Integrand size = 72, antiderivative size = 24 \[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=\left (2-\frac {1}{2} (630-x) x\right ) \log \left (\frac {x}{-1+\log (4 x)}\right ) \]
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\[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=\int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{x (-2+2 \log (4 x))} \, dx \\ & = \int \frac {8-1260 x+2 x^2-\left (4-630 x+x^2\right ) \log (4 x)-\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{2 x (1-\log (4 x))} \, dx \\ & = \frac {1}{2} \int \frac {8-1260 x+2 x^2-\left (4-630 x+x^2\right ) \log (4 x)-\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{x (1-\log (4 x))} \, dx \\ & = \frac {1}{2} \int \left (\frac {\left (4-630 x+x^2\right ) (-2+\log (4 x))}{x (-1+\log (4 x))}+2 (-315+x) \log \left (\frac {x}{-1+\log (4 x)}\right )\right ) \, dx \\ & = \frac {1}{2} \int \frac {\left (4-630 x+x^2\right ) (-2+\log (4 x))}{x (-1+\log (4 x))} \, dx+\int (-315+x) \log \left (\frac {x}{-1+\log (4 x)}\right ) \, dx \\ & = -315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} \int \left (\frac {4-630 x+x^2}{x}+\frac {-4+630 x-x^2}{x (-1+\log (4 x))}\right ) \, dx-\int \frac {(630-x) (-2+\log (4 x))}{2 (1-\log (4 x))} \, dx \\ & = -315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} \int \frac {4-630 x+x^2}{x} \, dx-\frac {1}{2} \int \frac {(630-x) (-2+\log (4 x))}{1-\log (4 x)} \, dx+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx \\ & = -315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} \int \left (-630+\frac {4}{x}+x\right ) \, dx-\frac {1}{2} \int \left (-630+x+\frac {630-x}{-1+\log (4 x)}\right ) \, dx+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx \\ & = 2 \log (x)-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )-\frac {1}{2} \int \frac {630-x}{-1+\log (4 x)} \, dx+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx \\ & = 2 \log (x)-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )-\frac {1}{2} \int \left (\frac {630}{-1+\log (4 x)}-\frac {x}{-1+\log (4 x)}\right ) \, dx+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx \\ & = 2 \log (x)-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} \int \frac {x}{-1+\log (4 x)} \, dx+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx-315 \int \frac {1}{-1+\log (4 x)} \, dx \\ & = 2 \log (x)-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{32} \text {Subst}\left (\int \frac {e^{2 x}}{-1+x} \, dx,x,\log (4 x)\right )+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx-\frac {315}{4} \text {Subst}\left (\int \frac {e^x}{-1+x} \, dx,x,\log (4 x)\right ) \\ & = \frac {1}{32} e^2 \operatorname {ExpIntegralEi}(-2 (1-\log (4 x)))-\frac {315}{4} e \operatorname {ExpIntegralEi}(-1+\log (4 x))+2 \log (x)-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=\frac {1}{2} \left (4 \log (x)-4 \log (1-\log (4 x))+(-630+x) x \log \left (\frac {x}{-1+\log (4 x)}\right )\right ) \]
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Time = 0.96 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.88
method | result | size |
norman | \(2 \ln \left (\frac {x}{\ln \left (4 x \right )-1}\right )-315 x \ln \left (\frac {x}{\ln \left (4 x \right )-1}\right )+\frac {\ln \left (\frac {x}{\ln \left (4 x \right )-1}\right ) x^{2}}{2}\) | \(45\) |
parallelrisch | \(\frac {\ln \left (\frac {x}{\ln \left (4 x \right )-1}\right ) x^{2}}{2}-315 x \ln \left (\frac {x}{\ln \left (4 x \right )-1}\right )+2 \ln \left (x \right )-2 \ln \left (\ln \left (4 x \right )-1\right )\) | \(45\) |
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Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=\frac {1}{2} \, {\left (x^{2} - 630 \, x + 4\right )} \log \left (\frac {x}{\log \left (4 \, x\right ) - 1}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=\left (\frac {x^{2}}{2} - 315 x\right ) \log {\left (\frac {x}{\log {\left (4 x \right )} - 1} \right )} + 2 \log {\left (x \right )} - 2 \log {\left (\log {\left (4 x \right )} - 1 \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (20) = 40\).
Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=\frac {1}{2} \, {\left (x^{2} - 630 \, x + 4\right )} \log \left (x\right ) - \frac {1}{2} \, {\left (x^{2} - 630 \, x - 4\right )} \log \left (2 \, \log \left (2\right ) + \log \left (x\right ) - 1\right ) - 4 \, \log \left (2 \, \log \left (2\right ) + \log \left (x\right ) - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (20) = 40\).
Time = 0.31 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=\frac {1}{2} \, {\left (x^{2} - 630 \, x\right )} \log \left (x\right ) - \frac {1}{2} \, {\left (x^{2} - 630 \, x\right )} \log \left (\log \left (4 \, x\right ) - 1\right ) + 2 \, \log \left (x\right ) - 2 \, \log \left (2 \, \log \left (2\right ) + \log \left (x\right ) - 1\right ) \]
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Time = 12.85 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=2\,\ln \left (x\right )-2\,\ln \left (\ln \left (4\,x\right )-1\right )-\ln \left (\frac {x}{\ln \left (4\,x\right )-1}\right )\,\left (315\,x-\frac {x^2}{2}\right ) \]
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