Optimal. Leaf size=27 \[ e^{-e^{-x} x}+\frac {e^{e^x}+x+\log (5)}{4 x^2} \]
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Rubi [F] time = 1.41, 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^{-x-e^{-x} x} \left (-4 x^3+4 x^4+e^{e^x+e^{-x} x} \left (-2 e^x+e^{2 x} x\right )+e^{x+e^{-x} x} (-x-2 \log (5))\right )}{4 x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {e^{-x-e^{-x} x} \left (-4 x^3+4 x^4+e^{e^x+e^{-x} x} \left (-2 e^x+e^{2 x} x\right )+e^{x+e^{-x} x} (-x-2 \log (5))\right )}{x^3} \, dx\\ &=\frac {1}{4} \int \frac {-2 e^{e^x}-x+e^{e^x+x} x+4 e^{\left (-1-e^{-x}\right ) x} (-1+x) x^3-2 \log (5)}{x^3} \, dx\\ &=\frac {1}{4} \int \left (4 e^{-e^{-x} \left (1+e^x\right ) x} (-1+x)+\frac {-2 e^{e^x}-x+e^{e^x+x} x-\log (25)}{x^3}\right ) \, dx\\ &=\frac {1}{4} \int \frac {-2 e^{e^x}-x+e^{e^x+x} x-\log (25)}{x^3} \, dx+\int e^{-e^{-x} \left (1+e^x\right ) x} (-1+x) \, dx\\ &=\frac {1}{4} \int \left (\frac {e^{e^x+x}}{x^2}-\frac {2 e^{e^x}+x+\log (25)}{x^3}\right ) \, dx+\int \left (-e^{-e^{-x} \left (1+e^x\right ) x}+e^{-e^{-x} \left (1+e^x\right ) x} x\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^{e^x+x}}{x^2} \, dx-\frac {1}{4} \int \frac {2 e^{e^x}+x+\log (25)}{x^3} \, dx-\int e^{-e^{-x} \left (1+e^x\right ) x} \, dx+\int e^{-e^{-x} \left (1+e^x\right ) x} x \, dx\\ &=\frac {1}{4} \int \frac {e^{e^x+x}}{x^2} \, dx-\frac {1}{4} \int \left (\frac {2 e^{e^x}}{x^3}+\frac {x+\log (25)}{x^3}\right ) \, dx-\int e^{-e^{-x} \left (1+e^x\right ) x} \, dx+\int e^{-e^{-x} \left (1+e^x\right ) x} x \, dx\\ &=\frac {1}{4} \int \frac {e^{e^x+x}}{x^2} \, dx-\frac {1}{4} \int \frac {x+\log (25)}{x^3} \, dx-\frac {1}{2} \int \frac {e^{e^x}}{x^3} \, dx-\int e^{-e^{-x} \left (1+e^x\right ) x} \, dx+\int e^{-e^{-x} \left (1+e^x\right ) x} x \, dx\\ &=\frac {(x+\log (25))^2}{8 x^2 \log (25)}+\frac {1}{4} \int \frac {e^{e^x+x}}{x^2} \, dx-\frac {1}{2} \int \frac {e^{e^x}}{x^3} \, dx-\int e^{-e^{-x} \left (1+e^x\right ) x} \, dx+\int e^{-e^{-x} \left (1+e^x\right ) x} x \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.23, size = 38, normalized size = 1.41 \begin {gather*} \frac {1}{4} \left (4 e^{-e^{-x} x}+\frac {e^{e^x}}{x^2}+\frac {1}{x}+\frac {\log (25)}{2 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 57, normalized size = 2.11 \begin {gather*} \frac {{\left (4 \, x^{2} e^{x} + {\left (x + \log \relax (5)\right )} e^{\left ({\left (x e^{x} + x\right )} e^{\left (-x\right )}\right )} + e^{\left ({\left (x + e^{\left (2 \, x\right )}\right )} e^{\left (-x\right )} + x\right )}\right )} e^{\left (-{\left (x e^{x} + x\right )} e^{\left (-x\right )}\right )}}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 25, normalized size = 0.93 \begin {gather*} \frac {4 \, x^{2} e^{\left (-x e^{\left (-x\right )}\right )} + x + e^{\left (e^{x}\right )} + \log \relax (5)}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 27, normalized size = 1.00
method | result | size |
risch | \(\frac {\ln \relax (5)+x}{4 x^{2}}+\frac {{\mathrm e}^{{\mathrm e}^{x}}}{4 x^{2}}+{\mathrm e}^{-x \,{\mathrm e}^{-x}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 42, normalized size = 1.56 \begin {gather*} \frac {{\left (4 \, x^{2} + e^{\left (x e^{\left (-x\right )} + e^{x}\right )}\right )} e^{\left (-x e^{\left (-x\right )}\right )}}{4 \, x^{2}} + \frac {1}{4 \, x} + \frac {\log \relax (5)}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.66, size = 26, normalized size = 0.96 \begin {gather*} {\mathrm {e}}^{-x\,{\mathrm {e}}^{-x}}+\frac {{\mathrm {e}}^{{\mathrm {e}}^x}}{4\,x^2}+\frac {x+\ln \relax (5)}{4\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 29, normalized size = 1.07 \begin {gather*} e^{- x e^{- x}} - \frac {- x - \log {\relax (5 )}}{4 x^{2}} + \frac {e^{e^{x}}}{4 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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