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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 }

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, 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, 106, 108, 109, 111, 113, 115, 118, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139 }

B grade: { }

C grade: { 105, 107, 110, 112, 114, 116, 117, 119 }

F grade: { 140}

2.1.3 Maple

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 29, 30, 31, 32, 33, 34, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 80, 82, 83, 84, 85, 86, 87, 88, 91, 93, 95, 96, 97, 106, 107, 108, 109, 111, 113, 114, 115, 116, 117, 120, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139 }

B grade: { 19, 24, 25, 26, 27, 28, 35, 37, 47, 48, 65, 72, 79, 81, 89, 90, 92, 94, 98, 99, 100, 101, 102, 103, 105, 110, 112, 118, 119 }

C grade: { }

F grade: { 104, 121, 122, 140 }

2.1.4 Maxima

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 137

B grade: { }

C grade: { }

F grade: { 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, 72, 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, 138, 139, 140 }

2.1.5 FriCAS

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 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, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 80, 82, 84, 86, 88, 106, 108, 109, 111, 113, 115, 120, 121, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139 }

B grade: { 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 60, 72, 79, 81, 83, 85, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 118 }

C grade: { }

F grade: { 104, 105, 107, 110, 112, 114, 116, 117, 119, 122, 140 }

2.1.6 Sympy

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 81, 83, 85, 87, 90, 92, 94, 96, 98, 100, 102 }

B grade: { 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 78, 80, 82, 84, 86, 88, 89, 91, 93, 95, 97, 99, 101, 103 }

C grade: { }

F grade: { 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, 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 }

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, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 38, 39, 40, 41, 42, 50, 51, 52, 56, 57, 58, 59, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 80, 82, 84, 86, 88, 89, 91, 93, 95, 97, 99, 101, 103, 106, 113, 126, 127, 128, 129, 130, 131, 132, 139 }

B grade: { 65, 72, 111 }

C grade: { 79, 81, 83, 85, 87 }

F grade: { 33, 34, 35, 36, 37, 43, 44, 45, 46, 47, 48, 49, 53, 54, 55, 60, 61, 62, 63, 64, 90, 92, 94, 96, 98, 100, 102, 104, 105, 107, 108, 109, 110, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 133, 134, 135, 136, 137, 138, 140 }

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 25 25 25 20 26 47 19 26
normalized size 1 1. 1. 0.8 1.04 1.88 0.76 1.04
time (sec) N/A 0.007 0.002 0. 1.099 1.279 0.058 1.099
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 25 25 25 20 26 47 19 26
normalized size 1 1. 1. 0.8 1.04 1.88 0.76 1.04
time (sec) N/A 0.007 0.002 0. 1.158 1.301 0.058 1.101
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 25 25 25 20 26 47 19 26
normalized size 1 1. 1. 0.8 1.04 1.88 0.76 1.04
time (sec) N/A 0.004 0. 0. 1.076 1.365 0.058 1.081
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 25 25 25 20 26 47 19 26
normalized size 1 1. 1. 0.8 1.04 1.88 0.76 1.04
time (sec) N/A 0.006 0.001 0.001 1.071 1.442 0.057 1.083
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 20 20 20 17 22 39 15 22
normalized size 1 1. 1. 0.85 1.1 1.95 0.75 1.1
time (sec) N/A 0.006 0.001 0.002 1.118 1.468 0.056 1.075
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 54 54 54 45 59 116 48 62
normalized size 1 1. 1. 0.83 1.09 2.15 0.89 1.15
time (sec) N/A 0.054 0.01 0.001 1.165 1.268 0.071 1.071
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 54 54 54 45 59 115 49 62
normalized size 1 1. 1. 0.83 1.09 2.13 0.91 1.15
time (sec) N/A 0.029 0.007 0. 1.127 1.289 0.07 1.081
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 54 54 54 45 65 112 48 62
normalized size 1 1. 1. 0.83 1.2 2.07 0.89 1.15
time (sec) N/A 0.027 0.006 0. 1.337 1.365 0.073 1.096
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 54 45 59 107 49 62
normalized size 1 1. 1. 0.83 1.09 1.98 0.91 1.15
time (sec) N/A 0.032 0.006 0.002 1.125 1.425 0.07 1.099
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 54 54 54 45 59 107 48 62
normalized size 1 1. 1. 0.83 1.09 1.98 0.89 1.15
time (sec) N/A 0.032 0.007 0. 1.153 1.455 0.072 1.072
Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 89 89 84 132 0 653 381 116
normalized size 1 1. 0.94 1.48 0. 7.34 4.28 1.3
time (sec) N/A 0.094 0.11 0.003 0. 1.59 0.784 1.083
Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 70 70 73 101 0 537 306 90
normalized size 1 1. 1.04 1.44 0. 7.67 4.37 1.29
time (sec) N/A 0.059 0.064 0.003 0. 1.688 0.647 1.083
Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 56 56 57 56 0 427 216 74
normalized size 1 1. 1.02 1. 0. 7.62 3.86 1.32
time (sec) N/A 0.04 0.032 0.001 0. 1.559 0.302 1.105
Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 34 34 38 35 0 277 124 46
normalized size 1 1. 1.12 1.03 0. 8.15 3.65 1.35
time (sec) N/A 0.026 0.006 0.002 0. 1.571 0.201 1.113
Problem 15 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 62 62 61 62 0 494 564 84
normalized size 1 1. 0.98 1. 0. 7.97 9.1 1.35
time (sec) N/A 0.047 0.073 0.006 0. 1.762 1.927 1.083
Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 81 81 77 112 0 626 862 107
normalized size 1 1. 0.95 1.38 0. 7.73 10.64 1.32
time (sec) N/A 0.098 0.079 0.006 0. 1.776 3.507 1.093
Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 104 104 102 150 0 798 1525 142
normalized size 1 1. 0.98 1.44 0. 7.67 14.66 1.37
time (sec) N/A 0.146 0.138 0.007 0. 1.946 5.173 1.132
Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 137 137 131 214 0 979 2105 184
normalized size 1 1. 0.96 1.56 0. 7.15 15.36 1.34
time (sec) N/A 0.203 0.114 0.009 0. 2.039 8.804 1.093
Problem 19 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 150 150 132 352 0 1760 842 217
normalized size 1 1. 0.88 2.35 0. 11.73 5.61 1.45
time (sec) N/A 0.159 0.193 0.008 0. 1.598 1.625 1.116
Problem 20 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 114 114 109 209 0 1374 729 169
normalized size 1 1. 0.96 1.83 0. 12.05 6.39 1.48
time (sec) N/A 0.103 0.153 0.009 0. 1.658 1.209 1.097
Problem 21 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 67 67 81 97 0 826 280 119
normalized size 1 1. 1.21 1.45 0. 12.33 4.18 1.78
time (sec) N/A 0.039 0.094 0.006 0. 1.597 0.756 1.117
Problem 22 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 66 66 69 70 0 740 252 103
normalized size 1 1. 1.05 1.06 0. 11.21 3.82 1.56
time (sec) N/A 0.038 0.068 0.002 0. 1.53 0.705 1.14
Problem 23 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 66 66 70 68 0 745 265 103
normalized size 1 1. 1.06 1.03 0. 11.29 4.02 1.56
time (sec) N/A 0.035 0.079 0.003 0. 1.551 0.722 1.106
Problem 24 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 108 108 107 237 0 1685 2236 170
normalized size 1 1. 0.99 2.19 0. 15.6 20.7 1.57
time (sec) N/A 0.148 0.236 0.012 0. 2.092 7.645 1.131
Problem 25 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 148 148 131 328 0 2049 2672 231
normalized size 1 1. 0.89 2.22 0. 13.84 18.05 1.56
time (sec) N/A 0.198 0.305 0.013 0. 2.612 11.941 1.089
Problem 26 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 202 202 175 418 0 2606 4083 309
normalized size 1 1. 0.87 2.07 0. 12.9 20.21 1.53
time (sec) N/A 0.25 0.428 0.016 0. 3.117 19.174 1.121
Problem 27 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 252 252 218 515 0 2989 4774 381
normalized size 1 1. 0.87 2.04 0. 11.86 18.94 1.51
time (sec) N/A 0.323 0.341 0.016 0. 4.108 31.024 1.107
Problem 28 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 318 318 272 619 0 3563 6181 468
normalized size 1 1. 0.86 1.95 0. 11.2 19.44 1.47
time (sec) N/A 0.392 0.413 0.018 0. 5.725 47.472 1.123
Problem 29 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 257 257 180 310 0 898 0 382
normalized size 1 1. 0.7 1.21 0. 3.49 0. 1.49
time (sec) N/A 0.588 0.241 0.007 0. 1.736 0. 1.182
Problem 30 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 205 205 150 265 0 737 0 311
normalized size 1 1. 0.73 1.29 0. 3.6 0. 1.52
time (sec) N/A 0.37 0.193 0.008 0. 1.646 0. 1.211
Problem 31 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 163 163 119 167 0 591 0 224
normalized size 1 1. 0.73 1.02 0. 3.63 0. 1.37
time (sec) N/A 0.058 0.22 0.006 0. 1.615 0. 1.144
Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 119 119 100 146 0 502 0 169
normalized size 1 1. 0.84 1.23 0. 4.22 0. 1.42
time (sec) N/A 0.078 0.146 0.004 0. 1.669 0. 1.166
Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 173 173 134 126 0 1457 0 0
normalized size 1 1. 0.77 0.73 0. 8.42 0. 0.
time (sec) N/A 0.126 0.11 0.004 0. 1.842 0. 0.
Problem 34 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 173 173 131 174 0 1489 0 0
normalized size 1 1. 0.76 1.01 0. 8.61 0. 0.
time (sec) N/A 0.124 0.127 0.006 0. 1.963 0. 0.
Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F A F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 114 114 112 207 0 517 0 0
normalized size 1 1. 0.98 1.82 0. 4.54 0. 0.
time (sec) N/A 0.148 0.1 0.006 0. 1.692 0. 0.
Problem 36 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 155 155 131 234 0 609 0 0
normalized size 1 1. 0.85 1.51 0. 3.93 0. 0.
time (sec) N/A 0.256 0.144 0.006 0. 1.841 0. 0.
Problem 37 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F A F F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 205 205 160 387 0 757 0 0
normalized size 1 1. 0.78 1.89 0. 3.69 0. 0.
time (sec) N/A 0.386 0.225 0.006 0. 2.129 0. 0.
Problem 38 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 422 422 236 649 0 1600 0 703
normalized size 1 1. 0.56 1.54 0. 3.79 0. 1.67
time (sec) N/A 1.203 0.426 0.008 0. 2.031 0. 1.294
Problem 39 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 364 364 197 479 0 1299 0 579
normalized size 1 1. 0.54 1.32 0. 3.57 0. 1.59
time (sec) N/A 1.039 0.267 0.009 0. 1.985 0. 1.302
Problem 40 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 288 288 180 431 0 1099 0 493
normalized size 1 1. 0.62 1.5 0. 3.82 0. 1.71
time (sec) N/A 0.519 0.242 0.009 0. 1.835 0. 1.325
Problem 41 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 198 198 163 289 0 878 0 383
normalized size 1 1. 0.82 1.46 0. 4.43 0. 1.93
time (sec) N/A 0.178 0.182 0.006 0. 1.695 0. 1.273
Problem 42 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 165 165 132 265 0 725 0 313
normalized size 1 1. 0.8 1.61 0. 4.39 0. 1.9
time (sec) N/A 0.128 0.064 0.006 0. 1.646 0. 1.28
Problem 43 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 227 227 166 222 0 1825 0 0
normalized size 1 1. 0.73 0.98 0. 8.04 0. 0.
time (sec) N/A 0.255 0.238 0.005 0. 2.562 0. 0.
Problem 44 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 219 219 158 254 0 1735 0 0
normalized size 1 1. 0.72 1.16 0. 7.92 0. 0.
time (sec) N/A 0.242 0.184 0.006 0. 2.218 0. 0.
Problem 45 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 219 219 162 338 0 1737 0 0
normalized size 1 1. 0.74 1.54 0. 7.93 0. 0.
time (sec) N/A 0.238 0.199 0.006 0. 2.242 0. 0.
Problem 46 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 257 257 175 435 0 1871 0 0
normalized size 1 1. 0.68 1.69 0. 7.28 0. 0.
time (sec) N/A 0.351 0.307 0.007 0. 2.5 0. 0.
Problem 47 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F A F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 197 197 141 501 0 743 0 0
normalized size 1 1. 0.72 2.54 0. 3.77 0. 0.
time (sec) N/A 0.363 0.117 0.009 0. 2.341 0. 0.
Problem 48 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F A F(-1) F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 249 249 177 534 0 898 0 0
normalized size 1 1. 0.71 2.14 0. 3.61 0. 0.
time (sec) N/A 0.504 0.176 0.008 0. 2.71 0. 0.
Problem 49 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 143 143 105 144 0 514 0 0
normalized size 1 1. 0.73 1.01 0. 3.59 0. 0.
time (sec) N/A 0.174 0.137 0.006 0. 1.569 0. 0.
Problem 50 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 103 103 89 88 0 429 0 146
normalized size 1 1. 0.86 0.85 0. 4.17 0. 1.42
time (sec) N/A 0.078 0.054 0.006 0. 1.591 0. 1.156
Problem 51 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 71 71 66 65 0 298 0 50
normalized size 1 1. 0.93 0.92 0. 4.2 0. 0.7
time (sec) N/A 0.037 0.041 0.006 0. 1.546 0. 1.159
Problem 52 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 45 45 70 66 0 300 0 80
normalized size 1 1. 1.56 1.47 0. 6.67 0. 1.78
time (sec) N/A 0.016 0.021 0.005 0. 1.616 0. 1.157
Problem 53 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 77 77 89 88 0 444 0 0
normalized size 1 1. 1.16 1.14 0. 5.77 0. 0.
time (sec) N/A 0.054 0.056 0.006 0. 1.732 0. 0.
Problem 54 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 119 119 112 152 0 527 0 0
normalized size 1 1. 0.94 1.28 0. 4.43 0. 0.
time (sec) N/A 0.149 0.096 0.006 0. 1.791 0. 0.
Problem 55 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 262 262 183 283 0 1316 0 0
normalized size 1 1. 0.7 1.08 0. 5.02 0. 0.
time (sec) N/A 0.506 0.258 0.009 0. 2.276 0. 0.
Problem 56 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 201 201 141 199 0 1026 0 263
normalized size 1 1. 0.7 0.99 0. 5.1 0. 1.31
time (sec) N/A 0.305 0.159 0.006 0. 2.09 0. 1.17
Problem 57 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 153 153 112 166 0 873 0 149
normalized size 1 1. 0.73 1.08 0. 5.71 0. 0.97
time (sec) N/A 0.175 0.123 0.006 0. 2.159 0. 1.192
Problem 58 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 40 40 37 53 0 150 0 61
normalized size 1 1. 0.92 1.32 0. 3.75 0. 1.52
time (sec) N/A 0.04 0.074 0.004 0. 1.985 0. 1.192
Problem 59 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 39 39 36 52 0 151 0 61
normalized size 1 1. 0.92 1.33 0. 3.87 0. 1.56
time (sec) N/A 0.04 0.024 0.003 0. 1.946 0. 1.165
Problem 60 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F B F F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 94 94 109 164 0 869 0 0
normalized size 1 1. 1.16 1.74 0. 9.24 0. 0.
time (sec) N/A 0.068 0.123 0.006 0. 2.065 0. 0.
Problem 61 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 144 144 138 201 0 1041 0 0
normalized size 1 1. 0.96 1.4 0. 7.23 0. 0.
time (sec) N/A 0.163 0.106 0.006 0. 1.876 0. 0.
Problem 62 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 209 209 181 292 0 1328 0 0
normalized size 1 1. 0.87 1.4 0. 6.35 0. 0.
time (sec) N/A 0.287 0.167 0.007 0. 2.208 0. 0.
Problem 63 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 271 271 225 340 0 1550 0 0
normalized size 1 1. 0.83 1.25 0. 5.72 0. 0.
time (sec) N/A 0.452 0.265 0.009 0. 2.913 0. 0.
Problem 64 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 343 343 272 446 0 1901 0 0
normalized size 1 1. 0.79 1.3 0. 5.54 0. 0.
time (sec) N/A 0.621 0.287 0.008 0. 4.085 0. 0.
Problem 65 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F(-2) A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 37 37 34 77 0 161 280 144
normalized size 1 1. 0.92 2.08 0. 4.35 7.57 3.89
time (sec) N/A 0.013 0.029 0.003 0. 1.313 1.09 1.112
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 25 25 25 20 26 47 19 26
normalized size 1 1. 1. 0.8 1.04 1.88 0.76 1.04
time (sec) N/A 0.008 0.002 0.002 1.127 1.051 0.056 1.098
Problem 67 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 25 25 25 20 26 47 19 26
normalized size 1 1. 1. 0.8 1.04 1.88 0.76 1.04
time (sec) N/A 0.007 0.002 0. 1.072 1.087 0.056 1.091
Problem 68 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 25 25 25 20 26 47 19 26
normalized size 1 1. 1. 0.8 1.04 1.88 0.76 1.04
time (sec) N/A 0.004 0. 0. 1.114 1.049 0.056 1.1
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 20 20 20 17 22 39 15 22
normalized size 1 1. 1. 0.85 1.1 1.95 0.75 1.1
time (sec) N/A 0.006 0.001 0. 1.131 1.23 0.056 1.1
Problem 70 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 21 21 21 18 23 46 17 27
normalized size 1 1. 1. 0.86 1.1 2.19 0.81 1.29
time (sec) N/A 0.008 0.002 0.003 1.142 1.224 0.089 1.121
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 18 18 18 17 22 42 12 22
normalized size 1 1. 1. 0.94 1.22 2.33 0.67 1.22
time (sec) N/A 0.007 0.002 0.003 1.112 1.191 0.248 1.091
Problem 72 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 76 76 69 300 0 602 1377 539
normalized size 1 1. 0.91 3.95 0. 7.92 18.12 7.09
time (sec) N/A 0.046 0.079 0.006 0. 1.41 3.99 1.127
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 54 54 54 45 59 117 51 62
normalized size 1 1. 1. 0.83 1.09 2.17 0.94 1.15
time (sec) N/A 0.036 0.007 0.001 1.119 1.015 0.07 1.086
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 54 54 48 45 59 116 46 62
normalized size 1 1. 0.89 0.83 1.09 2.15 0.85 1.15
time (sec) N/A 0.054 0.009 0.001 1.09 1.123 0.07 1.09
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 54 54 54 45 65 115 51 62
normalized size 1 1. 1. 0.83 1.2 2.13 0.94 1.15
time (sec) N/A 0.026 0.008 0.002 1.12 1.118 0.07 1.098
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 48 45 59 109 46 62
normalized size 1 1. 0.89 0.83 1.09 2.02 0.85 1.15
time (sec) N/A 0.046 0.009 0.002 1.022 1.309 0.069 1.089
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 42 55 99 48 58
normalized size 1 1. 1. 0.86 1.12 2.02 0.98 1.18
time (sec) N/A 0.027 0.005 0.001 1.128 1.099 0.072 1.078
Problem 78 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 100 100 93 142 0 675 391 124
normalized size 1 1. 0.93 1.42 0. 6.75 3.91 1.24
time (sec) N/A 0.122 0.094 0.005 0. 1.329 1.892 1.1
Problem 79 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 203 203 250 467 0 3217 194 5466
normalized size 1 1. 1.23 2.3 0. 15.85 0.96 26.93
time (sec) N/A 0.602 0.172 0.023 0. 1.54 2.207 2.487
Problem 80 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 81 81 78 111 0 556 316 101
normalized size 1 1. 0.96 1.37 0. 6.86 3.9 1.25
time (sec) N/A 0.087 0.048 0.003 0. 1.278 1.541 1.095
Problem 81 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 179 179 202 343 0 2168 129 4593
normalized size 1 1. 1.13 1.92 0. 12.11 0.72 25.66
time (sec) N/A 0.227 0.145 0.013 0. 1.412 1.604 2.336
Problem 82 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 63 63 62 60 0 443 223 80
normalized size 1 1. 0.98 0.95 0. 7.03 3.54 1.27
time (sec) N/A 0.067 0.028 0.003 0. 1.316 0.813 1.089
Problem 83 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F B A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 150 150 165 208 0 1206 75 5350
normalized size 1 1. 1.1 1.39 0. 8.04 0.5 35.67
time (sec) N/A 0.094 0.098 0.011 0. 1.522 0.768 2.206
Problem 84 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 36 36 39 36 0 290 131 47
normalized size 1 1. 1.08 1. 0. 8.06 3.64 1.31
time (sec) N/A 0.043 0.009 0. 0. 1.539 0.445 1.116
Problem 85 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F B A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 150 150 129 116 0 1323 87 1365
normalized size 1 1. 0.86 0.77 0. 8.82 0.58 9.1
time (sec) N/A 0.08 0.083 0.01 0. 1.571 0.896 1.359
Problem 86 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 69 69 113 66 0 510 253 92
normalized size 1 1. 1.64 0.96 0. 7.39 3.67 1.33
time (sec) N/A 0.072 0.073 0.006 0. 1.588 2.868 1.096
Problem 87 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F B A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 174 174 191 232 0 2279 148 4581
normalized size 1 1. 1.1 1.33 0. 13.1 0.85 26.33
time (sec) N/A 0.195 0.423 0.019 0. 1.625 1.808 2.354
Problem 88 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 89 89 135 119 0 664 345 127
normalized size 1 1. 1.52 1.34 0. 7.46 3.88 1.43
time (sec) N/A 0.133 0.125 0.006 0. 2.017 7.246 1.095
Problem 89 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 166 166 151 383 0 1806 877 217
normalized size 1 1. 0.91 2.31 0. 10.88 5.28 1.31
time (sec) N/A 0.223 0.209 0.013 0. 1.671 4.751 21.874
Problem 90 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B A F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 331 331 327 844 0 6311 450 0
normalized size 1 1. 0.99 2.55 0. 19.07 1.36 0.
time (sec) N/A 0.703 0.713 0.033 0. 2.158 6.661 0.
Problem 91 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F B B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 132 132 121 222 0 1412 745 205
normalized size 1 1. 0.92 1.68 0. 10.7 5.64 1.55
time (sec) N/A 0.152 0.186 0.011 0. 1.385 3.505 25.605
Problem 92 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B A F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 271 271 282 602 0 4806 379 0
normalized size 1 1. 1.04 2.22 0. 17.73 1.4 0.
time (sec) N/A 0.526 0.572 0.029 0. 1.723 4.632 0.
Problem 93 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 78 78 93 104 0 864 282 130
normalized size 1 1. 1.19 1.33 0. 11.08 3.62 1.67
time (sec) N/A 0.072 0.091 0.009 0. 1.366 1.746 29.874
Problem 94 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B A F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 237 237 235 452 0 3584 294 0
normalized size 1 1. 0.99 1.91 0. 15.12 1.24 0.
time (sec) N/A 0.358 0.441 0.026 0. 1.395 3.359 0.
Problem 95 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 75 75 79 77 0 778 267 111
normalized size 1 1. 1.05 1.03 0. 10.37 3.56 1.48
time (sec) N/A 0.069 0.069 0.004 0. 1.282 1.667 25.713
Problem 96 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F B A F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 221 221 222 342 0 3623 298 0
normalized size 1 1. 1. 1.55 0. 16.39 1.35 0.
time (sec) N/A 0.24 0.462 0.063 0. 1.612 3.489 0.
Problem 97 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 74 74 79 75 0 783 267 111
normalized size 1 1. 1.07 1.01 0. 10.58 3.61 1.5
time (sec) N/A 0.065 0.085 0.004 0. 1.378 1.574 31.512
Problem 98 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B A F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 252 252 243 733 0 4918 394 0
normalized size 1 1. 0.96 2.91 0. 19.52 1.56 0.
time (sec) N/A 0.461 0.461 0.044 0. 1.767 4.473 0.
Problem 99 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 122 122 207 253 0 1728 772 224
normalized size 1 1. 1.7 2.07 0. 14.16 6.33 1.84
time (sec) N/A 0.188 0.322 0.014 0. 1.784 40.968 22.145
Problem 100 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B A F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 308 308 302 712 0 6460 481 0
normalized size 1 1. 0.98 2.31 0. 20.97 1.56 0.
time (sec) N/A 1.352 0.668 0.03 0. 2.108 7.145 0.
Problem 101 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 162 162 248 352 0 2103 906 246
normalized size 1 1. 1.53 2.17 0. 12.98 5.59 1.52
time (sec) N/A 0.25 0.275 0.017 0. 2.194 72.952 25.533
Problem 102 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B A F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 361 361 344 913 0 7960 566 0
normalized size 1 1. 0.95 2.53 0. 22.05 1.57 0.
time (sec) N/A 3.077 0.788 0.036 0. 2.942 12.022 0.
Problem 103 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 219 219 328 443 0 2627 1074 370
normalized size 1 1. 1.5 2.02 0. 12. 4.9 1.69
time (sec) N/A 0.312 0.366 0.019 0. 2.724 131.412 23.764
Problem 104 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F F F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 142 142 170 0 0 0 0 0
normalized size 1 1. 1.2 0. 0. 0. 0. 0.
time (sec) N/A 0.171 0.095 0.017 0. 0. 0. 0.
Problem 105 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 380 380 486 1042 0 0 0 0
normalized size 1 1. 1.28 2.74 0. 0. 0. 0.
time (sec) N/A 0.287 1.52 0.069 0. 0. 0. 0.
Problem 106 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 129 129 126 157 0 547 0 171
normalized size 1 1. 0.98 1.22 0. 4.24 0. 1.33
time (sec) N/A 0.093 0.078 0.016 0. 1.397 0. 1.194
Problem 107 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C A F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 347 347 452 508 0 0 0 0
normalized size 1 1. 1.3 1.46 0. 0. 0. 0.
time (sec) N/A 0.224 0.971 0.019 0. 0. 0. 0.
Problem 108 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 194 194 155 136 0 1575 0 0
normalized size 1 1. 0.8 0.7 0. 8.12 0. 0.
time (sec) N/A 0.209 0.066 0.017 0. 1.641 0. 0.
Problem 109 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 244 244 192 369 0 921 0 0
normalized size 1 1. 0.79 1.51 0. 3.77 0. 0.
time (sec) N/A 0.357 0.212 0.026 0. 1.444 0. 0.
Problem 110 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 487 487 609 1878 0 0 0 0
normalized size 1 1. 1.25 3.86 0. 0. 0. 0.
time (sec) N/A 0.457 2.321 0.024 0. 0. 0. 0.
Problem 111 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 177 177 152 295 0 768 0 624
normalized size 1 1. 0.86 1.67 0. 4.34 0. 3.53
time (sec) N/A 0.138 0.114 0.02 0. 1.511 0. 1.518
Problem 112 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 425 425 540 1394 0 0 0 0
normalized size 1 1. 1.27 3.28 0. 0. 0. 0.
time (sec) N/A 0.449 1.748 0.023 0. 0. 0. 0.
Problem 113 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 82 82 82 72 0 325 0 54
normalized size 1 1. 1. 0.88 0. 3.96 0. 0.66
time (sec) N/A 0.062 0.018 0.012 0. 1.469 0. 1.314
Problem 114 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C A F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 121 121 193 177 0 0 0 0
normalized size 1 1. 1.6 1.46 0. 0. 0. 0.
time (sec) N/A 0.046 0.126 0.016 0. 0. 0. 0.
Problem 115 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 51 51 83 72 0 327 0 0
normalized size 1 1. 1.63 1.41 0. 6.41 0. 0.
time (sec) N/A 0.029 0.019 0.014 0. 1.4 0. 0.
Problem 116 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C A F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 330 330 303 508 0 0 0 0
normalized size 1 1. 0.92 1.54 0. 0. 0. 0.
time (sec) N/A 0.184 0.487 0.023 0. 0. 0. 0.
Problem 117 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C A F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 391 391 463 533 0 0 0 0
normalized size 1 1. 1.18 1.36 0. 0. 0. 0.
time (sec) N/A 0.248 1.083 0.027 0. 0. 0. 0.
Problem 118 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A B F B F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 103 103 126 179 0 918 0 0
normalized size 1 1. 1.22 1.74 0. 8.91 0. 0.
time (sec) N/A 0.073 0.083 0.019 0. 1.777 0. 0.
Problem 119 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A C B F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 468 468 519 1136 0 0 0 0
normalized size 1 1. 1.11 2.43 0. 0. 0. 0.
time (sec) N/A 0.409 1.351 0.029 0. 0. 0. 0.
Problem 120 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 154 154 160 220 0 1084 0 0
normalized size 1 1. 1.04 1.43 0. 7.04 0. 0.
time (sec) N/A 0.175 0.082 0.019 0. 1.854 0. 0.
Problem 121 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F A F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 51 51 46 0 0 186 0 0
normalized size 1 1. 0.9 0. 0. 3.65 0. 0.
time (sec) N/A 0.05 0.088 0.074 0. 1.657 0. 0.
Problem 122 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A F F F F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 287 287 239 0 0 0 0 0
normalized size 1 1. 0.83 0. 0. 0. 0. 0.
time (sec) N/A 0.399 5.135 0.02 0. 0. 0. 0.
Problem 123 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 45 45 73 58 0 142 0 0
normalized size 1 1. 1.62 1.29 0. 3.16 0. 0.
time (sec) N/A 0.009 0.018 0.011 0. 1.268 0. 0.
Problem 124 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 45 45 73 58 0 142 0 0
normalized size 1 1. 1.62 1.29 0. 3.16 0. 0.
time (sec) N/A 0.012 0.003 0.008 0. 1.37 0. 0.
Problem 125 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 45 45 73 58 0 142 0 0
normalized size 1 1. 1.62 1.29 0. 3.16 0. 0.
time (sec) N/A 0.012 0.004 0.003 0. 1.284 0. 0.
Problem 126 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 86 86 70 81 0 157 0 93
normalized size 1 1. 0.81 0.94 0. 1.83 0. 1.08
time (sec) N/A 0.041 0.034 0.007 0. 1.296 0. 1.089
Problem 127 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 86 86 70 81 0 157 0 93
normalized size 1 1. 0.81 0.94 0. 1.83 0. 1.08
time (sec) N/A 0.042 0.009 0.003 0. 1.288 0. 1.092
Problem 128 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 86 86 70 81 0 157 0 93
normalized size 1 1. 0.81 0.94 0. 1.83 0. 1.08
time (sec) N/A 0.041 0.003 0.003 0. 1.293 0. 1.096
Problem 129 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F(-2) A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 38 38 37 35 0 273 0 47
normalized size 1 1. 0.97 0.92 0. 7.18 0. 1.24
time (sec) N/A 0.016 0.009 0.003 0. 1.361 0. 1.097
Problem 130 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 45 45 70 64 0 300 0 80
normalized size 1 1. 1.56 1.42 0. 6.67 0. 1.78
time (sec) N/A 0.022 0.023 0.008 0. 1.394 0. 1.104
Problem 131 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 47 47 72 64 0 315 0 47
normalized size 1 1. 1.53 1.36 0. 6.7 0. 1.
time (sec) N/A 0.076 0.033 0.011 0. 1.704 0. 1.145
Problem 132 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 49 49 74 66 0 325 0 47
normalized size 1 1. 1.51 1.35 0. 6.63 0. 0.96
time (sec) N/A 0.09 0.028 0.006 0. 1.707 0. 1.157
Problem 133 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 44 44 44 39 0 294 0 0
normalized size 1 1. 1. 0.89 0. 6.68 0. 0.
time (sec) N/A 0.036 0.012 0.004 0. 1.467 0. 0.
Problem 134 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 49 49 81 72 0 311 0 0
normalized size 1 1. 1.65 1.47 0. 6.35 0. 0.
time (sec) N/A 0.015 0.016 0.005 0. 1.325 0. 0.
Problem 135 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 51 51 83 72 0 327 0 0
normalized size 1 1. 1.63 1.41 0. 6.41 0. 0.
time (sec) N/A 0.067 0.02 0.008 0. 1.372 0. 0.
Problem 136 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 53 53 85 74 0 338 0 0
normalized size 1 1. 1.6 1.4 0. 6.38 0. 0.
time (sec) N/A 0.077 0.019 0.007 0. 1.362 0. 0.
Problem 137 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 40 40 40 31 27 131 0 0
normalized size 1 1. 1. 0.78 0.68 3.28 0. 0.
time (sec) N/A 0.027 0.009 0.003 1.646 1.311 0. 0.
Problem 138 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 45 45 73 58 0 142 0 0
normalized size 1 1. 1.62 1.29 0. 3.16 0. 0.
time (sec) N/A 0.012 0.008 0. 0. 1.289 0. 0.
Problem 139 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A A A F A F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 43 43 62 50 0 132 0 69
normalized size 1 1. 1.44 1.16 0. 3.07 0. 1.6
time (sec) N/A 0.047 0.02 0.011 0. 1.393 0. 1.144
Problem 140 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac
grade A A F F F F(-2) F(-1) F
verified N/A Yes N/A TBD TBD TBD TBD TBD
size 70 70 0 0 0 0 0 0
normalized size 1 1. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.067 0.393 0.121 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 [86] had the largest ratio of [ 0.5 ]

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. 20 0.05
2 A 2 1 1. 18 0.056
3 A 1 0 1. 16 0.
4 A 2 1 1. 20 0.05
5 A 2 1 1. 20 0.05
6 A 3 2 1. 22 0.091
7 A 3 2 1. 20 0.1
8 A 3 2 1. 18 0.111
9 A 3 2 1. 22 0.091
10 A 3 2 1. 22 0.091
11 A 7 6 1. 22 0.273
12 A 6 6 1. 22 0.273
13 A 5 5 1. 22 0.227
14 A 3 3 1. 22 0.136
15 A 7 7 1. 20 0.35
16 A 8 7 1. 18 0.389
17 A 8 7 1. 22 0.318
18 A 8 7 1. 22 0.318
19 A 8 7 1. 22 0.318
20 A 7 7 1. 22 0.318
21 A 4 4 1. 22 0.182
22 A 4 4 1. 22 0.182
23 A 4 4 1. 22 0.182
24 A 8 7 1. 22 0.318
25 A 8 7 1. 22 0.318
26 A 8 7 1. 20 0.35
27 A 8 7 1. 18 0.389
28 A 8 7 1. 22 0.318
29 A 8 6 1. 24 0.25
30 A 7 6 1. 22 0.273
31 A 5 5 1. 20 0.25
32 A 4 4 1. 24 0.167
33 A 7 6 1. 24 0.25
34 A 7 6 1. 24 0.25
35 A 5 5 1. 24 0.208
36 A 6 5 1. 24 0.208
37 A 7 5 1. 24 0.208
38 A 10 7 1. 22 0.318
39 A 10 7 1. 20 0.35
40 A 8 7 1. 24 0.292
41 A 6 5 1. 24 0.208
42 A 5 4 1. 24 0.167
43 A 8 7 1. 24 0.292
44 A 8 7 1. 24 0.292
45 A 8 7 1. 24 0.292
46 A 9 8 1. 24 0.333
47 A 7 6 1. 24 0.25
48 A 8 6 1. 24 0.25
49 A 6 6 1. 24 0.25
50 A 4 4 1. 24 0.167
51 A 3 3 1. 22 0.136
52 A 2 2 1. 20 0.1
53 A 3 3 1. 24 0.125
54 A 5 5 1. 24 0.208
55 A 8 6 1. 24 0.25
56 A 7 6 1. 24 0.25
57 A 6 6 1. 24 0.25
58 A 1 1 1. 24 0.042
59 A 1 1 1. 24 0.042
60 A 3 3 1. 24 0.125
61 A 5 5 1. 22 0.227
62 A 6 5 1. 20 0.25
63 A 7 5 1. 24 0.208
64 A 8 5 1. 24 0.208
65 A 2 1 1. 18 0.056
66 A 2 1 1. 18 0.056
67 A 2 1 1. 16 0.062
68 A 1 0 1. 14 0.
69 A 2 1 1. 18 0.056
70 A 2 1 1. 18 0.056
71 A 2 1 1. 18 0.056
72 A 3 2 1. 20 0.1
73 A 3 2 1. 20 0.1
74 A 4 3 1. 18 0.167
75 A 3 2 1. 16 0.125
76 A 4 3 1. 20 0.15
77 A 3 2 1. 20 0.1
78 A 8 7 1. 20 0.35
79 A 6 5 1. 20 0.25
80 A 7 7 1. 20 0.35
81 A 5 4 1. 20 0.2
82 A 6 6 1. 20 0.3
83 A 4 3 1. 20 0.15
84 A 4 4 1. 20 0.2
85 A 4 3 1. 18 0.167
86 A 8 8 1. 16 0.5
87 A 5 4 1. 20 0.2
88 A 9 8 1. 20 0.4
89 A 9 8 1. 20 0.4
90 A 7 5 1. 20 0.25
91 A 8 8 1. 20 0.4
92 A 6 5 1. 20 0.25
93 A 5 5 1. 20 0.25
94 A 5 4 1. 20 0.2
95 A 5 5 1. 20 0.25
96 A 5 4 1. 20 0.2
97 A 5 5 1. 20 0.25
98 A 5 4 1. 20 0.2
99 A 9 8 1. 18 0.444
100 A 6 5 1. 16 0.312
101 A 9 8 1. 20 0.4
102 A 7 5 1. 20 0.25
103 A 9 8 1. 20 0.4
104 A 3 3 1. 20 0.15
105 A 5 5 1. 24 0.208
106 A 5 5 1. 24 0.208
107 A 5 5 1. 24 0.208
108 A 8 7 1. 24 0.292
109 A 8 8 1. 24 0.333
110 A 6 6 1. 24 0.25
111 A 6 5 1. 24 0.208
112 A 6 6 1. 24 0.25
113 A 4 4 1. 24 0.167
114 A 2 2 1. 24 0.083
115 A 2 2 1. 24 0.083
116 A 6 6 1. 24 0.25
117 A 5 5 1. 24 0.208
118 A 3 3 1. 24 0.125
119 A 6 6 1. 24 0.25
120 A 5 5 1. 24 0.208
121 A 1 1 1. 34 0.029
122 A 7 4 1. 27 0.148
123 A 2 2 1. 18 0.111
124 A 3 3 1. 18 0.167
125 A 3 3 1. 17 0.176
126 A 5 5 1. 18 0.278
127 A 6 6 1. 18 0.333
128 A 6 6 1. 17 0.353
129 A 2 2 1. 18 0.111
130 A 3 3 1. 18 0.167
131 A 3 3 1. 22 0.136
132 A 3 3 1. 24 0.125
133 A 3 3 1. 20 0.15
134 A 3 3 1. 20 0.15
135 A 3 3 1. 24 0.125
136 A 3 3 1. 26 0.115
137 A 3 3 1. 18 0.167
138 A 3 3 1. 18 0.167
139 A 3 3 1. 20 0.15
140 A 2 2 1. 36 0.056

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