Optimal. Leaf size=20 \[ -x \left (\frac {1}{2}+x\right )+\log \left (3 e^{-e^x} x\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 18, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {12, 14, 2194} \begin {gather*} -x^2-\frac {x}{2}-e^x+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2194
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {2-x-2 e^x x-4 x^2}{x} \, dx\\ &=\frac {1}{2} \int \left (-2 e^x+\frac {2-x-4 x^2}{x}\right ) \, dx\\ &=\frac {1}{2} \int \frac {2-x-4 x^2}{x} \, dx-\int e^x \, dx\\ &=-e^x+\frac {1}{2} \int \left (-1+\frac {2}{x}-4 x\right ) \, dx\\ &=-e^x-\frac {x}{2}-x^2+\log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 0.90 \begin {gather*} -e^x-\frac {x}{2}-x^2+\log (x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 15, normalized size = 0.75 \begin {gather*} -x^{2} - \frac {1}{2} \, x - e^{x} + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 15, normalized size = 0.75 \begin {gather*} -x^{2} - \frac {1}{2} \, x - e^{x} + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 16, normalized size = 0.80
method | result | size |
default | \(-x^{2}-\frac {x}{2}+\ln \relax (x )-{\mathrm e}^{x}\) | \(16\) |
norman | \(-x^{2}-\frac {x}{2}+\ln \relax (x )-{\mathrm e}^{x}\) | \(16\) |
risch | \(-x^{2}-\frac {x}{2}+\ln \relax (x )-{\mathrm e}^{x}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 15, normalized size = 0.75 \begin {gather*} -x^{2} - \frac {1}{2} \, x - e^{x} + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.15, size = 15, normalized size = 0.75 \begin {gather*} \ln \relax (x)-{\mathrm {e}}^x-\frac {x}{2}-x^2 \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 = 0.60 \begin {gather*} - x^{2} - \frac {x}{2} - e^{x} + \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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