Chapter 2detailed summary tables of results

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 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 }

2.1.2 Mathematica

A grade: { 27, 28, 29, 30, 31, 36, 37, 38, 39, 40, 47, 48, 49, 50 }

C 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, 32, 33, 34, 35, 41, 42, 43, 44, 45, 46, 51 }

2.1.3 Maple

A grade: { 29, 32, 33, 36, 37, 38, 41, 42, 43 }

B grade: { 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 45, 46, 47, 48, 51 }

C grade: { 7, 16 }

F grade: { 25, 26, 30, 31, 34, 35, 39, 40, 44, 49, 50 }

2.1.4 Maxima

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 }

2.1.5 FriCAS

A grade: { 27, 28, 29, 30, 31, 36, 37, 38, 39, 40 }

B grade: { 45, 46, 47, 48, 49, 50 }

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, 32, 33, 34, 35, 41, 42, 43, 44, 51 }

2.1.6 Sympy

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 }

2.1.7 Giac

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 }

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 }

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 C B F F(-1) F F(-1) F(-1) veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 975 975 690 17767396 0 0 0 0 -1 normalized size 1 1.00 0.71 18222.97 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 24.463 6.246 2.127 0.000 0.000 0.000 0.000 0.000
 Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F F(-1) F F(-1) F(-1) veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 889 889 582 17247437 0 0 0 0 -1 normalized size 1 1.00 0.65 19400.94 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 23.995 5.412 0.971 0.000 0.000 0.000 0.000 0.000
 Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F F(-1) F F(-1) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 748 748 451 17766851 0 0 0 0 -1 normalized size 1 1.00 0.60 23752.47 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 23.630 2.527 0.997 0.000 0.000 0.000 0.000 0.000
 Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F F(-1) F F(-1) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 676 676 405 17246975 0 0 0 0 -1 normalized size 1 1.00 0.60 25513.28 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 26.571 0.642 0.946 0.000 0.000 0.000 0.000 0.000
 Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F F(-1) F F(-1) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 601 601 250 17766655 0 0 0 0 -1 normalized size 1 1.00 0.42 29561.82 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 23.288 0.307 0.778 0.000 0.000 0.000 0.000 0.000
 Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F(-2) F(-1) F F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 574 574 228 17246812 0 0 0 0 -1 normalized size 1 1.00 0.40 30046.71 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 23.269 0.198 0.831 0.000 0.000 0.000 0.000 0.000
 Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C C F(-2) F(-1) F F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 571 571 223 53065 0 0 0 0 -1 normalized size 1 1.00 0.39 92.93 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 23.583 0.361 9.102 0.000 0.000 0.000 0.000 0.000
 Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F F(-1) F F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 612 612 261 1165398 0 0 0 0 -1 normalized size 1 1.00 0.43 1904.25 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 23.744 1.219 27.166 0.000 0.000 0.000 0.000 0.000
 Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F F(-1) F F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 690 690 289 2844525 0 0 0 0 -1 normalized size 1 1.00 0.42 4122.50 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 23.733 1.920 152.533 0.000 0.000 0.000 0.000 0.000
 Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F(-1) F(-1) F F(-2) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 548 548 456 9581348 0 0 0 0 -1 normalized size 1 1.00 0.83 17484.21 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 1.085 6.135 0.844 0.000 0.000 0.000 0.000 0.000
 Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F(-1) F(-1) F F(-2) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 495 495 283 7491876 0 0 0 0 -1 normalized size 1 1.00 0.57 15135.10 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.820 2.757 0.823 0.000 0.000 0.000 0.000 0.000
 Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F F(-1) F F(-2) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 383 383 252 9581103 0 0 0 0 -1 normalized size 1 1.00 0.66 25015.93 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.731 0.942 0.813 0.000 0.000 0.000 0.000 0.000
 Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F F(-1) F F(-2) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 352 352 228 7491751 0 0 0 0 -1 normalized size 1 1.00 0.65 21283.38 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.373 0.205 0.790 0.000 0.000 0.000 0.000 0.000
 Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F(-2) F(-1) F F(-2) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 294 294 173 9338543 0 0 0 0 -1 normalized size 1 1.00 0.59 31763.75 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.286 0.115 0.834 0.000 0.000 0.000 0.000 0.000
 Problem 15 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F(-2) F(-1) F F(-2) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 298 298 173 7300213 0 0 0 0 -1 normalized size 1 1.00 0.58 24497.36 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.311 0.141 0.786 0.000 0.000 0.000 0.000 0.000
 Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C C F F(-1) F F(-2) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 350 350 223 28519 0 0 0 0 -1 normalized size 1 1.00 0.64 81.48 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.701 0.404 4.291 0.000 0.000 0.000 0.000 0.000
 Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F F(-1) F F(-2) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 395 395 264 636822 0 0 0 0 -1 normalized size 1 1.00 0.67 1612.21 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.765 1.575 21.380 0.000 0.000 0.000 0.000 0.000
 Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F F(-1) F F(-2) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 500 500 315 2045168 0 0 0 0 -1 normalized size 1 1.00 0.63 4090.34 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.800 5.976 52.267 0.000 0.000 0.000 0.000 0.000
 Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F(-1) F(-1) F F(-2) F(-1) veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 1190 1190 884 13067521 0 0 0 0 -1 normalized size 1 1.00 0.74 10981.11 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 6.507 25.135 0.924 0.000 0.000 0.000 0.000 0.000
 Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F(-1) F(-1) F F(-2) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 864 864 611 13066795 0 0 0 0 -1 normalized size 1 1.00 0.71 15123.61 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 4.790 21.301 0.890 0.000 0.000 0.000 0.000 0.000
 Problem 21 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F(-2) F(-1) F F(-2) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 686 686 286 13066487 0 0 0 0 -1 normalized size 1 1.00 0.42 19047.36 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 4.514 17.333 0.858 0.000 0.000 0.000 0.000 0.000
 Problem 22 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F(-2) F(-1) F F(-2) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 638 638 328 11847956 0 0 0 0 -1 normalized size 1 1.00 0.51 18570.46 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 3.950 4.992 0.827 0.000 0.000 0.000 0.000 0.000
 Problem 23 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F(-2) F(-1) F F(-2) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 635 635 318 13066297 0 0 0 0 -1 normalized size 1 1.00 0.50 20576.85 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 3.791 4.751 0.805 0.000 0.000 0.000 0.000 0.000
 Problem 24 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F(-1) F(-1) F F(-2) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 750 750 450 15830985 0 0 0 0 -1 normalized size 1 1.00 0.60 21107.98 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 4.578 5.192 168.926 0.000 0.000 0.000 0.000 0.000
 Problem 25 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C F(-1) F(-1) F(-1) F F(-2) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 829 829 583 0 0 0 0 0 -1 normalized size 1 1.00 0.70 0.00 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 4.939 6.171 180.000 0.000 0.000 0.000 0.000 0.000
 Problem 26 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C F(-1) F(-1) F(-1) F F(-2) F(-1) veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 1007 1007 786 0 0 0 0 0 -1 normalized size 1 1.00 0.78 0.00 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 4.928 6.208 180.000 0.000 0.000 0.000 0.000 0.000
 Problem 27 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A B F A F F(-1) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 270 270 467 684 0 1405 0 0 -1 normalized size 1 1.00 1.73 2.53 0.00 5.20 0.00 0.00 -0.00 time (sec) N/A 0.613 6.133 0.674 0.000 8.732 0.000 0.000 0.000
 Problem 28 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A B F A F F(-1) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 209 209 208 467 0 1199 0 0 -1 normalized size 1 1.00 1.00 2.23 0.00 5.74 0.00 0.00 -0.00 time (sec) N/A 0.349 1.566 0.435 0.000 6.366 0.000 0.000 0.000
 Problem 29 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F A F F(-1) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 179 179 180 289 0 1057 0 0 -1 normalized size 1 1.00 1.01 1.61 0.00 5.91 0.00 0.00 -0.01 time (sec) N/A 0.224 0.300 0.396 0.000 3.292 0.000 0.000 0.000
 Problem 30 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F A F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 203 203 192 0 0 2097 0 0 -1 normalized size 1 1.00 0.95 0.00 0.00 10.33 0.00 0.00 -0.00 time (sec) N/A 0.292 1.178 1.494 0.000 1.953 0.000 0.000 0.000
 Problem 31 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F A F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 435 435 187 0 0 1186 0 0 -1 normalized size 1 1.00 0.43 0.00 0.00 2.73 0.00 0.00 -0.00 time (sec) N/A 0.544 1.199 1.371 0.000 5.566 0.000 0.000 0.000
 Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C A F F F F(-1) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 1254 1254 639 1945 0 0 0 0 -1 normalized size 1 1.00 0.51 1.55 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.993 29.608 0.417 0.000 1.672 0.000 0.000 0.000
 Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C A F F(-1) F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 829 829 428 1497 0 0 0 0 -1 normalized size 1 1.00 0.52 1.81 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.552 1.981 0.480 0.000 0.000 0.000 0.000 0.000
 Problem 34 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C F F F(-1) F F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 861 861 1258 0 0 0 0 0 -1 normalized size 1 1.00 1.46 0.00 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.585 27.042 1.374 0.000 0.000 0.000 0.000 0.000
 Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C F F F(-1) F F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 943 943 1590 0 0 0 0 0 -1 normalized size 1 1.00 1.69 0.00 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.717 31.415 1.316 0.000 0.000 0.000 0.000 0.000
 Problem 36 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F(-1) A F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 182 182 173 240 0 1226 0 0 -1 normalized size 1 1.00 0.95 1.32 0.00 6.74 0.00 0.00 -0.01 time (sec) N/A 0.353 2.488 0.476 0.000 2.923 0.000 0.000 0.000
 Problem 37 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F A F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 141 141 136 155 0 993 0 0 -1 normalized size 1 1.00 0.96 1.10 0.00 7.04 0.00 0.00 -0.01 time (sec) N/A 0.210 0.251 0.437 0.000 2.201 0.000 0.000 0.000
 Problem 38 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F A F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 79 79 79 102 0 299 0 0 -1 normalized size 1 1.00 1.00 1.29 0.00 3.78 0.00 0.00 -0.01 time (sec) N/A 0.114 0.114 0.394 0.000 0.820 0.000 0.000 0.000
 Problem 39 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F A F F(-2) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 142 142 136 0 0 1015 0 0 -1 normalized size 1 1.00 0.96 0.00 0.00 7.15 0.00 0.00 -0.01 time (sec) N/A 0.228 0.752 1.733 0.000 1.924 0.000 0.000 0.000
 Problem 40 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F(-1) A F F(-1) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 249 249 188 0 0 1350 0 0 -1 normalized size 1 1.00 0.76 0.00 0.00 5.42 0.00 0.00 -0.00 time (sec) N/A 0.318 4.247 1.721 0.000 3.025 0.000 0.000 0.000
 Problem 41 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C A F(-1) F(-1) F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 662 662 533 646 0 0 0 0 -1 normalized size 1 1.00 0.81 0.98 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.439 23.158 0.456 0.000 0.000 0.000 0.000 0.000
 Problem 42 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C A F F(-1) F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 436 436 311 402 0 0 0 0 -1 normalized size 1 1.00 0.71 0.92 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.341 11.188 0.424 0.000 0.000 0.000 0.000 0.000
 Problem 43 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C A F F(-1) F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 436 436 235 231 0 0 0 0 -1 normalized size 1 1.00 0.54 0.53 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.306 0.739 0.573 0.000 0.000 0.000 0.000 0.000
 Problem 44 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C F F(-1) F(-1) F F(-1) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 707 707 683 0 0 0 0 0 -1 normalized size 1 1.00 0.97 0.00 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.728 28.157 1.475 0.000 0.000 0.000 0.000 0.000
 Problem 45 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F(-2) B F F(-1) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 235 235 182725 826 0 3773 0 0 -1 normalized size 1 1.00 777.55 3.51 0.00 16.06 0.00 0.00 -0.00 time (sec) N/A 0.549 34.860 0.552 0.000 6.310 0.000 0.000 0.000
 Problem 46 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F B F F(-1) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 159 159 57597 601 0 1095 0 0 -1 normalized size 1 1.00 362.25 3.78 0.00 6.89 0.00 0.00 -0.01 time (sec) N/A 0.394 36.240 0.446 0.000 1.277 0.000 0.000 0.000
 Problem 47 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A B F B F F(-1) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 154 154 155 508 0 1077 0 0 -1 normalized size 1 1.00 1.01 3.30 0.00 6.99 0.00 0.00 -0.01 time (sec) N/A 0.285 2.895 0.451 0.000 1.151 0.000 0.000 0.000
 Problem 48 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A B F B F F(-1) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 155 155 156 417 0 1099 0 0 -1 normalized size 1 1.00 1.01 2.69 0.00 7.09 0.00 0.00 -0.01 time (sec) N/A 0.218 2.978 0.380 0.000 1.063 0.000 0.000 0.000
 Problem 49 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F(-2) B F F(-2) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 280 280 278 0 0 3951 0 0 -1 normalized size 1 1.00 0.99 0.00 0.00 14.11 0.00 0.00 -0.00 time (sec) N/A 0.387 3.449 1.549 0.000 7.467 0.000 0.000 0.000
 Problem 50 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F(-1) B F F(-1) F(-1) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 477 477 555 0 0 5189 0 0 -1 normalized size 1 1.00 1.16 0.00 0.00 10.88 0.00 0.00 -0.00 time (sec) N/A 0.552 6.069 1.580 0.000 9.152 0.000 0.000 0.000
 Problem 51 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A C B F F(-1) F F(-1) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD TBD size 981 981 831 3598 0 0 0 0 -1 normalized size 1 1.00 0.85 3.67 0.00 0.00 0.00 0.00 -0.00 time (sec) N/A 0.876 34.453 0.560 0.000 0.000 0.000 0.000 0.000

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 [1] had the largest ratio of [.4242]

 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 21 14 1.00 33 0.424

2 A 19 12 1.00 33 0.364

3 A 16 12 1.00 33 0.364

4 A 10 9 1.00 33 0.273

5 A 10 9 1.00 31 0.290

6 A 9 7 1.00 24 0.292

7 A 18 13 1.00 31 0.419

8 A 17 12 1.00 33 0.364

9 A 21 14 1.00 33 0.424

10 A 15 10 1.00 33 0.303

11 A 14 9 1.00 33 0.273

12 A 11 8 1.00 33 0.242

13 A 9 7 1.00 33 0.212

14 A 6 4 1.00 31 0.129

15 A 6 3 1.00 24 0.125

16 A 10 7 1.00 31 0.226

17 A 11 8 1.00 33 0.242

18 A 14 9 1.00 33 0.273

19 A 20 12 1.00 33 0.364

20 A 14 11 1.00 33 0.333

21 A 10 7 1.00 33 0.212

22 A 7 5 1.00 33 0.152

23 A 7 5 1.00 31 0.161

24 A 13 10 1.00 31 0.323

25 A 13 10 1.00 33 0.303

26 A 18 12 1.00 33 0.364

27 A 9 8 1.00 35 0.229

28 A 8 7 1.00 35 0.200

29 A 8 7 1.00 33 0.212

30 A 10 7 1.00 33 0.212

31 A 22 9 1.00 35 0.257

32 A 14 9 1.00 35 0.257

33 A 8 6 1.00 26 0.231

34 A 9 8 1.00 35 0.229

35 A 10 8 1.00 35 0.229

36 A 8 7 1.00 35 0.200

37 A 7 6 1.00 35 0.171

38 A 4 4 1.00 33 0.121

39 A 8 5 1.00 33 0.152

40 A 11 6 1.00 35 0.171

41 A 5 5 1.00 35 0.143

42 A 4 4 1.00 35 0.114

43 A 4 3 1.00 26 0.115

44 A 7 7 1.00 35 0.200

45 A 8 7 1.00 35 0.200

46 A 6 6 1.00 35 0.171

47 A 6 6 1.00 35 0.171

48 A 6 6 1.00 33 0.182

49 A 12 7 1.00 33 0.212

50 A 16 8 1.00 35 0.229

51 A 9 8 1.00 35 0.229