Optimal. Leaf size=31 \[ e^{-e^{e+x}} \left (-x+\frac {\log \left (\frac {e^{e^e}}{4}+x\right )}{\log (x)}\right ) \]
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Rubi [F]
time = 4.52, 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^{e+x}} \left (4 x \log (x)+e^{e^e} \left (-x+e^{e+x} x^2\right ) \log ^2(x)+\left (-4 x^2+4 e^{e+x} x^3\right ) \log ^2(x)+\left (-4 x-4 e^{e+x} x^2 \log (x)+e^{e^e} \left (-1-e^{e+x} x \log (x)\right )\right ) \log \left (\frac {1}{4} \left (e^{e^e}+4 x\right )\right )\right )}{e^{e^e} x \log ^2(x)+4 x^2 \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 e^{-e^{e+x}} \left (-1+e^{e+x} x-\frac {\log \left (\frac {e^{e^e}}{4}+x\right )}{x \log ^2(x)}+\frac {\frac {4}{e^{e^e}+4 x}-e^{e+x} \log \left (\frac {e^{e^e}}{4}+x\right )}{\log (x)}\right ) \, dx\\ &=\int \left (-e^{-e^{e+x}}+e^{e-e^{e+x}+x} x-\frac {e^{-e^{e+x}} \log \left (\frac {e^{e^e}}{4}+x\right )}{x \log ^2(x)}-\frac {e^{-e^{e+x}} \left (-4+e^{e+e^e+x} \log \left (\frac {e^{e^e}}{4}+x\right )+4 e^{e+x} x \log \left (\frac {e^{e^e}}{4}+x\right )\right )}{\left (e^{e^e}+4 x\right ) \log (x)}\right ) \, dx\\ &=-\int e^{-e^{e+x}} \, dx+\int e^{e-e^{e+x}+x} x \, dx-\int \frac {e^{-e^{e+x}} \log \left (\frac {e^{e^e}}{4}+x\right )}{x \log ^2(x)} \, dx-\int \frac {e^{-e^{e+x}} \left (-4+e^{e+e^e+x} \log \left (\frac {e^{e^e}}{4}+x\right )+4 e^{e+x} x \log \left (\frac {e^{e^e}}{4}+x\right )\right )}{\left (e^{e^e}+4 x\right ) \log (x)} \, dx\\ &=\int e^{e-e^{e+x}+x} x \, dx-\int \frac {e^{-e^{e+x}} \log \left (\frac {e^{e^e}}{4}+x\right )}{x \log ^2(x)} \, dx-\int \frac {e^{-e^{e+x}} \left (-4+e^{e+x} \left (e^{e^e}+4 x\right ) \log \left (\frac {e^{e^e}}{4}+x\right )\right )}{\left (e^{e^e}+4 x\right ) \log (x)} \, dx-\text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,e^{e+x}\right )\\ &=-\text {Ei}\left (-e^{e+x}\right )+\int e^{e-e^{e+x}+x} x \, dx-\int \frac {e^{-e^{e+x}} \log \left (\frac {e^{e^e}}{4}+x\right )}{x \log ^2(x)} \, dx-\int \left (-\frac {4 e^{-e^{e+x}}}{\left (e^{e^e}+4 x\right ) \log (x)}+\frac {e^{e-e^{e+x}+x} \log \left (\frac {e^{e^e}}{4}+x\right )}{\log (x)}\right ) \, dx\\ &=-\text {Ei}\left (-e^{e+x}\right )+4 \int \frac {e^{-e^{e+x}}}{\left (e^{e^e}+4 x\right ) \log (x)} \, dx+\int e^{e-e^{e+x}+x} x \, dx-\int \frac {e^{-e^{e+x}} \log \left (\frac {e^{e^e}}{4}+x\right )}{x \log ^2(x)} \, dx-\int \frac {e^{e-e^{e+x}+x} \log \left (\frac {e^{e^e}}{4}+x\right )}{\log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.90, size = 31, normalized size = 1.00 \begin {gather*} e^{-e^{e+x}} \left (-x+\frac {\log \left (\frac {e^{e^e}}{4}+x\right )}{\log (x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 31, normalized size = 1.00
method | result | size |
risch | \(-\frac {\left (x \ln \left (x \right )-\ln \left (\frac {{\mathrm e}^{{\mathrm e}^{{\mathrm e}}}}{4}+x \right )\right ) {\mathrm e}^{-{\mathrm e}^{x +{\mathrm e}}}}{\ln \left (x \right )}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 44, normalized size = 1.42 \begin {gather*} -\frac {{\left (x \log \left (x\right ) + 2 \, \log \left (2\right )\right )} e^{\left (-e^{\left (x + e\right )}\right )} - e^{\left (-e^{\left (x + e\right )}\right )} \log \left (4 \, x + e^{\left (e^{e}\right )}\right )}{\log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 38, normalized size = 1.23 \begin {gather*} -\frac {x e^{\left (-e^{\left (x + e\right )}\right )} \log \left (x\right ) - e^{\left (-e^{\left (x + e\right )}\right )} \log \left (x + \frac {1}{4} \, e^{\left (e^{e}\right )}\right )}{\log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 27, normalized size = 0.87 \begin {gather*} \frac {\left (- x \log {\left (x \right )} + \log {\left (x + \frac {e^{e^{e}}}{4} \right )}\right ) e^{- e^{x + e}}}{\log {\left (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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.55, size = 28, normalized size = 0.90 \begin {gather*} \frac {{\mathrm {e}}^{-{\mathrm {e}}^{\mathrm {e}}\,{\mathrm {e}}^x}\,\left (\ln \left (x+\frac {{\mathrm {e}}^{{\mathrm {e}}^{\mathrm {e}}}}{4}\right )-x\,\ln \left (x\right )\right )}{\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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