Integrand size = 168, antiderivative size = 29 \[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\frac {4 \log (3)}{\log \left (x^2 \left (x+\frac {4}{\log \left (\frac {x}{3}\right )}-\log (x)\right )^2\right )} \]
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\[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {8 \log (3) \left (4 (1+\log (3))-4 \log (x)+\log ^2\left (\frac {x}{3}\right ) (1-2 x+\log (x))\right )}{x \log \left (\frac {x}{3}\right ) \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx \\ & = (8 \log (3)) \int \frac {4 (1+\log (3))-4 \log (x)+\log ^2\left (\frac {x}{3}\right ) (1-2 x+\log (x))}{x \log \left (\frac {x}{3}\right ) \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx \\ & = (8 \log (3)) \int \left (\frac {4 (1+\log (3))}{x \log \left (\frac {x}{3}\right ) \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}-\frac {2 \log \left (\frac {x}{3}\right )}{\left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}+\frac {\log \left (\frac {x}{3}\right )}{x \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}-\frac {4 \log (x)}{x \log \left (\frac {x}{3}\right ) \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}+\frac {\log \left (\frac {x}{3}\right ) \log (x)}{x \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}\right ) \, dx \\ & = (8 \log (3)) \int \frac {\log \left (\frac {x}{3}\right )}{x \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx+(8 \log (3)) \int \frac {\log \left (\frac {x}{3}\right ) \log (x)}{x \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx-(16 \log (3)) \int \frac {\log \left (\frac {x}{3}\right )}{\left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx-(32 \log (3)) \int \frac {\log (x)}{x \log \left (\frac {x}{3}\right ) \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx+(32 \log (3) (1+\log (3))) \int \frac {1}{x \log \left (\frac {x}{3}\right ) \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx \\ \end{align*}
\[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(27)=54\).
Time = 54.44 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.31
method | result | size |
parallelrisch | \(\frac {4 \ln \left (3\right )}{\ln \left (\frac {x^{2} \left (\ln \left (x \right )^{2} \ln \left (\frac {x}{3}\right )^{2}-2 x \ln \left (\frac {x}{3}\right )^{2} \ln \left (x \right )+x^{2} \ln \left (\frac {x}{3}\right )^{2}-8 \ln \left (\frac {x}{3}\right ) \ln \left (x \right )+8 x \ln \left (\frac {x}{3}\right )+16\right )}{\ln \left (\frac {x}{3}\right )^{2}}\right )}\) | \(67\) |
default | \(\frac {4 \ln \left (3\right )}{\ln \left (\frac {x^{2} \left (\ln \left (x \right )^{4}-2 \ln \left (x \right )^{3} \ln \left (3\right )+\ln \left (3\right )^{2} \ln \left (x \right )^{2}-2 x \ln \left (x \right )^{3}+4 x \ln \left (3\right ) \ln \left (x \right )^{2}-2 x \ln \left (x \right ) \ln \left (3\right )^{2}+x^{2} \ln \left (x \right )^{2}-2 x^{2} \ln \left (3\right ) \ln \left (x \right )+x^{2} \ln \left (3\right )^{2}-8 \ln \left (x \right )^{2}+8 \ln \left (3\right ) \ln \left (x \right )+8 x \ln \left (x \right )-8 x \ln \left (3\right )+16\right )}{\left (\ln \left (x \right )-\ln \left (3\right )\right )^{2}}\right )}\) | \(116\) |
risch | \(\text {Expression too large to display}\) | \(1002\) |
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.34 \[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\frac {4 \, \log \left (3\right )}{\log \left (\frac {x^{2} \log \left (\frac {1}{3} \, x\right )^{4} - 2 \, {\left (x^{3} - x^{2} \log \left (3\right )\right )} \log \left (\frac {1}{3} \, x\right )^{3} + {\left (x^{4} - 2 \, x^{3} \log \left (3\right ) + x^{2} \log \left (3\right )^{2} - 8 \, x^{2}\right )} \log \left (\frac {1}{3} \, x\right )^{2} + 16 \, x^{2} + 8 \, {\left (x^{3} - x^{2} \log \left (3\right )\right )} \log \left (\frac {1}{3} \, x\right )}{\log \left (\frac {1}{3} \, x\right )^{2}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (22) = 44\).
Time = 0.53 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.10 \[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\frac {4 \log {\left (3 \right )}}{\log {\left (\frac {x^{4} \left (\log {\left (x \right )} - \log {\left (3 \right )}\right )^{2} + 8 x^{3} \left (\log {\left (x \right )} - \log {\left (3 \right )}\right ) + x^{2} \left (\log {\left (x \right )} - \log {\left (3 \right )}\right )^{2} \log {\left (x \right )}^{2} + 16 x^{2} + \left (- 2 x^{3} \left (\log {\left (x \right )} - \log {\left (3 \right )}\right )^{2} - 8 x^{2} \left (\log {\left (x \right )} - \log {\left (3 \right )}\right )\right ) \log {\left (x \right )}}{\left (\log {\left (x \right )} - \log {\left (3 \right )}\right )^{2}} \right )}} \]
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Time = 0.38 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\frac {2 \, \log \left (3\right )}{\log \left (x \log \left (3\right ) - {\left (x + \log \left (3\right )\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 4\right ) + \log \left (x\right ) - \log \left (-\log \left (3\right ) + \log \left (x\right )\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (27) = 54\).
Time = 5.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 4.31 \[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\frac {4 \, \log \left (3\right )}{\log \left (x^{2} \log \left (3\right )^{2} - 2 \, x^{2} \log \left (3\right ) \log \left (x\right ) - 2 \, x \log \left (3\right )^{2} \log \left (x\right ) + x^{2} \log \left (x\right )^{2} + 4 \, x \log \left (3\right ) \log \left (x\right )^{2} + \log \left (3\right )^{2} \log \left (x\right )^{2} - 2 \, x \log \left (x\right )^{3} - 2 \, \log \left (3\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4} - 8 \, x \log \left (3\right ) + 8 \, x \log \left (x\right ) + 8 \, \log \left (3\right ) \log \left (x\right ) - 8 \, \log \left (x\right )^{2} + 16\right ) - \log \left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right ) + 2 \, \log \left (x\right )} \]
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Time = 13.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.69 \[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\frac {4\,\ln \left (3\right )}{\ln \left (\frac {8\,x^3\,\ln \left (\frac {x}{3}\right )+16\,x^2+x^4\,{\ln \left (\frac {x}{3}\right )}^2-\ln \left (x\right )\,\left (2\,x^3\,{\ln \left (\frac {x}{3}\right )}^2+8\,x^2\,\ln \left (\frac {x}{3}\right )\right )+x^2\,{\ln \left (\frac {x}{3}\right )}^2\,{\ln \left (x\right )}^2}{{\ln \left (\frac {x}{3}\right )}^2}\right )} \]
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