Optimal. Leaf size=24 \[ 1+x+\frac {1}{2} \left (-22-x-\frac {1}{2} e^{\frac {1}{x}} x \log (x)\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 18, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {12, 14, 2288} \begin {gather*} \frac {x}{2}-\frac {1}{4} e^{\frac {1}{x}} x \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2288
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {2 x-e^{\frac {1}{x}} x+e^{\frac {1}{x}} (1-x) \log (x)}{x} \, dx\\ &=\frac {1}{4} \int \left (2-\frac {e^{\frac {1}{x}} (x-\log (x)+x \log (x))}{x}\right ) \, dx\\ &=\frac {x}{2}-\frac {1}{4} \int \frac {e^{\frac {1}{x}} (x-\log (x)+x \log (x))}{x} \, dx\\ &=\frac {x}{2}-\frac {1}{4} e^{\frac {1}{x}} x \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 18, normalized size = 0.75 \begin {gather*} \frac {1}{4} \left (2 x-e^{\frac {1}{x}} x \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 13, normalized size = 0.54 \begin {gather*} -\frac {1}{4} \, x e^{\frac {1}{x}} \log \relax (x) + \frac {1}{2} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 13, normalized size = 0.54 \begin {gather*} -\frac {1}{4} \, x e^{\frac {1}{x}} \log \relax (x) + \frac {1}{2} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 14, normalized size = 0.58
method | result | size |
norman | \(\frac {x}{2}-\frac {x \,{\mathrm e}^{\frac {1}{x}} \ln \relax (x )}{4}\) | \(14\) |
risch | \(\frac {x}{2}-\frac {x \,{\mathrm e}^{\frac {1}{x}} \ln \relax (x )}{4}\) | \(14\) |
derivativedivides | \(\frac {x}{2}-\frac {\left (\frac {\left (\ln \left (\frac {1}{x}\right )+\ln \relax (x )\right ) {\mathrm e}^{\frac {1}{x}}}{x}-\frac {{\mathrm e}^{\frac {1}{x}} \ln \left (\frac {1}{x}\right )}{x}\right ) x^{2}}{4}\) | \(39\) |
default | \(\frac {x}{2}-\frac {\left (\frac {\left (\ln \left (\frac {1}{x}\right )+\ln \relax (x )\right ) {\mathrm e}^{\frac {1}{x}}}{x}-\frac {{\mathrm e}^{\frac {1}{x}} \ln \left (\frac {1}{x}\right )}{x}\right ) x^{2}}{4}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {1}{4} \, x e^{\frac {1}{x}} \log \relax (x) + \frac {1}{2} \, x + \frac {1}{4} \, \Gamma \left (-1, -\frac {1}{x}\right ) + \frac {1}{4} \, \int e^{\frac {1}{x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 12, normalized size = 0.50 \begin {gather*} -\frac {x\,\left ({\mathrm {e}}^{1/x}\,\ln \relax (x)-2\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 14, normalized size = 0.58 \begin {gather*} - \frac {x e^{\frac {1}{x}} \log {\relax (x )}}{4} + \frac {x}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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