Integrand size = 86, antiderivative size = 29 \[ \int \frac {e^{\frac {e^{e^x}-e^x-2 x-x^2+x^2 \log (\log (x))}{x}} \left (x^2+e^{e^x} \left (-1+e^x x\right ) \log (x)+\left (e^x (1-x)-x^2\right ) \log (x)+x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x)} \, dx=e^{\frac {e^{e^x}-e^x-2 x}{x}-x+x \log (\log (x))} \]
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Time = 2.50 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {6838} \[ \int \frac {e^{\frac {e^{e^x}-e^x-2 x-x^2+x^2 \log (\log (x))}{x}} \left (x^2+e^{e^x} \left (-1+e^x x\right ) \log (x)+\left (e^x (1-x)-x^2\right ) \log (x)+x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x)} \, dx=e^{\frac {-x^2-2 x+e^{e^x}-e^x}{x}} \log ^x(x) \]
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Rule 6838
Rubi steps \begin{align*} \text {integral}& = e^{\frac {e^{e^x}-e^x-2 x-x^2}{x}} \log ^x(x) \\ \end{align*}
Time = 5.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {e^{e^x}-e^x-2 x-x^2+x^2 \log (\log (x))}{x}} \left (x^2+e^{e^x} \left (-1+e^x x\right ) \log (x)+\left (e^x (1-x)-x^2\right ) \log (x)+x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x)} \, dx=e^{-2+\frac {e^{e^x}}{x}-\frac {e^x}{x}-x} \log ^x(x) \]
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Time = 7.54 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\ln \left (x \right )^{x} {\mathrm e}^{-\frac {x^{2}+{\mathrm e}^{x}-{\mathrm e}^{{\mathrm e}^{x}}+2 x}{x}}\) | \(26\) |
parallelrisch | \({\mathrm e}^{\frac {x^{2} \ln \left (\ln \left (x \right )\right )+{\mathrm e}^{{\mathrm e}^{x}}-{\mathrm e}^{x}-x^{2}-2 x}{x}}\) | \(29\) |
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Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {e^{e^x}-e^x-2 x-x^2+x^2 \log (\log (x))}{x}} \left (x^2+e^{e^x} \left (-1+e^x x\right ) \log (x)+\left (e^x (1-x)-x^2\right ) \log (x)+x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x)} \, dx=e^{\left (\frac {x^{2} \log \left (\log \left (x\right )\right ) - x^{2} - 2 \, x - e^{x} + e^{\left (e^{x}\right )}}{x}\right )} \]
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Time = 4.82 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {e^{e^x}-e^x-2 x-x^2+x^2 \log (\log (x))}{x}} \left (x^2+e^{e^x} \left (-1+e^x x\right ) \log (x)+\left (e^x (1-x)-x^2\right ) \log (x)+x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x)} \, dx=e^{\frac {x^{2} \log {\left (\log {\left (x \right )} \right )} - x^{2} - 2 x - e^{x} + e^{e^{x}}}{x}} \]
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Time = 0.45 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\frac {e^{e^x}-e^x-2 x-x^2+x^2 \log (\log (x))}{x}} \left (x^2+e^{e^x} \left (-1+e^x x\right ) \log (x)+\left (e^x (1-x)-x^2\right ) \log (x)+x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x)} \, dx=e^{\left (x \log \left (\log \left (x\right )\right ) - x - \frac {e^{x}}{x} + \frac {e^{\left (e^{x}\right )}}{x} - 2\right )} \]
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\[ \int \frac {e^{\frac {e^{e^x}-e^x-2 x-x^2+x^2 \log (\log (x))}{x}} \left (x^2+e^{e^x} \left (-1+e^x x\right ) \log (x)+\left (e^x (1-x)-x^2\right ) \log (x)+x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x)} \, dx=\int { \frac {{\left (x^{2} \log \left (x\right ) \log \left (\log \left (x\right )\right ) + {\left (x e^{x} - 1\right )} e^{\left (e^{x}\right )} \log \left (x\right ) + x^{2} - {\left (x^{2} + {\left (x - 1\right )} e^{x}\right )} \log \left (x\right )\right )} e^{\left (\frac {x^{2} \log \left (\log \left (x\right )\right ) - x^{2} - 2 \, x - e^{x} + e^{\left (e^{x}\right )}}{x}\right )}}{x^{2} \log \left (x\right )} \,d x } \]
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Time = 8.88 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {e^{e^x}-e^x-2 x-x^2+x^2 \log (\log (x))}{x}} \left (x^2+e^{e^x} \left (-1+e^x x\right ) \log (x)+\left (e^x (1-x)-x^2\right ) \log (x)+x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x)} \, dx={\mathrm {e}}^{-x}\,{\mathrm {e}}^{-2}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^x}{x}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{{\mathrm {e}}^x}}{x}}\,{\ln \left (x\right )}^x \]
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