Integrand size = 126, antiderivative size = 21 \[ \int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx=x+\frac {\log \left (e^{e^{3 e^{-x}}}+x\right )}{\log (x)} \]
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\[ \int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx=\int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (1+\frac {1-3 e^{e^{3 e^{-x}}+3 e^{-x}-x}}{\left (e^{e^{3 e^{-x}}}+x\right ) \log (x)}-\frac {\log \left (e^{e^{3 e^{-x}}}+x\right )}{x \log ^2(x)}\right ) \, dx \\ & = x+\int \frac {1-3 e^{e^{3 e^{-x}}+3 e^{-x}-x}}{\left (e^{e^{3 e^{-x}}}+x\right ) \log (x)} \, dx-\int \frac {\log \left (e^{e^{3 e^{-x}}}+x\right )}{x \log ^2(x)} \, dx \\ & = x+\int \left (\frac {1}{\left (e^{e^{3 e^{-x}}}+x\right ) \log (x)}-\frac {3 e^{e^{3 e^{-x}}+3 e^{-x}-x}}{\left (e^{e^{3 e^{-x}}}+x\right ) \log (x)}\right ) \, dx-\int \frac {\log \left (e^{e^{3 e^{-x}}}+x\right )}{x \log ^2(x)} \, dx \\ & = x-3 \int \frac {e^{e^{3 e^{-x}}+3 e^{-x}-x}}{\left (e^{e^{3 e^{-x}}}+x\right ) \log (x)} \, dx+\int \frac {1}{\left (e^{e^{3 e^{-x}}}+x\right ) \log (x)} \, dx-\int \frac {\log \left (e^{e^{3 e^{-x}}}+x\right )}{x \log ^2(x)} \, dx \\ \end{align*}
Time = 0.63 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx=x+\frac {\log \left (e^{e^{3 e^{-x}}}+x\right )}{\log (x)} \]
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Time = 48.35 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{3 \,{\mathrm e}^{-x}}}+x \right )}{\ln \left (x \right )}+x\) | \(19\) |
parallelrisch | \(-\frac {-2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (x \right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{3 \,{\mathrm e}^{-x}}}+x \right )}{2 \ln \left (x \right )}\) | \(28\) |
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Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx=\frac {x \log \left (x\right ) + \log \left ({\left (x e^{x} + e^{\left (x + e^{\left (3 \, e^{\left (-x\right )}\right )}\right )}\right )} e^{\left (-x\right )}\right )}{\log \left (x\right )} \]
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Exception generated. \[ \int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx=\frac {x \log \left (x\right ) + \log \left (x + e^{\left (e^{\left (3 \, e^{\left (-x\right )}\right )}\right )}\right )}{\log \left (x\right )} \]
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Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx=\frac {x \log \left (x\right ) + \log \left ({\left (x e^{x} + e^{\left (x + e^{\left (3 \, e^{\left (-x\right )}\right )}\right )}\right )} e^{\left (-x\right )}\right )}{\log \left (x\right )} \]
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Timed out. \[ \int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx=-\int \frac {\ln \left (x+{\mathrm {e}}^{{\mathrm {e}}^{3\,{\mathrm {e}}^{-x}}}\right )\,\left ({\mathrm {e}}^{x+{\mathrm {e}}^{3\,{\mathrm {e}}^{-x}}}+x\,{\mathrm {e}}^x\right )+{\mathrm {e}}^{{\mathrm {e}}^{3\,{\mathrm {e}}^{-x}}}\,\left (3\,x\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-x}}\,\ln \left (x\right )-x\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2\right )-x^2\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2-x\,{\mathrm {e}}^x\,\ln \left (x\right )}{x^2\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2+x\,{\mathrm {e}}^{x+{\mathrm {e}}^{3\,{\mathrm {e}}^{-x}}}\,{\ln \left (x\right )}^2} \,d x \]
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