Integrand size = 235, antiderivative size = 30 \[ \int \frac {e^{\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx=e^{1-x-x^2+\frac {x}{\log \left (\left (e^x+e^{2 x}\right )^2+x\right )}} \]
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\[ \int \frac {e^{\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx=\int \frac {\exp \left (\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\exp \left (\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) x \left (-1+2 e^{2 x}+2 e^{3 x}+4 x\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}+\frac {\exp \left (\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) \left (-4 x+\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )-\log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )-2 x \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) \, dx \\ & = \int \frac {\exp \left (\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) x \left (-1+2 e^{2 x}+2 e^{3 x}+4 x\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx+\int \frac {\exp \left (\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) \left (-4 x+\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )-\log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )-2 x \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx \\ & = \int \frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x \left (-1+2 e^{2 x}+2 e^{3 x}+4 x\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx+\int \frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-4 x+\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )-(1+2 x) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx \\ & = \int \left (-\frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}+\frac {2 e^{1+x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}+\frac {2 \exp \left (1+2 x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) x}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}+\frac {4 e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x^2}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) \, dx+\int \left (-e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}}-2 e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x-\frac {4 e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x}{\log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}+\frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}}}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) \, dx \\ & = -\left (2 \int e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x \, dx\right )+2 \int \frac {e^{1+x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx+2 \int \frac {\exp \left (1+2 x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}\right ) x}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx-4 \int \frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x}{\log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx+4 \int \frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x^2}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx-\int e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \, dx-\int \frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} x}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx+\int \frac {e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}}}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx=e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(27)=54\).
Time = 84.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.00
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {\left (-x^{2}-x +1\right ) \ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{x} {\mathrm e}^{2 x}+{\mathrm e}^{2 x}+x \right )+x}{\ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{x} {\mathrm e}^{2 x}+{\mathrm e}^{2 x}+x \right )}}\) | \(60\) |
risch | \({\mathrm e}^{-\frac {\ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+{\mathrm e}^{2 x}+x \right ) x^{2}+\ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+{\mathrm e}^{2 x}+x \right ) x -\ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+{\mathrm e}^{2 x}+x \right )-x}{\ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+{\mathrm e}^{2 x}+x \right )}}\) | \(86\) |
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Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67 \[ \int \frac {e^{\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx=e^{\left (-\frac {{\left (x^{2} + x - 1\right )} \log \left (x + e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )}\right ) - x}{\log \left (x + e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )}\right )}\right )} \]
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Timed out. \[ \int \frac {e^{\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {e^{\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx=\text {Exception raised: RuntimeError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (27) = 54\).
Time = 0.65 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.83 \[ \int \frac {e^{\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx=e^{\left (-\frac {x^{2} \log \left (x + e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )}\right ) + x \log \left (x + e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )}\right ) - x - \log \left (x + e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )}\right )}{\log \left (x + e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )}\right )}\right )} \]
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Time = 13.78 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {e^{\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx={\mathrm {e}}^{-x}\,\mathrm {e}\,{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^{\frac {x}{\ln \left (x+{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}\right )}} \]
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