Integrand size = 100, antiderivative size = 18 \[ \int \frac {e^{9 x+3 x^2+\left (3 x+x^2\right ) \log \left (-1+e^{e^x}+x\right )} \left (-9+6 x+7 x^2+e^{e^x} \left (9+6 x+e^x \left (3 x+x^2\right )\right )+\left (-3+x+2 x^2+e^{e^x} (3+2 x)\right ) \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x} \, dx=e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \]
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\[ \int \frac {e^{9 x+3 x^2+\left (3 x+x^2\right ) \log \left (-1+e^{e^x}+x\right )} \left (-9+6 x+7 x^2+e^{e^x} \left (9+6 x+e^x \left (3 x+x^2\right )\right )+\left (-3+x+2 x^2+e^{e^x} (3+2 x)\right ) \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x} \, dx=\int \frac {\exp \left (9 x+3 x^2+\left (3 x+x^2\right ) \log \left (-1+e^{e^x}+x\right )\right ) \left (-9+6 x+7 x^2+e^{e^x} \left (9+6 x+e^x \left (3 x+x^2\right )\right )+\left (-3+x+2 x^2+e^{e^x} (3+2 x)\right ) \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \left (9-6 x-7 x^2-e^{e^x} \left (9+6 x+e^x \left (3 x+x^2\right )\right )-\left (-3+x+2 x^2+e^{e^x} (3+2 x)\right ) \log \left (-1+e^{e^x}+x\right )\right )}{1-e^{e^x}-x} \, dx \\ & = \int \left (\frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x (3+x)}{-1+e^{e^x}+x}+\frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \left (-9+9 e^{e^x}+6 x+6 e^{e^x} x+7 x^2-3 \log \left (-1+e^{e^x}+x\right )+3 e^{e^x} \log \left (-1+e^{e^x}+x\right )+x \log \left (-1+e^{e^x}+x\right )+2 e^{e^x} x \log \left (-1+e^{e^x}+x\right )+2 x^2 \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x}\right ) \, dx \\ & = \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x (3+x)}{-1+e^{e^x}+x} \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \left (-9+9 e^{e^x}+6 x+6 e^{e^x} x+7 x^2-3 \log \left (-1+e^{e^x}+x\right )+3 e^{e^x} \log \left (-1+e^{e^x}+x\right )+x \log \left (-1+e^{e^x}+x\right )+2 e^{e^x} x \log \left (-1+e^{e^x}+x\right )+2 x^2 \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x} \, dx \\ & = \int \left (\frac {3 e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x}+\frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x}\right ) \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \left (9-6 x-7 x^2-e^{e^x} (9+6 x)-\left (-1+e^{e^x}+x\right ) (3+2 x) \log \left (-1+e^{e^x}+x\right )\right )}{1-e^{e^x}-x} \, dx \\ & = 3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \left (\frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x (3+x)}{-1+e^{e^x}+x}+e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )\right ) \, dx \\ & = 3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x (3+x)}{-1+e^{e^x}+x} \, dx+\int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x) \left (3+\log \left (-1+e^{e^x}+x\right )\right ) \, dx \\ & = 3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \left (\frac {3 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x}+\frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x}\right ) \, dx+\int \left (3 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x)+e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x) \log \left (-1+e^{e^x}+x\right )\right ) \, dx \\ & = 3 \int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x) \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} (3+2 x) \log \left (-1+e^{e^x}+x\right ) \, dx \\ & = 3 \int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int \left (3 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )}+2 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x\right ) \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \left (3 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \log \left (-1+e^{e^x}+x\right )+2 e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x \log \left (-1+e^{e^x}+x\right )\right ) \, dx \\ & = 2 \int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x \log \left (-1+e^{e^x}+x\right ) \, dx+3 \int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x}{-1+e^{e^x}+x} \, dx+3 \int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \log \left (-1+e^{e^x}+x\right ) \, dx+6 \int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x \, dx+9 \int e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} \, dx+\int \frac {e^{x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx+\int \frac {e^{e^x+x+x (3+x) \left (3+\log \left (-1+e^{e^x}+x\right )\right )} x^2}{-1+e^{e^x}+x} \, dx \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {e^{9 x+3 x^2+\left (3 x+x^2\right ) \log \left (-1+e^{e^x}+x\right )} \left (-9+6 x+7 x^2+e^{e^x} \left (9+6 x+e^x \left (3 x+x^2\right )\right )+\left (-3+x+2 x^2+e^{e^x} (3+2 x)\right ) \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x} \, dx=e^{3 x (3+x)} \left (-1+e^{e^x}+x\right )^{x (3+x)} \]
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Time = 3.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17
method | result | size |
risch | \(\left ({\mathrm e}^{{\mathrm e}^{x}}+x -1\right )^{\left (3+x \right ) x} {\mathrm e}^{3 \left (3+x \right ) x}\) | \(21\) |
parallelrisch | \({\mathrm e}^{\left (x^{2}+3 x \right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}+x -1\right )+3 x^{2}+9 x}\) | \(26\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {e^{9 x+3 x^2+\left (3 x+x^2\right ) \log \left (-1+e^{e^x}+x\right )} \left (-9+6 x+7 x^2+e^{e^x} \left (9+6 x+e^x \left (3 x+x^2\right )\right )+\left (-3+x+2 x^2+e^{e^x} (3+2 x)\right ) \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x} \, dx=e^{\left (3 \, x^{2} + {\left (x^{2} + 3 \, x\right )} \log \left (x + e^{\left (e^{x}\right )} - 1\right ) + 9 \, x\right )} \]
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Timed out. \[ \int \frac {e^{9 x+3 x^2+\left (3 x+x^2\right ) \log \left (-1+e^{e^x}+x\right )} \left (-9+6 x+7 x^2+e^{e^x} \left (9+6 x+e^x \left (3 x+x^2\right )\right )+\left (-3+x+2 x^2+e^{e^x} (3+2 x)\right ) \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).
Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {e^{9 x+3 x^2+\left (3 x+x^2\right ) \log \left (-1+e^{e^x}+x\right )} \left (-9+6 x+7 x^2+e^{e^x} \left (9+6 x+e^x \left (3 x+x^2\right )\right )+\left (-3+x+2 x^2+e^{e^x} (3+2 x)\right ) \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x} \, dx=e^{\left (x^{2} \log \left (x + e^{\left (e^{x}\right )} - 1\right ) + 3 \, x^{2} + 3 \, x \log \left (x + e^{\left (e^{x}\right )} - 1\right ) + 9 \, x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).
Time = 0.68 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {e^{9 x+3 x^2+\left (3 x+x^2\right ) \log \left (-1+e^{e^x}+x\right )} \left (-9+6 x+7 x^2+e^{e^x} \left (9+6 x+e^x \left (3 x+x^2\right )\right )+\left (-3+x+2 x^2+e^{e^x} (3+2 x)\right ) \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x} \, dx=e^{\left (x^{2} \log \left (x + e^{\left (e^{x}\right )} - 1\right ) + 3 \, x^{2} + 3 \, x \log \left (x + e^{\left (e^{x}\right )} - 1\right ) + 9 \, x\right )} \]
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Time = 10.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {e^{9 x+3 x^2+\left (3 x+x^2\right ) \log \left (-1+e^{e^x}+x\right )} \left (-9+6 x+7 x^2+e^{e^x} \left (9+6 x+e^x \left (3 x+x^2\right )\right )+\left (-3+x+2 x^2+e^{e^x} (3+2 x)\right ) \log \left (-1+e^{e^x}+x\right )\right )}{-1+e^{e^x}+x} \, dx={\mathrm {e}}^{3\,x^2+9\,x}\,{\left (x+{\mathrm {e}}^{{\mathrm {e}}^x}-1\right )}^{x^2+3\,x} \]
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