Integrand size = 126, antiderivative size = 24 \[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=2-\frac {x \left (4+x+\frac {65536}{(-x+\log (\log (x)))^2}\right )}{\sqrt [3]{e}} \]
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\[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=\int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (-65536-\log (x) \left (x \left (-32768+2 x^2+x^3\right )-\left (32768+6 x^2+3 x^3\right ) \log (\log (x))+3 x (2+x) \log ^2(\log (x))-(2+x) \log ^3(\log (x))\right )\right )}{\sqrt [3]{e} \log (x) (x-\log (\log (x)))^3} \, dx \\ & = \frac {2 \int \frac {-65536-\log (x) \left (x \left (-32768+2 x^2+x^3\right )-\left (32768+6 x^2+3 x^3\right ) \log (\log (x))+3 x (2+x) \log ^2(\log (x))-(2+x) \log ^3(\log (x))\right )}{\log (x) (x-\log (\log (x)))^3} \, dx}{\sqrt [3]{e}} \\ & = \frac {2 \int \left (-2-x+\frac {65536 (-1+x \log (x))}{\log (x) (x-\log (\log (x)))^3}-\frac {32768}{(x-\log (\log (x)))^2}\right ) \, dx}{\sqrt [3]{e}} \\ & = -\frac {4 x}{\sqrt [3]{e}}-\frac {x^2}{\sqrt [3]{e}}-\frac {65536 \int \frac {1}{(x-\log (\log (x)))^2} \, dx}{\sqrt [3]{e}}+\frac {131072 \int \frac {-1+x \log (x)}{\log (x) (x-\log (\log (x)))^3} \, dx}{\sqrt [3]{e}} \\ & = -\frac {4 x}{\sqrt [3]{e}}-\frac {x^2}{\sqrt [3]{e}}-\frac {65536 \int \frac {1}{(x-\log (\log (x)))^2} \, dx}{\sqrt [3]{e}}+\frac {131072 \int \left (\frac {x}{(x-\log (\log (x)))^3}-\frac {1}{\log (x) (x-\log (\log (x)))^3}\right ) \, dx}{\sqrt [3]{e}} \\ & = -\frac {4 x}{\sqrt [3]{e}}-\frac {x^2}{\sqrt [3]{e}}-\frac {65536 \int \frac {1}{(x-\log (\log (x)))^2} \, dx}{\sqrt [3]{e}}+\frac {131072 \int \frac {x}{(x-\log (\log (x)))^3} \, dx}{\sqrt [3]{e}}-\frac {131072 \int \frac {1}{\log (x) (x-\log (\log (x)))^3} \, dx}{\sqrt [3]{e}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=-\frac {x \left (4+x+\frac {65536}{(x-\log (\log (x)))^2}\right )}{\sqrt [3]{e}} \]
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Time = 0.86 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\left (4+x \right ) x \,{\mathrm e}^{-\frac {1}{3}}-\frac {65536 x \,{\mathrm e}^{-\frac {1}{3}}}{\left (x -\ln \left (\ln \left (x \right )\right )\right )^{2}}\) | \(24\) |
parallelrisch | \(\frac {\left (-x^{4}+2 x^{3} \ln \left (\ln \left (x \right )\right )-x^{2} \ln \left (\ln \left (x \right )\right )^{2}-4 x^{3}+8 x^{2} \ln \left (\ln \left (x \right )\right )-4 x \ln \left (\ln \left (x \right )\right )^{2}-65536 x \right ) {\mathrm e}^{-\frac {1}{3}}}{\ln \left (\ln \left (x \right )\right )^{2}-2 x \ln \left (\ln \left (x \right )\right )+x^{2}}\) | \(71\) |
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (21) = 42\).
Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.75 \[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=-\frac {x^{4} + 4 \, x^{3} + {\left (x^{2} + 4 \, x\right )} \log \left (\log \left (x\right )\right )^{2} - 2 \, {\left (x^{3} + 4 \, x^{2}\right )} \log \left (\log \left (x\right )\right ) + 65536 \, x}{x^{2} e^{\frac {1}{3}} - 2 \, x e^{\frac {1}{3}} \log \left (\log \left (x\right )\right ) + e^{\frac {1}{3}} \log \left (\log \left (x\right )\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (20) = 40\).
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.25 \[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=- \frac {x^{2}}{e^{\frac {1}{3}}} - \frac {4 x}{e^{\frac {1}{3}}} - \frac {65536 x}{x^{2} e^{\frac {1}{3}} - 2 x e^{\frac {1}{3}} \log {\left (\log {\left (x \right )} \right )} + e^{\frac {1}{3}} \log {\left (\log {\left (x \right )} \right )}^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (21) = 42\).
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.75 \[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=-\frac {x^{4} + 4 \, x^{3} + {\left (x^{2} + 4 \, x\right )} \log \left (\log \left (x\right )\right )^{2} - 2 \, {\left (x^{3} + 4 \, x^{2}\right )} \log \left (\log \left (x\right )\right ) + 65536 \, x}{x^{2} e^{\frac {1}{3}} - 2 \, x e^{\frac {1}{3}} \log \left (\log \left (x\right )\right ) + e^{\frac {1}{3}} \log \left (\log \left (x\right )\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (21) = 42\).
Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.29 \[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=-\frac {x^{4} e^{\left (-\frac {1}{3}\right )} - 2 \, x^{3} e^{\left (-\frac {1}{3}\right )} \log \left (\log \left (x\right )\right ) + x^{2} e^{\left (-\frac {1}{3}\right )} \log \left (\log \left (x\right )\right )^{2} + 4 \, x^{3} e^{\left (-\frac {1}{3}\right )} - 8 \, x^{2} e^{\left (-\frac {1}{3}\right )} \log \left (\log \left (x\right )\right ) + 4 \, x e^{\left (-\frac {1}{3}\right )} \log \left (\log \left (x\right )\right )^{2} + 65536 \, x e^{\left (-\frac {1}{3}\right )}}{x^{2} - 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}} \]
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Time = 11.18 (sec) , antiderivative size = 236, normalized size of antiderivative = 9.83 \[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=\frac {32768\,{\mathrm {e}}^{-\frac {1}{3}}}{x\,\left (x\,\ln \left (x\right )-1\right )}-\frac {\frac {32768\,x\,{\mathrm {e}}^{-\frac {1}{3}}\,\ln \left (x\right )\,\left (x^2\,{\ln \left (x\right )}^2-x\,\ln \left (x\right )+x+1\right )}{{\left (x\,\ln \left (x\right )-1\right )}^3}-\frac {32768\,x\,\ln \left (\ln \left (x\right )\right )\,{\mathrm {e}}^{-\frac {1}{3}}\,\ln \left (x\right )\,\left (\ln \left (x\right )+1\right )}{{\left (x\,\ln \left (x\right )-1\right )}^3}}{x-\ln \left (\ln \left (x\right )\right )}-x^2\,{\mathrm {e}}^{-\frac {1}{3}}-\frac {\frac {32768\,x\,{\mathrm {e}}^{-\frac {1}{3}}\,\left (x\,\ln \left (x\right )-2\right )}{x\,\ln \left (x\right )-1}+\frac {32768\,x\,\ln \left (\ln \left (x\right )\right )\,{\mathrm {e}}^{-\frac {1}{3}}\,\ln \left (x\right )}{x\,\ln \left (x\right )-1}}{x^2-2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}-4\,x\,{\mathrm {e}}^{-\frac {1}{3}}-\frac {32768\,{\mathrm {e}}^{-\frac {1}{3}}\,\left (x^3+2\,x^2+x\right )}{x^2\,\left (x+1\right )\,\left (-x^3\,{\ln \left (x\right )}^3+3\,x^2\,{\ln \left (x\right )}^2-3\,x\,\ln \left (x\right )+1\right )}+\frac {32768\,{\mathrm {e}}^{-\frac {1}{3}}\,\left (x^2+3\,x+2\right )}{x\,\left (x+1\right )\,\left (x^2\,{\ln \left (x\right )}^2-2\,x\,\ln \left (x\right )+1\right )} \]
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