Optimal. Leaf size=24 \[ e^{e^{-\frac {2 e^{-x}}{\log (x)}}} \left (5-e^x+x\right ) \]
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Rubi [F] time = 11.39, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}} \left (10-2 e^x+2 x+\left (10 x-2 e^x x+2 x^2\right ) \log (x)+e^{\frac {2 e^{-x}}{\log (x)}} \left (e^x x-e^{2 x} x\right ) \log ^2(x)\right )}{x \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-e^{e^{-\frac {2 e^{-x}}{\log (x)}}} \left (-1+e^x\right )+\frac {2 e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}} \left (5-e^x+x\right ) (1+x \log (x))}{x \log ^2(x)}\right ) \, dx\\ &=2 \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}} \left (5-e^x+x\right ) (1+x \log (x))}{x \log ^2(x)} \, dx-\int e^{e^{-\frac {2 e^{-x}}{\log (x)}}} \left (-1+e^x\right ) \, dx\\ &=2 \int \left (-\frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-\frac {2 e^{-x}}{\log (x)}} (1+x \log (x))}{x \log ^2(x)}+\frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}} (5+x) (1+x \log (x))}{x \log ^2(x)}\right ) \, dx-\int \left (-e^{e^{-\frac {2 e^{-x}}{\log (x)}}}+e^{e^{-\frac {2 e^{-x}}{\log (x)}}+x}\right ) \, dx\\ &=-\left (2 \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-\frac {2 e^{-x}}{\log (x)}} (1+x \log (x))}{x \log ^2(x)} \, dx\right )+2 \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}} (5+x) (1+x \log (x))}{x \log ^2(x)} \, dx+\int e^{e^{-\frac {2 e^{-x}}{\log (x)}}} \, dx-\int e^{e^{-\frac {2 e^{-x}}{\log (x)}}+x} \, dx\\ &=-\left (2 \int \left (\frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-\frac {2 e^{-x}}{\log (x)}}}{x \log ^2(x)}+\frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-\frac {2 e^{-x}}{\log (x)}}}{\log (x)}\right ) \, dx\right )+2 \int \left (\frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}} (5+x)}{x \log ^2(x)}+\frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}} (5+x)}{\log (x)}\right ) \, dx+\int e^{e^{-\frac {2 e^{-x}}{\log (x)}}} \, dx-\int e^{e^{-\frac {2 e^{-x}}{\log (x)}}+x} \, dx\\ &=-\left (2 \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-\frac {2 e^{-x}}{\log (x)}}}{x \log ^2(x)} \, dx\right )+2 \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}} (5+x)}{x \log ^2(x)} \, dx-2 \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-\frac {2 e^{-x}}{\log (x)}}}{\log (x)} \, dx+2 \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}} (5+x)}{\log (x)} \, dx+\int e^{e^{-\frac {2 e^{-x}}{\log (x)}}} \, dx-\int e^{e^{-\frac {2 e^{-x}}{\log (x)}}+x} \, dx\\ &=2 \int \left (\frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}}}{\log ^2(x)}+\frac {5 e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}}}{x \log ^2(x)}\right ) \, dx+2 \int \left (\frac {5 e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}}}{\log (x)}+\frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}} x}{\log (x)}\right ) \, dx-2 \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-\frac {2 e^{-x}}{\log (x)}}}{x \log ^2(x)} \, dx-2 \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-\frac {2 e^{-x}}{\log (x)}}}{\log (x)} \, dx+\int e^{e^{-\frac {2 e^{-x}}{\log (x)}}} \, dx-\int e^{e^{-\frac {2 e^{-x}}{\log (x)}}+x} \, dx\\ &=2 \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}}}{\log ^2(x)} \, dx-2 \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-\frac {2 e^{-x}}{\log (x)}}}{x \log ^2(x)} \, dx-2 \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-\frac {2 e^{-x}}{\log (x)}}}{\log (x)} \, dx+2 \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}} x}{\log (x)} \, dx+10 \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}}}{x \log ^2(x)} \, dx+10 \int \frac {e^{e^{-\frac {2 e^{-x}}{\log (x)}}-x-\frac {2 e^{-x}}{\log (x)}}}{\log (x)} \, dx+\int e^{e^{-\frac {2 e^{-x}}{\log (x)}}} \, dx-\int e^{e^{-\frac {2 e^{-x}}{\log (x)}}+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.16, size = 25, normalized size = 1.04 \begin {gather*} -e^{e^{-\frac {2 e^{-x}}{\log (x)}}} \left (-5+e^x-x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 74, normalized size = 3.08 \begin {gather*} {\left ({\left (x + 5\right )} e^{x} - e^{\left (2 \, x\right )}\right )} e^{\left (-\frac {{\left ({\left (x e^{x} \log \relax (x) + 2\right )} e^{\left (\frac {2 \, e^{\left (-x\right )}}{\log \relax (x)}\right )} - e^{x} \log \relax (x)\right )} e^{\left (-x - \frac {2 \, e^{\left (-x\right )}}{\log \relax (x)}\right )}}{\log \relax (x)} + \frac {2 \, e^{\left (-x\right )}}{\log \relax (x)}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 21, normalized size = 0.88
method | result | size |
risch | \({\mathrm e}^{{\mathrm e}^{-\frac {2 \,{\mathrm e}^{-x}}{\ln \relax (x )}}} \left (x -{\mathrm e}^{x}+5\right )\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left ({\left (x e^{\left (2 \, x\right )} - x e^{x}\right )} e^{\left (\frac {2 \, e^{\left (-x\right )}}{\log \relax (x)}\right )} \log \relax (x)^{2} - 2 \, {\left (x^{2} - x e^{x} + 5 \, x\right )} \log \relax (x) - 2 \, x + 2 \, e^{x} - 10\right )} e^{\left (-x - \frac {2 \, e^{\left (-x\right )}}{\log \relax (x)} + e^{\left (-\frac {2 \, e^{\left (-x\right )}}{\log \relax (x)}\right )}\right )}}{x \log \relax (x)^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^{-x}}{\ln \relax (x)}}\,{\mathrm {e}}^{{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^{-x}}{\ln \relax (x)}}}\,\left (2\,x-2\,{\mathrm {e}}^x+\ln \relax (x)\,\left (10\,x-2\,x\,{\mathrm {e}}^x+2\,x^2\right )-{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{-x}}{\ln \relax (x)}}\,{\ln \relax (x)}^2\,\left (x\,{\mathrm {e}}^{2\,x}-x\,{\mathrm {e}}^x\right )+10\right )}{x\,{\ln \relax (x)}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.01, size = 27, normalized size = 1.12 \begin {gather*} \left (x e^{- x} - 1 + 5 e^{- x}\right ) e^{x} e^{e^{- \frac {2 e^{- x}}{\log {\relax (x )}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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