Integrand size = 71, antiderivative size = 28 \[ \int \frac {e^3 (-1-x)-x^2-3 x^3+(-1-x) \log (4)-2 x^2 \log (x)+e^{e^x} \left (x+2 x^2+e^x x^3+\left (x+e^x x^2\right ) \log (x)\right )}{x} \, dx=-3+\left (-e^3+\left (e^{e^x}-x\right ) x-\log (4)\right ) (x+\log (x)) \]
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\[ \int \frac {e^3 (-1-x)-x^2-3 x^3+(-1-x) \log (4)-2 x^2 \log (x)+e^{e^x} \left (x+2 x^2+e^x x^3+\left (x+e^x x^2\right ) \log (x)\right )}{x} \, dx=\int \frac {e^3 (-1-x)-x^2-3 x^3+(-1-x) \log (4)-2 x^2 \log (x)+e^{e^x} \left (x+2 x^2+e^x x^3+\left (x+e^x x^2\right ) \log (x)\right )}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-x^2-3 x^3+(-1-x) \left (e^3+\log (4)\right )-2 x^2 \log (x)+e^{e^x} \left (x+2 x^2+e^x x^3+\left (x+e^x x^2\right ) \log (x)\right )}{x} \, dx \\ & = \int \left (e^{e^x+x} x (x+\log (x))+\frac {e^{e^x} x-x^2+2 e^{e^x} x^2-3 x^3-e^3 \left (1+\frac {\log (4)}{e^3}\right )-e^3 x \left (1+\frac {\log (4)}{e^3}\right )+e^{e^x} x \log (x)-2 x^2 \log (x)}{x}\right ) \, dx \\ & = \int e^{e^x+x} x (x+\log (x)) \, dx+\int \frac {e^{e^x} x-x^2+2 e^{e^x} x^2-3 x^3-e^3 \left (1+\frac {\log (4)}{e^3}\right )-e^3 x \left (1+\frac {\log (4)}{e^3}\right )+e^{e^x} x \log (x)-2 x^2 \log (x)}{x} \, dx \\ & = \int \left (e^{e^x+x} x^2+e^{e^x+x} x \log (x)\right ) \, dx+\int \left (e^{e^x} (1+2 x+\log (x))+\frac {-x^2-3 x^3-e^3 \left (1+\frac {\log (4)}{e^3}\right )-e^3 x \left (1+\frac {\log (4)}{e^3}\right )-2 x^2 \log (x)}{x}\right ) \, dx \\ & = \int e^{e^x+x} x^2 \, dx+\int e^{e^x+x} x \log (x) \, dx+\int e^{e^x} (1+2 x+\log (x)) \, dx+\int \frac {-x^2-3 x^3-e^3 \left (1+\frac {\log (4)}{e^3}\right )-e^3 x \left (1+\frac {\log (4)}{e^3}\right )-2 x^2 \log (x)}{x} \, dx \\ & = \log (x) \int e^{e^x+x} x \, dx+\int e^{e^x+x} x^2 \, dx+\int \left (e^{e^x}+2 e^{e^x} x+e^{e^x} \log (x)\right ) \, dx+\int \left (\frac {-e^3-x^2-3 x^3-\log (4)-x \left (e^3+\log (4)\right )}{x}-2 x \log (x)\right ) \, dx-\int \frac {\int e^{e^x+x} x \, dx}{x} \, dx \\ & = 2 \int e^{e^x} x \, dx-2 \int x \log (x) \, dx+\log (x) \int e^{e^x+x} x \, dx+\int e^{e^x} \, dx+\int e^{e^x+x} x^2 \, dx+\int \frac {-e^3-x^2-3 x^3-\log (4)-x \left (e^3+\log (4)\right )}{x} \, dx+\int e^{e^x} \log (x) \, dx-\int \frac {\int e^{e^x+x} x \, dx}{x} \, dx \\ & = \frac {x^2}{2}-x^2 \log (x)+\text {Ei}\left (e^x\right ) \log (x)+2 \int e^{e^x} x \, dx+\log (x) \int e^{e^x+x} x \, dx+\int e^{e^x+x} x^2 \, dx-\int \frac {\text {Ei}\left (e^x\right )}{x} \, dx+\int \left (-e^3-x-3 x^2+\frac {-e^3-\log (4)}{x}-\log (4)\right ) \, dx-\int \frac {\int e^{e^x+x} x \, dx}{x} \, dx+\text {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^x\right ) \\ & = -x^3+\text {Ei}\left (e^x\right )-x \left (e^3+\log (4)\right )-x^2 \log (x)+\text {Ei}\left (e^x\right ) \log (x)-\left (e^3+\log (4)\right ) \log (x)+2 \int e^{e^x} x \, dx+\log (x) \int e^{e^x+x} x \, dx+\int e^{e^x+x} x^2 \, dx-\int \frac {\text {Ei}\left (e^x\right )}{x} \, dx-\int \frac {\int e^{e^x+x} x \, dx}{x} \, dx \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {e^3 (-1-x)-x^2-3 x^3+(-1-x) \log (4)-2 x^2 \log (x)+e^{e^x} \left (x+2 x^2+e^x x^3+\left (x+e^x x^2\right ) \log (x)\right )}{x} \, dx=-\left (\left (e^3-e^{e^x} x+x^2+\log (4)\right ) (x+\log (x))\right ) \]
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Time = 1.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71
method | result | size |
risch | \(-x^{2} \ln \left (x \right )-x^{3}-2 \ln \left (2\right ) \ln \left (x \right )-2 x \ln \left (2\right )-\ln \left (x \right ) {\mathrm e}^{3}-x \,{\mathrm e}^{3}+\left (x \ln \left (x \right )+x^{2}\right ) {\mathrm e}^{{\mathrm e}^{x}}\) | \(48\) |
parallelrisch | \(x \ln \left (x \right ) {\mathrm e}^{{\mathrm e}^{x}}-x^{2} \ln \left (x \right )+{\mathrm e}^{{\mathrm e}^{x}} x^{2}-x^{3}-2 \ln \left (2\right ) \ln \left (x \right )-2 x \ln \left (2\right )-\ln \left (x \right ) {\mathrm e}^{3}-x \,{\mathrm e}^{3}\) | \(50\) |
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {e^3 (-1-x)-x^2-3 x^3+(-1-x) \log (4)-2 x^2 \log (x)+e^{e^x} \left (x+2 x^2+e^x x^3+\left (x+e^x x^2\right ) \log (x)\right )}{x} \, dx=-x^{3} - x e^{3} + {\left (x^{2} + x \log \left (x\right )\right )} e^{\left (e^{x}\right )} - 2 \, x \log \left (2\right ) - {\left (x^{2} + e^{3} + 2 \, \log \left (2\right )\right )} \log \left (x\right ) \]
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Time = 11.96 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {e^3 (-1-x)-x^2-3 x^3+(-1-x) \log (4)-2 x^2 \log (x)+e^{e^x} \left (x+2 x^2+e^x x^3+\left (x+e^x x^2\right ) \log (x)\right )}{x} \, dx=- x^{3} - x^{2} \log {\left (x \right )} - x \left (2 \log {\left (2 \right )} + e^{3}\right ) + \left (x^{2} + x \log {\left (x \right )}\right ) e^{e^{x}} - \left (2 \log {\left (2 \right )} + e^{3}\right ) \log {\left (x \right )} \]
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\[ \int \frac {e^3 (-1-x)-x^2-3 x^3+(-1-x) \log (4)-2 x^2 \log (x)+e^{e^x} \left (x+2 x^2+e^x x^3+\left (x+e^x x^2\right ) \log (x)\right )}{x} \, dx=\int { -\frac {3 \, x^{3} + 2 \, x^{2} \log \left (x\right ) + x^{2} + {\left (x + 1\right )} e^{3} - {\left (x^{3} e^{x} + 2 \, x^{2} + {\left (x^{2} e^{x} + x\right )} \log \left (x\right ) + x\right )} e^{\left (e^{x}\right )} + 2 \, {\left (x + 1\right )} \log \left (2\right )}{x} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.46 \[ \int \frac {e^3 (-1-x)-x^2-3 x^3+(-1-x) \log (4)-2 x^2 \log (x)+e^{e^x} \left (x+2 x^2+e^x x^3+\left (x+e^x x^2\right ) \log (x)\right )}{x} \, dx=-{\left (x^{3} e^{x} + x^{2} e^{x} \log \left (x\right ) - x^{2} e^{\left (x + e^{x}\right )} + 2 \, x e^{x} \log \left (2\right ) - x e^{\left (x + e^{x}\right )} \log \left (x\right ) + 2 \, e^{x} \log \left (2\right ) \log \left (x\right ) + x e^{\left (x + 3\right )} + e^{\left (x + 3\right )} \log \left (x\right )\right )} e^{\left (-x\right )} \]
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Time = 11.80 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {e^3 (-1-x)-x^2-3 x^3+(-1-x) \log (4)-2 x^2 \log (x)+e^{e^x} \left (x+2 x^2+e^x x^3+\left (x+e^x x^2\right ) \log (x)\right )}{x} \, dx={\mathrm {e}}^{{\mathrm {e}}^x}\,\left (x\,\ln \left (x\right )+x^2\right )-x^2\,\ln \left (x\right )-\ln \left (x\right )\,\left ({\mathrm {e}}^3+\ln \left (4\right )\right )-x\,\left ({\mathrm {e}}^3+\ln \left (4\right )\right )-x^3 \]
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