Optimal. Leaf size=22 \[ e^{e^{-x} x}+\frac {e^x}{4 x}+\log (3) \]
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Rubi [F]
time = 0.50, 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} \left (e^{2 x} (-1+x)+e^{e^{-x} x} \left (4 x^2-4 x^3\right )\right )}{4 x^2} \, 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} \left (e^{2 x} (-1+x)+e^{e^{-x} x} \left (4 x^2-4 x^3\right )\right )}{x^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{-x} (1-x) \left (-e^{2 x}+4 e^{e^{-x} x} x^2\right )}{x^2} \, dx\\ &=\frac {1}{4} \int \left (-4 e^{-x+e^{-x} x} (-1+x)+\frac {e^x (-1+x)}{x^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^x (-1+x)}{x^2} \, dx-\int e^{-x+e^{-x} x} (-1+x) \, dx\\ &=\frac {e^x}{4 x}-\int e^{-e^{-x} \left (-1+e^x\right ) x} (-1+x) \, dx\\ &=\frac {e^x}{4 x}-\int \left (-e^{-e^{-x} \left (-1+e^x\right ) x}+e^{-e^{-x} \left (-1+e^x\right ) x} x\right ) \, dx\\ &=\frac {e^x}{4 x}+\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 = 0.09, size = 23, normalized size = 1.05 \begin {gather*} \frac {1}{4} \left (4 e^{e^{-x} x}+\frac {e^x}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 16, normalized size = 0.73
method | result | size |
risch | \(\frac {{\mathrm e}^{x}}{4 x}+{\mathrm e}^{x \,{\mathrm e}^{-x}}\) | \(16\) |
norman | \(\frac {\left ({\mathrm e}^{x} x \,{\mathrm e}^{x \,{\mathrm e}^{-x}}+\frac {{\mathrm e}^{2 x}}{4}\right ) {\mathrm e}^{-x}}{x}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.34, size = 19, normalized size = 0.86 \begin {gather*} \frac {1}{4} \, {\rm Ei}\left (x\right ) + e^{\left (x e^{\left (-x\right )}\right )} - \frac {1}{4} \, \Gamma \left (-1, -x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 23, normalized size = 1.05 \begin {gather*} \frac {{\left (4 \, x e^{\left (x e^{\left (-x\right )} - x\right )} + 1\right )} e^{x}}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 12, normalized size = 0.55 \begin {gather*} e^{x e^{- x}} + \frac {e^{x}}{4 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 23, normalized size = 1.05 \begin {gather*} \frac {{\left (4 \, x e^{\left (x e^{\left (-x\right )} - x\right )} + 1\right )} e^{x}}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.49, size = 15, normalized size = 0.68 \begin {gather*} {\mathrm {e}}^{x\,{\mathrm {e}}^{-x}}+\frac {{\mathrm {e}}^x}{4\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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