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, 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, 98, 99, 100 }

B grade: { 38, 97 }

C grade: { 39 }

F grade: { }

2.1.2 Mathematica

A grade: { 8, 9, 10, 11, 12, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 32, 34, 39, 40, 41, 42, 43, 44, 47, 51, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 80, 82, 83, 84, 85, 86, 87, 89, 91, 93, 94, 95, 96, 98, 99, 100 }

B grade: { 4, 6, 7, 29, 31, 36, 38, 97 }

C grade: { 1, 2, 3, 5, 13, 14, 18, 33, 35, 37, 45, 46, 48, 49, 50, 52, 54, 63, 65, 67, 77, 79, 81, 90, 92 }

F grade: { 19, 88}

2.1.3 Maple

A grade: { 1, 2, 3, 4, 9, 10, 11, 13, 14, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 66, 68, 70, 71, 72, 73, 74, 75, 89, 91, 93, 95, 97, 98, 99 }

B grade: { 6, 7, 8, 80 }

C grade: { 5, 33, 35, 37, 51, 55, 63, 65, 67, 69, 76, 78, 90, 92, 94, 96 }

F grade: { 12, 15, 16, 17, 18, 19, 30, 56, 57, 58, 59, 60, 61, 62, 64, 77, 79, 81, 82, 83, 84, 85, 86, 87, 88, 100 }

2.1.4 Maxima

A grade: { 4, 20, 21, 22, 23, 25, 26, 27, 28, 32, 34, 39, 41, 43, 47, 66, 71, 73, 75, 99, 100

B grade: { }

C grade: { }

F grade: { 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 24, 29, 30, 31, 33, 35, 36, 37, 38, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 72, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98 }

2.1.5 FriCAS

A grade: { 1, 20, 21, 22, 23, 25, 26, 27, 28, 32, 33, 34, 35, 37, 39, 40, 41, 42, 43, 44, 45, 47, 49, 55, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 91, 94, 96, 97 }

B grade: { 2, 3, 4, 6, 7, 8, 36, 38, 51, 53, 90, 92, 93, 95, 99, 100 }

C grade: { }

F grade: { 5, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 24, 29, 30, 31, 46, 48, 50, 52, 54, 56, 57, 58, 59, 60, 61, 62, 64, 88, 98 }

2.1.6 Sympy

A grade: { 93 }

B grade: { }

C grade: { }

F 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, 94, 95, 96, 97, 98, 99, 100 }

2.1.7 Giac

A grade: { 49, 71 }

B grade: { 45, 47 }

C grade: { }

F 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, 46, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 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 }

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 C A F A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 203 203 225 250 0 772 0 0
normalized size 1 1. 1.11 1.23 0. 3.8 0. 0.
time (sec) N/A 0.155 0.494 0.292 0. 2.526 0. 0.


















Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C A F B F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 153 153 200 189 0 720 0 0
normalized size 1 1. 1.31 1.24 0. 4.71 0. 0.
time (sec) N/A 0.099 0.258 0.244 0. 2.351 0. 0.


















Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C A F B F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 107 107 176 111 0 687 0 0
normalized size 1 1. 1.64 1.04 0. 6.42 0. 0.
time (sec) N/A 0.062 0.161 0.234 0. 2.263 0. 0.


















Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A B A A B F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 44 44 95 50 42 567 0 0
normalized size 1 1. 2.16 1.14 0.95 12.89 0. 0.
time (sec) N/A 0.056 0.196 0.213 0.978 2.147 0. 0.


















Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C C F F F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 170 170 332 886 0 0 0 0
normalized size 1 1. 1.95 5.21 0. 0. 0. 0.
time (sec) N/A 0.29 0.192 0.352 0. 0. 0. 0.


















Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A B B F B F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 70 70 244 542 0 1462 0 0
normalized size 1 1. 3.49 7.74 0. 20.89 0. 0.
time (sec) N/A 0.106 0.215 0.252 0. 2.199 0. 0.


















Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A B B F B F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 133 133 315 1233 0 1912 0 0
normalized size 1 1. 2.37 9.27 0. 14.38 0. 0.
time (sec) N/A 0.204 0.907 0.259 0. 2.481 0. 0.


















Problem 8 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 197 197 368 1946 0 2163 0 0
normalized size 1 1. 1.87 9.88 0. 10.98 0. 0.
time (sec) N/A 0.316 0.402 0.256 0. 2.668 0. 0.


















Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F F F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 279 279 305 655 0 0 0 0
normalized size 1 1. 1.09 2.35 0. 0. 0. 0.
time (sec) N/A 0.242 1.915 0.483 0. 0. 0. 0.


















Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F F F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 149 149 172 396 0 0 0 0
normalized size 1 1. 1.15 2.66 0. 0. 0. 0.
time (sec) N/A 0.138 0.418 0.392 0. 0. 0. 0.


















Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F F F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 80 80 105 209 0 0 0 0
normalized size 1 1. 1.31 2.61 0. 0. 0. 0.
time (sec) N/A 0.068 0.21 0.285 0. 0. 0. 0.


















Problem 12 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 274 274 280 0 0 0 0 0
normalized size 1 1. 1.02 0. 0. 0. 0. 0.
time (sec) N/A 0.463 0.295 0.392 0. 0. 0. 0.


















Problem 13 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 224 224 678 367 0 0 0 0
normalized size 1 1. 3.03 1.64 0. 0. 0. 0.
time (sec) N/A 0.389 2.625 0.357 0. 0. 0. 0.


















Problem 14 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 537 537 1439 1026 0 0 0 0
normalized size 1 1. 2.68 1.91 0. 0. 0. 0.
time (sec) N/A 0.75 7.763 0.563 0. 0. 0. 0.


















Problem 15 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 260 260 254 0 0 0 0 0
normalized size 1 1. 0.98 0. 0. 0. 0. 0.
time (sec) N/A 0.257 0.552 0.559 0. 0. 0. 0.


















Problem 16 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 136 136 153 0 0 0 0 0
normalized size 1 1. 1.12 0. 0. 0. 0. 0.
time (sec) N/A 0.105 0.14 0.353 0. 0. 0. 0.


















Problem 17 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 378 378 384 0 0 0 0 0
normalized size 1 1. 1.02 0. 0. 0. 0. 0.
time (sec) N/A 0.511 0.319 0.373 0. 0. 0. 0.


















Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C F F F F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 330 330 1849 0 0 0 0 0
normalized size 1 1. 5.6 0. 0. 0. 0. 0.
time (sec) N/A 0.575 45.653 0.458 0. 0. 0. 0.


















Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A F F F F F F
verified N/A Yes N/A TBD TBD TBD TBD TBD
size 965 965 0 0 0 0 0 0
normalized size 1 1. 0. 0. 0. 0. 0. 0.
time (sec) N/A 1.338 9.65 0.694 0. 0. 0. 0.


















Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 164 164 84 54 78 144 0 0
normalized size 1 1. 0.51 0.33 0.48 0.88 0. 0.
time (sec) N/A 0.03 0.041 0.139 0.982 1.838 0. 0.


















Problem 21 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 126 126 72 49 62 131 0 0
normalized size 1 1. 0.57 0.39 0.49 1.04 0. 0.
time (sec) N/A 0.027 0.04 0.126 0.983 1.872 0. 0.


















Problem 22 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 88 88 56 42 46 116 0 0
normalized size 1 1. 0.64 0.48 0.52 1.32 0. 0.
time (sec) N/A 0.018 0.026 0.13 0.987 1.865 0. 0.


















Problem 23 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 43 43 67 36 26 92 0 0
normalized size 1 1. 1.56 0.84 0.6 2.14 0. 0.
time (sec) N/A 0.007 0.096 0.123 0.972 1.806 0. 0.


















Problem 24 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F F F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 46 46 45 65 0 0 0 0
normalized size 1 1. 0.98 1.41 0. 0. 0. 0.
time (sec) N/A 0.095 0.036 0.16 0. 0. 0. 0.


















Problem 25 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 98 98 111 64 88 111 0 0
normalized size 1 1. 1.13 0.65 0.9 1.13 0. 0.
time (sec) N/A 0.023 0.077 0.134 0.98 1.878 0. 0.


















Problem 26 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 136 136 125 79 124 134 0 0
normalized size 1 1. 0.92 0.58 0.91 0.99 0. 0.
time (sec) N/A 0.033 0.112 0.133 0.993 1.885 0. 0.


















Problem 27 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 172 172 140 91 157 153 0 0
normalized size 1 1. 0.81 0.53 0.91 0.89 0. 0.
time (sec) N/A 0.04 0.128 0.141 0.99 1.93 0. 0.


















Problem 28 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 21 21 25 29 22 57 0 0
normalized size 1 1. 1.19 1.38 1.05 2.71 0. 0.
time (sec) N/A 0.007 0.024 0.101 1.024 1.811 0. 0.


















Problem 29 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A B A F F(-2) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 61 61 219 116 0 0 0 0
normalized size 1 1. 3.59 1.9 0. 0. 0. 0.
time (sec) N/A 0.105 0.957 0.333 0. 0. 0. 0.


















Problem 30 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 54 54 49 0 0 0 0 0
normalized size 1 1. 0.91 0. 0. 0. 0. 0.
time (sec) N/A 0.107 0.046 0.284 0. 0. 0. 0.


















Problem 31 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A B A F F(-2) F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 77 77 249 140 0 0 0 0
normalized size 1 1. 3.23 1.82 0. 0. 0. 0.
time (sec) N/A 0.101 1.171 0.361 0. 0. 0. 0.


















Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A F F
verified N/A NO NO TBD TBD TBD TBD TBD
size 64 83 65 64 63 135 0 0
normalized size 1 1.3 1.02 1. 0.98 2.11 0. 0.
time (sec) N/A 0.038 0.09 0.189 1.078 1.714 0. 0.


















Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C C F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 84 84 97 118 0 203 0 0
normalized size 1 1. 1.15 1.4 0. 2.42 0. 0.
time (sec) N/A 0.032 0.127 0.179 0. 1.778 0. 0.


















Problem 34 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A F F
verified N/A NO NO TBD TBD TBD TBD TBD
size 38 52 48 54 51 111 0 0
normalized size 1 1.37 1.26 1.42 1.34 2.92 0. 0.
time (sec) N/A 0.024 0.059 0.173 1.07 1.821 0. 0.


















Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C C F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 53 53 75 92 0 171 0 0
normalized size 1 1. 1.42 1.74 0. 3.23 0. 0.
time (sec) N/A 0.018 0.076 0.184 0. 1.808 0. 0.


















Problem 36 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A B A F B F F
verified N/A NO NO TBD TBD TBD TBD TBD
size 24 39 79 79 0 258 0 0
normalized size 1 1.62 3.29 3.29 0. 10.75 0. 0.
time (sec) N/A 0.137 0.04 0.184 0. 1.832 0. 0.


















Problem 37 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C C F A F F
verified N/A NO NO TBD TBD TBD TBD TBD
size 48 64 75 92 0 165 0 0
normalized size 1 1.33 1.56 1.92 0. 3.44 0. 0.
time (sec) N/A 0.031 0.05 0.189 0. 2.024 0. 0.


















Problem 38 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A B B A F B F F
verified N/A NO NO TBD TBD TBD TBD TBD
size 35 99 93 90 0 278 0 0
normalized size 1 2.83 2.66 2.57 0. 7.94 0. 0.
time (sec) N/A 0.041 0.06 0.204 0. 1.81 0. 0.


















Problem 39 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A C A A A A F F
verified N/A NO Yes TBD TBD TBD TBD TBD
size 55 84 43 53 58 108 0 0
normalized size 1 1.53 0.78 0.96 1.05 1.96 0. 0.
time (sec) N/A 0.037 0.043 0.178 1.045 1.835 0. 0.


















Problem 40 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 132 132 110 110 0 298 0 0
normalized size 1 1. 0.83 0.83 0. 2.26 0. 0.
time (sec) N/A 0.059 0.073 0.184 0. 1.809 0. 0.


















Problem 41 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 115 115 60 63 69 128 0 0
normalized size 1 1. 0.52 0.55 0.6 1.11 0. 0.
time (sec) N/A 0.052 0.067 0.179 1.04 1.922 0. 0.


















Problem 42 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 163 163 129 132 0 324 0 0
normalized size 1 1. 0.79 0.81 0. 1.99 0. 0.
time (sec) N/A 0.077 0.104 0.184 0. 2.194 0. 0.


















Problem 43 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 146 146 76 71 81 151 0 0
normalized size 1 1. 0.52 0.49 0.55 1.03 0. 0.
time (sec) N/A 0.071 0.085 0.191 1.037 2.129 0. 0.


















Problem 44 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 194 194 145 152 0 348 0 0
normalized size 1 1. 0.75 0.78 0. 1.79 0. 0.
time (sec) N/A 0.098 0.133 0.203 0. 2.133 0. 0.


















Problem 45 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C A F A F(-1) B
verified N/A Yes NO TBD TBD TBD TBD TBD
size 111 111 111 137 0 251 0 360
normalized size 1 1. 1. 1.23 0. 2.26 0. 3.24
time (sec) N/A 0.062 0.181 0.322 0. 2.125 0. 1.195


















Problem 46 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C A F F F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 115 115 139 114 0 0 0 0
normalized size 1 1. 1.21 0.99 0. 0. 0. 0.
time (sec) N/A 0.051 0.332 0.19 0. 0. 0. 0.


















Problem 47 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A F(-1) B
verified N/A NO NO TBD TBD TBD TBD TBD
size 58 71 56 60 57 126 0 240
normalized size 1 1.22 0.97 1.03 0.98 2.17 0. 4.14
time (sec) N/A 0.041 0.094 0.178 1.068 2.061 0. 1.182


















Problem 48 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C A F F F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 112 112 140 136 0 0 0 0
normalized size 1 1. 1.25 1.21 0. 0. 0. 0.
time (sec) N/A 0.067 0.435 0.181 0. 0. 0. 0.


















Problem 49 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C A F A F A
verified N/A Yes NO TBD TBD TBD TBD TBD
size 63 63 92 112 0 225 0 107
normalized size 1 1. 1.46 1.78 0. 3.57 0. 1.7
time (sec) N/A 0.037 0.104 0.33 0. 2.053 0. 1.163


















Problem 50 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C A F F F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 67 67 116 102 0 0 0 0
normalized size 1 1. 1.73 1.52 0. 0. 0. 0.
time (sec) N/A 0.028 0.189 0.204 0. 0. 0. 0.


















Problem 51 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A C F B F F(-2)
verified N/A Yes NO TBD TBD TBD TBD TBD
size 68 68 100 127 0 298 0 0
normalized size 1 1. 1.47 1.87 0. 4.38 0. 0.
time (sec) N/A 0.043 0.063 0.312 0. 2.099 0. 0.


















Problem 52 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C A F F F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 147 147 135 132 0 0 0 0
normalized size 1 1. 0.92 0.9 0. 0. 0. 0.
time (sec) N/A 0.068 0.282 0.187 0. 0. 0. 0.


















Problem 53 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F B F F(-2)
verified N/A NO NO TBD TBD TBD TBD TBD
size 80 93 113 103 0 230 0 0
normalized size 1 1.16 1.41 1.29 0. 2.88 0. 0.
time (sec) N/A 0.053 0.144 0.284 0. 2.013 0. 0.


















Problem 54 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C A F F F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 115 115 123 104 0 0 0 0
normalized size 1 1. 1.07 0.9 0. 0. 0. 0.
time (sec) N/A 0.051 0.192 0.208 0. 0. 0. 0.


















Problem 55 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A C F A F F(-2)
verified N/A Yes NO TBD TBD TBD TBD TBD
size 118 118 105 129 0 319 0 0
normalized size 1 1. 0.89 1.09 0. 2.7 0. 0.
time (sec) N/A 0.061 0.256 0.299 0. 2.034 0. 0.


















Problem 56 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A F F(-2) F F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 109 109 159 0 0 0 0 0
normalized size 1 1. 1.46 0. 0. 0. 0. 0.
time (sec) N/A 0.068 2.495 0.23 0. 0. 0. 0.


















Problem 57 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A F F(-2) F F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 107 107 159 0 0 0 0 0
normalized size 1 1. 1.49 0. 0. 0. 0. 0.
time (sec) N/A 0.056 2.303 0.194 0. 0. 0. 0.


















Problem 58 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A F F(-2) F F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 91 91 145 0 0 0 0 0
normalized size 1 1. 1.59 0. 0. 0. 0. 0.
time (sec) N/A 0.04 0.302 0.174 0. 0. 0. 0.


















Problem 59 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 109 109 139 0 0 0 0 0
normalized size 1 1. 1.28 0. 0. 0. 0. 0.
time (sec) N/A 0.075 0.825 0.643 0. 0. 0. 0.


















Problem 60 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A F F(-2) F(-2) F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 133 133 186 0 0 0 0 0
normalized size 1 1. 1.4 0. 0. 0. 0. 0.
time (sec) N/A 0.083 4.953 1.224 0. 0. 0. 0.


















Problem 61 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A F F(-2) F(-2) F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 119 119 159 0 0 0 0 0
normalized size 1 1. 1.34 0. 0. 0. 0. 0.
time (sec) N/A 0.053 0.297 1.102 0. 0. 0. 0.


















Problem 62 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A F F(-2) F(-2) F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 105 105 139 0 0 0 0 0
normalized size 1 1. 1.32 0. 0. 0. 0. 0.
time (sec) N/A 0.047 0.386 1.154 0. 0. 0. 0.


















Problem 63 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C C F(-2) A F F
verified N/A NO NO TBD TBD TBD TBD TBD
size 87 106 96 145 0 197 0 0
normalized size 1 1.22 1.1 1.67 0. 2.26 0. 0.
time (sec) N/A 0.069 0.137 0.326 0. 2.454 0. 0.


















Problem 64 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A F F(-2) F(-2) F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 107 107 156 0 0 0 0 0
normalized size 1 1. 1.46 0. 0. 0. 0. 0.
time (sec) N/A 0.06 0.259 1.182 0. 0. 0. 0.


















Problem 65 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C C F A F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 203 203 105 123 0 225 0 0
normalized size 1 1. 0.52 0.61 0. 1.11 0. 0.
time (sec) N/A 0.7 0.141 0.206 0. 2.091 0. 0.


















Problem 66 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 117 117 52 72 57 131 0 0
normalized size 1 1. 0.44 0.62 0.49 1.12 0. 0.
time (sec) N/A 0.546 0.068 0.181 1.045 1.946 0. 0.


















Problem 67 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C C F A F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 169 169 86 97 0 192 0 0
normalized size 1 1. 0.51 0.57 0. 1.14 0. 0.
time (sec) N/A 0.465 0.07 0.182 0. 2.116 0. 0.


















Problem 68 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F A F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 85 85 89 89 0 281 0 0
normalized size 1 1. 1.05 1.05 0. 3.31 0. 0.
time (sec) N/A 0.434 0.06 0.208 0. 2.106 0. 0.


















Problem 69 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A C F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 57 57 75 98 0 186 0 0
normalized size 1 1. 1.32 1.72 0. 3.26 0. 0.
time (sec) N/A 0.174 0.095 0.209 0. 2.301 0. 0.


















Problem 70 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 86 86 86 97 0 306 0 0
normalized size 1 1. 1. 1.13 0. 3.56 0. 0.
time (sec) N/A 0.446 0.066 0.187 0. 2.149 0. 0.


















Problem 71 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 57 57 52 73 62 130 0 126
normalized size 1 1. 0.91 1.28 1.09 2.28 0. 2.21
time (sec) N/A 0.39 0.066 0.194 1.043 2.055 0. 1.152


















Problem 72 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 147 147 121 131 0 316 0 0
normalized size 1 1. 0.82 0.89 0. 2.15 0. 0.
time (sec) N/A 0.453 0.123 0.192 0. 2.123 0. 0.


















Problem 73 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 183 183 69 84 76 150 0 0
normalized size 1 1. 0.38 0.46 0.42 0.82 0. 0.
time (sec) N/A 0.505 0.079 0.196 1.046 2.166 0. 0.


















Problem 74 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 267 267 137 153 0 344 0 0
normalized size 1 1. 0.51 0.57 0. 1.29 0. 0.
time (sec) N/A 0.543 0.143 0.227 0. 2.116 0. 0.


















Problem 75 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 301 301 85 92 88 174 0 0
normalized size 1 1. 0.28 0.31 0.29 0.58 0. 0.
time (sec) N/A 0.576 0.099 0.24 1.078 2.082 0. 0.


















Problem 76 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A C F A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 147 147 65 531 0 135 0 0
normalized size 1 1. 0.44 3.61 0. 0.92 0. 0.
time (sec) N/A 0.619 0.09 0.54 0. 2.037 0. 0.


















Problem 77 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C F F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 163 163 97 0 0 203 0 0
normalized size 1 1. 0.6 0. 0. 1.25 0. 0.
time (sec) N/A 0.569 0.113 180. 0. 2.07 0. 0.


















Problem 78 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A C F A F F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 75 75 48 269 0 111 0 0
normalized size 1 1. 0.64 3.59 0. 1.48 0. 0.
time (sec) N/A 0.513 0.055 0.496 0. 1.91 0. 0.


















Problem 79 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C F F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 94 94 75 0 0 173 0 0
normalized size 1 1. 0.8 0. 0. 1.84 0. 0.
time (sec) N/A 0.309 0.068 180. 0. 2.024 0. 0.


















Problem 80 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A B F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 65 65 72 2632 0 259 0 0
normalized size 1 1. 1.11 40.49 0. 3.98 0. 0.
time (sec) N/A 0.166 0.031 0.475 0. 2.081 0. 0.


















Problem 81 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C F F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 46 46 74 0 0 163 0 0
normalized size 1 1. 1.61 0. 0. 3.54 0. 0.
time (sec) N/A 0.401 0.036 180. 0. 2.303 0. 0.


















Problem 82 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A F F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 72 72 92 0 0 279 0 0
normalized size 1 1. 1.28 0. 0. 3.88 0. 0.
time (sec) N/A 0.383 0.057 180. 0. 2.028 0. 0.


















Problem 83 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A F F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 116 116 43 0 0 109 0 0
normalized size 1 1. 0.37 0. 0. 0.94 0. 0.
time (sec) N/A 0.424 0.052 180. 0. 2.066 0. 0.


















Problem 84 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A F F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 200 200 110 0 0 300 0 0
normalized size 1 1. 0.55 0. 0. 1.5 0. 0.
time (sec) N/A 0.5 0.1 180. 0. 2.077 0. 0.


















Problem 85 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A F F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 233 233 60 0 0 130 0 0
normalized size 1 1. 0.26 0. 0. 0.56 0. 0.
time (sec) N/A 0.5 0.069 180. 0. 2.032 0. 0.


















Problem 86 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A F F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 320 320 129 0 0 325 0 0
normalized size 1 1. 0.4 0. 0. 1.02 0. 0.
time (sec) N/A 0.567 0.141 180. 0. 2.096 0. 0.


















Problem 87 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A F F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 353 353 76 0 0 153 0 0
normalized size 1 1. 0.22 0. 0. 0.43 0. 0.
time (sec) N/A 0.605 0.097 180. 0. 2.094 0. 0.


















Problem 88 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A F F F F F F
verified N/A Yes N/A TBD TBD TBD TBD TBD
size 89 89 0 0 0 0 0 0
normalized size 1 1. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.272 0.635 0.434 0. 0. 0. 0.


















Problem 89 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F A F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 88 88 69 69 0 149 0 0
normalized size 1 1. 0.78 0.78 0. 1.69 0. 0.
time (sec) N/A 0.181 0.189 0.211 0. 2.067 0. 0.


















Problem 90 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C C F B F(-1) F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 75 75 110 117 0 213 0 0
normalized size 1 1. 1.47 1.56 0. 2.84 0. 0.
time (sec) N/A 0.165 0.154 0.213 0. 2.093 0. 0.


















Problem 91 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 44 52 0 111 0 0
normalized size 1 1. 0.98 1.16 0. 2.47 0. 0.
time (sec) N/A 0.125 0.09 0.2 0. 2.117 0. 0.


















Problem 92 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A C C F B F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 37 37 68 92 0 131 0 0
normalized size 1 1. 1.84 2.49 0. 3.54 0. 0.
time (sec) N/A 0.114 0.044 0.227 0. 2.102 0. 0.


















Problem 93 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F B A F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 71 71 73 87 0 213 48 0
normalized size 1 1. 1.03 1.23 0. 3. 0.68 0.
time (sec) N/A 0.119 0.04 0.204 0. 2.076 15.488 0.


















Problem 94 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A C F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 42 42 59 61 0 144 0 0
normalized size 1 1. 1.4 1.45 0. 3.43 0. 0.
time (sec) N/A 0.143 0.164 0.21 0. 2.162 0. 0.


















Problem 95 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F B F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 108 108 108 111 0 344 0 0
normalized size 1 1. 1. 1.03 0. 3.19 0. 0.
time (sec) N/A 0.176 0.151 0.227 0. 2.12 0. 0.


















Problem 96 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A C F A F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 85 85 90 86 0 197 0 0
normalized size 1 1. 1.06 1.01 0. 2.32 0. 0.
time (sec) N/A 0.165 0.242 0.22 0. 2.206 0. 0.


















Problem 97 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A B B A F A F F
verified N/A NO NO TBD TBD TBD TBD TBD
size 12 26 28 36 0 69 0 0
normalized size 1 2.17 2.33 3. 0. 5.75 0. 0.
time (sec) N/A 1.052 0.244 0.15 0. 2.042 0. 0.


















Problem 98 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A F F F F
verified N/A Yes NO TBD TBD TBD TBD TBD
size 61 61 52 104 0 0 0 0
normalized size 1 1. 0.85 1.7 0. 0. 0. 0.
time (sec) N/A 0.098 0.055 0.283 0. 0. 0. 0.


















Problem 99 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A A A B F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 57 57 84 62 51 613 0 0
normalized size 1 1. 1.47 1.09 0.89 10.75 0. 0.
time (sec) N/A 0.115 0.202 0.289 0.994 2.473 0. 0.


















Problem 100 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac









grade A A A F A B F(-1) F
verified N/A Yes Yes TBD TBD TBD TBD TBD
size 58 58 84 0 54 1138 0 0
normalized size 1 1. 1.45 0. 0.93 19.62 0. 0.
time (sec) N/A 0.113 0.248 0.401 0.999 2.608 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 [15] had the largest ratio of [ 1.2 ]

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 8 7 1. 10 0.7







2 A 7 6 1. 10 0.6







3 A 6 6 1. 8 0.75







4 A 4 4 1. 6 0.667







5 A 14 8 1. 10 0.8







6 A 5 5 1. 10 0.5







7 A 7 7 1. 10 0.7







8 A 8 8 1. 10 0.8







9 A 17 9 1. 12 0.75







10 A 11 8 1. 10 0.8







11 A 8 6 1. 8 0.75







12 A 17 9 1. 12 0.75







13 A 12 8 1. 12 0.667







14 A 23 11 1. 12 0.917







15 A 16 12 1. 10 1.2







16 A 10 7 1. 8 0.875







17 A 20 10 1. 12 0.833







18 A 14 9 1. 12 0.75







19 A 32 13 1. 12 1.083







20 A 4 3 1. 10 0.3







21 A 4 3 1. 10 0.3







22 A 4 3 1. 8 0.375







23 A 3 3 1. 6 0.5







24 A 7 6 1. 10 0.6







25 A 5 5 1. 10 0.5







26 A 6 5 1. 10 0.5







27 A 7 5 1. 10 0.5







28 A 3 3 1. 4 0.75







29 A 7 6 1. 10 0.6







30 A 7 6 1. 10 0.6







31 A 7 7 1. 10 0.7







32 A 5 5 1.3 10 0.5







33 A 5 5 1. 10 0.5







34 A 3 3 1.37 10 0.3







35 A 4 4 1. 8 0.5







36 A 3 3 1.62 6 0.5







37 A 5 5 1.33 10 0.5







38 B 6 6 2.83 10 0.6







39 C 5 5 1.53 10 0.5







40 A 8 6 1. 10 0.6







41 A 7 5 1. 10 0.5







42 A 10 6 1. 10 0.6







43 A 9 5 1. 10 0.5







44 A 12 6 1. 10 0.6







45 A 6 6 1. 12 0.5







46 A 5 5 1. 12 0.417







47 A 4 4 1.22 12 0.333







48 A 7 7 1. 12 0.583







49 A 5 5 1. 12 0.417







50 A 4 4 1. 12 0.333







51 A 6 6 1. 10 0.6







52 A 8 8 1. 8 1.







53 A 5 5 1.16 12 0.417







54 A 5 5 1. 12 0.417







55 A 7 7 1. 12 0.583







56 A 4 4 1. 12 0.333







57 A 4 4 1. 12 0.333







58 A 4 4 1. 10 0.4







59 A 5 5 1. 12 0.417







60 A 4 4 1. 12 0.333







61 A 4 4 1. 10 0.4







62 A 4 4 1. 8 0.5







63 A 6 6 1.22 12 0.5







64 A 4 4 1. 12 0.333







65 A 9 6 1. 12 0.5







66 A 8 6 1. 12 0.5







67 A 7 6 1. 12 0.5







68 A 8 5 1. 10 0.5







69 A 7 5 1. 8 0.625







70 A 5 3 1. 12 0.25







71 A 4 2 1. 12 0.167







72 A 5 3 1. 12 0.25







73 A 4 2 1. 12 0.167







74 A 5 3 1. 12 0.25







75 A 4 2 1. 12 0.167







76 A 8 5 1. 12 0.417







77 A 7 5 1. 12 0.417







78 A 6 5 1. 12 0.417







79 A 5 4 1. 10 0.4







80 A 6 4 1. 8 0.5







81 A 5 3 1. 12 0.25







82 A 5 3 1. 12 0.25







83 A 4 2 1. 12 0.167







84 A 5 3 1. 12 0.25







85 A 4 2 1. 12 0.167







86 A 5 3 1. 12 0.25







87 A 4 2 1. 12 0.167







88 A 5 4 1. 24 0.167







89 A 8 7 1. 22 0.318







90 A 7 7 1. 22 0.318







91 A 4 4 1. 22 0.182







92 A 5 5 1. 20 0.25







93 A 8 8 1. 19 0.421







94 A 5 5 1. 22 0.227







95 A 9 8 1. 22 0.364







96 A 8 7 1. 22 0.318







97 B 7 5 2.17 25 0.2







98 A 8 8 1. 19 0.421







99 A 5 5 1. 12 0.417







100 A 5 5 1. 14 0.357