Integrand size = 72, antiderivative size = 22 \[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=1+\frac {\log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{x-\log (x)} \]
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\[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=\int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{x (x-\log (x))^2 \log \left (2 x^2\right )} \, dx \\ & = \int \left (\frac {-2+\log \left (2 x^2\right )}{x (x-\log (x)) \log \left (2 x^2\right )}-\frac {(-1+x) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{x (x-\log (x))^2}\right ) \, dx \\ & = \int \frac {-2+\log \left (2 x^2\right )}{x (x-\log (x)) \log \left (2 x^2\right )} \, dx-\int \frac {(-1+x) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{x (x-\log (x))^2} \, dx \\ & = \int \left (\frac {1}{x (x-\log (x))}-\frac {2}{x (x-\log (x)) \log \left (2 x^2\right )}\right ) \, dx-\int \left (\frac {\log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{(x-\log (x))^2}-\frac {\log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{x (x-\log (x))^2}\right ) \, dx \\ & = -\left (2 \int \frac {1}{x (x-\log (x)) \log \left (2 x^2\right )} \, dx\right )+\int \frac {1}{x (x-\log (x))} \, dx-\int \frac {\log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{(x-\log (x))^2} \, dx+\int \frac {\log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{x (x-\log (x))^2} \, dx \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=\frac {\log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{x-\log (x)} \]
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Time = 2.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
parallelrisch | \(\frac {\ln \left (\frac {x}{\ln \left (2 x^{2}\right )}\right )}{x -\ln \left (x \right )}\) | \(21\) |
risch | \(\text {Expression too large to display}\) | \(878\) |
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Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=\frac {\log \left (\frac {x}{\log \left (2\right ) + 2 \, \log \left (x\right )}\right )}{x - \log \left (x\right )} \]
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Exception generated. \[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=\frac {x - \log \left (\log \left (2\right ) + 2 \, \log \left (x\right )\right )}{x - \log \left (x\right )} \]
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Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=\frac {x}{x - \log \left (x\right )} - \frac {\log \left (\log \left (2 \, x^{2}\right )\right )}{x - \log \left (x\right )} \]
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Time = 8.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {-2 x+2 \log (x)+(x-\log (x)) \log \left (2 x^2\right )+(1-x) \log \left (2 x^2\right ) \log \left (\frac {x}{\log \left (2 x^2\right )}\right )}{\left (x^3-2 x^2 \log (x)+x \log ^2(x)\right ) \log \left (2 x^2\right )} \, dx=\frac {\ln \left (\frac {x}{\ln \left (2\,x^2\right )}\right )}{x-\ln \left (x\right )} \]
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