 0. 0.55 0. 0.
Problem 91 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 187 187 216 169 0 765 0 0
normalized size 1 1. 1.16 0.9 0. 4.09 0. 0.
time (sec) N/A 0.367 0.967 0.379 0. 0.549 0. 0.
Problem 92 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 211 211 251 191 0 927 0 0
normalized size 1 1. 1.19 0.91 0. 4.39 0. 0.
time (sec) N/A 0.419 2.253 0.39 0. 0.557 0. 0.
Problem 93 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 377 377 490 338 0 1840 0 0
normalized size 1 1. 1.3 0.9 0. 4.88 0. 0.
time (sec) N/A 0.655 6.619 0.613 0. 0.668 0. 0.
Problem 94 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 176 176 155 170 0 487 0 0
normalized size 1 1. 0.88 0.97 0. 2.77 0. 0.
time (sec) N/A 0.335 0.332 0.234 0. 0.498 0. 0.
Problem 95 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 231 231 204 217 0 617 0 0
normalized size 1 1. 0.88 0.94 0. 2.67 0. 0.
time (sec) N/A 0.383 0.716 0.414 0. 0.506 0. 0.
Problem 96 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 354 354 391 338 0 973 0 0
normalized size 1 1. 1.1 0.95 0. 2.75 0. 0.
time (sec) N/A 0.49 1.004 0.453 0. 0.526 0. 0.
Problem 97 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 193 193 230 180 0 778 0 0
normalized size 1 1. 1.19 0.93 0. 4.03 0. 0.
time (sec) N/A 0.385 0.968 0.378 0. 0.544 0. 0.
Problem 98 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 245 245 299 227 0 1025 0 0
normalized size 1 1. 1.22 0.93 0. 4.18 0. 0.
time (sec) N/A 0.462 3.09 0.424 0. 0.56 0. 0.
Problem 99 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B A F(-2) B F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 386 386 3291 358 0 1854 0 0
normalized size 1 1. 8.53 0.93 0. 4.8 0. 0.
time (sec) N/A 0.573 7.012 0.629 0. 0.672 0. 0.
Problem 100 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 212 212 347 216 0 938 0 0
normalized size 1 1. 1.64 1.02 0. 4.42 0. 0.
time (sec) N/A 0.566 2.181 0.289 0. 0.565 0. 0.
Problem 101 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B A F(-2) B F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 268 268 1120 263 0 1204 0 0
normalized size 1 1. 4.18 0.98 0. 4.49 0. 0.
time (sec) N/A 0.646 6.725 0.362 0. 0.575 0. 0.
Problem 102 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B A F(-2) B F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 430 430 3835 430 0 2196 0 0
normalized size 1 1. 8.92 1. 0. 5.11 0. 0.
time (sec) N/A 0.911 7.248 0.611 0. 0.698 0. 0.
Problem 103 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 213 213 324 218 0 938 0 0
normalized size 1 1. 1.52 1.02 0. 4.4 0. 0.
time (sec) N/A 0.795 1.853 0.454 0. 0.558 0. 0.
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 36 36 26 35 34 69 114 45
normalized size 1 1. 0.72 0.97 0.94 1.92 3.17 1.25
time (sec) N/A 0.012 0.05 0.008 1.094 0.461 2.475 1.137
Problem 105 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B B F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 115 115 109 86 378 549 0 171
normalized size 1 1. 0.95 0.75 3.29 4.77 0. 1.49
time (sec) N/A 0.099 0.16 0.054 2.107 0.491 0. 1.143
Problem 106 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B B F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 144 144 135 115 420 647 0 198
normalized size 1 1. 0.94 0.8 2.92 4.49 0. 1.38
time (sec) N/A 0.169 0.254 0.05 2.361 0.5 0. 1.152
Problem 107 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A A A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 77 77 82 54 70 127 0 0
normalized size 1 1. 1.06 0.7 0.91 1.65 0. 0.
time (sec) N/A 0.057 0.083 0.002 1.163 0.475 0. 0.
Problem 108 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 83 83 107 60 0 198 0 0
normalized size 1 1. 1.29 0.72 0. 2.39 0. 0.
time (sec) N/A 0.079 0.194 0.043 0. 0.479 0. 0.
Problem 109 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 151 151 166 125 0 431 0 0
normalized size 1 1. 1.1 0.83 0. 2.85 0. 0.
time (sec) N/A 0.172 0.577 0.066 0. 0.503 0. 0.
Problem 110 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 142 142 133 114 0 749 0 405
normalized size 1 1. 0.94 0.8 0. 5.27 0. 2.85
time (sec) N/A 0.169 0.239 0.066 0. 0.511 0. 1.302
Problem 111 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 157 157 158 139 0 768 0 703
normalized size 1 1. 1.01 0.89 0. 4.89 0. 4.48
time (sec) N/A 0.173 1.092 0.109 0. 0.516 0. 1.347
Problem 112 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 298 298 267 235 0 1517 0 803
normalized size 1 1. 0.9 0.79 0. 5.09 0. 2.69
time (sec) N/A 0.33 0.899 0.217 0. 0.56 0. 1.469
Problem 113 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 162 162 163 150 0 868 0 518
normalized size 1 1. 1.01 0.93 0. 5.36 0. 3.2
time (sec) N/A 0.255 0.39 0.071 0. 0.512 0. 1.326
Problem 114 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 179 179 245 175 0 909 0 817
normalized size 1 1. 1.37 0.98 0. 5.08 0. 4.56
time (sec) N/A 0.284 1.143 0.14 0. 0.522 0. 1.38
Problem 115 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 340 340 322 307 0 1771 0 1030
normalized size 1 1. 0.95 0.9 0. 5.21 0. 3.03
time (sec) N/A 0.52 1.611 0.252 0. 0.559 0. 1.607
Problem 116 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 147 147 116 121 0 409 0 0
normalized size 1 1. 0.79 0.82 0. 2.78 0. 0.
time (sec) N/A 0.168 0.154 0.066 0. 0.492 0. 0.
Problem 117 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 171 171 131 145 0 452 0 0
normalized size 1 1. 0.77 0.85 0. 2.64 0. 0.
time (sec) N/A 0.2 0.257 0.092 0. 0.497 0. 0.
Problem 118 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 293 293 218 242 0 815 0 0
normalized size 1 1. 0.74 0.83 0. 2.78 0. 0.
time (sec) N/A 0.331 0.422 0.202 0. 0.528 0. 0.
Problem 119 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 103 103 170 82 0 301 0 0
normalized size 1 1. 1.65 0.8 0. 2.92 0. 0.
time (sec) N/A 0.154 0.477 0.063 0. 0.504 0. 0.
Problem 120 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 140 140 189 107 0 479 0 0
normalized size 1 1. 1.35 0.76 0. 3.42 0. 0.
time (sec) N/A 0.188 0.827 0.082 0. 0.52 0. 0.
Problem 121 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 205 205 389 162 0 869 0 0
normalized size 1 1. 1.9 0.79 0. 4.24 0. 0.
time (sec) N/A 0.288 2.21 0.195 0. 0.548 0. 0.
Problem 122 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 183 183 217 167 0 765 0 0
normalized size 1 1. 1.19 0.91 0. 4.18 0. 0.
time (sec) N/A 0.304 0.942 0.09 0. 0.548 0. 0.
Problem 123 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 211 211 252 191 0 927 0 0
normalized size 1 1. 1.19 0.91 0. 4.39 0. 0.
time (sec) N/A 0.358 2.298 0.138 0. 0.56 0. 0.
Problem 124 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B A F(-2) B F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 369 369 2997 334 0 1839 0 0
normalized size 1 1. 8.12 0.91 0. 4.98 0. 0.
time (sec) N/A 0.583 7.018 0.31 0. 0.662 0. 0.
Problem 125 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 172 172 151 168 0 483 0 0
normalized size 1 1. 0.88 0.98 0. 2.81 0. 0.
time (sec) N/A 0.258 0.327 0.07 0. 0.496 0. 0.
Problem 126 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 231 231 204 217 0 618 0 0
normalized size 1 1. 0.88 0.94 0. 2.68 0. 0.
time (sec) N/A 0.282 0.639 0.106 0. 0.508 0. 0.
Problem 127 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 346 346 386 334 0 964 0 0
normalized size 1 1. 1.12 0.97 0. 2.79 0. 0.
time (sec) N/A 0.416 0.967 0.237 0. 0.521 0. 0.
Problem 128 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 189 189 231 178 0 778 0 0
normalized size 1 1. 1.22 0.94 0. 4.12 0. 0.
time (sec) N/A 0.265 0.983 0.089 0. 0.548 0. 0.
Problem 129 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 245 245 301 227 0 1026 0 0
normalized size 1 1. 1.23 0.93 0. 4.19 0. 0.
time (sec) N/A 0.406 3.096 0.137 0. 0.56 0. 0.
Problem 130 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B A F(-2) B F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 378 378 3285 354 0 1852 0 0
normalized size 1 1. 8.69 0.94 0. 4.9 0. 0.
time (sec) N/A 0.522 6.973 0.316 0. 0.66 0. 0.
Problem 131 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 208 208 348 214 0 938 0 0
normalized size 1 1. 1.67 1.03 0. 4.51 0. 0.
time (sec) N/A 0.395 2.126 0.1 0. 0.561 0. 0.
Problem 132 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B A F(-2) B F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 268 268 1118 263 0 1206 0 0
normalized size 1 1. 4.17 0.98 0. 4.5 0. 0.
time (sec) N/A 0.461 6.735 0.176 0. 0.575 0. 0.
Problem 133 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A B A F(-2) B F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 422 422 3829 426 0 2195 0 0
normalized size 1 1. 9.07 1.01 0. 5.2 0. 0.
time (sec) N/A 0.665 7.183 0.422 0. 0.695 0. 0.
Problem 134 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) B F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 209 209 325 216 0 938 0 0
normalized size 1 1. 1.56 1.03 0. 4.49 0. 0.
time (sec) N/A 0.482 1.796 0.099 0. 0.554 0. 0.
Problem 135 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 245 245 180 368 784 581 1760 2395
normalized size 1 1. 0.73 1.5 3.2 2.37 7.18 9.78
time (sec) N/A 0.358 1.604 0.099 1.184 0.511 162.303 1.454
Problem 136 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 99 99 83 183 294 197 408 1270
normalized size 1 1. 0.84 1.85 2.97 1.99 4.12 12.83
time (sec) N/A 0.159 0.578 0.014 1.059 0.487 12.888 1.263
Problem 137 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 80 80 128 0 0 0 0 0
normalized size 1 1. 1.6 0. 0. 0. 0. 0.
time (sec) N/A 0.066 1.699 0.119 0. 0. 0. 0.
Problem 138 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 184 184 240 0 0 0 0 0
normalized size 1 1. 1.3 0. 0. 0. 0. 0.
time (sec) N/A 0.1 3.23 0.373 0. 0. 0. 0.
Problem 139 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 245 245 228 371 780 536 1760 2392
normalized size 1 1. 0.93 1.51 3.18 2.19 7.18 9.76
time (sec) N/A 0.326 0.629 0.076 1.288 0.499 169.745 1.388
Problem 140 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A B A A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 98 98 82 186 292 188 408 1266
normalized size 1 1. 0.84 1.9 2.98 1.92 4.16 12.92
time (sec) N/A 0.148 0.235 0.012 1.135 0.479 14.178 1.255
Problem 141 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 79 79 80 0 0 0 0 0
normalized size 1 1. 1.01 0. 0. 0. 0. 0.
time (sec) N/A 0.06 0.05 0.056 0. 0. 0. 0.
Problem 142 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F(-1) F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 169 169 145 0 0 0 0 0
normalized size 1 1. 0.86 0. 0. 0. 0. 0.
time (sec) N/A 0.097 0.393 0.114 0. 0. 0. 0.

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 [48] had the largest ratio of [ 0.5556 ]

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 2 1. 18 0.111
2 A 2 2 1. 18 0.111
3 A 2 2 1. 18 0.111
4 A 1 1 1. 16 0.062
5 A 1 1 1. 16 0.062
6 A 1 1 1. 18 0.056
7 A 2 2 1. 18 0.111
8 A 2 2 1. 18 0.111
9 A 3 2 1. 8 0.25
10 A 2 2 1. 18 0.111
11 A 2 2 1. 18 0.111
12 A 2 2 1. 18 0.111
13 A 1 1 1. 16 0.062
14 A 1 1 1. 16 0.062
15 A 1 1 1. 18 0.056
16 A 2 2 1. 18 0.111
17 A 2 2 1. 18 0.111
18 A 3 2 1. 8 0.25
19 A 6 3 1. 18 0.167
20 A 5 3 1. 18 0.167
21 A 4 3 1. 16 0.188
22 A 4 3 1. 16 0.188
23 A 5 3 1. 18 0.167
24 A 6 3 1. 18 0.167
25 A 5 4 1. 27 0.148
26 A 2 2 1. 18 0.111
27 A 2 2 1. 18 0.111
28 F 0 0 N/A 0 N/A
29 A 0 0 0. 0 0.
30 A 0 0 0. 0 0.
31 A 11 5 1. 44 0.114
32 F 0 0 N/A 0 N/A
33 A 7 4 1. 43 0.093
34 B 14 6 19.24 35 0.171
35 A 1 1 1. 30 0.033
36 A 6 3 1. 38 0.079
37 A 10 4 1. 38 0.105
38 A 3 3 1. 20 0.15
39 A 4 2 1. 22 0.091
40 A 4 2 1. 22 0.091
41 A 4 2 1. 22 0.091
42 A 4 3 1. 24 0.125
43 A 5 2 1. 24 0.083
44 A 4 2 1. 22 0.091
45 A 5 2 1. 24 0.083
46 A 4 2 1. 24 0.083
47 A 4 3 1. 7 0.429
48 A 11 5 1. 9 0.556
49 A 4 3 1. 7 0.429
50 A 11 5 1. 9 0.556
51 A 4 3 1. 19 0.158
52 A 3 1 1. 15 0.067
53 A 5 4 1. 26 0.154
54 A 5 4 1. 27 0.148
55 A 5 4 1. 26 0.154
56 A 5 4 1. 27 0.148
57 A 6 3 1. 10 0.3
58 A 6 3 1. 10 0.3
59 A 6 3 1. 12 0.25
60 A 6 3 1. 12 0.25
61 A 2 2 1. 15 0.133
62 A 2 2 1. 8 0.25
63 A 3 3 1. 12 0.25
64 A 2 2 1. 8 0.25
65 A 2 2 1. 12 0.167
66 A 2 2 1. 12 0.167
67 A 3 3 1. 10 0.3
68 A 4 3 1. 26 0.115
69 A 2 2 1. 8 0.25
70 A 2 2 1. 8 0.25
71 A 3 3 1. 12 0.25
72 A 3 3 1. 10 0.3
73 A 1 1 1. 10 0.1
74 A 6 4 1. 12 0.333
75 A 6 4 1. 15 0.267
76 A 6 3 1. 12 0.25
77 A 4 2 1. 14 0.143
78 A 6 3 1. 17 0.176
79 A 8 5 1. 16 0.312
80 A 9 6 1. 18 0.333
81 A 14 5 1. 18 0.278
82 A 8 5 1. 19 0.263
83 A 9 6 1. 21 0.286
84 A 14 5 1. 21 0.238
85 A 8 4 1. 16 0.25
86 A 9 4 1. 18 0.222
87 A 14 4 1. 18 0.222
88 A 6 4 1. 18 0.222
89 A 7 4 1. 20 0.2
90 A 10 4 1. 20 0.2
91 A 8 5 1. 21 0.238
92 A 9 5 1. 23 0.217
93 A 14 5 1. 23 0.217
94 A 8 4 1. 19 0.21
95 A 10 4 1. 21 0.19
96 A 14 4 1. 21 0.19
97 A 8 5 1. 21 0.238
98 A 10 5 1. 23 0.217
99 A 14 5 1. 23 0.217
100 A 8 5 1. 24 0.208
101 A 10 5 1. 26 0.192
102 A 14 5 1. 26 0.192
103 A 8 5 1. 24 0.208
104 A 1 1 1. 10 0.1
105 A 6 4 1. 12 0.333
106 A 6 4 1. 15 0.267
107 A 6 3 1. 12 0.25
108 A 4 2 1. 14 0.143
109 A 6 3 1. 17 0.176
110 A 8 5 1. 16 0.312
111 A 9 6 1. 18 0.333
112 A 14 5 1. 18 0.278
113 A 8 5 1. 19 0.263
114 A 9 6 1. 21 0.286
115 A 14 5 1. 21 0.238
116 A 8 4 1. 16 0.25
117 A 9 4 1. 18 0.222
118 A 14 4 1. 18 0.222
119 A 6 4 1. 18 0.222
120 A 7 4 1. 20 0.2
121 A 10 4 1. 20 0.2
122 A 8 5 1. 21 0.238
123 A 9 5 1. 23 0.217
124 A 14 5 1. 23 0.217
125 A 8 4 1. 19 0.21
126 A 10 4 1. 21 0.19
127 A 14 4 1. 21 0.19
128 A 8 5 1. 21 0.238
129 A 10 5 1. 23 0.217
130 A 14 5 1. 23 0.217
131 A 8 5 1. 24 0.208
132 A 10 5 1. 26 0.192
133 A 14 5 1. 26 0.192
134 A 8 5 1. 24 0.208
135 A 8 6 1. 22 0.273
136 A 6 5 1. 20 0.25
137 A 2 2 1. 22 0.091
138 A 3 3 1. 22 0.136
139 A 8 6 1. 22 0.273
140 A 6 5 1. 20 0.25
141 A 2 2 1. 22 0.091
142 A 3 3 1. 22 0.136

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