Integrand size = 62, antiderivative size = 18 \[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=2+\frac {1}{3+\frac {3 \log (x) \log (\log (x))}{2+x}} \]
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\[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=\int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-2-x-(2+x-x \log (x)) \log (\log (x))}{3 x (2+x+\log (x) \log (\log (x)))^2} \, dx \\ & = \frac {1}{3} \int \frac {-2-x-(2+x-x \log (x)) \log (\log (x))}{x (2+x+\log (x) \log (\log (x)))^2} \, dx \\ & = \frac {1}{3} \int \left (-\frac {(2+x) (-2-x+\log (x)+x \log (x))}{x \log (x) (2+x+\log (x) \log (\log (x)))^2}+\frac {-2-x+x \log (x)}{x \log (x) (2+x+\log (x) \log (\log (x)))}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {(2+x) (-2-x+\log (x)+x \log (x))}{x \log (x) (2+x+\log (x) \log (\log (x)))^2} \, dx\right )+\frac {1}{3} \int \frac {-2-x+x \log (x)}{x \log (x) (2+x+\log (x) \log (\log (x)))} \, dx \\ & = -\left (\frac {1}{3} \int \left (\frac {-2-x+\log (x)+x \log (x)}{\log (x) (2+x+\log (x) \log (\log (x)))^2}+\frac {2 (-2-x+\log (x)+x \log (x))}{x \log (x) (2+x+\log (x) \log (\log (x)))^2}\right ) \, dx\right )+\frac {1}{3} \int \left (\frac {1}{2+x+\log (x) \log (\log (x))}-\frac {1}{\log (x) (2+x+\log (x) \log (\log (x)))}-\frac {2}{x \log (x) (2+x+\log (x) \log (\log (x)))}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {-2-x+\log (x)+x \log (x)}{\log (x) (2+x+\log (x) \log (\log (x)))^2} \, dx\right )+\frac {1}{3} \int \frac {1}{2+x+\log (x) \log (\log (x))} \, dx-\frac {1}{3} \int \frac {1}{\log (x) (2+x+\log (x) \log (\log (x)))} \, dx-\frac {2}{3} \int \frac {-2-x+\log (x)+x \log (x)}{x \log (x) (2+x+\log (x) \log (\log (x)))^2} \, dx-\frac {2}{3} \int \frac {1}{x \log (x) (2+x+\log (x) \log (\log (x)))} \, dx \\ & = \frac {1}{3} \int \frac {1}{2+x+\log (x) \log (\log (x))} \, dx-\frac {1}{3} \int \frac {1}{\log (x) (2+x+\log (x) \log (\log (x)))} \, dx-\frac {1}{3} \int \left (\frac {1}{(2+x+\log (x) \log (\log (x)))^2}+\frac {x}{(2+x+\log (x) \log (\log (x)))^2}-\frac {2}{\log (x) (2+x+\log (x) \log (\log (x)))^2}-\frac {x}{\log (x) (2+x+\log (x) \log (\log (x)))^2}\right ) \, dx-\frac {2}{3} \int \frac {1}{x \log (x) (2+x+\log (x) \log (\log (x)))} \, dx-\frac {2}{3} \int \left (\frac {1}{(2+x+\log (x) \log (\log (x)))^2}+\frac {1}{x (2+x+\log (x) \log (\log (x)))^2}-\frac {1}{\log (x) (2+x+\log (x) \log (\log (x)))^2}-\frac {2}{x \log (x) (2+x+\log (x) \log (\log (x)))^2}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {1}{(2+x+\log (x) \log (\log (x)))^2} \, dx\right )-\frac {1}{3} \int \frac {x}{(2+x+\log (x) \log (\log (x)))^2} \, dx+\frac {1}{3} \int \frac {x}{\log (x) (2+x+\log (x) \log (\log (x)))^2} \, dx+\frac {1}{3} \int \frac {1}{2+x+\log (x) \log (\log (x))} \, dx-\frac {1}{3} \int \frac {1}{\log (x) (2+x+\log (x) \log (\log (x)))} \, dx-\frac {2}{3} \int \frac {1}{(2+x+\log (x) \log (\log (x)))^2} \, dx-\frac {2}{3} \int \frac {1}{x (2+x+\log (x) \log (\log (x)))^2} \, dx+2 \left (\frac {2}{3} \int \frac {1}{\log (x) (2+x+\log (x) \log (\log (x)))^2} \, dx\right )-\frac {2}{3} \int \frac {1}{x \log (x) (2+x+\log (x) \log (\log (x)))} \, dx+\frac {4}{3} \int \frac {1}{x \log (x) (2+x+\log (x) \log (\log (x)))^2} \, dx \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=-\frac {-2-x}{3 (2+x+\log (x) \log (\log (x)))} \]
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Time = 0.72 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {2+x}{3 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+3 x +6}\) | \(17\) |
risch | \(\frac {2+x}{3 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+3 x +6}\) | \(17\) |
parallelrisch | \(\frac {2+x}{3 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+3 x +6}\) | \(17\) |
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Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=\frac {x + 2}{3 \, {\left (\log \left (x\right ) \log \left (\log \left (x\right )\right ) + x + 2\right )}} \]
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Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=\frac {x + 2}{3 x + 3 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )} + 6} \]
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Time = 0.21 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=\frac {x + 2}{3 \, {\left (\log \left (x\right ) \log \left (\log \left (x\right )\right ) + x + 2\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=\frac {x + 2}{3 \, {\left (\log \left (x\right ) \log \left (\log \left (x\right )\right ) + x + 2\right )}} \]
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Time = 14.47 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.50 \[ \int \frac {-2-x+(-2-x+x \log (x)) \log (\log (x))}{12 x+12 x^2+3 x^3+\left (12 x+6 x^2\right ) \log (x) \log (\log (x))+3 x \log ^2(x) \log ^2(\log (x))} \, dx=\frac {\frac {4\,x}{3}-\ln \left (x\right )\,\left (\frac {x^3}{3}+x^2+\frac {2\,x}{3}\right )+\frac {4\,x^2}{3}+\frac {x^3}{3}}{\left (x+\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )+2\right )\,\left (2\,x-x^2\,\ln \left (x\right )-x\,\ln \left (x\right )+x^2\right )} \]
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