Optimal. Leaf size=21 \[ \frac {x}{\log \left (e^{1+x+e^{-e^x} x}+x\right )} \]
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Rubi [F] time = 9.35, 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^x} x+e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )} \left (-x-e^{e^x} x+e^x x^2\right )+\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log \left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )}{\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log ^2\left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-\frac {e^{-e^x} x \left (e^{e^x}+e^{1+x+e^{-e^x} x}+e^{1+e^x+x+e^{-e^x} x}-e^{1+\left (2+e^{-e^x}\right ) x} x\right )}{e^{1+x+e^{-e^x} x}+x}+\log \left (e^{1+x+e^{-e^x} x}+x\right )}{\log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx\\ &=\int \left (\frac {e^{-e^x} x \left (-e^{e^x}+x+e^{e^x} x+e^{1+\left (2+e^{-e^x}\right ) x} x\right )}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )}+\frac {e^{-e^x} \left (-x-e^{e^x} x+e^{e^x} \log \left (e^{1+x+e^{-e^x} x}+x\right )\right )}{\log ^2\left (e^{1+x+e^{-e^x} x}+x\right )}\right ) \, dx\\ &=\int \frac {e^{-e^x} x \left (-e^{e^x}+x+e^{e^x} x+e^{1+\left (2+e^{-e^x}\right ) x} x\right )}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \frac {e^{-e^x} \left (-x-e^{e^x} x+e^{e^x} \log \left (e^{1+x+e^{-e^x} x}+x\right )\right )}{\log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx\\ &=\int \left (-\frac {x}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )}+\frac {x^2}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )}+\frac {e^{-e^x} x^2}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )}+\frac {e^{1-e^x+\left (2+e^{-e^x}\right ) x} x^2}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )}\right ) \, dx+\int \frac {-x-e^{-e^x} x+\log \left (e^{1+x+e^{-e^x} x}+x\right )}{\log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx\\ &=-\int \frac {x}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \frac {x^2}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \frac {e^{-e^x} x^2}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \frac {e^{1-e^x+\left (2+e^{-e^x}\right ) x} x^2}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \left (-\frac {e^{-e^x} x}{\log ^2\left (e^{1+x+e^{-e^x} x}+x\right )}+\frac {-x+\log \left (e^{1+x+e^{-e^x} x}+x\right )}{\log ^2\left (e^{1+x+e^{-e^x} x}+x\right )}\right ) \, dx\\ &=-\int \frac {e^{-e^x} x}{\log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx-\int \frac {x}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \frac {x^2}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \frac {e^{-e^x} x^2}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \frac {e^{1-e^x+\left (2+e^{-e^x}\right ) x} x^2}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \frac {-x+\log \left (e^{1+x+e^{-e^x} x}+x\right )}{\log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx\\ &=\int \left (-\frac {x}{\log ^2\left (e^{1+x+e^{-e^x} x}+x\right )}+\frac {1}{\log \left (e^{1+x+e^{-e^x} x}+x\right )}\right ) \, dx-\int \frac {e^{-e^x} x}{\log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx-\int \frac {x}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \frac {x^2}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \frac {e^{-e^x} x^2}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \frac {e^{1-e^x+\left (2+e^{-e^x}\right ) x} x^2}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx\\ &=-\int \frac {x}{\log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx-\int \frac {e^{-e^x} x}{\log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx-\int \frac {x}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \frac {x^2}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \frac {e^{-e^x} x^2}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \frac {e^{1-e^x+\left (2+e^{-e^x}\right ) x} x^2}{\left (e^{1+x+e^{-e^x} x}+x\right ) \log ^2\left (e^{1+x+e^{-e^x} x}+x\right )} \, dx+\int \frac {1}{\log \left (e^{1+x+e^{-e^x} x}+x\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.31, size = 21, normalized size = 1.00 \begin {gather*} \frac {x}{\log \left (e^{1+x+e^{-e^x} x}+x\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 35, normalized size = 1.67 \begin {gather*} \frac {x}{\log \left ({\left (x e^{\left (e^{x}\right )} + e^{\left ({\left ({\left (x + e^{x} + 1\right )} e^{\left (e^{x}\right )} + x\right )} e^{\left (-e^{x}\right )}\right )}\right )} e^{\left (-e^{x}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 194, normalized size = 9.24 \begin {gather*} \frac {x^{2} e^{\left (2 \, x e^{\left (-e^{x}\right )} + 3 \, x + 1\right )} - x e^{\left (2 \, x e^{\left (-e^{x}\right )} + 2 \, x + e^{x} + 1\right )} - x e^{\left (2 \, x e^{\left (-e^{x}\right )} + 2 \, x + 1\right )} - x e^{\left (x e^{\left (-e^{x}\right )} + x + e^{x}\right )}}{x e^{\left (2 \, x e^{\left (-e^{x}\right )} + 3 \, x + 1\right )} \log \left (x + e^{\left (x e^{\left (-e^{x}\right )} + x + 1\right )}\right ) - e^{\left (2 \, x e^{\left (-e^{x}\right )} + 2 \, x + e^{x} + 1\right )} \log \left (x + e^{\left (x e^{\left (-e^{x}\right )} + x + 1\right )}\right ) - e^{\left (2 \, x e^{\left (-e^{x}\right )} + 2 \, x + 1\right )} \log \left (x + e^{\left (x e^{\left (-e^{x}\right )} + x + 1\right )}\right ) - e^{\left (x e^{\left (-e^{x}\right )} + x + e^{x}\right )} \log \left (x + e^{\left (x e^{\left (-e^{x}\right )} + x + 1\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 25, normalized size = 1.19
method | result | size |
risch | \(\frac {x}{\ln \left ({\mathrm e}^{\left (x \,{\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{{\mathrm e}^{x}}+x \right ) {\mathrm e}^{-{\mathrm e}^{x}}}+x \right )}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 18, normalized size = 0.86 \begin {gather*} \frac {x}{\log \left (x + e^{\left (x e^{\left (-e^{x}\right )} + x + 1\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.34, size = 451, normalized size = 21.48 \begin {gather*} \frac {x-\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,\ln \left (x+\mathrm {e}\,{\mathrm {e}}^x\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-{\mathrm {e}}^x}}\right )\,\left (x+{\mathrm {e}}^{x+x\,{\mathrm {e}}^{-{\mathrm {e}}^x}+1}\right )}{{\mathrm {e}}^{{\mathrm {e}}^x}-x\,{\mathrm {e}}^{2\,x+x\,{\mathrm {e}}^{-{\mathrm {e}}^x}+1}+2\,{\mathrm {e}}^{x+\frac {{\mathrm {e}}^x}{2}+x\,{\mathrm {e}}^{-{\mathrm {e}}^x}+1}\,\mathrm {cosh}\left (\frac {{\mathrm {e}}^x}{2}\right )}}{\ln \left (x+\mathrm {e}\,{\mathrm {e}}^x\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-{\mathrm {e}}^x}}\right )}-\frac {{\mathrm {e}}^{4\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{5\,{\mathrm {e}}^x}+{\mathrm {e}}^{6\,{\mathrm {e}}^x}-{\mathrm {e}}^{2\,x+4\,{\mathrm {e}}^x}\,\left (4\,x^3+2\,x^2\right )-3\,x^3\,{\mathrm {e}}^{2\,x+3\,{\mathrm {e}}^x}+x^3\,{\mathrm {e}}^{3\,x+4\,{\mathrm {e}}^x}+x^4\,{\mathrm {e}}^{3\,x+3\,{\mathrm {e}}^x}+{\mathrm {e}}^{2\,x+5\,{\mathrm {e}}^x}\,\left (x-x^2\right )+3\,x^2\,{\mathrm {e}}^{x+3\,{\mathrm {e}}^x}+7\,x^2\,{\mathrm {e}}^{x+4\,{\mathrm {e}}^x}-{\mathrm {e}}^{x+5\,{\mathrm {e}}^x}\,\left (-4\,x^2+x+2\right )-6\,x\,{\mathrm {e}}^{\frac {9\,{\mathrm {e}}^x}{2}}\,\mathrm {cosh}\left (\frac {{\mathrm {e}}^x}{2}\right )-2\,x\,{\mathrm {e}}^{\frac {9\,{\mathrm {e}}^x}{2}}\,\mathrm {cosh}\left (\frac {3\,{\mathrm {e}}^x}{2}\right )}{\left ({\mathrm {e}}^{{\mathrm {e}}^x}+{\mathrm {e}}^{x+x\,{\mathrm {e}}^{-{\mathrm {e}}^x}+1}\,\left (2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2}}\,\mathrm {cosh}\left (\frac {{\mathrm {e}}^x}{2}\right )-x\,{\mathrm {e}}^x\right )\right )\,\left (2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2}}\,\mathrm {cosh}\left (\frac {{\mathrm {e}}^x}{2}\right )-x\,{\mathrm {e}}^x\right )\,\left (2\,{\mathrm {e}}^{3\,{\mathrm {e}}^x}-{\mathrm {e}}^{x+3\,{\mathrm {e}}^x}\,\left (3\,x+2\right )+x^2\,{\mathrm {e}}^{2\,x+2\,{\mathrm {e}}^x}+2\,\mathrm {cosh}\left ({\mathrm {e}}^x\right )\,{\mathrm {e}}^{3\,{\mathrm {e}}^x}-2\,x\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^x}+x\,{\mathrm {e}}^{2\,x+3\,{\mathrm {e}}^x}\right )}-\frac {{\mathrm {e}}^x\,\left (x\,{\mathrm {e}}^x-1\right )\,\left (2\,x\,\mathrm {sinh}\left (\frac {x}{2}\right )\,{\mathrm {e}}^{x/2}-2\right )}{\left (2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2}}\,\mathrm {cosh}\left (\frac {{\mathrm {e}}^x}{2}\right )-x\,{\mathrm {e}}^x\right )\,\left (2\,{\mathrm {e}}^x-2\,x\,\mathrm {sinh}\left (\frac {x}{2}\right )\,{\mathrm {e}}^{\frac {3\,x}{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.07, size = 20, normalized size = 0.95 \begin {gather*} \frac {x}{\log {\left (x + e^{\left (x + \left (x + 1\right ) e^{e^{x}}\right ) e^{- e^{x}}} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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