Integrand size = 126, antiderivative size = 36 \[ \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx=1-\frac {5-x}{x}+x^2+x \left (\frac {1}{4}+\log ^2\left (\log \left (\frac {2}{x}-x^2\right )\right )\right ) \]
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\[ \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx=\int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{x^2 \left (-8+4 x^3\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx \\ & = \int \left (\frac {1}{4}+\frac {5}{x^2}+2 x+\frac {4 \left (1+x^3\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-2+x^3\right ) \log \left (\frac {2-x^3}{x}\right )}+\log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )\right ) \, dx \\ & = -\frac {5}{x}+\frac {x}{4}+x^2+4 \int \frac {\left (1+x^3\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-2+x^3\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx+\int \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right ) \, dx \\ & = -\frac {5}{x}+\frac {x}{4}+x^2+4 \int \left (\frac {\log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\log \left (\frac {2-x^3}{x}\right )}+\frac {3 \log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-2+x^3\right ) \log \left (\frac {2-x^3}{x}\right )}\right ) \, dx+\int \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right ) \, dx \\ & = -\frac {5}{x}+\frac {x}{4}+x^2+4 \int \frac {\log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\log \left (\frac {2-x^3}{x}\right )} \, dx+12 \int \frac {\log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-2+x^3\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx+\int \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right ) \, dx \\ & = -\frac {5}{x}+\frac {x}{4}+x^2+4 \int \frac {\log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\log \left (\frac {2-x^3}{x}\right )} \, dx+12 \int \left (-\frac {\log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{3\ 2^{2/3} \left (\sqrt [3]{2}-x\right ) \log \left (\frac {2-x^3}{x}\right )}-\frac {\log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{3\ 2^{2/3} \left (\sqrt [3]{2}+\sqrt [3]{-1} x\right ) \log \left (\frac {2-x^3}{x}\right )}-\frac {\log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{3\ 2^{2/3} \left (\sqrt [3]{2}-(-1)^{2/3} x\right ) \log \left (\frac {2-x^3}{x}\right )}\right ) \, dx+\int \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right ) \, dx \\ & = -\frac {5}{x}+\frac {x}{4}+x^2+4 \int \frac {\log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\log \left (\frac {2-x^3}{x}\right )} \, dx-\left (2 \sqrt [3]{2}\right ) \int \frac {\log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (\sqrt [3]{2}-x\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx-\left (2 \sqrt [3]{2}\right ) \int \frac {\log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (\sqrt [3]{2}+\sqrt [3]{-1} x\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx-\left (2 \sqrt [3]{2}\right ) \int \frac {\log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (\sqrt [3]{2}-(-1)^{2/3} x\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx+\int \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right ) \, dx \\ \end{align*}
\[ \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx=\int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx \]
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Time = 1.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03
method | result | size |
parallelrisch | \(-\frac {-16 {\ln \left (\ln \left (-\frac {x^{3}-2}{x}\right )\right )}^{2} x^{2}+80-16 x^{3}-4 x^{2}}{16 x}\) | \(37\) |
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Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx=\frac {4 \, x^{2} \log \left (\log \left (-\frac {x^{3} - 2}{x}\right )\right )^{2} + 4 \, x^{3} + x^{2} - 20}{4 \, x} \]
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Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.61 \[ \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx=x^{2} + x \log {\left (\log {\left (\frac {2 - x^{3}}{x} \right )} \right )}^{2} + \frac {x}{4} - \frac {5}{x} \]
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Time = 0.38 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx=x \log \left (\log \left (-x^{3} + 2\right ) - \log \left (x\right )\right )^{2} + x^{2} + \frac {1}{4} \, x - \frac {5}{x} \]
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\[ \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx=\int { \frac {4 \, {\left (x^{5} - 2 \, x^{2}\right )} \log \left (-\frac {x^{3} - 2}{x}\right ) \log \left (\log \left (-\frac {x^{3} - 2}{x}\right )\right )^{2} + {\left (8 \, x^{6} + x^{5} + 4 \, x^{3} - 2 \, x^{2} - 40\right )} \log \left (-\frac {x^{3} - 2}{x}\right ) + 16 \, {\left (x^{5} + x^{2}\right )} \log \left (\log \left (-\frac {x^{3} - 2}{x}\right )\right )}{4 \, {\left (x^{5} - 2 \, x^{2}\right )} \log \left (-\frac {x^{3} - 2}{x}\right )} \,d x } \]
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Time = 11.78 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78 \[ \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx=\frac {x}{4}+x\,{\ln \left (\ln \left (-\frac {x^3-2}{x}\right )\right )}^2-\frac {5}{x}+x^2 \]
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