Integrand size = 191, antiderivative size = 24 \[ \int \frac {-3 x+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx=\frac {3 x}{\log (x) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))} \]
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\[ \int \frac {-3 x+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx=\int \frac {-3 x+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {3 \left (\log ^2(x) \log (3 (\log (x)+x (-1+\log (\log (x)))))-x (1-x-\log (3 (\log (x)+x (-1+\log (\log (x)))))+\log (\log (x)) (x+\log (3 (\log (x)+x (-1+\log (\log (x)))))))-\log (x) (1+(1+x) \log (3 (\log (x)+x (-1+\log (\log (x)))))+\log (\log (x)) (x-x \log (3 (\log (x)+x (-1+\log (\log (x)))))))\right )}{\log ^2(x) (\log (x)+x (-1+\log (\log (x)))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2} \, dx \\ & = 3 \int \frac {\log ^2(x) \log (3 (\log (x)+x (-1+\log (\log (x)))))-x (1-x-\log (3 (\log (x)+x (-1+\log (\log (x)))))+\log (\log (x)) (x+\log (3 (\log (x)+x (-1+\log (\log (x)))))))-\log (x) (1+(1+x) \log (3 (\log (x)+x (-1+\log (\log (x)))))+\log (\log (x)) (x-x \log (3 (\log (x)+x (-1+\log (\log (x)))))))}{\log ^2(x) (\log (x)+x (-1+\log (\log (x)))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2} \, dx \\ & = 3 \int \left (\frac {-x-\log (x)+x \log (x)+x^2 \log (x)-x \log ^2(x)-x \log (x) \log (\log (x))-x^2 \log (x) \log (\log (x))}{\log ^2(x) (-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2}+\frac {-1+\log (x)}{\log ^2(x) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))}\right ) \, dx \\ & = 3 \int \frac {-x-\log (x)+x \log (x)+x^2 \log (x)-x \log ^2(x)-x \log (x) \log (\log (x))-x^2 \log (x) \log (\log (x))}{\log ^2(x) (-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2} \, dx+3 \int \frac {-1+\log (x)}{\log ^2(x) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))} \, dx \\ & = 3 \int \frac {-x-x \log ^2(x)+\log (x) \left (-1+x+x^2-x (1+x) \log (\log (x))\right )}{\log ^2(x) (\log (x)+x (-1+\log (\log (x)))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2} \, dx+3 \int \left (-\frac {1}{\log ^2(x) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))}+\frac {1}{\log (x) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))}\right ) \, dx \\ & = -\left (3 \int \frac {1}{\log ^2(x) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))} \, dx\right )+3 \int \frac {1}{\log (x) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))} \, dx+3 \int \left (-\frac {x}{(-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2}-\frac {x}{\log ^2(x) (-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2}-\frac {1}{\log (x) (-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2}+\frac {x}{\log (x) (-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2}+\frac {x^2}{\log (x) (-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2}-\frac {x \log (\log (x))}{\log (x) (-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2}-\frac {x^2 \log (\log (x))}{\log (x) (-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2}\right ) \, dx \\ & = -\left (3 \int \frac {x}{(-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2} \, dx\right )-3 \int \frac {x}{\log ^2(x) (-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2} \, dx-3 \int \frac {1}{\log (x) (-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2} \, dx+3 \int \frac {x}{\log (x) (-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2} \, dx+3 \int \frac {x^2}{\log (x) (-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2} \, dx-3 \int \frac {x \log (\log (x))}{\log (x) (-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2} \, dx-3 \int \frac {x^2 \log (\log (x))}{\log (x) (-x+\log (x)+x \log (\log (x))) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))^2} \, dx-3 \int \frac {1}{\log ^2(x) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))} \, dx+3 \int \frac {1}{\log (x) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))} \, dx \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-3 x+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx=\frac {3 x}{\log (x) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))} \]
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Time = 25.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {3 x}{\ln \left (x \right ) \left (\ln \left (3\right )+x +\ln \left (x \ln \left (\ln \left (x \right )\right )+\ln \left (x \right )-x \right )\right )}\) | \(26\) |
risch | \(\frac {3 x}{\ln \left (x \right ) \left (\ln \left (3 x \ln \left (\ln \left (x \right )\right )+3 \ln \left (x \right )-3 x \right )+x \right )}\) | \(27\) |
parallelrisch | \(\frac {3 x}{\ln \left (x \right ) \left (\ln \left (3 x \ln \left (\ln \left (x \right )\right )+3 \ln \left (x \right )-3 x \right )+x \right )}\) | \(27\) |
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Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-3 x+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx=\frac {3 \, x}{x \log \left (x\right ) + \log \left (3 \, x \log \left (\log \left (x\right )\right ) - 3 \, x + 3 \, \log \left (x\right )\right ) \log \left (x\right )} \]
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Time = 0.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {-3 x+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx=\frac {3 x}{x \log {\left (x \right )} + \log {\left (x \right )} \log {\left (3 x \log {\left (\log {\left (x \right )} \right )} - 3 x + 3 \log {\left (x \right )} \right )}} \]
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Time = 0.37 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-3 x+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx=\frac {3 \, x}{{\left (x + \log \left (3\right )\right )} \log \left (x\right ) + \log \left (x \log \left (\log \left (x\right )\right ) - x + \log \left (x\right )\right ) \log \left (x\right )} \]
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Time = 0.75 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-3 x+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx=\frac {3 \, x}{x \log \left (x\right ) + \log \left (3 \, x \log \left (\log \left (x\right )\right ) - 3 \, x + 3 \, \log \left (x\right )\right ) \log \left (x\right )} \]
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Timed out. \[ \int \frac {-3 x+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx=\int -\frac {3\,x+3\,\ln \left (x\right )+\ln \left (\ln \left (x\right )\right )\,\left (3\,x\,\ln \left (x\right )+3\,x^2\right )-\ln \left (3\,\ln \left (x\right )-3\,x+3\,x\,\ln \left (\ln \left (x\right )\right )\right )\,\left (3\,x+3\,{\ln \left (x\right )}^2-\ln \left (x\right )\,\left (3\,x+3\right )-\ln \left (\ln \left (x\right )\right )\,\left (3\,x-3\,x\,\ln \left (x\right )\right )\right )-3\,x^2}{\ln \left (3\,\ln \left (x\right )-3\,x+3\,x\,\ln \left (\ln \left (x\right )\right )\right )\,\left (2\,x\,{\ln \left (x\right )}^3-2\,x^2\,{\ln \left (x\right )}^2+2\,x^2\,\ln \left (\ln \left (x\right )\right )\,{\ln \left (x\right )}^2\right )+x^2\,{\ln \left (x\right )}^3-x^3\,{\ln \left (x\right )}^2+{\ln \left (3\,\ln \left (x\right )-3\,x+3\,x\,\ln \left (\ln \left (x\right )\right )\right )}^2\,\left ({\ln \left (x\right )}^3-x\,{\ln \left (x\right )}^2+x\,\ln \left (\ln \left (x\right )\right )\,{\ln \left (x\right )}^2\right )+x^3\,\ln \left (\ln \left (x\right )\right )\,{\ln \left (x\right )}^2} \,d x \]
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