Integrand size = 195, antiderivative size = 26 \[ \int \frac {8 x^2-24 x^3-16 x^5+16 x^6+16 x^7-80 x^8+320 x^9-640 x^{10}+640 x^{11}-320 x^{12}+64 x^{13}+\left (6 x-14 x^2-16 x^5+32 x^6-40 x^7+120 x^8-240 x^9+240 x^{10}-120 x^{11}+24 x^{12}\right ) \log (2)+\left (1-2 x+x^3-5 x^4+7 x^5-5 x^6+10 x^7-20 x^8+20 x^9-10 x^{10}+2 x^{11}\right ) \log ^2(2)}{-4 x^5+20 x^6-40 x^7+40 x^8-20 x^9+4 x^{10}} \, dx=\left (2 x+\frac {1}{\left (x-x^2\right )^2}\right )^2 \left (x+\frac {\log (2)}{4}\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(163\) vs. \(2(26)=52\).
Time = 0.31 (sec) , antiderivative size = 163, normalized size of antiderivative = 6.27, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.005, Rules used = {2099} \[ \int \frac {8 x^2-24 x^3-16 x^5+16 x^6+16 x^7-80 x^8+320 x^9-640 x^{10}+640 x^{11}-320 x^{12}+64 x^{13}+\left (6 x-14 x^2-16 x^5+32 x^6-40 x^7+120 x^8-240 x^9+240 x^{10}-120 x^{11}+24 x^{12}\right ) \log (2)+\left (1-2 x+x^3-5 x^4+7 x^5-5 x^6+10 x^7-20 x^8+20 x^9-10 x^{10}+2 x^{11}\right ) \log ^2(2)}{-4 x^5+20 x^6-40 x^7+40 x^8-20 x^9+4 x^{10}} \, dx=4 x^4+\frac {\log ^2(2)}{16 x^4}+\frac {\log ^2(2)+\log (4)}{4 x^3}+\frac {1}{3} x^3 \log (64)+\frac {1}{4} x^2 \log ^2(2)+\frac {8+5 \log ^2(2)+16 \log (2)}{8 x^2}+\frac {8+\log ^2(2)+\log (64)}{4 (1-x)^3}+\frac {56+7 \log ^2(2)+40 \log (2)}{8 (1-x)^2}+\frac {8+3 \log ^2(2)+\log (1024)}{2 x}+\frac {\log (2) (10+\log (8))}{2 (1-x)}+\frac {(4+\log (2))^2}{16 (1-x)^4} \]
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Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \left (16 x^3-\frac {\log ^2(2)}{4 x^5}+\frac {1}{2} x \log ^2(2)-\frac {(4+\log (2))^2}{4 (-1+x)^5}+\frac {-56-40 \log (2)-7 \log ^2(2)}{4 (-1+x)^3}+\frac {-8-16 \log (2)-5 \log ^2(2)}{4 x^3}-\frac {3 \left (\log ^2(2)+\log (4)\right )}{4 x^4}+\frac {\log (2) (10+\log (8))}{2 (-1+x)^2}+x^2 \log (64)+\frac {3 \left (8+\log ^2(2)+\log (64)\right )}{4 (-1+x)^4}+\frac {-8-3 \log ^2(2)-\log (1024)}{2 x^2}\right ) \, dx \\ & = 4 x^4+\frac {\log ^2(2)}{16 x^4}+\frac {1}{4} x^2 \log ^2(2)+\frac {(4+\log (2))^2}{16 (1-x)^4}+\frac {8+16 \log (2)+5 \log ^2(2)}{8 x^2}+\frac {56+40 \log (2)+7 \log ^2(2)}{8 (1-x)^2}+\frac {\log ^2(2)+\log (4)}{4 x^3}+\frac {\log (2) (10+\log (8))}{2 (1-x)}+\frac {1}{3} x^3 \log (64)+\frac {8+\log ^2(2)+\log (64)}{4 (1-x)^3}+\frac {8+3 \log ^2(2)+\log (1024)}{2 x} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {8 x^2-24 x^3-16 x^5+16 x^6+16 x^7-80 x^8+320 x^9-640 x^{10}+640 x^{11}-320 x^{12}+64 x^{13}+\left (6 x-14 x^2-16 x^5+32 x^6-40 x^7+120 x^8-240 x^9+240 x^{10}-120 x^{11}+24 x^{12}\right ) \log (2)+\left (1-2 x+x^3-5 x^4+7 x^5-5 x^6+10 x^7-20 x^8+20 x^9-10 x^{10}+2 x^{11}\right ) \log ^2(2)}{-4 x^5+20 x^6-40 x^7+40 x^8-20 x^9+4 x^{10}} \, dx=\frac {\left (1+2 x^3-4 x^4+2 x^5\right )^2 (4 x+\log (2))^2}{16 (-1+x)^4 x^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(119\) vs. \(2(24)=48\).
Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.62
method | result | size |
risch | \(\frac {x^{2} \ln \left (2\right )^{2}}{4}+2 x^{3} \ln \left (2\right )+4 x^{4}+\frac {4 x^{7}+\frac {\left (-16+4 \ln \left (2\right )\right ) x^{6}}{2}+\frac {\left (\frac {\ln \left (2\right )^{2}}{2}-8 \ln \left (2\right )+8\right ) x^{5}}{2}+\frac {\left (-\ln \left (2\right )^{2}+4 \ln \left (2\right )\right ) x^{4}}{2}+\frac {x^{3} \ln \left (2\right )^{2}}{4}+x^{2}+\frac {x \ln \left (2\right )}{2}+\frac {\ln \left (2\right )^{2}}{16}}{x^{4} \left (x^{4}-4 x^{3}+6 x^{2}-4 x +1\right )}\) | \(120\) |
norman | \(\frac {x^{2}+\left (-16+2 \ln \left (2\right )\right ) x^{11}+\left (24-8 \ln \left (2\right )+\frac {\ln \left (2\right )^{2}}{4}\right ) x^{10}+\left (-\ln \left (2\right )^{2}+12 \ln \left (2\right )-16\right ) x^{9}+\left (-2 \ln \left (2\right )^{2}+10 \ln \left (2\right )-4\right ) x^{4}+\left (5 \ln \left (2\right )^{2}-30 \ln \left (2\right )+20\right ) x^{7}+\left (-\frac {35 \ln \left (2\right )^{2}}{4}+50 \ln \left (2\right )-32\right ) x^{6}+\left (\frac {25 \ln \left (2\right )^{2}}{4}+20-36 \ln \left (2\right )\right ) x^{5}+4 x^{12}+\frac {\ln \left (2\right )^{2}}{16}+\frac {x \ln \left (2\right )}{2}+\frac {x^{3} \ln \left (2\right )^{2}}{4}}{x^{4} \left (-1+x \right )^{4}}\) | \(145\) |
default | \(4 x^{4}+2 x^{3} \ln \left (2\right )+\frac {x^{2} \ln \left (2\right )^{2}}{4}-\frac {-6 \ln \left (2\right )^{2}-20 \ln \left (2\right )-16}{4 x}-\frac {-5 \ln \left (2\right )^{2}-16 \ln \left (2\right )-8}{8 x^{2}}+\frac {\ln \left (2\right )^{2}}{16 x^{4}}+\frac {\ln \left (2\right ) \left (\ln \left (2\right )+2\right )}{4 x^{3}}-\frac {-7 \ln \left (2\right )^{2}-40 \ln \left (2\right )-56}{8 \left (-1+x \right )^{2}}-\frac {-\ln \left (2\right )^{2}-8 \ln \left (2\right )-16}{16 \left (-1+x \right )^{4}}-\frac {3 \ln \left (2\right )^{2}+18 \ln \left (2\right )+24}{12 \left (-1+x \right )^{3}}-\frac {\ln \left (2\right ) \left (3 \ln \left (2\right )+10\right )}{2 \left (-1+x \right )}\) | \(149\) |
gosper | \(\frac {100 x^{5} \ln \left (2\right )^{2}+80 x^{7} \ln \left (2\right )^{2}-480 x^{7} \ln \left (2\right )-32 x^{4} \ln \left (2\right )^{2}+4 x^{3} \ln \left (2\right )^{2}+4 \ln \left (2\right )^{2} x^{10}+32 \ln \left (2\right ) x^{11}-16 \ln \left (2\right )^{2} x^{9}-128 \ln \left (2\right ) x^{10}+160 x^{4} \ln \left (2\right )-576 x^{5} \ln \left (2\right )+800 x^{6} \ln \left (2\right )+8 x \ln \left (2\right )-256 x^{11}+64 x^{12}-140 x^{6} \ln \left (2\right )^{2}+384 x^{10}-256 x^{9}+\ln \left (2\right )^{2}+320 x^{7}-64 x^{4}+16 x^{2}-512 x^{6}+320 x^{5}+192 \ln \left (2\right ) x^{9}}{16 x^{4} \left (x^{4}-4 x^{3}+6 x^{2}-4 x +1\right )}\) | \(193\) |
parallelrisch | \(\frac {100 x^{5} \ln \left (2\right )^{2}+80 x^{7} \ln \left (2\right )^{2}-480 x^{7} \ln \left (2\right )-32 x^{4} \ln \left (2\right )^{2}+4 x^{3} \ln \left (2\right )^{2}+4 \ln \left (2\right )^{2} x^{10}+32 \ln \left (2\right ) x^{11}-16 \ln \left (2\right )^{2} x^{9}-128 \ln \left (2\right ) x^{10}+160 x^{4} \ln \left (2\right )-576 x^{5} \ln \left (2\right )+800 x^{6} \ln \left (2\right )+8 x \ln \left (2\right )-256 x^{11}+64 x^{12}-140 x^{6} \ln \left (2\right )^{2}+384 x^{10}-256 x^{9}+\ln \left (2\right )^{2}+320 x^{7}-64 x^{4}+16 x^{2}-512 x^{6}+320 x^{5}+192 \ln \left (2\right ) x^{9}}{16 x^{4} \left (x^{4}-4 x^{3}+6 x^{2}-4 x +1\right )}\) | \(193\) |
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Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 165, normalized size of antiderivative = 6.35 \[ \int \frac {8 x^2-24 x^3-16 x^5+16 x^6+16 x^7-80 x^8+320 x^9-640 x^{10}+640 x^{11}-320 x^{12}+64 x^{13}+\left (6 x-14 x^2-16 x^5+32 x^6-40 x^7+120 x^8-240 x^9+240 x^{10}-120 x^{11}+24 x^{12}\right ) \log (2)+\left (1-2 x+x^3-5 x^4+7 x^5-5 x^6+10 x^7-20 x^8+20 x^9-10 x^{10}+2 x^{11}\right ) \log ^2(2)}{-4 x^5+20 x^6-40 x^7+40 x^8-20 x^9+4 x^{10}} \, dx=\frac {64 \, x^{12} - 256 \, x^{11} + 384 \, x^{10} - 256 \, x^{9} + 64 \, x^{8} + 64 \, x^{7} - 128 \, x^{6} + 64 \, x^{5} + {\left (4 \, x^{10} - 16 \, x^{9} + 24 \, x^{8} - 16 \, x^{7} + 4 \, x^{6} + 4 \, x^{5} - 8 \, x^{4} + 4 \, x^{3} + 1\right )} \log \left (2\right )^{2} + 16 \, x^{2} + 8 \, {\left (4 \, x^{11} - 16 \, x^{10} + 24 \, x^{9} - 16 \, x^{8} + 4 \, x^{7} + 4 \, x^{6} - 8 \, x^{5} + 4 \, x^{4} + x\right )} \log \left (2\right )}{16 \, {\left (x^{8} - 4 \, x^{7} + 6 \, x^{6} - 4 \, x^{5} + x^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (20) = 40\).
Time = 5.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.69 \[ \int \frac {8 x^2-24 x^3-16 x^5+16 x^6+16 x^7-80 x^8+320 x^9-640 x^{10}+640 x^{11}-320 x^{12}+64 x^{13}+\left (6 x-14 x^2-16 x^5+32 x^6-40 x^7+120 x^8-240 x^9+240 x^{10}-120 x^{11}+24 x^{12}\right ) \log (2)+\left (1-2 x+x^3-5 x^4+7 x^5-5 x^6+10 x^7-20 x^8+20 x^9-10 x^{10}+2 x^{11}\right ) \log ^2(2)}{-4 x^5+20 x^6-40 x^7+40 x^8-20 x^9+4 x^{10}} \, dx=4 x^{4} + 2 x^{3} \log {\left (2 \right )} + \frac {x^{2} \log {\left (2 \right )}^{2}}{4} + \frac {64 x^{7} + x^{6} \left (-128 + 32 \log {\left (2 \right )}\right ) + x^{5} \left (- 64 \log {\left (2 \right )} + 4 \log {\left (2 \right )}^{2} + 64\right ) + x^{4} \left (- 8 \log {\left (2 \right )}^{2} + 32 \log {\left (2 \right )}\right ) + 4 x^{3} \log {\left (2 \right )}^{2} + 16 x^{2} + 8 x \log {\left (2 \right )} + \log {\left (2 \right )}^{2}}{16 x^{8} - 64 x^{7} + 96 x^{6} - 64 x^{5} + 16 x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (25) = 50\).
Time = 0.20 (sec) , antiderivative size = 115, normalized size of antiderivative = 4.42 \[ \int \frac {8 x^2-24 x^3-16 x^5+16 x^6+16 x^7-80 x^8+320 x^9-640 x^{10}+640 x^{11}-320 x^{12}+64 x^{13}+\left (6 x-14 x^2-16 x^5+32 x^6-40 x^7+120 x^8-240 x^9+240 x^{10}-120 x^{11}+24 x^{12}\right ) \log (2)+\left (1-2 x+x^3-5 x^4+7 x^5-5 x^6+10 x^7-20 x^8+20 x^9-10 x^{10}+2 x^{11}\right ) \log ^2(2)}{-4 x^5+20 x^6-40 x^7+40 x^8-20 x^9+4 x^{10}} \, dx=4 \, x^{4} + 2 \, x^{3} \log \left (2\right ) + \frac {1}{4} \, x^{2} \log \left (2\right )^{2} + \frac {64 \, x^{7} + 32 \, x^{6} {\left (\log \left (2\right ) - 4\right )} + 4 \, {\left (\log \left (2\right )^{2} - 16 \, \log \left (2\right ) + 16\right )} x^{5} - 8 \, {\left (\log \left (2\right )^{2} - 4 \, \log \left (2\right )\right )} x^{4} + 4 \, x^{3} \log \left (2\right )^{2} + 16 \, x^{2} + 8 \, x \log \left (2\right ) + \log \left (2\right )^{2}}{16 \, {\left (x^{8} - 4 \, x^{7} + 6 \, x^{6} - 4 \, x^{5} + x^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 111, normalized size of antiderivative = 4.27 \[ \int \frac {8 x^2-24 x^3-16 x^5+16 x^6+16 x^7-80 x^8+320 x^9-640 x^{10}+640 x^{11}-320 x^{12}+64 x^{13}+\left (6 x-14 x^2-16 x^5+32 x^6-40 x^7+120 x^8-240 x^9+240 x^{10}-120 x^{11}+24 x^{12}\right ) \log (2)+\left (1-2 x+x^3-5 x^4+7 x^5-5 x^6+10 x^7-20 x^8+20 x^9-10 x^{10}+2 x^{11}\right ) \log ^2(2)}{-4 x^5+20 x^6-40 x^7+40 x^8-20 x^9+4 x^{10}} \, dx=4 \, x^{4} + 2 \, x^{3} \log \left (2\right ) + \frac {1}{4} \, x^{2} \log \left (2\right )^{2} + \frac {64 \, x^{7} + 32 \, x^{6} \log \left (2\right ) + 4 \, x^{5} \log \left (2\right )^{2} - 128 \, x^{6} - 64 \, x^{5} \log \left (2\right ) - 8 \, x^{4} \log \left (2\right )^{2} + 64 \, x^{5} + 32 \, x^{4} \log \left (2\right ) + 4 \, x^{3} \log \left (2\right )^{2} + 16 \, x^{2} + 8 \, x \log \left (2\right ) + \log \left (2\right )^{2}}{16 \, {\left (x^{2} - x\right )}^{4}} \]
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Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 5.04 \[ \int \frac {8 x^2-24 x^3-16 x^5+16 x^6+16 x^7-80 x^8+320 x^9-640 x^{10}+640 x^{11}-320 x^{12}+64 x^{13}+\left (6 x-14 x^2-16 x^5+32 x^6-40 x^7+120 x^8-240 x^9+240 x^{10}-120 x^{11}+24 x^{12}\right ) \log (2)+\left (1-2 x+x^3-5 x^4+7 x^5-5 x^6+10 x^7-20 x^8+20 x^9-10 x^{10}+2 x^{11}\right ) \log ^2(2)}{-4 x^5+20 x^6-40 x^7+40 x^8-20 x^9+4 x^{10}} \, dx=\frac {x^2\,{\ln \left (2\right )}^2}{4}+\frac {16\,x^7+\left (2\,\ln \left (16\right )-32\right )\,x^6+\left ({\ln \left (2\right )}^2-\frac {4\,\ln \left (256\right )}{3}-\frac {4\,\ln \left (16\right )}{3}+16\right )\,x^5+\left (\frac {4\,\ln \left (16\right )}{3}+\frac {\ln \left (256\right )}{3}-2\,{\ln \left (2\right )}^2\right )\,x^4+{\ln \left (2\right )}^2\,x^3+4\,x^2+2\,\ln \left (2\right )\,x+\frac {{\ln \left (2\right )}^2}{4}}{4\,x^8-16\,x^7+24\,x^6-16\,x^5+4\,x^4}+2\,x^3\,\ln \left (2\right )+4\,x^4 \]
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