Integrand size = 60, antiderivative size = 24 \[ \int \frac {-3+x-4 x \log (x)+(6-x) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )}{\left (4 x^2-x^3\right ) \log (x)+\left (-3 x+x^2\right ) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )} \, dx=\log \left (\frac {x-(-3+x) \left (x-\log \left (\frac {5 \log (x)}{e^3}\right )\right )}{x^2}\right ) \]
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\[ \int \frac {-3+x-4 x \log (x)+(6-x) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )}{\left (4 x^2-x^3\right ) \log (x)+\left (-3 x+x^2\right ) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )} \, dx=\int \frac {-3+x-4 x \log (x)+(6-x) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )}{\left (4 x^2-x^3\right ) \log (x)+\left (-3 x+x^2\right ) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-3+x-4 x \log (x)+(6-x) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )}{x \log (x) \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )} \, dx \\ & = \int \left (\frac {-3+x-4 x \log (x)}{x \log (x) \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )}+\frac {(6-x) \left (-3 \left (1-\frac {\log (5)}{3}\right )+\log (\log (x))\right )}{x \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )}\right ) \, dx \\ & = \int \frac {-3+x-4 x \log (x)}{x \log (x) \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )} \, dx+\int \frac {(6-x) \left (-3 \left (1-\frac {\log (5)}{3}\right )+\log (\log (x))\right )}{x \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )} \, dx \\ & = \int \left (\frac {6 \left (-3 \left (1-\frac {\log (5)}{3}\right )+\log (\log (x))\right )}{x \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )}+\frac {3-\log (5 \log (x))}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))}\right ) \, dx+\int \left (\frac {1}{\log (x) \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )}+\frac {4}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))}+\frac {3}{x \log (x) \left (-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))\right )}\right ) \, dx \\ & = 3 \int \frac {1}{x \log (x) \left (-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))\right )} \, dx+4 \int \frac {1}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+6 \int \frac {-3 \left (1-\frac {\log (5)}{3}\right )+\log (\log (x))}{x \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )} \, dx+\int \frac {1}{\log (x) \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )} \, dx+\int \frac {3-\log (5 \log (x))}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))} \, dx \\ & = 3 \int \frac {1}{x \log (x) \left (-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))\right )} \, dx+4 \int \frac {1}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+6 \int \frac {-3+\log (5 \log (x))}{x \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )} \, dx+\int \left (\frac {1}{3}+\frac {x (-1+x-\log (5)-\log (\log (x)))}{3 \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )}\right ) \, dx+\int \frac {1}{\log (x) \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )} \, dx \\ & = \frac {x}{3}+\frac {1}{3} \int \frac {x (-1+x-\log (5)-\log (\log (x)))}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))} \, dx+3 \int \frac {1}{x \log (x) \left (-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))\right )} \, dx+4 \int \frac {1}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+6 \int \left (-\frac {1}{3 x}+\frac {1-x+\log (5 \log (x))}{3 \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )}\right ) \, dx+\int \frac {1}{\log (x) \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )} \, dx \\ & = \frac {x}{3}-2 \log (x)+\frac {1}{3} \int \left (\frac {x^2}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))}+\frac {x (1+\log (5))}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))}+\frac {x \log (\log (x))}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))}\right ) \, dx+2 \int \frac {1-x+\log (5 \log (x))}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))} \, dx+3 \int \frac {1}{x \log (x) \left (-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))\right )} \, dx+4 \int \frac {1}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+\int \frac {1}{\log (x) \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )} \, dx \\ & = \frac {x}{3}-2 \log (x)+\frac {1}{3} \int \frac {x^2}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))} \, dx+\frac {1}{3} \int \frac {x \log (\log (x))}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+2 \int \left (-\frac {1}{3}+\frac {12-x^2-2 x \left (1-\frac {\log (5)}{2}\right )+x \log (\log (x))}{3 \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )}\right ) \, dx+3 \int \frac {1}{x \log (x) \left (-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))\right )} \, dx+4 \int \frac {1}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+\frac {1}{3} (1+\log (5)) \int \frac {x}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+\int \frac {1}{\log (x) \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )} \, dx \\ & = -\frac {x}{3}-2 \log (x)+\frac {1}{3} \int \frac {x^2}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))} \, dx+\frac {1}{3} \int \frac {x \log (\log (x))}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+\frac {2}{3} \int \frac {12-x^2-2 x \left (1-\frac {\log (5)}{2}\right )+x \log (\log (x))}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))} \, dx+3 \int \frac {1}{x \log (x) \left (-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))\right )} \, dx+4 \int \frac {1}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+\frac {1}{3} (1+\log (5)) \int \frac {x}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+\int \frac {1}{\log (x) \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )} \, dx \\ & = -\frac {x}{3}-2 \log (x)+\frac {1}{3} \int \frac {x^2}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))} \, dx+\frac {1}{3} \int \frac {x \log (\log (x))}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+\frac {2}{3} \int \left (\frac {12}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))}+\frac {x (-2+\log (5))}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))}+\frac {x \log (\log (x))}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))}+\frac {x^2}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))}\right ) \, dx+3 \int \frac {1}{x \log (x) \left (-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))\right )} \, dx+4 \int \frac {1}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+\frac {1}{3} (1+\log (5)) \int \frac {x}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+\int \frac {1}{\log (x) \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )} \, dx \\ & = -\frac {x}{3}-2 \log (x)+\frac {1}{3} \int \frac {x^2}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))} \, dx+\frac {1}{3} \int \frac {x \log (\log (x))}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+\frac {2}{3} \int \frac {x \log (\log (x))}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))} \, dx+\frac {2}{3} \int \frac {x^2}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+3 \int \frac {1}{x \log (x) \left (-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))\right )} \, dx+4 \int \frac {1}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+8 \int \frac {1}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))} \, dx-\frac {1}{3} (2 (2-\log (5))) \int \frac {x}{9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))} \, dx+\frac {1}{3} (1+\log (5)) \int \frac {x}{-9+x^2-x (1+\log (5))-x \log (\log (x))+3 \log (5 \log (x))} \, dx+\int \frac {1}{\log (x) \left (9-x^2+x (1+\log (5))+x \log (\log (x))-3 \log (5 \log (x))\right )} \, dx \\ \end{align*}
\[ \int \frac {-3+x-4 x \log (x)+(6-x) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )}{\left (4 x^2-x^3\right ) \log (x)+\left (-3 x+x^2\right ) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )} \, dx=\int \frac {-3+x-4 x \log (x)+(6-x) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )}{\left (4 x^2-x^3\right ) \log (x)+\left (-3 x+x^2\right ) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )} \, dx \]
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Time = 0.60 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25
method | result | size |
risch | \(-2 \ln \left (x \right )+\ln \left (-3+x \right )+\ln \left (\ln \left (5 \ln \left (x \right ) {\mathrm e}^{-3}\right )-\frac {\left (x -4\right ) x}{-3+x}\right )\) | \(30\) |
parallelrisch | \(\ln \left (x^{2}-\ln \left (5 \ln \left (x \right ) {\mathrm e}^{-3}\right ) x -4 x +3 \ln \left (5 \ln \left (x \right ) {\mathrm e}^{-3}\right )\right )-2 \ln \left (x \right )\) | \(37\) |
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Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-3+x-4 x \log (x)+(6-x) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )}{\left (4 x^2-x^3\right ) \log (x)+\left (-3 x+x^2\right ) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )} \, dx=\log \left (x - 3\right ) - 2 \, \log \left (x\right ) + \log \left (-\frac {x^{2} - {\left (x - 3\right )} \log \left (5 \, e^{\left (-3\right )} \log \left (x\right )\right ) - 4 \, x}{x - 3}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {-3+x-4 x \log (x)+(6-x) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )}{\left (4 x^2-x^3\right ) \log (x)+\left (-3 x+x^2\right ) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )} \, dx=- 2 \log {\left (x \right )} + \log {\left (x - 3 \right )} + \log {\left (\log {\left (\frac {5 \log {\left (x \right )}}{e^{3}} \right )} + \frac {- x^{2} + 4 x}{x - 3} \right )} \]
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Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {-3+x-4 x \log (x)+(6-x) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )}{\left (4 x^2-x^3\right ) \log (x)+\left (-3 x+x^2\right ) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )} \, dx=\log \left (x - 3\right ) - 2 \, \log \left (x\right ) + \log \left (-\frac {x^{2} - x {\left (\log \left (5\right ) + 1\right )} - {\left (x - 3\right )} \log \left (\log \left (x\right )\right ) + 3 \, \log \left (5\right ) - 9}{x - 3}\right ) \]
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Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-3+x-4 x \log (x)+(6-x) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )}{\left (4 x^2-x^3\right ) \log (x)+\left (-3 x+x^2\right ) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )} \, dx=\log \left (-x^{2} + x \log \left (5 \, \log \left (x\right )\right ) + x - 3 \, \log \left (5 \, \log \left (x\right )\right ) + 9\right ) - 2 \, \log \left (x\right ) \]
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Timed out. \[ \int \frac {-3+x-4 x \log (x)+(6-x) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )}{\left (4 x^2-x^3\right ) \log (x)+\left (-3 x+x^2\right ) \log (x) \log \left (\frac {5 \log (x)}{e^3}\right )} \, dx=\int -\frac {4\,x\,\ln \left (x\right )-x+\ln \left (5\,{\mathrm {e}}^{-3}\,\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (x-6\right )+3}{\ln \left (x\right )\,\left (4\,x^2-x^3\right )-\ln \left (5\,{\mathrm {e}}^{-3}\,\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (3\,x-x^2\right )} \,d x \]
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