Optimal. Leaf size=28 \[ e^{\frac {1}{4} x \left (4-\left (-e^{4+4 x}+x\right ) \log (\log (x+\log (x)))\right )} \]
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Rubi [F] time = 20.38, 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 {\exp \left (\frac {1}{4} \left (4 x+\left (e^{4+4 x} x-x^2\right ) \log (\log (x+\log (x)))\right )\right ) \left (-x-x^2+e^{4+4 x} (1+x)+(4 x+4 \log (x)) \log (x+\log (x))+\left (-2 x^2+e^{4+4 x} \left (x+4 x^2\right )+\left (-2 x+e^{4+4 x} (1+4 x)\right ) \log (x)\right ) \log (x+\log (x)) \log (\log (x+\log (x)))\right )}{(4 x+4 \log (x)) \log (x+\log (x))} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) \left (\left (e^{4+4 x}-x\right ) (1+x)+(x+\log (x)) \log (x+\log (x)) \left (4+\left (-2 x+e^{4+4 x} (1+4 x)\right ) \log (\log (x+\log (x)))\right )\right )}{4 (x+\log (x))} \, dx\\ &=\frac {1}{4} \int \frac {e^x \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) \left (\left (e^{4+4 x}-x\right ) (1+x)+(x+\log (x)) \log (x+\log (x)) \left (4+\left (-2 x+e^{4+4 x} (1+4 x)\right ) \log (\log (x+\log (x)))\right )\right )}{x+\log (x)} \, dx\\ &=\frac {1}{4} \int \left (\frac {e^x \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) \left (-x-x^2+4 x \log (x+\log (x))+4 \log (x) \log (x+\log (x))-2 x^2 \log (x+\log (x)) \log (\log (x+\log (x)))-2 x \log (x) \log (x+\log (x)) \log (\log (x+\log (x)))\right )}{x+\log (x)}+\frac {e^{4+5 x} \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) \left (1+x+x \log (x+\log (x)) \log (\log (x+\log (x)))+4 x^2 \log (x+\log (x)) \log (\log (x+\log (x)))+\log (x) \log (x+\log (x)) \log (\log (x+\log (x)))+4 x \log (x) \log (x+\log (x)) \log (\log (x+\log (x)))\right )}{x+\log (x)}\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^x \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) \left (-x-x^2+4 x \log (x+\log (x))+4 \log (x) \log (x+\log (x))-2 x^2 \log (x+\log (x)) \log (\log (x+\log (x)))-2 x \log (x) \log (x+\log (x)) \log (\log (x+\log (x)))\right )}{x+\log (x)} \, dx+\frac {1}{4} \int \frac {e^{4+5 x} \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) \left (1+x+x \log (x+\log (x)) \log (\log (x+\log (x)))+4 x^2 \log (x+\log (x)) \log (\log (x+\log (x)))+\log (x) \log (x+\log (x)) \log (\log (x+\log (x)))+4 x \log (x) \log (x+\log (x)) \log (\log (x+\log (x)))\right )}{x+\log (x)} \, dx\\ &=\frac {1}{4} \int \frac {e^{4+5 x} \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) (1+x+(1+4 x) (x+\log (x)) \log (x+\log (x)) \log (\log (x+\log (x))))}{x+\log (x)} \, dx+\frac {1}{4} \int \frac {e^x \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) (-x (1+x)-2 (x+\log (x)) \log (x+\log (x)) (-2+x \log (\log (x+\log (x)))))}{x+\log (x)} \, dx\\ &=\frac {1}{4} \int \left (\frac {e^x \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) \left (-x-x^2+4 x \log (x+\log (x))+4 \log (x) \log (x+\log (x))\right )}{x+\log (x)}-2 e^x x \log ^{1+\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) \log (\log (x+\log (x)))\right ) \, dx+\frac {1}{4} \int \left (\frac {e^{4+5 x} (1+x) \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)}+e^{4+5 x} (1+4 x) \log ^{1+\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) \log (\log (x+\log (x)))\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^{4+5 x} (1+x) \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)} \, dx+\frac {1}{4} \int \frac {e^x \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) \left (-x-x^2+4 x \log (x+\log (x))+4 \log (x) \log (x+\log (x))\right )}{x+\log (x)} \, dx+\frac {1}{4} \int e^{4+5 x} (1+4 x) \log ^{1+\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) \log (\log (x+\log (x))) \, dx-\frac {1}{2} \int e^x x \log ^{1+\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) \log (\log (x+\log (x))) \, dx\\ &=\frac {1}{4} \int \frac {e^x \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) (-x (1+x)+4 (x+\log (x)) \log (x+\log (x)))}{x+\log (x)} \, dx+\frac {1}{4} \int \left (\frac {e^{4+5 x} \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)}+\frac {e^{4+5 x} x \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)}\right ) \, dx+\frac {1}{4} \int e^{4+5 x} (1+4 x) \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \log (\log (x+\log (x))) \, dx-\frac {1}{2} \int e^x x \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \log (\log (x+\log (x))) \, dx\\ &=\frac {1}{4} \int \frac {e^{4+5 x} \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)} \, dx+\frac {1}{4} \int \frac {e^{4+5 x} x \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)} \, dx+\frac {1}{4} \int \left (-\frac {e^x x (1+x) \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)}+4 e^x \log ^{1+\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))\right ) \, dx+\frac {1}{4} \int \left (e^{4+5 x} \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \log (\log (x+\log (x)))+4 e^{4+5 x} x \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \log (\log (x+\log (x)))\right ) \, dx-\frac {1}{2} \int e^x x \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \log (\log (x+\log (x))) \, dx\\ &=\frac {1}{4} \int \frac {e^{4+5 x} \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)} \, dx+\frac {1}{4} \int \frac {e^{4+5 x} x \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)} \, dx-\frac {1}{4} \int \frac {e^x x (1+x) \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)} \, dx+\frac {1}{4} \int e^{4+5 x} \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \log (\log (x+\log (x))) \, dx-\frac {1}{2} \int e^x x \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \log (\log (x+\log (x))) \, dx+\int e^x \log ^{1+\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x)) \, dx+\int e^{4+5 x} x \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \log (\log (x+\log (x))) \, dx\\ &=\frac {1}{4} \int \frac {e^{4+5 x} \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)} \, dx+\frac {1}{4} \int \frac {e^{4+5 x} x \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)} \, dx-\frac {1}{4} \int \left (\frac {e^x x \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)}+\frac {e^x x^2 \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)}\right ) \, dx+\frac {1}{4} \int e^{4+5 x} \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \log (\log (x+\log (x))) \, dx-\frac {1}{2} \int e^x x \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \log (\log (x+\log (x))) \, dx+\int e^x \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \, dx+\int e^{4+5 x} x \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \log (\log (x+\log (x))) \, dx\\ &=\frac {1}{4} \int \frac {e^{4+5 x} \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)} \, dx-\frac {1}{4} \int \frac {e^x x \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)} \, dx+\frac {1}{4} \int \frac {e^{4+5 x} x \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)} \, dx-\frac {1}{4} \int \frac {e^x x^2 \log ^{\frac {1}{4} \left (-4+e^{4+4 x} x-x^2\right )}(x+\log (x))}{x+\log (x)} \, dx+\frac {1}{4} \int e^{4+5 x} \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \log (\log (x+\log (x))) \, dx-\frac {1}{2} \int e^x x \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \log (\log (x+\log (x))) \, dx+\int e^x \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \, dx+\int e^{4+5 x} x \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \log (\log (x+\log (x))) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.23, size = 26, normalized size = 0.93 \begin {gather*} e^x \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 24, normalized size = 0.86 \begin {gather*} e^{\left (-\frac {1}{4} \, {\left (x^{2} - x e^{\left (4 \, x + 4\right )}\right )} \log \left (\log \left (x + \log \relax (x)\right )\right ) + 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 = 0.08, size = 23, normalized size = 0.82
method | result | size |
risch | \(\ln \left (x +\ln \relax (x )\right )^{\frac {\left ({\mathrm e}^{4 x +4}-x \right ) x}{4}} {\mathrm e}^{x}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 29, normalized size = 1.04 \begin {gather*} e^{\left (-\frac {1}{4} \, x^{2} \log \left (\log \left (x + \log \relax (x)\right )\right ) + \frac {1}{4} \, x e^{\left (4 \, x + 4\right )} \log \left (\log \left (x + \log \relax (x)\right )\right ) + x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.79, size = 24, normalized size = 0.86 \begin {gather*} {\ln \left (x+\ln \relax (x)\right )}^{\frac {x\,{\mathrm {e}}^{4\,x+4}}{4}-\frac {x^2}{4}}\,{\mathrm {e}}^x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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