Optimal. Leaf size=25 \[ e^{2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \]
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Rubi [F] time = 41.96, 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^{-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \left (e^8 x \log ^2(x)+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}} \left (e^8 (4-x)+e^{8+\frac {x}{e^8}} x+\left (e^8 x+e^{\frac {x}{e^8}} \left (-e^8 x-x^2\right )\right ) \log (x)\right )\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^{2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x}+\frac {\exp \left (-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right ) \left (4 e^8-e^8 x+e^{8+\frac {x}{e^8}} x+e^8 x \log (x)-e^{8+\frac {x}{e^8}} x \log (x)-e^{\frac {x}{e^8}} x^2 \log (x)\right )}{x \log ^2(x)}\right ) \, dx\\ &=\int e^{2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \, dx+\int \frac {\exp \left (-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right ) \left (4 e^8-e^8 x+e^{8+\frac {x}{e^8}} x+e^8 x \log (x)-e^{8+\frac {x}{e^8}} x \log (x)-e^{\frac {x}{e^8}} x^2 \log (x)\right )}{x \log ^2(x)} \, dx\\ &=\int e^{2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \, dx+\int \left (\frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right ) (4-x+x \log (x))}{x \log ^2(x)}-\frac {\exp \left (-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x+\frac {x}{e^8}-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right ) \left (-e^8+e^8 \log (x)+x \log (x)\right )}{\log ^2(x)}\right ) \, dx\\ &=\int e^{2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \, dx+\int \frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right ) (4-x+x \log (x))}{x \log ^2(x)} \, dx-\int \frac {\exp \left (-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x+\frac {x}{e^8}-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right ) \left (-e^8+e^8 \log (x)+x \log (x)\right )}{\log ^2(x)} \, dx\\ &=\int e^{2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \, dx+\int \left (\frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right ) (4-x)}{x \log ^2(x)}+\frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{\log (x)}\right ) \, dx-\int \frac {\exp \left (-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+\left (1+\frac {1}{e^8}\right ) x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right ) \left (-e^8+e^8 \log (x)+x \log (x)\right )}{\log ^2(x)} \, dx\\ &=\int e^{2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \, dx-\int \left (-\frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+\left (1+\frac {1}{e^8}\right ) x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{\log ^2(x)}+\frac {\exp \left (-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+\left (1+\frac {1}{e^8}\right ) x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right ) \left (e^8+x\right )}{\log (x)}\right ) \, dx+\int \frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right ) (4-x)}{x \log ^2(x)} \, dx+\int \frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{\log (x)} \, dx\\ &=\int e^{2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \, dx+\int \left (-\frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{\log ^2(x)}+\frac {4 \exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{x \log ^2(x)}\right ) \, dx+\int \frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+\left (1+\frac {1}{e^8}\right ) x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{\log ^2(x)} \, dx+\int \frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{\log (x)} \, dx-\int \frac {\exp \left (-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+\left (1+\frac {1}{e^8}\right ) x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right ) \left (e^8+x\right )}{\log (x)} \, dx\\ &=4 \int \frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{x \log ^2(x)} \, dx+\int e^{2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \, dx-\int \left (\frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+\left (1+\frac {1}{e^8}\right ) x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{\log (x)}+\frac {\exp \left (-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+\left (1+\frac {1}{e^8}\right ) x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right ) x}{\log (x)}\right ) \, dx-\int \frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{\log ^2(x)} \, dx+\int \frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+\left (1+\frac {1}{e^8}\right ) x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{\log ^2(x)} \, dx+\int \frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{\log (x)} \, dx\\ &=4 \int \frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{x \log ^2(x)} \, dx+\int e^{2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \, dx-\int \frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{\log ^2(x)} \, dx+\int \frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+\left (1+\frac {1}{e^8}\right ) x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{\log ^2(x)} \, dx+\int \frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{\log (x)} \, dx-\int \frac {\exp \left (2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+\left (1+\frac {1}{e^8}\right ) x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right )}{\log (x)} \, dx-\int \frac {\exp \left (-6+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+\left (1+\frac {1}{e^8}\right ) x-\frac {4-x+e^{\frac {x}{e^8}} x}{\log (x)}\right ) x}{\log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.37, size = 25, normalized size = 1.00 \begin {gather*} e^{2+e^{\frac {-4+x-e^{\frac {x}{e^8}} x}{\log (x)}}+x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 32, normalized size = 1.28 \begin {gather*} e^{\left (x + e^{\left (\frac {{\left ({\left (x - 4\right )} e^{8} - x e^{\left ({\left (x + 8 \, e^{8}\right )} e^{\left (-8\right )}\right )}\right )} e^{\left (-8\right )}}{\log \relax (x)}\right )} + 2\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 = 0.18, size = 24, normalized size = 0.96
method | result | size |
risch | \({\mathrm e}^{{\mathrm e}^{-\frac {x \,{\mathrm e}^{x \,{\mathrm e}^{-8}}-x +4}{\ln \relax (x )}}+2+x}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 30, normalized size = 1.20 \begin {gather*} e^{\left (x + e^{\left (-\frac {x e^{\left (x e^{\left (-8\right )}\right )}}{\log \relax (x)} + \frac {x}{\log \relax (x)} - \frac {4}{\log \relax (x)}\right )} + 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.22, size = 34, normalized size = 1.36 \begin {gather*} {\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^{-\frac {4}{\ln \relax (x)}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-8}}}{\ln \relax (x)}}\,{\mathrm {e}}^{\frac {x}{\ln \relax (x)}}}\,{\mathrm {e}}^x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 64.39, size = 20, normalized size = 0.80 \begin {gather*} e^{x + e^{\frac {- x e^{\frac {x}{e^{8}}} + x - 4}{\log {\relax (x )}}} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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