Integrand size = 115, antiderivative size = 21 \[ \int \frac {x-x \log (x)+\left (-4 x^6 \log (x)+4 x^5 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )+\left (x \log (x)-\log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )\right )}{\left (-x^3 \log (x)+x^2 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )} \, dx=x^4+\frac {\log \left (\log \left (5 \left (1-\frac {x}{\log (x)}\right )\right )\right )}{x} \]
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\[ \int \frac {x-x \log (x)+\left (-4 x^6 \log (x)+4 x^5 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )+\left (x \log (x)-\log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )\right )}{\left (-x^3 \log (x)+x^2 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )} \, dx=\int \frac {x-x \log (x)+\left (-4 x^6 \log (x)+4 x^5 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )+\left (x \log (x)-\log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )\right )}{\left (-x^3 \log (x)+x^2 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-x+x \log (x)-\left (-4 x^6 \log (x)+4 x^5 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )-\left (x \log (x)-\log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )\right )}{x^2 (x-\log (x)) \log (x) \log \left (-\frac {5 (x-\log (x))}{\log (x)}\right )} \, dx \\ & = \int \left (\frac {-1+\log (x)+4 x^5 \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )-4 x^4 \log ^2(x) \log \left (5-\frac {5 x}{\log (x)}\right )}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )}-\frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2}\right ) \, dx \\ & = \int \frac {-1+\log (x)+4 x^5 \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )-4 x^4 \log ^2(x) \log \left (5-\frac {5 x}{\log (x)}\right )}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )} \, dx-\int \frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2} \, dx \\ & = \int \left (4 x^3+\frac {-1+\log (x)}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )}\right ) \, dx-\int \frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2} \, dx \\ & = x^4+\int \frac {-1+\log (x)}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )} \, dx-\int \frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2} \, dx \\ & = x^4+\int \left (\frac {1}{x (x-\log (x)) \log \left (5-\frac {5 x}{\log (x)}\right )}-\frac {1}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )}\right ) \, dx-\int \frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2} \, dx \\ & = x^4+\int \frac {1}{x (x-\log (x)) \log \left (5-\frac {5 x}{\log (x)}\right )} \, dx-\int \frac {1}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )} \, dx-\int \frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2} \, dx \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {x-x \log (x)+\left (-4 x^6 \log (x)+4 x^5 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )+\left (x \log (x)-\log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )\right )}{\left (-x^3 \log (x)+x^2 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )} \, dx=x^4+\frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x} \]
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Time = 9.48 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33
method | result | size |
parallelrisch | \(\frac {2 x^{5}+2 \ln \left (\ln \left (\frac {5 \ln \left (x \right )-5 x}{\ln \left (x \right )}\right )\right )}{2 x}\) | \(28\) |
risch | \(\frac {\ln \left (\ln \left (5\right )+i \pi -\ln \left (\ln \left (x \right )\right )+\ln \left (x -\ln \left (x \right )\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right ) \left (\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )\right ) \left (\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right )-\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )\right )\right )}{2}+i \pi \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right )^{2} \left (-\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right )-1\right )\right )}{x}+x^{4}\) | \(135\) |
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Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {x-x \log (x)+\left (-4 x^6 \log (x)+4 x^5 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )+\left (x \log (x)-\log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )\right )}{\left (-x^3 \log (x)+x^2 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )} \, dx=\frac {x^{5} + \log \left (\log \left (-\frac {5 \, {\left (x - \log \left (x\right )\right )}}{\log \left (x\right )}\right )\right )}{x} \]
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Time = 0.49 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {x-x \log (x)+\left (-4 x^6 \log (x)+4 x^5 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )+\left (x \log (x)-\log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )\right )}{\left (-x^3 \log (x)+x^2 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )} \, dx=x^{4} + \frac {\log {\left (\log {\left (\frac {- 5 x + 5 \log {\left (x \right )}}{\log {\left (x \right )}} \right )} \right )}}{x} \]
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Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {x-x \log (x)+\left (-4 x^6 \log (x)+4 x^5 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )+\left (x \log (x)-\log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )\right )}{\left (-x^3 \log (x)+x^2 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )} \, dx=\frac {x^{5} + \log \left (\log \left (5\right ) + \log \left (-x + \log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )\right )}{x} \]
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Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {x-x \log (x)+\left (-4 x^6 \log (x)+4 x^5 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )+\left (x \log (x)-\log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )\right )}{\left (-x^3 \log (x)+x^2 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )} \, dx=x^{4} + \frac {\log \left (\log \left (-5 \, x + 5 \, \log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )\right )}{x} \]
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Time = 11.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {x-x \log (x)+\left (-4 x^6 \log (x)+4 x^5 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )+\left (x \log (x)-\log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )\right )}{\left (-x^3 \log (x)+x^2 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )} \, dx=\frac {\ln \left (\ln \left (-\frac {5\,\left (x-\ln \left (x\right )\right )}{\ln \left (x\right )}\right )\right )}{x}+x^4 \]
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