Integrand size = 144, antiderivative size = 26 \[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=x \left (-x+\frac {x}{5+x \left (6 x+\log \left (\frac {36}{x}\right )\right )+\log (x)}\right ) \]
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\[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=\int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (-41+x-120 x^2-72 x^4-2 x^2 \log ^2\left (\frac {36}{x}\right )-6 \left (3+4 x^2\right ) \log (x)-2 \log ^2(x)-x \log \left (\frac {36}{x}\right ) \left (19+24 x^2+4 \log (x)\right )\right )}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2} \, dx \\ & = \int \left (-2 x-\frac {x \left (1-x+12 x^2+x \log \left (\frac {36}{x}\right )\right )}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2}+\frac {2 x}{5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)}\right ) \, dx \\ & = -x^2+2 \int \frac {x}{5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)} \, dx-\int \frac {x \left (1-x+12 x^2+x \log \left (\frac {36}{x}\right )\right )}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2} \, dx \\ & = -x^2+2 \int \frac {x}{5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)} \, dx-\int \left (\frac {x}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2}-\frac {x^2}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2}+\frac {12 x^3}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2}+\frac {x^2 \log \left (\frac {36}{x}\right )}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2}\right ) \, dx \\ & = -x^2+2 \int \frac {x}{5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)} \, dx-12 \int \frac {x^3}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2} \, dx-\int \frac {x}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2} \, dx+\int \frac {x^2}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2} \, dx-\int \frac {x^2 \log \left (\frac {36}{x}\right )}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2} \, dx \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=-x^2+\frac {x^2}{5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)} \]
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Time = 2.41 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54
method | result | size |
risch | \(-x^{2}+\frac {2 x^{2}}{10+4 x \ln \left (3\right )+4 x \ln \left (2\right )+12 x^{2}-2 x \ln \left (x \right )+2 \ln \left (x \right )}\) | \(40\) |
parallelrisch | \(\frac {-x^{3} \ln \left (\frac {36}{x}\right )-6 x^{4}-4 x^{2}-x^{2} \ln \left (x \right )}{x \ln \left (\frac {36}{x}\right )+6 x^{2}+\ln \left (x \right )+5}\) | \(50\) |
default | \(\frac {\left (-6+\frac {\ln \left (\frac {1}{x}\right )}{x^{2}}+\frac {-4-\ln \left (x \right )-\ln \left (\frac {1}{x}\right )}{x^{2}}-\frac {\ln \left (\frac {1}{x}\right )}{x}-\frac {2 \ln \left (6\right )}{x}\right ) x^{2}}{-\frac {\ln \left (\frac {1}{x}\right )}{x^{2}}+\frac {\ln \left (x \right )+\ln \left (\frac {1}{x}\right )}{x^{2}}+6+\frac {2 \ln \left (6\right )}{x}+\frac {\ln \left (\frac {1}{x}\right )}{x}+\frac {5}{x^{2}}}\) | \(91\) |
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (26) = 52\).
Time = 0.23 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.27 \[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=-\frac {6 \, x^{4} + 2 \, x^{2} \log \left (6\right ) + 4 \, x^{2} + {\left (x^{3} - x^{2}\right )} \log \left (\frac {36}{x}\right )}{6 \, x^{2} + {\left (x - 1\right )} \log \left (\frac {36}{x}\right ) + 2 \, \log \left (6\right ) + 5} \]
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Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=- x^{2} - \frac {x^{2}}{- 6 x^{2} - 2 x \log {\left (6 \right )} + \left (x - 1\right ) \log {\left (x \right )} - 5} \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).
Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=-\frac {6 \, x^{4} + 2 \, x^{3} {\left (\log \left (3\right ) + \log \left (2\right )\right )} + 4 \, x^{2} - {\left (x^{3} - x^{2}\right )} \log \left (x\right )}{6 \, x^{2} + 2 \, x {\left (\log \left (3\right ) + \log \left (2\right )\right )} - {\left (x - 1\right )} \log \left (x\right ) + 5} \]
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Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=-x^{2} + \frac {x^{2}}{6 \, x^{2} + 2 \, x \log \left (6\right ) - x \log \left (x\right ) + \log \left (x\right ) + 5} \]
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Time = 12.61 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=-\frac {x^2\,\left (\ln \left (x\right )+x\,\ln \left (\frac {36}{x}\right )+6\,x^2+4\right )}{\ln \left (x\right )+x\,\ln \left (\frac {36}{x}\right )+6\,x^2+5} \]
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