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 }

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, 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, 68, 70, 71, 72, 87 }

C grade: { 10, 11, 47 }

F grade: { 27, 32, 40, 42, 67, 69 }

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, 36, 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, 43, 45, 70, 71, 72 }

C grade: { 47 }

F grade: { 32, 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









grade A A A A A B A B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 12 12 18 14 22 126 19 46
normalized size 1 1. 1.5 1.17 1.83 10.5 1.58 3.83
time (sec) N/A 0.01 0.003 0. 1.414 1.655 0.066 1.063


















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 13 13 13 12 20 53 14 20
normalized size 1 1. 1. 0.92 1.54 4.08 1.08 1.54
time (sec) N/A 0.005 0.004 0.005 1.413 1.506 0.105 1.079


















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 19 19 14 11 15 38 15 15
normalized size 1 1. 0.74 0.58 0.79 2. 0.79 0.79
time (sec) N/A 0.003 0.003 0.001 0.936 1.5 0.223 1.09


















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 2 2 2 3 3 11 2 3
normalized size 1 1. 1. 1.5 1.5 5.5 1. 1.5
time (sec) N/A 0.002 0.001 0. 0.927 1.582 0.053 1.093


















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 9 9 9 7 8 18 5 8
normalized size 1 1. 1. 0.78 0.89 2. 0.56 0.89
time (sec) N/A 0.006 0.001 0. 0.927 1.633 0.079 1.079


















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 8 8 8 7 8 19 7 8
normalized size 1 1. 1. 0.88 1. 2.38 0.88 1.
time (sec) N/A 0.011 0.004 0.005 0.936 1.599 0.066 1.066


















Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 13 13 13 10 12 28 22 12
normalized size 1 1. 1. 0.77 0.92 2.15 1.69 0.92
time (sec) N/A 0.002 0.002 0.002 0.925 1.568 0.19 1.085


















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 19 19 14 14 15 46 15 15
normalized size 1 1. 0.74 0.74 0.79 2.42 0.79 0.79
time (sec) N/A 0.007 0.014 0. 0.934 1.641 0.298 1.098


















Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A B A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 8 8 8 7 8 39 8 8
normalized size 1 1. 1. 0.88 1. 4.88 1. 1.
time (sec) N/A 0.012 0.004 0.013 0.94 1.607 0.064 1.072


















Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A C A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 4 4 4 4 20 26 3 4
normalized size 1 1. 1. 1. 5. 6.5 0.75 1.
time (sec) N/A 0.012 0.005 0.004 1.065 1.931 0.641 1.06


















Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A C A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 2 2 2 3 18 23 2 3
normalized size 1 1. 1. 1.5 9. 11.5 1. 1.5
time (sec) N/A 0.011 0.013 0.001 1.055 1.86 0.54 1.117


















Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 8 8 8 8 9 20 5 9
normalized size 1 1. 1. 1. 1.12 2.5 0.62 1.12
time (sec) N/A 0.003 0.003 0.002 0.946 1.82 0.072 1.07


















Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 7 7 7 7 8 15 5 8
normalized size 1 1. 1. 1. 1.14 2.14 0.71 1.14
time (sec) N/A 0.02 0.005 0.002 0.935 1.886 0.08 1.07


















Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 28 28 26 22 26 53 20 26
normalized size 1 1. 0.93 0.79 0.93 1.89 0.71 0.93
time (sec) N/A 0.02 0.014 0.002 0.928 1.802 0.087 1.098


















Problem 15 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 22 22 22 17 22 42 15 22
normalized size 1 1. 1. 0.77 1. 1.91 0.68 1.
time (sec) N/A 0.005 0.002 0.001 0.933 1.878 0.083 1.074


















Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 8 8 8 7 8 20 5 8
normalized size 1 1. 1. 0.88 1. 2.5 0.62 1.
time (sec) N/A 0.007 0.001 0.001 0.913 1.915 0.054 1.081


















Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 9 9 9 7 8 18 5 8
normalized size 1 1. 1. 0.78 0.89 2. 0.56 0.89
time (sec) N/A 0.006 0.001 0. 0.925 1.913 0.075 1.07


















Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 13 13 13 10 12 28 22 12
normalized size 1 1. 1. 0.77 0.92 2.15 1.69 0.92
time (sec) N/A 0.002 0. 0. 0.93 1.701 0.18 1.06


















Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 6 6 6 6 7 19 5 7
normalized size 1 1. 1. 1. 1.17 3.17 0.83 1.17
time (sec) N/A 0.015 0.004 0.001 0.922 1.682 0.085 1.094


















Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 9 9 9 6 7 18 7 7
normalized size 1 1. 1. 0.67 0.78 2. 0.78 0.78
time (sec) N/A 0. 0.001 0.001 0.927 1.739 0.052 1.081


















Problem 21 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 9 11 24 7 11
normalized size 1 1. 1. 0.9 1.1 2.4 0.7 1.1
time (sec) N/A 0.003 0.004 0.005 0.925 1.88 0.112 1.101


















Problem 22 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 8 8 8 8 9 18 7 9
normalized size 1 1. 1. 1. 1.12 2.25 0.88 1.12
time (sec) N/A 0.003 0.001 0.002 0.933 1.862 0.06 1.071


















Problem 23 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 19 8 9 23 8 9
normalized size 1 1. 1.9 0.8 0.9 2.3 0.8 0.9
time (sec) N/A 0.027 0.02 0.004 0.923 1.89 2.086 1.081


















Problem 24 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 23 23 16 13 20 46 34 20
normalized size 1 1. 0.7 0.57 0.87 2. 1.48 0.87
time (sec) N/A 0.004 0.004 0.003 0.925 1.819 0.848 1.1


















Problem 25 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 13 13 25 10 23 68 17 26
normalized size 1 1. 1.92 0.77 1.77 5.23 1.31 2.
time (sec) N/A 0.003 0.004 0. 1.418 1.883 0.115 1.087


















Problem 26 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 14 18 50 15 18
normalized size 1 1. 1. 0.78 1. 2.78 0.83 1.
time (sec) N/A 0.02 0.007 0.004 1.412 1.949 0.107 1.093


















Problem 27 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F(-2) A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 31 31 31 20 0 186 22 26
normalized size 1 1. 1. 0.65 0. 6. 0.71 0.84
time (sec) N/A 0.035 0.008 0.007 0. 1.932 0.151 1.085


















Problem 28 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 8 8 8 9 11 30 7 11
normalized size 1 1. 1. 1.12 1.38 3.75 0.88 1.38
time (sec) N/A 0.034 0.005 0.001 0.932 1.947 0.096 1.082


















Problem 29 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 12 12 12 13 16 27 8 16
normalized size 1 1. 1. 1.08 1.33 2.25 0.67 1.33
time (sec) N/A 0.006 0.004 0.01 0.931 1.967 0.087 1.08


















Problem 30 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 46 15 22
normalized size 1 1. 1. 0.85 1.1 2.3 0.75 1.1
time (sec) N/A 0.019 0.002 0. 0.933 1.99 0.548 1.073


















Problem 31 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A B A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 49 49 260 41 54 124 46 54
normalized size 1 1. 5.31 0.84 1.1 2.53 0.94 1.1
time (sec) N/A 0.036 0.109 0. 1.407 1.843 0.159 1.096


















Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A C F F A F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 115 115 142 41 0 0 54 0
normalized size 1 1. 1.23 0.36 0. 0. 0.47 0.
time (sec) N/A 0.056 0.293 0.089 0. 0. 5.489 0.


















Problem 33 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 22 22 22 17 22 55 20 22
normalized size 1 1. 1. 0.77 1. 2.5 0.91 1.
time (sec) N/A 0.01 0.014 0.01 0.931 1.83 0.295 1.079


















Problem 34 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 23 23 16 13 20 46 34 20
normalized size 1 1. 0.7 0.57 0.87 2. 1.48 0.87
time (sec) N/A 0.004 0.003 0. 0.927 1.79 0.84 1.092


















Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A B A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 32 32 32 92 32 76 0 32
normalized size 1 1. 1. 2.88 1. 2.38 0. 1.
time (sec) N/A 0.014 0.013 0. 0.93 1.731 0. 1.076


















Problem 36 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A B B A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 44 44 71 75 108 151 0 82
normalized size 1 1. 1.61 1.7 2.45 3.43 0. 1.86
time (sec) N/A 0.016 0.036 0.01 1.406 1.831 0. 1.105


















Problem 37 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A B B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 35 35 26 30 59 146 105 39
normalized size 1 1. 0.74 0.86 1.69 4.17 3. 1.11
time (sec) N/A 0.009 0.024 0.004 1.425 1.835 2.606 1.083


















Problem 38 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A B A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 75 75 41 70 153 140 190 111
normalized size 1 1. 0.55 0.93 2.04 1.87 2.53 1.48
time (sec) N/A 0.013 0.019 0.003 0.943 1.796 11.904 1.278


















Problem 39 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A B B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 35 35 26 30 59 146 105 39
normalized size 1 1. 0.74 0.86 1.69 4.17 3. 1.11
time (sec) N/A 0.008 0.004 0. 1.419 1.87 2.568 1.099


















Problem 40 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C B F(-2) B F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 51 51 134 262 0 285 0 0
normalized size 1 1. 2.63 5.14 0. 5.59 0. 0.
time (sec) N/A 0.029 0.029 0.022 0. 2.025 0. 0.


















Problem 41 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 14 14 14 11 14 38 10 14
normalized size 1 1. 1. 0.79 1. 2.71 0.71 1.
time (sec) N/A 0.006 0.002 0. 0.925 1.985 0.057 1.137


















Problem 42 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A B C F(-2) B F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 49 57 99 149 0 635 0 0
normalized size 1 1.16 2.02 3.04 0. 12.96 0. 0.
time (sec) N/A 0.085 0.106 0.096 0. 2.998 0. 0.


















Problem 43 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 9 9 6 5 12 28 3 41
normalized size 1 1. 0.67 0.56 1.33 3.11 0.33 4.56
time (sec) N/A 0.008 0.004 0. 0.918 1.945 0.179 1.109


















Problem 44 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 11 11 7 7 8 18 5 8
normalized size 1 1. 0.64 0.64 0.73 1.64 0.45 0.73
time (sec) N/A 0.007 0.001 0. 0.929 1.931 0.074 1.092


















Problem 45 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 9 9 9 9 11 18 5 26
normalized size 1 1. 1. 1. 1.22 2. 0.56 2.89
time (sec) N/A 0.026 0.023 0.002 0.942 1.746 0.085 1.104


















Problem 46 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 8 15 5 8
normalized size 1 1. 1. 1. 1.14 2.14 0.71 1.14
time (sec) N/A 0.032 0.005 0.001 0.938 1.975 0.081 1.067


















Problem 47 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A C A A C
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 11 11 11 8 12 30 8 12
normalized size 1 1. 1. 0.73 1.09 2.73 0.73 1.09
time (sec) N/A 0.002 0.002 0. 0.927 2.029 0.231 1.085


















Problem 48 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 2 2 2 8 3 9 2 3
normalized size 1 1. 1. 4. 1.5 4.5 1. 1.5
time (sec) N/A 0.01 0.004 0.002 1.013 1.869 0.736 1.076


















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 41 41 40 35 46 112 41 47
normalized size 1 1. 0.98 0.85 1.12 2.73 1. 1.15
time (sec) N/A 0.022 0.009 0. 1.405 1.981 0.122 1.072


















Problem 50 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 47 73 75 66 88 230 83 90
normalized size 1 1.55 1.6 1.4 1.87 4.89 1.77 1.91
time (sec) N/A 0.105 0.013 0.004 1.417 1.842 0.233 1.069


















Problem 51 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A B A A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 21 21 21 44 53 62 70 55
normalized size 1 1. 1. 2.1 2.52 2.95 3.33 2.62
time (sec) N/A 0.009 0.004 0.005 0.931 1.813 0.271 1.076


















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 17 17 17 14 18 35 12 18
normalized size 1 1. 1. 0.82 1.06 2.06 0.71 1.06
time (sec) N/A 0.004 0.001 0. 0.924 2.002 0.08 1.091


















Problem 53 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 40 40 29 34 45 68 32 51
normalized size 1 1. 0.72 0.85 1.12 1.7 0.8 1.27
time (sec) N/A 0.024 0.012 0. 1.402 2.045 0.321 1.075


















Problem 54 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 8 9 16 5 9
normalized size 1 1. 1. 1.14 1.29 2.29 0.71 1.29
time (sec) N/A 0.001 0.001 0.002 0.925 2.021 0.071 1.084


















Problem 55 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 8 8 8 9 11 22 5 11
normalized size 1 1. 1. 1.12 1.38 2.75 0.62 1.38
time (sec) N/A 0.044 0.013 0.023 0.931 1.956 0.082 1.076


















Problem 56 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 3 3 3 4 4 22 15 4
normalized size 1 1. 1. 1.33 1.33 7.33 5. 1.33
time (sec) N/A 0.02 0.011 0.002 1.405 1.924 0.123 1.082


















Problem 57 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A F B A A A A
verified N/A Yes N/A TBD TBD TBD TBD TBD
size 2 2 0 9 4 23 2 4
normalized size 1 1. 0. 4.5 2. 11.5 1. 2.
time (sec) N/A 0.002 0.002 0. 1.022 1.801 0.439 1.12


















Problem 58 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 14 14 14 15 19 46 15 19
normalized size 1 1. 1. 1.07 1.36 3.29 1.07 1.36
time (sec) N/A 0.021 0.005 0.011 0.938 1.856 0.166 1.069


















Problem 59 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 17 17 13 16 15 32 8 15
normalized size 1 1. 0.76 0.94 0.88 1.88 0.47 0.88
time (sec) N/A 0.031 0.01 0.001 0.932 1.878 0.081 1.099


















Problem 60 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 38 38 29 29 32 66 26 32
normalized size 1 1. 0.76 0.76 0.84 1.74 0.68 0.84
time (sec) N/A 0.033 0.026 0.002 0.937 1.921 0.088 1.066


















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 7 7 7 8 9 26 7 9
normalized size 1 1. 1. 1.14 1.29 3.71 1. 1.29
time (sec) N/A 0.009 0.002 0. 0.935 1.905 0.167 1.061


















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 22 22 22 17 22 55 20 22
normalized size 1 1. 1. 0.77 1. 2.5 0.91 1.
time (sec) N/A 0.012 0.009 0. 0.941 2.054 0.293 1.07


















Problem 63 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 8 9 26 7 9
normalized size 1 1. 1. 1.14 1.29 3.71 1. 1.29
time (sec) N/A 0.009 0.002 0. 0.932 2.008 0.161 1.06


















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 28 28 28 23 23 61 22 30
normalized size 1 1. 1. 0.82 0.82 2.18 0.79 1.07
time (sec) N/A 0.01 0.001 0.002 0.926 1.817 0.095 1.059


















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 11 11 11 10 12 51 8 12
normalized size 1 1. 1. 0.91 1.09 4.64 0.73 1.09
time (sec) N/A 0.012 0.003 0.013 0.917 2.014 0.536 1.048


















Problem 66 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 3 3 3 4 4 22 15 4
normalized size 1 1. 1. 1.33 1.33 7.33 5. 1.33
time (sec) N/A 0.019 0.008 0. 1.417 1.798 0.124 1.04


















Problem 67 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 3 3 3 4 0 26 3 4
normalized size 1 1. 1. 1.33 0. 8.67 1. 1.33
time (sec) N/A 0.047 0.055 0.005 0. 2.154 0.418 1.064


















Problem 68 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A B A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 16 16 16 21 72 51 12 28
normalized size 1 1. 1. 1.31 4.5 3.19 0.75 1.75
time (sec) N/A 0.041 0.023 0.023 1.412 2.275 0.132 1.09


















Problem 69 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C B F(-2) B F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 53 53 127 262 0 284 0 0
normalized size 1 1. 2.4 4.94 0. 5.36 0. 0.
time (sec) N/A 0.065 0.055 0.006 0. 2.626 0. 0.


















Problem 70 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A B B B B F(-1) B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 16 16 35 127 93 177 0 112
normalized size 1 1. 2.19 7.94 5.81 11.06 0. 7.
time (sec) N/A 0.086 0.09 0.036 1.417 2.736 0. 1.117


















Problem 71 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A B B B A C B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 16 16 35 121 92 76 422 111
normalized size 1 1. 2.19 7.56 5.75 4.75 26.38 6.94
time (sec) N/A 0.124 0.018 0.007 1.41 2.162 1.301 1.097


















Problem 72 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A B B B A C B
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 16 16 35 121 88 76 422 107
normalized size 1 1. 2.19 7.56 5.5 4.75 26.38 6.69
time (sec) N/A 0.019 0.013 0.005 1.41 2.432 1.275 1.076


















Problem 73 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A B B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 35 35 26 30 59 146 105 39
normalized size 1 1. 0.74 0.86 1.69 4.17 3. 1.11
time (sec) N/A 0.008 0.021 0. 1.413 2.289 2.558 1.065


















Problem 74 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A B A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 14 14 18 13 16 74 19 16
normalized size 1 1. 1.29 0.93 1.14 5.29 1.36 1.14
time (sec) N/A 0.009 0.007 0.007 1.418 2.252 0.061 1.066


















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 13 13 13 12 15 34 8 15
normalized size 1 1. 1. 0.92 1.15 2.62 0.62 1.15
time (sec) N/A 0.005 0.004 0.001 1.403 2.098 0.081 1.06


















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 7 7 7 7 8 15 5 8
normalized size 1 1. 1. 1. 1.14 2.14 0.71 1.14
time (sec) N/A 0.03 0.004 0. 0.989 2.099 0.08 1.055


















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 24 24 20 24 31 55 20 41
normalized size 1 1. 0.83 1. 1.29 2.29 0.83 1.71
time (sec) N/A 0.357 0.069 0.005 1.547 2.173 0.108 1.062


















Problem 78 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 9 9 9 9 11 18 7 11
normalized size 1 1. 1. 1. 1.22 2. 0.78 1.22
time (sec) N/A 0.003 0.003 0. 0.936 2.05 0.068 1.064


















Problem 79 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 25 25 25 20 32 55 19 26
normalized size 1 1. 1. 0.8 1.28 2.2 0.76 1.04
time (sec) N/A 0.045 0.002 0.002 0.927 2.145 0.085 1.051


















Problem 80 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 8 8 8 7 8 23 5 8
normalized size 1 1. 1. 0.88 1. 2.88 0.62 1.
time (sec) N/A 0.003 0.003 0.002 1.409 2.189 0.087 1.065


















Problem 81 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 16 16 16 13 16 36 10 16
normalized size 1 1. 1. 0.81 1. 2.25 0.62 1.
time (sec) N/A 0.008 0.003 0.002 1.408 2.053 0.09 1.059


















Problem 82 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 14 14 14 17 22 62 36 22
normalized size 1 1. 1. 1.21 1.57 4.43 2.57 1.57
time (sec) N/A 0.015 0.003 0.01 1.407 2.281 0.384 1.053


















Problem 83 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A B B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 35 35 26 30 59 146 105 39
normalized size 1 1. 0.74 0.86 1.69 4.17 3. 1.11
time (sec) N/A 0.011 0.003 0. 1.413 2.105 2.556 1.062


















Problem 84 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A B B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 17 17 32 16 20 103 34 28
normalized size 1 1. 1.88 0.94 1.18 6.06 2. 1.65
time (sec) N/A 0.005 0.007 0.004 1.41 2.169 0.526 1.071


















Problem 85 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 19 19 14 14 15 46 15 15
normalized size 1 1. 0.74 0.74 0.79 2.42 0.79 0.79
time (sec) N/A 0.008 0.014 0. 0.928 2.231 0.28 1.046


















Problem 86 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 2 2 2 3 3 11 2 4
normalized size 1 1. 1. 1.5 1.5 5.5 1. 2.
time (sec) N/A 0. 0. 0. 0.93 2.241 0.05 1.057


















Problem 87 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A B A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 45 45 73 31 346 271 0 45
normalized size 1 1. 1.62 0.69 7.69 6.02 0. 1.
time (sec) N/A 0.123 0.181 0.142 1.78 2.618 0. 1.104


















Problem 88 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 14 14 14 11 14 36 10 22
normalized size 1 1. 1. 0.79 1. 2.57 0.71 1.57
time (sec) N/A 0.006 0.002 0.002 0.928 2.206 0.061 1.065


















Problem 89 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 17 17 14 11 15 31 14 15
normalized size 1 1. 0.82 0.65 0.88 1.82 0.82 0.88
time (sec) N/A 0.003 0.003 0.003 0.924 1.775 0.203 1.059


















Problem 90 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 23 23 25 20 26 77 0 36
normalized size 1 1. 1.09 0.87 1.13 3.35 0. 1.57
time (sec) N/A 0.01 0.005 0.004 1.409 1.805 0. 1.059


















Problem 91 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 8 8 8 7 8 38 5 8
normalized size 1 1. 1. 0.88 1. 4.75 0.62 1.
time (sec) N/A 0.013 0.001 0.002 0.919 2.072 0.058 1.059


















Problem 92 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 6 6 6 6 7 19 5 7
normalized size 1 1. 1. 1. 1.17 3.17 0.83 1.17
time (sec) N/A 0.016 0.004 0. 0.934 2.045 0.072 1.063


















Problem 93 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 12 12 12 11 14 27 8 14
normalized size 1 1. 1. 0.92 1.17 2.25 0.67 1.17
time (sec) N/A 0.026 0.008 0.002 0.93 2.125 0.08 1.055


















Problem 94 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 12 12 8 9 14 30 7 11
normalized size 1 1. 0.67 0.75 1.17 2.5 0.58 0.92
time (sec) N/A 0.009 0.008 0. 0.927 2.21 0.354 1.055


















Problem 95 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 8 8 8 7 8 19 7 8
normalized size 1 1. 1. 0.88 1. 2.38 0.88 1.
time (sec) N/A 0.012 0.004 0. 0.927 1.911 0.064 1.06


















Problem 96 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 17 17 17 14 18 35 12 18
normalized size 1 1. 1. 0.82 1.06 2.06 0.71 1.06
time (sec) N/A 0.004 0.001 0. 0.936 1.806 0.082 1.059


















Problem 97 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 8 8 8 7 8 20 5 8
normalized size 1 1. 1. 0.88 1. 2.5 0.62 1.
time (sec) N/A 0.007 0.001 0. 0.923 1.909 0.054 1.068


















Problem 98 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A F A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 24 24 27 21 30 68 0 27
normalized size 1 1. 1.12 0.88 1.25 2.83 0. 1.12
time (sec) N/A 0.009 0.032 0.006 1.415 1.802 0. 1.077


















Problem 99 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 20 20 20 14 18 72 15 18
normalized size 1 1. 1. 0.7 0.9 3.6 0.75 0.9
time (sec) N/A 0.023 0.005 0. 1.421 1.922 0.106 1.066


















Problem 100 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A B B A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 35 35 26 30 59 146 105 39
normalized size 1 1. 0.74 0.86 1.69 4.17 3. 1.11
time (sec) N/A 0.009 0.003 0. 1.407 1.876 2.557 1.078


















Problem 101 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 20 20 18 15 19 49 24 19
normalized size 1 1. 0.9 0.75 0.95 2.45 1.2 0.95
time (sec) N/A 0.024 0.008 0.003 1.416 1.919 0.111 1.048


















Problem 102 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 33 33 33 27 35 103 31 35
normalized size 1 1. 1. 0.82 1.06 3.12 0.94 1.06
time (sec) N/A 0.011 0.012 0. 1.412 1.894 0.126 1.049


















Problem 103 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 21 21 21 20 26 55 15 26
normalized size 1 1. 1. 0.95 1.24 2.62 0.71 1.24
time (sec) N/A 0.016 0. 0.002 0.932 1.74 0.052 1.046


















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 26 26 26 25 32 74 19 31
normalized size 1 1. 1. 0.96 1.23 2.85 0.73 1.19
time (sec) N/A 0.019 0. 0.002 0.936 1.789 0.062 1.053


















Problem 105 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 27 27 27 26 34 85 22 34
normalized size 1 1. 1. 0.96 1.26 3.15 0.81 1.26
time (sec) N/A 0.023 0. 0. 0.928 1.865 0.055 1.053


















Problem 106 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 32 32 32 31 41 104 26 39
normalized size 1 1. 1. 0.97 1.28 3.25 0.81 1.22
time (sec) N/A 0.026 0. 0.001 0.933 1.813 0.055 1.047


















Problem 107 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 24 24 25 24 31 80 20 31
normalized size 1 1. 1.04 1. 1.29 3.33 0.83 1.29
time (sec) N/A 0.027 0. 0. 0.931 1.875 0.053 1.242


















Problem 108 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 29 29 30 29 38 99 24 36
normalized size 1 1. 1.03 1. 1.31 3.41 0.83 1.24
time (sec) N/A 0.032 0. 0. 0.929 1.817 0.054 1.078


















Problem 109 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 19 19 19 18 23 34 15 23
normalized size 1 1. 1. 0.95 1.21 1.79 0.79 1.21
time (sec) N/A 0.009 0. 0. 0.93 1.807 0.051 1.049


















Problem 110 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 24 24 24 23 30 42 19 28
normalized size 1 1. 1. 0.96 1.25 1.75 0.79 1.17
time (sec) N/A 0.011 0. 0.002 0.937 1.817 0.053 1.05


















Problem 111 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 24 31 85 22 31
normalized size 1 1. 1. 0.96 1.24 3.4 0.88 1.24
time (sec) N/A 0.013 0. 0.002 0.931 1.881 0.053 1.06


















Problem 112 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 30 29 38 101 26 36
normalized size 1 1. 1. 0.97 1.27 3.37 0.87 1.2
time (sec) N/A 0.015 0. 0. 0.929 1.706 0.055 1.07


















Problem 113 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 27 27 28 27 35 53 24 34
normalized size 1 1. 1.04 1. 1.3 1.96 0.89 1.26
time (sec) N/A 0.006 0. 0.001 0.926 1.812 0.054 1.057









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 [ 0.8571 ]

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 3 2 1. 4 0.5







2 A 3 2 1. 11 0.182







3 A 2 1 1. 11 0.091







4 A 1 1 1. 2 0.5







5 A 1 1 1. 7 0.143







6 A 2 2 1. 7 0.286







7 A 1 1 1. 11 0.091







8 A 1 1 1. 6 0.167







9 A 2 2 1. 7 0.286







10 A 2 2 1. 4 0.5







11 A 1 1 1. 6 0.167







12 A 3 2 1. 6 0.333







13 A 4 2 1. 16 0.125







14 A 5 3 1. 7 0.429







15 A 3 1 1. 14 0.071







16 A 2 2 1. 5 0.4







17 A 1 1 1. 7 0.143







18 A 1 1 1. 11 0.091







19 A 2 2 1. 11 0.182







20 A 1 1 1. 5 0.2







21 A 1 1 1. 6 0.167







22 A 2 2 1. 9 0.222







23 A 3 3 1. 14 0.214







24 A 2 1 1. 9 0.111







25 A 3 3 1. 7 0.429







26 A 2 2 1. 15 0.133







27 A 2 2 1. 17 0.118







28 A 3 3 1. 13 0.231







29 A 1 1 1. 5 0.2







30 A 3 3 1. 8 0.375







31 A 7 7 1. 11 0.636







32 A 3 2 1. 12 0.167







33 A 3 3 1. 6 0.5







34 A 2 1 1. 9 0.111







35 A 4 3 1. 13 0.231







36 A 4 4 1. 15 0.267







37 A 3 2 1. 15 0.133







38 A 6 3 1. 13 0.231







39 A 3 2 1. 15 0.133







40 A 5 5 1. 29 0.172







41 A 2 2 1. 4 0.5







42 A 6 6 1.16 19 0.316







43 A 1 1 1. 6 0.167







44 A 2 2 1. 5 0.4







45 A 1 1 1. 10 0.1







46 A 5 3 1. 13 0.231







47 A 1 1 1. 5 0.2







48 A 1 1 1. 7 0.143







49 A 6 6 1. 9 0.667







50 A 10 6 1.55 7 0.857







51 A 1 1 1. 27 0.037







52 A 1 1 1. 4 0.25







53 A 4 3 1. 6 0.5







54 A 2 2 1. 10 0.2







55 A 7 4 1. 9 0.444







56 A 2 1 1. 12 0.083







57 A 1 1 1. 4 0.25







58 A 6 4 1. 7 0.571







59 A 4 3 1. 11 0.273







60 A 6 3 1. 9 0.333







61 A 2 2 1. 4 0.5







62 A 3 3 1. 6 0.5







63 A 2 2 1. 4 0.5







64 A 2 2 1. 6 0.333







65 A 2 1 1. 9 0.111







66 A 2 1 1. 12 0.083







67 A 2 2 1. 20 0.1







68 A 2 2 1. 10 0.2







69 A 6 6 1. 30 0.2







70 A 5 5 1. 39 0.128







71 A 6 5 1. 45 0.111







72 A 4 4 1. 31 0.129







73 A 3 2 1. 15 0.133







74 A 3 2 1. 4 0.5







75 A 3 2 1. 11 0.182







76 A 5 3 1. 13 0.231







77 A 10 6 1. 33 0.182







78 A 1 1 1. 7 0.143







79 A 5 5 1. 8 0.625







80 A 2 2 1. 9 0.222







81 A 3 3 1. 11 0.273







82 A 3 3 1. 8 0.375







83 A 3 2 1. 15 0.133







84 A 2 2 1. 16 0.125







85 A 1 1 1. 6 0.167







86 A 1 1 1. 3 0.333







87 A 4 2 1. 17 0.118







88 A 2 2 1. 4 0.5







89 A 2 1 1. 11 0.091







90 A 3 3 1. 14 0.214







91 A 2 2 1. 7 0.286







92 A 2 2 1. 11 0.182







93 A 3 2 1. 13 0.154







94 A 1 1 1. 8 0.125







95 A 2 2 1. 7 0.286







96 A 1 1 1. 4 0.25







97 A 2 2 1. 5 0.4







98 A 3 3 1. 17 0.176







99 A 3 3 1. 16 0.188







100 A 3 2 1. 15 0.133







101 A 3 3 1. 15 0.2







102 A 3 3 1. 8 0.375







103 A 1 1 1. 20 0.05







104 A 1 1 1. 25 0.04







105 A 1 1 1. 26 0.038







106 A 1 1 1. 31 0.032







107 A 1 1 1. 24 0.042







108 A 1 1 1. 29 0.034







109 A 1 1 1. 18 0.056







110 A 1 1 1. 23 0.043







111 A 1 1 1. 24 0.042







112 A 1 1 1. 29 0.034







113 A 1 1 1. 27 0.037