Optimal. Leaf size=18 \[ 1-x+x^2+x \left (x+\log \left (e^x+x\right )\right ) \]
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Rubi [A] time = 0.48, antiderivative size = 17, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6742, 2548} \begin {gather*} 2 x^2-x+x \log \left (x+e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2548
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+5 x-\frac {(-1+x) x}{e^x+x}+\log \left (e^x+x\right )\right ) \, dx\\ &=-x+\frac {5 x^2}{2}-\int \frac {(-1+x) x}{e^x+x} \, dx+\int \log \left (e^x+x\right ) \, dx\\ &=-x+\frac {5 x^2}{2}+x \log \left (e^x+x\right )-\int \frac {\left (1+e^x\right ) x}{e^x+x} \, dx-\int \left (-\frac {x}{e^x+x}+\frac {x^2}{e^x+x}\right ) \, dx\\ &=-x+\frac {5 x^2}{2}+x \log \left (e^x+x\right )+\int \frac {x}{e^x+x} \, dx-\int \frac {x^2}{e^x+x} \, dx-\int \left (x-\frac {(-1+x) x}{e^x+x}\right ) \, dx\\ &=-x+2 x^2+x \log \left (e^x+x\right )+\int \frac {x}{e^x+x} \, dx+\int \frac {(-1+x) x}{e^x+x} \, dx-\int \frac {x^2}{e^x+x} \, dx\\ &=-x+2 x^2+x \log \left (e^x+x\right )+\int \frac {x}{e^x+x} \, dx-\int \frac {x^2}{e^x+x} \, dx+\int \left (-\frac {x}{e^x+x}+\frac {x^2}{e^x+x}\right ) \, dx\\ &=-x+2 x^2+x \log \left (e^x+x\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.09, size = 17, normalized size = 0.94 \begin {gather*} -x+2 x^2+x \log \left (e^x+x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 16, normalized size = 0.89 \begin {gather*} 2 \, x^{2} + x \log \left (x + e^{x}\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 16, normalized size = 0.89 \begin {gather*} 2 \, x^{2} + x \log \left (x + e^{x}\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 17, normalized size = 0.94
method | result | size |
norman | \(\ln \left ({\mathrm e}^{x}+x \right ) x -x +2 x^{2}\) | \(17\) |
risch | \(\ln \left ({\mathrm e}^{x}+x \right ) x -x +2 x^{2}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 16, normalized size = 0.89 \begin {gather*} 2 \, x^{2} + x \log \left (x + e^{x}\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 12, normalized size = 0.67 \begin {gather*} x\,\left (2\,x+\ln \left (x+{\mathrm {e}}^x\right )-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 14, normalized size = 0.78 \begin {gather*} 2 x^{2} + x \log {\left (x + e^{x} \right )} - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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