Integrand size = 96, antiderivative size = 33 \[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\frac {1}{3} \left (-\frac {-2+e^{1+x}}{x}+x\right ) \left (-1+\frac {x}{2}+\frac {x^2}{\log (x)}\right ) \]
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\[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx \\ & = \frac {1}{6} \int \left (\frac {\left (2-e^{1+x}+2 x\right ) \left (2-2 x+x^2\right )}{x^2}+\frac {2 \left (-2+e^{1+x}-x^2\right )}{\log ^2(x)}+\frac {4+6 x^2-2 e^{1+x} (1+x)}{\log (x)}\right ) \, dx \\ & = \frac {1}{6} \int \frac {\left (2-e^{1+x}+2 x\right ) \left (2-2 x+x^2\right )}{x^2} \, dx+\frac {1}{6} \int \frac {4+6 x^2-2 e^{1+x} (1+x)}{\log (x)} \, dx+\frac {1}{3} \int \frac {-2+e^{1+x}-x^2}{\log ^2(x)} \, dx \\ & = \frac {1}{6} \int \left (-\frac {e^{1+x} \left (2-2 x+x^2\right )}{x^2}+\frac {2 \left (2-x^2+x^3\right )}{x^2}\right ) \, dx+\frac {1}{6} \int \left (-\frac {2 e^{1+x} (1+x)}{\log (x)}+\frac {2 \left (2+3 x^2\right )}{\log (x)}\right ) \, dx+\frac {1}{3} \int \left (\frac {e^{1+x}}{\log ^2(x)}+\frac {-2-x^2}{\log ^2(x)}\right ) \, dx \\ & = -\left (\frac {1}{6} \int \frac {e^{1+x} \left (2-2 x+x^2\right )}{x^2} \, dx\right )+\frac {1}{3} \int \frac {2-x^2+x^3}{x^2} \, dx+\frac {1}{3} \int \frac {e^{1+x}}{\log ^2(x)} \, dx+\frac {1}{3} \int \frac {-2-x^2}{\log ^2(x)} \, dx-\frac {1}{3} \int \frac {e^{1+x} (1+x)}{\log (x)} \, dx+\frac {1}{3} \int \frac {2+3 x^2}{\log (x)} \, dx \\ & = -\left (\frac {1}{6} \int \left (e^{1+x}+\frac {2 e^{1+x}}{x^2}-\frac {2 e^{1+x}}{x}\right ) \, dx\right )+\frac {1}{3} \int \left (-1+\frac {2}{x^2}+x\right ) \, dx+\frac {1}{3} \int \left (-\frac {2}{\log ^2(x)}-\frac {x^2}{\log ^2(x)}\right ) \, dx-\frac {1}{3} \int \left (\frac {e^{1+x}}{\log (x)}+\frac {e^{1+x} x}{\log (x)}\right ) \, dx+\frac {1}{3} \int \left (\frac {2}{\log (x)}+\frac {3 x^2}{\log (x)}\right ) \, dx+\frac {1}{3} \int \frac {e^{1+x}}{\log ^2(x)} \, dx \\ & = -\frac {2}{3 x}-\frac {x}{3}+\frac {x^2}{6}-\frac {1}{6} \int e^{1+x} \, dx-\frac {1}{3} \int \frac {e^{1+x}}{x^2} \, dx+\frac {1}{3} \int \frac {e^{1+x}}{x} \, dx+\frac {1}{3} \int \frac {e^{1+x}}{\log ^2(x)} \, dx-\frac {1}{3} \int \frac {x^2}{\log ^2(x)} \, dx-\frac {1}{3} \int \frac {e^{1+x}}{\log (x)} \, dx-\frac {1}{3} \int \frac {e^{1+x} x}{\log (x)} \, dx-\frac {2}{3} \int \frac {1}{\log ^2(x)} \, dx+\frac {2}{3} \int \frac {1}{\log (x)} \, dx+\int \frac {x^2}{\log (x)} \, dx \\ & = -\frac {e^{1+x}}{6}-\frac {2}{3 x}+\frac {e^{1+x}}{3 x}-\frac {x}{3}+\frac {x^2}{6}+\frac {e \operatorname {ExpIntegralEi}(x)}{3}+\frac {2 x}{3 \log (x)}+\frac {x^3}{3 \log (x)}+\frac {2 \operatorname {LogIntegral}(x)}{3}-\frac {1}{3} \int \frac {e^{1+x}}{x} \, dx+\frac {1}{3} \int \frac {e^{1+x}}{\log ^2(x)} \, dx-\frac {1}{3} \int \frac {e^{1+x}}{\log (x)} \, dx-\frac {1}{3} \int \frac {e^{1+x} x}{\log (x)} \, dx-\frac {2}{3} \int \frac {1}{\log (x)} \, dx-\int \frac {x^2}{\log (x)} \, dx+\text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right ) \\ & = -\frac {e^{1+x}}{6}-\frac {2}{3 x}+\frac {e^{1+x}}{3 x}-\frac {x}{3}+\frac {x^2}{6}+\operatorname {ExpIntegralEi}(3 \log (x))+\frac {2 x}{3 \log (x)}+\frac {x^3}{3 \log (x)}+\frac {1}{3} \int \frac {e^{1+x}}{\log ^2(x)} \, dx-\frac {1}{3} \int \frac {e^{1+x}}{\log (x)} \, dx-\frac {1}{3} \int \frac {e^{1+x} x}{\log (x)} \, dx-\text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right ) \\ & = -\frac {e^{1+x}}{6}-\frac {2}{3 x}+\frac {e^{1+x}}{3 x}-\frac {x}{3}+\frac {x^2}{6}+\frac {2 x}{3 \log (x)}+\frac {x^3}{3 \log (x)}+\frac {1}{3} \int \frac {e^{1+x}}{\log ^2(x)} \, dx-\frac {1}{3} \int \frac {e^{1+x}}{\log (x)} \, dx-\frac {1}{3} \int \frac {e^{1+x} x}{\log (x)} \, dx \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\frac {1}{6} \left (e^x \left (-e+\frac {2 e}{x}\right )-\frac {4}{x}-2 x+x^2+\frac {2 x \left (2-e^{1+x}+x^2\right )}{\log (x)}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45
method | result | size |
risch | \(\frac {x^{3}-2 x^{2}-x \,{\mathrm e}^{1+x}+2 \,{\mathrm e}^{1+x}-4}{6 x}+\frac {x \left (x^{2}-{\mathrm e}^{1+x}+2\right )}{3 \ln \left (x \right )}\) | \(48\) |
parallelrisch | \(-\frac {-x^{3} \ln \left (x \right )-2 x^{4}+x \,{\mathrm e}^{1+x} \ln \left (x \right )+2 x^{2} \ln \left (x \right )+2 x^{2} {\mathrm e}^{1+x}-2 \ln \left (x \right ) {\mathrm e}^{1+x}-4 x^{2}+4 \ln \left (x \right )}{6 \ln \left (x \right ) x}\) | \(64\) |
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Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\frac {2 \, x^{4} - 2 \, x^{2} e^{\left (x + 1\right )} + 4 \, x^{2} + {\left (x^{3} - 2 \, x^{2} - {\left (x - 2\right )} e^{\left (x + 1\right )} - 4\right )} \log \left (x\right )}{6 \, x \log \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\frac {x^{2}}{6} - \frac {x}{3} + \frac {x^{3} + 2 x}{3 \log {\left (x \right )}} + \frac {\left (- 2 x^{2} - x \log {\left (x \right )} + 2 \log {\left (x \right )}\right ) e^{x + 1}}{6 x \log {\left (x \right )}} - \frac {2}{3 x} \]
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\[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\int { -\frac {2 \, x^{4} - 2 \, x^{2} e^{\left (x + 1\right )} - {\left (2 \, x^{3} - 2 \, x^{2} - {\left (x^{2} - 2 \, x + 2\right )} e^{\left (x + 1\right )} + 4\right )} \log \left (x\right )^{2} + 4 \, x^{2} - 2 \, {\left (3 \, x^{4} + 2 \, x^{2} - {\left (x^{3} + x^{2}\right )} e^{\left (x + 1\right )}\right )} \log \left (x\right )}{6 \, x^{2} \log \left (x\right )^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).
Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.91 \[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\frac {2 \, x^{4} + x^{3} \log \left (x\right ) - 2 \, x^{2} e^{\left (x + 1\right )} - 2 \, x^{2} \log \left (x\right ) - x e^{\left (x + 1\right )} \log \left (x\right ) + 4 \, x^{2} + 2 \, e^{\left (x + 1\right )} \log \left (x\right ) - 4 \, \log \left (x\right )}{6 \, x \log \left (x\right )} \]
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Time = 14.45 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\frac {2\,x}{3\,\ln \left (x\right )}-\frac {{\mathrm {e}}^{x+1}}{6}-\frac {x}{3}+\frac {{\mathrm {e}}^{x+1}}{3\,x}+\frac {x^3}{3\,\ln \left (x\right )}-\frac {2}{3\,x}+\frac {x^2}{6}-\frac {x\,{\mathrm {e}}^{x+1}}{3\,\ln \left (x\right )} \]
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