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Chapter 2
detailed summary tables of results

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

2.1 List of integrals sorted by grade for each CAS

2.1.1 Rubi

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113 }

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, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 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 }

B grade: { 31, 42, 70, 71, 72 }

C grade: { 40, 69 }

F grade: { 57 }

2.1.3 Maple

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 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 }

B grade: { 35, 36, 40, 48, 51, 57, 69, 70, 71, 72 }

C grade: { 32, 42 }

F grade: { }

2.1.4 Maxima

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 39, 41, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 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 }

B grade: { 36, 38, 40, 42, 68, 69, 70, 71, 72, 87 }

C grade: { 10, 11, 47 }

F grade: { 32, 67 }

2.1.5 FriCAS

A grade: { 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 41, 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, 71, 72, 75, 76, 77, 78, 79, 80, 81, 82, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113 }

B grade: { 1, 9, 37, 39, 40, 42, 69, 70, 73, 74, 83, 84, 100 }

C grade: { }

F grade: { 32 }

2.1.6 Sympy

A grade: { 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 38, 41, 43, 44, 45, 46, 47, 48, 49, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 74, 75, 76, 77, 78, 79, 80, 81, 85, 86, 88, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113 }

B grade: { 7, 18, 37, 39, 50, 51, 56, 66, 73, 82, 83, 84, 100 }

C grade: { 71, 72 }

F grade: { 35, 36, 40, 42, 69, 70, 87, 90, 98 }

2.1.7 Giac

A grade: { 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 44, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 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 }

B grade: { 1, 25, 36, 43, 45, 70, 71, 72 }

C grade: { 47 }

F grade: { 32, 40, 42, 69 }

2.1.8 Mupad

A grade: { }

B grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 39, 41, 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, 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 }

C grade: { }

F grade: { 10, 11, 32, 38, 40, 42, 69 }

2.2 Detailed conclusion table per each integral for all CAS systems

Detailed conclusion table per each integral is given by table below. The elapsed time is in seconds. For failed result it is given as F(-1) if the failure was due to timeout. It is given as F(-2) if the failure was due to an exception being raised, which could indicate a bug in the system. If the failure was due to integral not being evaluated within the time limit, then it is given just an F.

In this table,the column normalized size is defined as \(\frac {\text {antiderivative leaf size}}{\text {optimal antiderivative leaf size}}\)

Problem 1 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B A B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 12 12 18 14 16 48 19 34 10
normalized size 1 1.00 1.50 1.17 1.33 4.00 1.58 2.83 0.83
time (sec) N/A 0.010 0.003 0.000 1.284 0.431 0.073 0.937 0.004
Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 13 13 13 12 15 19 14 15 12
normalized size 1 1.00 1.00 0.92 1.15 1.46 1.08 1.15 0.92
time (sec) N/A 0.005 0.004 0.007 1.287 0.419 0.108 0.805 0.158
Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 19 19 14 11 11 14 15 11 10
normalized size 1 1.00 0.74 0.58 0.58 0.74 0.79 0.58 0.53
time (sec) N/A 0.003 0.004 0.003 0.500 0.411 0.220 0.943 0.030
Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 2 2 2 3 2 2 2 2 2
normalized size 1 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00
time (sec) N/A 0.002 0.001 0.000 0.511 0.428 0.060 1.005 0.028
Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 9 9 9 7 6 6 5 6 6
normalized size 1 1.00 1.00 0.78 0.67 0.67 0.56 0.67 0.67
time (sec) N/A 0.006 0.001 0.000 0.583 0.420 0.084 1.058 0.017
Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 8 8 8 7 6 6 7 6 6
normalized size 1 1.00 1.00 0.88 0.75 0.75 0.88 0.75 0.75
time (sec) N/A 0.011 0.005 0.018 0.553 0.430 0.070 1.039 0.022
Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 13 13 13 10 9 9 22 9 9
normalized size 1 1.00 1.00 0.77 0.69 0.69 1.69 0.69 0.69
time (sec) N/A 0.002 0.002 0.000 0.526 0.404 0.190 1.040 0.029
Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 19 19 14 14 11 13 15 11 11
normalized size 1 1.00 0.74 0.74 0.58 0.68 0.79 0.58 0.58
time (sec) N/A 0.007 0.015 0.000 0.572 0.420 0.300 1.052 0.019
Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 8 8 8 7 6 14 8 6 6
normalized size 1 1.00 1.00 0.88 0.75 1.75 1.00 0.75 0.75
time (sec) N/A 0.012 0.005 0.033 0.519 0.417 0.070 1.192 0.053
Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A C A A A F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 4 4 4 4 15 3 3 3 -1
normalized size 1 1.00 1.00 1.00 3.75 0.75 0.75 0.75 -0.25
time (sec) N/A 0.012 0.006 0.014 0.829 0.429 0.628 0.996 0.000
Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A C A A A F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 2 2 2 3 13 2 2 2 -1
normalized size 1 1.00 1.00 1.50 6.50 1.00 1.00 1.00 -0.50
time (sec) N/A 0.011 0.015 0.001 0.696 0.423 0.604 0.976 0.000
Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 8 8 8 8 7 7 5 7 7
normalized size 1 1.00 1.00 1.00 0.88 0.88 0.62 0.88 0.88
time (sec) N/A 0.003 0.003 0.003 0.484 0.434 0.078 1.082 0.042
Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 7 7 7 7 6 6 5 6 6
normalized size 1 1.00 1.00 1.00 0.86 0.86 0.71 0.86 0.86
time (sec) N/A 0.020 0.006 0.004 0.466 0.408 0.086 0.882 0.153
Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 28 28 26 22 19 19 20 19 21
normalized size 1 1.00 0.93 0.79 0.68 0.68 0.71 0.68 0.75
time (sec) N/A 0.020 0.015 0.007 0.476 0.412 0.094 0.930 0.054
Problem 15 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 22 22 22 17 16 16 15 16 16
normalized size 1 1.00 1.00 0.77 0.73 0.73 0.68 0.73 0.73
time (sec) N/A 0.005 0.002 0.001 0.524 0.423 0.088 1.081 0.047
Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 8 8 8 7 6 6 5 6 6
normalized size 1 1.00 1.00 0.88 0.75 0.75 0.62 0.75 0.75
time (sec) N/A 0.007 0.002 0.002 0.602 0.435 0.059 1.077 0.017
Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 9 9 9 7 6 6 5 6 6
normalized size 1 1.00 1.00 0.78 0.67 0.67 0.56 0.67 0.67
time (sec) N/A 0.006 0.001 0.000 0.681 0.408 0.084 0.915 0.002
Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 13 13 13 10 9 9 22 9 9
normalized size 1 1.00 1.00 0.77 0.69 0.69 1.69 0.69 0.69
time (sec) N/A 0.002 0.000 0.000 0.514 0.411 0.191 1.100 0.002
Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 6 6 6 6 5 5 5 5 5
normalized size 1 1.00 1.00 1.00 0.83 0.83 0.83 0.83 0.83
time (sec) N/A 0.015 0.005 0.002 0.562 0.409 0.077 0.905 0.029
Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 9 9 9 6 5 5 7 5 5
normalized size 1 1.00 1.00 0.67 0.56 0.56 0.78 0.56 0.56
time (sec) N/A 0.000 0.001 0.003 0.506 0.413 0.056 0.954 0.044
Problem 21 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 10 10 10 9 8 8 7 8 8
normalized size 1 1.00 1.00 0.90 0.80 0.80 0.70 0.80 0.80
time (sec) N/A 0.003 0.005 0.018 0.517 0.427 0.123 1.036 0.072
Problem 22 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 8 8 8 8 7 7 7 7 7
normalized size 1 1.00 1.00 1.00 0.88 0.88 0.88 0.88 0.88
time (sec) N/A 0.003 0.001 0.003 0.479 0.411 0.056 1.098 0.016
Problem 23 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 10 10 19 8 7 7 8 7 7
normalized size 1 1.00 1.90 0.80 0.70 0.70 0.80 0.70 0.70
time (sec) N/A 0.027 0.022 0.014 0.453 0.438 1.959 1.186 0.187
Problem 24 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 23 23 16 13 15 15 34 15 12
normalized size 1 1.00 0.70 0.57 0.65 0.65 1.48 0.65 0.52
time (sec) N/A 0.004 0.005 0.003 0.549 0.408 0.934 0.961 0.031
Problem 25 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 13 13 25 10 17 17 17 19 9
normalized size 1 1.00 1.92 0.77 1.31 1.31 1.31 1.46 0.69
time (sec) N/A 0.003 0.004 0.000 1.410 0.396 0.124 1.031 0.158
Problem 26 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 18 18 18 14 13 13 15 13 13
normalized size 1 1.00 1.00 0.78 0.72 0.72 0.83 0.72 0.72
time (sec) N/A 0.020 0.008 0.007 1.326 0.418 0.114 1.108 0.107
Problem 27 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 31 31 31 20 19 76 22 19 19
normalized size 1 1.00 1.00 0.65 0.61 2.45 0.71 0.61 0.61
time (sec) N/A 0.035 0.009 0.016 1.328 0.441 0.162 0.943 0.231
Problem 28 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 8 8 8 9 8 11 8 8 8
normalized size 1 1.00 1.00 1.12 1.00 1.38 1.00 1.00 1.00
time (sec) N/A 0.034 0.006 0.007 0.557 0.428 0.094 1.102 0.184
Problem 29 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 12 12 12 13 12 12 10 12 12
normalized size 1 1.00 1.00 1.08 1.00 1.00 0.83 1.00 1.00
time (sec) N/A 0.006 0.003 0.018 0.474 0.406 0.090 0.958 0.072
Problem 30 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 20 20 20 17 16 16 15 16 16
normalized size 1 1.00 1.00 0.85 0.80 0.80 0.75 0.80 0.80
time (sec) N/A 0.019 0.003 0.004 0.643 0.443 0.599 0.893 0.208
Problem 31 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A B A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 49 49 260 41 40 40 46 40 52
normalized size 1 1.00 5.31 0.84 0.82 0.82 0.94 0.82 1.06
time (sec) N/A 0.036 0.118 0.004 1.156 0.417 0.169 0.894 0.138
Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A C F F A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 115 115 142 41 0 0 54 0 -1
normalized size 1 1.00 1.23 0.36 0.00 0.00 0.47 0.00 -0.01
time (sec) N/A 0.056 0.298 0.164 0.000 0.432 2.895 0.000 0.000
Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 22 22 22 17 16 16 20 16 16
normalized size 1 1.00 1.00 0.77 0.73 0.73 0.91 0.73 0.73
time (sec) N/A 0.010 0.017 0.029 0.507 0.436 0.308 1.269 0.212
Problem 34 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 23 23 16 13 15 15 34 15 12
normalized size 1 1.00 0.70 0.57 0.65 0.65 1.48 0.65 0.52
time (sec) N/A 0.004 0.005 0.000 0.473 0.409 0.951 0.928 0.002
Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B A A F A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 32 32 32 92 24 24 0 24 24
normalized size 1 1.00 1.00 2.88 0.75 0.75 0.00 0.75 0.75
time (sec) N/A 0.014 0.013 0.002 0.507 0.418 0.000 0.917 0.002
Problem 36 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B B A F B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 44 44 71 76 80 55 0 61 57
normalized size 1 1.00 1.61 1.73 1.82 1.25 0.00 1.39 1.30
time (sec) N/A 0.016 0.037 0.012 1.314 0.428 0.000 1.055 0.206
Problem 37 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 35 35 26 30 44 63 105 29 91
normalized size 1 1.00 0.74 0.86 1.26 1.80 3.00 0.83 2.60
time (sec) N/A 0.009 0.025 0.006 1.217 0.402 2.139 0.929 0.151
Problem 38 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A B A A A F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 75 75 41 70 113 44 190 90 -1
normalized size 1 1.00 0.55 0.93 1.51 0.59 2.53 1.20 -0.01
time (sec) N/A 0.013 0.020 0.003 0.532 0.414 8.746 1.156 0.000
Problem 39 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 35 35 26 30 44 63 105 29 91
normalized size 1 1.00 0.74 0.86 1.26 1.80 3.00 0.83 2.60
time (sec) N/A 0.008 0.003 0.000 1.320 0.413 2.155 1.070 0.002
Problem 40 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C B B B F F(-2) F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 51 51 134 262 122 129 0 0 -1
normalized size 1 1.00 2.63 5.14 2.39 2.53 0.00 0.00 -0.02
time (sec) N/A 0.029 0.028 0.032 1.459 0.454 0.000 0.000 0.000
Problem 41 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 14 14 14 11 10 10 10 10 10
normalized size 1 1.00 1.00 0.79 0.71 0.71 0.71 0.71 0.71
time (sec) N/A 0.006 0.002 0.000 0.603 0.449 0.062 0.941 0.033
Problem 42 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A B C B B F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 49 57 99 149 116 244 0 0 -1
normalized size 1 1.16 2.02 3.04 2.37 4.98 0.00 0.00 -0.02
time (sec) N/A 0.085 0.112 0.193 1.357 0.550 0.000 0.000 0.000
Problem 43 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 9 9 6 5 9 9 3 30 4
normalized size 1 1.00 0.67 0.56 1.00 1.00 0.33 3.33 0.44
time (sec) N/A 0.008 0.004 0.015 0.549 0.413 0.192 0.820 0.182
Problem 44 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 11 11 7 7 6 6 5 6 6
normalized size 1 1.00 0.64 0.64 0.55 0.55 0.45 0.55 0.55
time (sec) N/A 0.007 0.001 0.000 0.570 0.406 0.078 1.092 0.017
Problem 45 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 9 9 9 9 8 8 5 30 8
normalized size 1 1.00 1.00 1.00 0.89 0.89 0.56 3.33 0.89
time (sec) N/A 0.026 0.024 0.003 0.462 0.396 0.087 0.917 0.086
Problem 46 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 7 7 7 7 6 6 5 6 6
normalized size 1 1.00 1.00 1.00 0.86 0.86 0.71 0.86 0.86
time (sec) N/A 0.032 0.006 0.003 0.480 0.411 0.087 0.827 0.145
Problem 47 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A C A A C B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 11 11 11 8 9 7 8 9 7
normalized size 1 1.00 1.00 0.73 0.82 0.64 0.73 0.82 0.64
time (sec) N/A 0.002 0.002 0.001 0.545 0.420 0.197 1.048 0.019
Problem 48 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 2 2 2 8 2 2 2 2 2
normalized size 1 1.00 1.00 4.00 1.00 1.00 1.00 1.00 1.00
time (sec) N/A 0.010 0.004 0.003 0.714 0.399 0.711 1.012 0.008
Problem 49 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 41 41 40 35 34 34 41 35 46
normalized size 1 1.00 0.98 0.85 0.83 0.83 1.00 0.85 1.12
time (sec) N/A 0.022 0.009 0.001 1.088 0.407 0.133 0.889 0.106
Problem 50 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 47 73 75 66 65 65 83 67 88
normalized size 1 1.55 1.60 1.40 1.38 1.38 1.77 1.43 1.87
time (sec) N/A 0.105 0.013 0.008 1.281 0.428 0.251 1.013 0.090
Problem 51 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A B A A B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 21 21 21 44 39 27 70 41 21
normalized size 1 1.00 1.00 2.10 1.86 1.29 3.33 1.95 1.00
time (sec) N/A 0.009 0.004 0.009 0.510 0.400 0.304 1.143 0.240
Problem 52 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 17 17 17 14 13 13 12 13 9
normalized size 1 1.00 1.00 0.82 0.76 0.76 0.71 0.76 0.53
time (sec) N/A 0.004 0.001 0.005 0.609 0.414 0.089 1.067 0.035
Problem 53 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 40 40 29 34 33 24 32 38 24
normalized size 1 1.00 0.72 0.85 0.82 0.60 0.80 0.95 0.60
time (sec) N/A 0.024 0.013 0.001 1.288 0.445 0.341 1.087 0.002
Problem 54 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 7 7 7 8 7 7 5 7 7
normalized size 1 1.00 1.00 1.14 1.00 1.00 0.71 1.00 1.00
time (sec) N/A 0.001 0.001 0.003 0.534 0.391 0.077 1.050 0.021
Problem 55 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 8 8 8 9 8 8 5 8 8
normalized size 1 1.00 1.00 1.12 1.00 1.00 0.62 1.00 1.00
time (sec) N/A 0.044 0.015 0.135 0.540 0.392 0.088 0.961 0.264
Problem 56 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 3 3 3 4 3 3 15 3 3
normalized size 1 1.00 1.00 1.33 1.00 1.00 5.00 1.00 1.00
time (sec) N/A 0.020 0.012 0.005 1.293 0.414 0.138 0.909 0.307
Problem 57 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A F B A A A A B
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 2 2 0 9 3 2 2 3 2
normalized size 1 1.00 0.00 4.50 1.50 1.00 1.00 1.50 1.00
time (sec) N/A 0.002 0.002 0.000 0.646 0.412 0.448 0.892 0.011
Problem 58 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 14 14 14 15 14 14 15 14 14
normalized size 1 1.00 1.00 1.07 1.00 1.00 1.07 1.00 1.00
time (sec) N/A 0.021 0.005 0.026 0.576 0.408 0.181 1.108 0.059
Problem 59 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 17 17 13 16 11 14 8 11 15
normalized size 1 1.00 0.76 0.94 0.65 0.82 0.47 0.65 0.88
time (sec) N/A 0.031 0.010 0.003 0.559 0.421 0.085 0.987 0.058
Problem 60 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 38 38 29 29 24 24 26 24 28
normalized size 1 1.00 0.76 0.76 0.63 0.63 0.68 0.63 0.74
time (sec) N/A 0.033 0.028 0.004 0.619 0.402 0.095 0.952 0.157
Problem 61 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 7 7 7 8 7 7 7 7 7
normalized size 1 1.00 1.00 1.14 1.00 1.00 1.00 1.00 1.00
time (sec) N/A 0.009 0.002 0.000 0.506 0.432 0.175 0.921 0.018
Problem 62 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 22 22 22 17 16 16 20 16 16
normalized size 1 1.00 1.00 0.77 0.73 0.73 0.91 0.73 0.73
time (sec) N/A 0.012 0.013 0.001 0.616 0.422 0.308 0.916 0.002
Problem 63 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 7 7 7 8 7 7 7 7 7
normalized size 1 1.00 1.00 1.14 1.00 1.00 1.00 1.00 1.00
time (sec) N/A 0.009 0.002 0.000 0.660 0.426 0.175 1.008 0.002
Problem 64 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 28 28 28 23 17 22 22 22 17
normalized size 1 1.00 1.00 0.82 0.61 0.79 0.79 0.79 0.61
time (sec) N/A 0.010 0.001 0.003 0.595 0.412 0.102 0.946 0.032
Problem 65 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 11 11 11 10 9 15 8 9 9
normalized size 1 1.00 1.00 0.91 0.82 1.36 0.73 0.82 0.82
time (sec) N/A 0.012 0.004 0.031 0.532 0.449 0.570 0.962 0.040
Problem 66 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 3 3 3 4 3 3 15 3 3
normalized size 1 1.00 1.00 1.33 1.00 1.00 5.00 1.00 1.00
time (sec) N/A 0.019 0.010 0.001 1.364 0.436 0.141 0.915 0.002
Problem 67 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A F A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 3 3 3 4 0 3 3 3 43
normalized size 1 1.00 1.00 1.33 0.00 1.00 1.00 1.00 14.33
time (sec) N/A 0.047 0.059 0.007 0.000 0.462 0.410 1.132 3.099
Problem 68 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A B A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 16 16 16 21 53 15 12 21 13
normalized size 1 1.00 1.00 1.31 3.31 0.94 0.75 1.31 0.81
time (sec) N/A 0.041 0.022 0.050 1.279 0.420 0.150 1.082 0.029
Problem 69 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A C B B B F F(-2) F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 53 53 127 262 123 128 0 0 -1
normalized size 1 1.00 2.40 4.94 2.32 2.42 0.00 0.00 -0.02
time (sec) N/A 0.065 0.057 0.008 1.495 0.436 0.000 0.000 0.000
Problem 70 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A B B B B F(-1) B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 16 16 35 127 69 67 0 83 360
normalized size 1 1.00 2.19 7.94 4.31 4.19 0.00 5.19 22.50
time (sec) N/A 0.086 0.098 0.074 1.513 0.488 0.000 1.232 0.499
Problem 71 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A B B B A C B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 16 16 35 121 68 26 422 82 352
normalized size 1 1.00 2.19 7.56 4.25 1.62 26.38 5.12 22.00
time (sec) N/A 0.124 0.019 0.010 1.414 0.431 1.927 1.158 0.236
Problem 72 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A B B B A C B B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 16 16 35 121 65 26 422 79 352
normalized size 1 1.00 2.19 7.56 4.06 1.62 26.38 4.94 22.00
time (sec) N/A 0.019 0.013 0.009 1.422 0.435 1.897 1.041 0.058
Problem 73 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 35 35 26 30 44 63 105 29 91
normalized size 1 1.00 0.74 0.86 1.26 1.80 3.00 0.83 2.60
time (sec) N/A 0.008 0.021 0.001 1.269 0.412 2.070 1.051 0.002
Problem 74 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 14 14 18 13 12 26 19 12 12
normalized size 1 1.00 1.29 0.93 0.86 1.86 1.36 0.86 0.86
time (sec) N/A 0.009 0.007 0.024 1.332 0.452 0.065 0.989 0.072
Problem 75 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 13 13 13 12 11 11 8 11 11
normalized size 1 1.00 1.00 0.92 0.85 0.85 0.62 0.85 0.85
time (sec) N/A 0.005 0.004 0.002 1.471 0.394 0.086 0.889 0.028
Problem 76 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 7 7 7 7 6 6 5 6 6
normalized size 1 1.00 1.00 1.00 0.86 0.86 0.71 0.86 0.86
time (sec) N/A 0.030 0.005 0.001 0.668 0.414 0.087 1.079 0.002
Problem 77 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 24 24 20 24 23 23 20 30 24
normalized size 1 1.00 0.83 1.00 0.96 0.96 0.83 1.25 1.00
time (sec) N/A 0.357 0.073 0.005 1.800 0.434 0.120 1.058 0.223
Problem 78 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 9 9 9 9 8 8 7 8 8
normalized size 1 1.00 1.00 1.00 0.89 0.89 0.78 0.89 0.89
time (sec) N/A 0.003 0.003 0.001 0.469 0.406 0.075 1.147 0.032
Problem 79 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 25 25 25 20 24 19 19 19 19
normalized size 1 1.00 1.00 0.80 0.96 0.76 0.76 0.76 0.76
time (sec) N/A 0.045 0.002 0.004 0.567 0.428 0.095 0.958 0.227
Problem 80 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 8 8 8 7 6 6 5 6 6
normalized size 1 1.00 1.00 0.88 0.75 0.75 0.62 0.75 0.75
time (sec) N/A 0.003 0.003 0.001 1.416 0.400 0.092 1.057 0.056
Problem 81 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 16 16 16 13 12 12 10 12 12
normalized size 1 1.00 1.00 0.81 0.75 0.75 0.62 0.75 0.75
time (sec) N/A 0.008 0.004 0.004 1.470 0.412 0.095 1.029 0.022
Problem 82 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 14 14 14 17 16 20 36 16 10
normalized size 1 1.00 1.00 1.21 1.14 1.43 2.57 1.14 0.71
time (sec) N/A 0.015 0.002 0.022 1.362 0.440 0.389 0.919 0.178
Problem 83 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 35 35 26 30 44 63 105 29 91
normalized size 1 1.00 0.74 0.86 1.26 1.80 3.00 0.83 2.60
time (sec) N/A 0.011 0.003 0.000 1.396 0.394 2.086 1.096 0.002
Problem 84 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 17 17 32 16 15 45 34 21 37
normalized size 1 1.00 1.88 0.94 0.88 2.65 2.00 1.24 2.18
time (sec) N/A 0.005 0.008 0.005 1.533 0.428 0.514 1.074 0.163
Problem 85 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 19 19 14 14 11 13 15 11 11
normalized size 1 1.00 0.74 0.74 0.58 0.68 0.79 0.58 0.58
time (sec) N/A 0.008 0.015 0.001 0.815 0.407 0.297 0.948 0.002
Problem 86 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 2 2 2 3 2 2 2 3 2
normalized size 1 1.00 1.00 1.50 1.00 1.00 1.00 1.50 1.00
time (sec) N/A 0.000 0.000 0.002 0.513 0.385 0.055 0.951 0.006
Problem 87 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A B A F A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 45 45 73 31 256 71 0 33 32
normalized size 1 1.00 1.62 0.69 5.69 1.58 0.00 0.73 0.71
time (sec) N/A 0.123 0.193 0.274 1.984 0.473 0.000 1.043 0.715
Problem 88 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 14 14 14 11 10 10 10 16 10
normalized size 1 1.00 1.00 0.79 0.71 0.71 0.71 1.14 0.71
time (sec) N/A 0.006 0.002 0.000 0.672 0.431 0.066 0.910 0.002
Problem 89 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 17 17 14 11 11 10 14 11 10
normalized size 1 1.00 0.82 0.65 0.65 0.59 0.82 0.65 0.59
time (sec) N/A 0.003 0.004 0.006 0.520 0.409 0.215 0.946 0.024
Problem 90 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A F A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 23 23 25 20 19 27 0 27 27
normalized size 1 1.00 1.09 0.87 0.83 1.17 0.00 1.17 1.17
time (sec) N/A 0.010 0.005 0.005 1.272 0.410 0.000 1.023 0.078
Problem 91 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 8 8 8 7 6 10 5 6 6
normalized size 1 1.00 1.00 0.88 0.75 1.25 0.62 0.75 0.75
time (sec) N/A 0.013 0.001 0.004 0.513 0.424 0.062 0.988 0.027
Problem 92 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 6 6 6 6 5 5 5 5 5
normalized size 1 1.00 1.00 1.00 0.83 0.83 0.83 0.83 0.83
time (sec) N/A 0.016 0.005 0.001 0.554 0.410 0.078 0.970 0.002
Problem 93 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 12 12 12 11 10 10 8 10 10
normalized size 1 1.00 1.00 0.92 0.83 0.83 0.67 0.83 0.83
time (sec) N/A 0.026 0.008 0.004 0.687 0.411 0.088 0.981 0.053
Problem 94 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 12 12 8 9 10 10 7 8 6
normalized size 1 1.00 0.67 0.75 0.83 0.83 0.58 0.67 0.50
time (sec) N/A 0.009 0.009 0.017 0.763 0.415 0.337 0.874 0.262
Problem 95 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 8 8 8 7 6 6 7 6 6
normalized size 1 1.00 1.00 0.88 0.75 0.75 0.88 0.75 0.75
time (sec) N/A 0.012 0.005 0.000 0.669 0.434 0.069 0.857 0.002
Problem 96 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 17 17 17 14 13 13 12 13 9
normalized size 1 1.00 1.00 0.82 0.76 0.76 0.71 0.76 0.53
time (sec) N/A 0.004 0.001 0.001 0.473 0.414 0.086 0.974 0.002
Problem 97 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 8 8 8 7 6 6 5 6 6
normalized size 1 1.00 1.00 0.88 0.75 0.75 0.62 0.75 0.75
time (sec) N/A 0.007 0.001 0.000 0.545 0.435 0.059 0.902 0.002
Problem 98 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A F A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 24 24 27 21 22 32 0 20 20
normalized size 1 1.00 1.12 0.88 0.92 1.33 0.00 0.83 0.83
time (sec) N/A 0.009 0.032 0.007 1.202 0.407 0.000 1.027 0.268
Problem 99 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 20 20 20 14 13 19 15 13 13
normalized size 1 1.00 1.00 0.70 0.65 0.95 0.75 0.65 0.65
time (sec) N/A 0.023 0.006 0.002 1.341 0.416 0.116 0.977 0.091
Problem 100 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A B B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 35 35 26 30 44 63 105 29 91
normalized size 1 1.00 0.74 0.86 1.26 1.80 3.00 0.83 2.60
time (sec) N/A 0.009 0.003 0.000 1.335 0.417 2.142 0.905 0.002
Problem 101 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 20 20 18 15 14 14 24 14 14
normalized size 1 1.00 0.90 0.75 0.70 0.70 1.20 0.70 0.70
time (sec) N/A 0.024 0.008 0.011 1.296 0.423 0.122 0.921 0.067
Problem 102 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 33 33 33 27 26 32 31 26 26
normalized size 1 1.00 1.00 0.82 0.79 0.97 0.94 0.79 0.79
time (sec) N/A 0.011 0.013 0.000 1.272 0.419 0.137 1.122 0.002
Problem 103 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 21 21 21 20 19 31 15 19 19
normalized size 1 1.00 1.00 0.95 0.90 1.48 0.71 0.90 0.90
time (sec) N/A 0.016 0.000 0.002 0.496 0.407 0.055 0.962 0.002
Problem 104 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 26 26 26 25 24 40 19 23 24
normalized size 1 1.00 1.00 0.96 0.92 1.54 0.73 0.88 0.92
time (sec) N/A 0.019 0.000 0.003 0.684 0.392 0.057 0.919 0.003
Problem 105 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 27 27 27 26 25 43 22 25 25
normalized size 1 1.00 1.00 0.96 0.93 1.59 0.81 0.93 0.93
time (sec) N/A 0.023 0.000 0.000 0.786 0.393 0.058 0.879 0.002
Problem 106 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 32 32 32 31 30 52 26 29 30
normalized size 1 1.00 1.00 0.97 0.94 1.62 0.81 0.91 0.94
time (sec) N/A 0.026 0.000 0.001 0.543 0.399 0.060 0.889 0.002
Problem 107 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 24 24 25 24 23 41 20 23 23
normalized size 1 1.00 1.04 1.00 0.96 1.71 0.83 0.96 0.96
time (sec) N/A 0.027 0.000 0.000 0.606 0.414 0.057 0.942 0.002
Problem 108 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 29 29 30 29 28 50 24 27 28
normalized size 1 1.00 1.03 1.00 0.97 1.72 0.83 0.93 0.97
time (sec) N/A 0.032 0.000 0.000 0.576 0.409 0.059 0.965 0.002
Problem 109 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 19 19 19 18 17 17 17 17 18
normalized size 1 1.00 1.00 0.95 0.89 0.89 0.89 0.89 0.95
time (sec) N/A 0.009 0.000 0.002 0.595 0.386 0.055 0.984 0.002
Problem 110 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 24 24 24 23 22 22 20 21 23
normalized size 1 1.00 1.00 0.96 0.92 0.92 0.83 0.88 0.96
time (sec) N/A 0.011 0.000 0.002 0.732 0.403 0.057 0.977 0.004
Problem 111 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 25 25 25 24 23 42 24 23 24
normalized size 1 1.00 1.00 0.96 0.92 1.68 0.96 0.92 0.96
time (sec) N/A 0.013 0.000 0.001 0.475 0.400 0.058 0.918 0.002
Problem 112 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 30 30 30 29 28 50 27 27 29
normalized size 1 1.00 1.00 0.97 0.93 1.67 0.90 0.90 0.97
time (sec) N/A 0.015 0.000 0.001 0.551 0.392 0.058 1.075 0.003
Problem 113 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad
grade A A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 27 27 28 27 26 26 24 25 26
normalized size 1 1.00 1.04 1.00 0.96 0.96 0.89 0.93 0.96
time (sec) N/A 0.006 0.000 0.001 0.649 0.404 0.059 1.284 0.003

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 [50] had the largest ratio of [.8571]

Table 2.1:Rubi specific breakdown of results for each integral
# grade
number of
steps
used
number of
unique
rules
normalized
antiderivative
leaf size
integrand
leaf size
\(\frac {\text {number of rules}}{\text {integrand leaf size}}\)
1 A 3 2 1.00 4 0.500
2 A 3 2 1.00 11 0.182
3 A 2 1 1.00 11 0.091
4 A 1 1 1.00 2 0.500
5 A 1 1 1.00 7 0.143
6 A 2 2 1.00 7 0.286
7 A 1 1 1.00 11 0.091
8 A 1 1 1.00 6 0.167
9 A 2 2 1.00 7 0.286
10 A 2 2 1.00 4 0.500
11 A 1 1 1.00 6 0.167
12 A 3 2 1.00 6 0.333
13 A 4 2 1.00 16 0.125
14 A 5 3 1.00 7 0.429
15 A 3 1 1.00 14 0.071
16 A 2 2 1.00 5 0.400
17 A 1 1 1.00 7 0.143
18 A 1 1 1.00 11 0.091
19 A 2 2 1.00 11 0.182
20 A 1 1 1.00 5 0.200
21 A 1 1 1.00 6 0.167
22 A 2 2 1.00 9 0.222
23 A 3 3 1.00 14 0.214
24 A 2 1 1.00 9 0.111
25 A 3 3 1.00 7 0.429
26 A 2 2 1.00 15 0.133
27 A 2 2 1.00 17 0.118
28 A 3 3 1.00 13 0.231
29 A 1 1 1.00 5 0.200
30 A 3 3 1.00 8 0.375
31 A 7 7 1.00 11 0.636
32 A 3 2 1.00 12 0.167
33 A 3 3 1.00 6 0.500
34 A 2 1 1.00 9 0.111
35 A 4 3 1.00 13 0.231
36 A 4 4 1.00 15 0.267
37 A 3 2 1.00 15 0.133
38 A 6 3 1.00 13 0.231
39 A 3 2 1.00 15 0.133
40 A 5 5 1.00 29 0.172
41 A 2 2 1.00 4 0.500
42 A 6 6 1.16 19 0.316
43 A 1 1 1.00 6 0.167
44 A 2 2 1.00 5 0.400
45 A 1 1 1.00 10 0.100
46 A 5 3 1.00 13 0.231
47 A 1 1 1.00 5 0.200
48 A 1 1 1.00 7 0.143
49 A 6 6 1.00 9 0.667
50 A 10 6 1.55 7 0.857
51 A 1 1 1.00 27 0.037
52 A 1 1 1.00 4 0.250
53 A 4 3 1.00 6 0.500
54 A 2 2 1.00 10 0.200
55 A 7 4 1.00 9 0.444
56 A 2 1 1.00 12 0.083
57 A 1 1 1.00 4 0.250
58 A 6 4 1.00 7 0.571
59 A 4 3 1.00 11 0.273
60 A 6 3 1.00 9 0.333
61 A 2 2 1.00 4 0.500
62 A 3 3 1.00 6 0.500
63 A 2 2 1.00 4 0.500
64 A 2 2 1.00 6 0.333
65 A 2 1 1.00 9 0.111
66 A 2 1 1.00 12 0.083
67 A 2 2 1.00 20 0.100
68 A 2 2 1.00 10 0.200
69 A 6 6 1.00 30 0.200
70 A 5 5 1.00 39 0.128
71 A 6 5 1.00 45 0.111
72 A 4 4 1.00 31 0.129
73 A 3 2 1.00 15 0.133
74 A 3 2 1.00 4 0.500
75 A 3 2 1.00 11 0.182
76 A 5 3 1.00 13 0.231
77 A 10 6 1.00 33 0.182
78 A 1 1 1.00 7 0.143
79 A 5 5 1.00 8 0.625
80 A 2 2 1.00 9 0.222
81 A 3 3 1.00 11 0.273
82 A 3 3 1.00 8 0.375
83 A 3 2 1.00 15 0.133
84 A 2 2 1.00 16 0.125
85 A 1 1 1.00 6 0.167
86 A 1 1 1.00 3 0.333
87 A 4 2 1.00 17 0.118
88 A 2 2 1.00 4 0.500
89 A 2 1 1.00 11 0.091
90 A 3 3 1.00 14 0.214
91 A 2 2 1.00 7 0.286
92 A 2 2 1.00 11 0.182
93 A 3 2 1.00 13 0.154
94 A 1 1 1.00 8 0.125
95 A 2 2 1.00 7 0.286
96 A 1 1 1.00 4 0.250
97 A 2 2 1.00 5 0.400
98 A 3 3 1.00 17 0.176
99 A 3 3 1.00 16 0.188
100 A 3 2 1.00 15 0.133
101 A 3 3 1.00 15 0.200
102 A 3 3 1.00 8 0.375
103 A 1 1 1.00 20 0.050
104 A 1 1 1.00 25 0.040
105 A 1 1 1.00 26 0.038
106 A 1 1 1.00 31 0.032
107 A 1 1 1.00 24 0.042
108 A 1 1 1.00 29 0.034
109 A 1 1 1.00 18 0.056
110 A 1 1 1.00 23 0.043
111 A 1 1 1.00 24 0.042
112 A 1 1 1.00 29 0.034
113 A 1 1 1.00 27 0.037

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