Integrand size = 160, antiderivative size = 32 \[ \int \frac {4 x^3 \log (x) \log ^2(\log (x))+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{\log (\log (x))}} \left (8+5 e^3-x+\left (-8-5 e^3+2 x\right ) \log (x) \log (\log (x))\right )+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{2 \log (\log (x))}} \left (8 x^2+5 e^3 x^2-x^3+\left (-8 x^2-5 e^3 x^2+2 x^3\right ) \log (x) \log (\log (x))+4 x \log (x) \log ^2(\log (x))\right )}{\log (x) \log ^2(\log (x))} \, dx=3+\left (e^{-5+\frac {x \left (-4+\frac {1}{2} \left (-5 e^3+x\right )\right )}{\log (\log (x))}}+x^2\right )^2 \]
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\[ \int \frac {4 x^3 \log (x) \log ^2(\log (x))+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{\log (\log (x))}} \left (8+5 e^3-x+\left (-8-5 e^3+2 x\right ) \log (x) \log (\log (x))\right )+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{2 \log (\log (x))}} \left (8 x^2+5 e^3 x^2-x^3+\left (-8 x^2-5 e^3 x^2+2 x^3\right ) \log (x) \log (\log (x))+4 x \log (x) \log ^2(\log (x))\right )}{\log (x) \log ^2(\log (x))} \, dx=\int \frac {4 x^3 \log (x) \log ^2(\log (x))+\exp \left (\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{\log (\log (x))}\right ) \left (8+5 e^3-x+\left (-8-5 e^3+2 x\right ) \log (x) \log (\log (x))\right )+\exp \left (\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{2 \log (\log (x))}\right ) \left (8 x^2+5 e^3 x^2-x^3+\left (-8 x^2-5 e^3 x^2+2 x^3\right ) \log (x) \log (\log (x))+4 x \log (x) \log ^2(\log (x))\right )}{\log (x) \log ^2(\log (x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (4 x^3+\frac {e^{-10+\frac {x \left (-8-5 e^3+x\right )}{\log (\log (x))}} \left (8 \left (1+\frac {5 e^3}{8}\right )-x-8 \left (1+\frac {5 e^3}{8}\right ) \log (x) \log (\log (x))+2 x \log (x) \log (\log (x))\right )}{\log (x) \log ^2(\log (x))}+\frac {e^{-5+\frac {x \left (-8-5 e^3+x\right )}{2 \log (\log (x))}} x \left (8 \left (1+\frac {5 e^3}{8}\right ) x-x^2-8 \left (1+\frac {5 e^3}{8}\right ) x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))+4 \log (x) \log ^2(\log (x))\right )}{\log (x) \log ^2(\log (x))}\right ) \, dx \\ & = x^4+\int \frac {e^{-10+\frac {x \left (-8-5 e^3+x\right )}{\log (\log (x))}} \left (8 \left (1+\frac {5 e^3}{8}\right )-x-8 \left (1+\frac {5 e^3}{8}\right ) \log (x) \log (\log (x))+2 x \log (x) \log (\log (x))\right )}{\log (x) \log ^2(\log (x))} \, dx+\int \frac {e^{-5+\frac {x \left (-8-5 e^3+x\right )}{2 \log (\log (x))}} x \left (8 \left (1+\frac {5 e^3}{8}\right ) x-x^2-8 \left (1+\frac {5 e^3}{8}\right ) x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))+4 \log (x) \log ^2(\log (x))\right )}{\log (x) \log ^2(\log (x))} \, dx \\ & = x^4+\frac {2 e^{-5-\frac {\left (8+5 e^3-x\right ) x}{2 \log (\log (x))}} x \left (\left (8+5 e^3\right ) x-x^2-\left (8+5 e^3\right ) x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (\frac {8+5 e^3-x}{\log (x) \log ^2(\log (x))}-\frac {8+5 e^3-x}{\log (\log (x))}+\frac {x}{\log (\log (x))}\right ) \log ^2(\log (x))}+\int \left (\frac {e^{-10+\frac {x \left (-8-5 e^3+x\right )}{\log (\log (x))}} \left (8+5 e^3-x\right )}{\log (x) \log ^2(\log (x))}+\frac {e^{-10+\frac {x \left (-8-5 e^3+x\right )}{\log (\log (x))}} \left (-8-5 e^3+2 x\right )}{\log (\log (x))}\right ) \, dx \\ & = x^4+\frac {2 e^{-5-\frac {\left (8+5 e^3-x\right ) x}{2 \log (\log (x))}} x \left (\left (8+5 e^3\right ) x-x^2-\left (8+5 e^3\right ) x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (\frac {8+5 e^3-x}{\log (x) \log ^2(\log (x))}-\frac {8+5 e^3-x}{\log (\log (x))}+\frac {x}{\log (\log (x))}\right ) \log ^2(\log (x))}+\int \frac {e^{-10+\frac {x \left (-8-5 e^3+x\right )}{\log (\log (x))}} \left (8+5 e^3-x\right )}{\log (x) \log ^2(\log (x))} \, dx+\int \frac {e^{-10+\frac {x \left (-8-5 e^3+x\right )}{\log (\log (x))}} \left (-8-5 e^3+2 x\right )}{\log (\log (x))} \, dx \\ & = x^4+\frac {2 e^{-5-\frac {\left (8+5 e^3-x\right ) x}{2 \log (\log (x))}} x \left (\left (8+5 e^3\right ) x-x^2-\left (8+5 e^3\right ) x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (\frac {8+5 e^3-x}{\log (x) \log ^2(\log (x))}-\frac {8+5 e^3-x}{\log (\log (x))}+\frac {x}{\log (\log (x))}\right ) \log ^2(\log (x))}+\int \left (\frac {8 e^{-10+\frac {x \left (-8-5 e^3+x\right )}{\log (\log (x))}} \left (1+\frac {5 e^3}{8}\right )}{\log (x) \log ^2(\log (x))}-\frac {e^{-10+\frac {x \left (-8-5 e^3+x\right )}{\log (\log (x))}} x}{\log (x) \log ^2(\log (x))}\right ) \, dx+\int \left (-\frac {8 e^{-10+\frac {x \left (-8-5 e^3+x\right )}{\log (\log (x))}} \left (1+\frac {5 e^3}{8}\right )}{\log (\log (x))}+\frac {2 e^{-10+\frac {x \left (-8-5 e^3+x\right )}{\log (\log (x))}} x}{\log (\log (x))}\right ) \, dx \\ & = x^4+\frac {2 e^{-5-\frac {\left (8+5 e^3-x\right ) x}{2 \log (\log (x))}} x \left (\left (8+5 e^3\right ) x-x^2-\left (8+5 e^3\right ) x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))\right )}{\log (x) \left (\frac {8+5 e^3-x}{\log (x) \log ^2(\log (x))}-\frac {8+5 e^3-x}{\log (\log (x))}+\frac {x}{\log (\log (x))}\right ) \log ^2(\log (x))}+2 \int \frac {e^{-10+\frac {x \left (-8-5 e^3+x\right )}{\log (\log (x))}} x}{\log (\log (x))} \, dx+\left (8+5 e^3\right ) \int \frac {e^{-10+\frac {x \left (-8-5 e^3+x\right )}{\log (\log (x))}}}{\log (x) \log ^2(\log (x))} \, dx-\left (8+5 e^3\right ) \int \frac {e^{-10+\frac {x \left (-8-5 e^3+x\right )}{\log (\log (x))}}}{\log (\log (x))} \, dx-\int \frac {e^{-10+\frac {x \left (-8-5 e^3+x\right )}{\log (\log (x))}} x}{\log (x) \log ^2(\log (x))} \, dx \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.94 \[ \int \frac {4 x^3 \log (x) \log ^2(\log (x))+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{\log (\log (x))}} \left (8+5 e^3-x+\left (-8-5 e^3+2 x\right ) \log (x) \log (\log (x))\right )+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{2 \log (\log (x))}} \left (8 x^2+5 e^3 x^2-x^3+\left (-8 x^2-5 e^3 x^2+2 x^3\right ) \log (x) \log (\log (x))+4 x \log (x) \log ^2(\log (x))\right )}{\log (x) \log ^2(\log (x))} \, dx=e^{-10-\frac {\left (8+5 e^3\right ) x}{\log (\log (x))}} \left (e^{\frac {x^2}{2 \log (\log (x))}}+e^{5+\frac {\left (8+5 e^3\right ) x}{2 \log (\log (x))}} x^2\right )^2 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(27)=54\).
Time = 28.66 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.00
method | result | size |
risch | \(x^{4}+2 x^{2} {\mathrm e}^{-\frac {5 x \,{\mathrm e}^{3}-x^{2}+10 \ln \left (\ln \left (x \right )\right )+8 x}{2 \ln \left (\ln \left (x \right )\right )}}+{\mathrm e}^{-\frac {5 x \,{\mathrm e}^{3}-x^{2}+10 \ln \left (\ln \left (x \right )\right )+8 x}{\ln \left (\ln \left (x \right )\right )}}\) | \(64\) |
parallelrisch | \(x^{4}+2 x^{2} {\mathrm e}^{-\frac {5 x \,{\mathrm e}^{3}-x^{2}+10 \ln \left (\ln \left (x \right )\right )+8 x}{2 \ln \left (\ln \left (x \right )\right )}}+{\mathrm e}^{-\frac {5 x \,{\mathrm e}^{3}-x^{2}+10 \ln \left (\ln \left (x \right )\right )+8 x}{\ln \left (\ln \left (x \right )\right )}}\) | \(66\) |
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.81 \[ \int \frac {4 x^3 \log (x) \log ^2(\log (x))+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{\log (\log (x))}} \left (8+5 e^3-x+\left (-8-5 e^3+2 x\right ) \log (x) \log (\log (x))\right )+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{2 \log (\log (x))}} \left (8 x^2+5 e^3 x^2-x^3+\left (-8 x^2-5 e^3 x^2+2 x^3\right ) \log (x) \log (\log (x))+4 x \log (x) \log ^2(\log (x))\right )}{\log (x) \log ^2(\log (x))} \, dx=x^{4} + 2 \, x^{2} e^{\left (\frac {x^{2} - 5 \, x e^{3} - 8 \, x - 10 \, \log \left (\log \left (x\right )\right )}{2 \, \log \left (\log \left (x\right )\right )}\right )} + e^{\left (\frac {x^{2} - 5 \, x e^{3} - 8 \, x - 10 \, \log \left (\log \left (x\right )\right )}{\log \left (\log \left (x\right )\right )}\right )} \]
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Exception generated. \[ \int \frac {4 x^3 \log (x) \log ^2(\log (x))+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{\log (\log (x))}} \left (8+5 e^3-x+\left (-8-5 e^3+2 x\right ) \log (x) \log (\log (x))\right )+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{2 \log (\log (x))}} \left (8 x^2+5 e^3 x^2-x^3+\left (-8 x^2-5 e^3 x^2+2 x^3\right ) \log (x) \log (\log (x))+4 x \log (x) \log ^2(\log (x))\right )}{\log (x) \log ^2(\log (x))} \, dx=\text {Exception raised: TypeError} \]
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Exception generated. \[ \int \frac {4 x^3 \log (x) \log ^2(\log (x))+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{\log (\log (x))}} \left (8+5 e^3-x+\left (-8-5 e^3+2 x\right ) \log (x) \log (\log (x))\right )+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{2 \log (\log (x))}} \left (8 x^2+5 e^3 x^2-x^3+\left (-8 x^2-5 e^3 x^2+2 x^3\right ) \log (x) \log (\log (x))+4 x \log (x) \log ^2(\log (x))\right )}{\log (x) \log ^2(\log (x))} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {4 x^3 \log (x) \log ^2(\log (x))+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{\log (\log (x))}} \left (8+5 e^3-x+\left (-8-5 e^3+2 x\right ) \log (x) \log (\log (x))\right )+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{2 \log (\log (x))}} \left (8 x^2+5 e^3 x^2-x^3+\left (-8 x^2-5 e^3 x^2+2 x^3\right ) \log (x) \log (\log (x))+4 x \log (x) \log ^2(\log (x))\right )}{\log (x) \log ^2(\log (x))} \, dx=\int { \frac {4 \, x^{3} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + {\left ({\left (2 \, x - 5 \, e^{3} - 8\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - x + 5 \, e^{3} + 8\right )} e^{\left (\frac {x^{2} - 5 \, x e^{3} - 8 \, x - 10 \, \log \left (\log \left (x\right )\right )}{\log \left (\log \left (x\right )\right )}\right )} + {\left (4 \, x \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - x^{3} + 5 \, x^{2} e^{3} + {\left (2 \, x^{3} - 5 \, x^{2} e^{3} - 8 \, x^{2}\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 8 \, x^{2}\right )} e^{\left (\frac {x^{2} - 5 \, x e^{3} - 8 \, x - 10 \, \log \left (\log \left (x\right )\right )}{2 \, \log \left (\log \left (x\right )\right )}\right )}}{\log \left (x\right ) \log \left (\log \left (x\right )\right )^{2}} \,d x } \]
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Time = 7.99 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.34 \[ \int \frac {4 x^3 \log (x) \log ^2(\log (x))+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{\log (\log (x))}} \left (8+5 e^3-x+\left (-8-5 e^3+2 x\right ) \log (x) \log (\log (x))\right )+e^{\frac {-8 x-5 e^3 x+x^2-10 \log (\log (x))}{2 \log (\log (x))}} \left (8 x^2+5 e^3 x^2-x^3+\left (-8 x^2-5 e^3 x^2+2 x^3\right ) \log (x) \log (\log (x))+4 x \log (x) \log ^2(\log (x))\right )}{\log (x) \log ^2(\log (x))} \, dx=x^4+{\mathrm {e}}^{\frac {x^2}{\ln \left (\ln \left (x\right )\right )}}\,{\mathrm {e}}^{-\frac {5\,x\,{\mathrm {e}}^3}{\ln \left (\ln \left (x\right )\right )}}\,{\mathrm {e}}^{-10}\,{\mathrm {e}}^{-\frac {8\,x}{\ln \left (\ln \left (x\right )\right )}}+2\,x^2\,{\mathrm {e}}^{\frac {x^2}{2\,\ln \left (\ln \left (x\right )\right )}}\,{\mathrm {e}}^{-\frac {5\,x\,{\mathrm {e}}^3}{2\,\ln \left (\ln \left (x\right )\right )}}\,{\mathrm {e}}^{-5}\,{\mathrm {e}}^{-\frac {4\,x}{\ln \left (\ln \left (x\right )\right )}} \]
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