Optimal. Leaf size=21 \[ \frac {1}{4} x \left (1+e^x+x\right )-\left (x+x^2\right ) \log (9) \]
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Rubi [B] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 2.05, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {12, 2176, 2194} \begin {gather*} \frac {x^2}{4}+\frac {x}{4}-\frac {e^x}{4}+\frac {1}{4} e^x (x+1)-\frac {1}{4} (2 x+1)^2 \log (9) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \left (1+2 x+e^x (1+x)+(-4-8 x) \log (9)\right ) \, dx\\ &=\frac {x}{4}+\frac {x^2}{4}-\frac {1}{4} (1+2 x)^2 \log (9)+\frac {1}{4} \int e^x (1+x) \, dx\\ &=\frac {x}{4}+\frac {x^2}{4}+\frac {1}{4} e^x (1+x)-\frac {1}{4} (1+2 x)^2 \log (9)-\frac {\int e^x \, dx}{4}\\ &=-\frac {e^x}{4}+\frac {x}{4}+\frac {x^2}{4}+\frac {1}{4} e^x (1+x)-\frac {1}{4} (1+2 x)^2 \log (9)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.24 \begin {gather*} \frac {1}{4} \left (x+e^x x+x^2-4 x \log (9)-4 x^2 \log (9)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 23, normalized size = 1.10 \begin {gather*} \frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{x} - 2 \, {\left (x^{2} + x\right )} \log \relax (3) + \frac {1}{4} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 23, normalized size = 1.10 \begin {gather*} \frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{x} - 2 \, {\left (x^{2} + x\right )} \log \relax (3) + \frac {1}{4} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 25, normalized size = 1.19
method | result | size |
norman | \(\left (-2 \ln \relax (3)+\frac {1}{4}\right ) x +\left (-2 \ln \relax (3)+\frac {1}{4}\right ) x^{2}+\frac {{\mathrm e}^{x} x}{4}\) | \(25\) |
default | \(\frac {x}{4}+\frac {x^{2}}{4}-2 x^{2} \ln \relax (3)-2 x \ln \relax (3)+\frac {{\mathrm e}^{x} x}{4}\) | \(27\) |
risch | \(\frac {x}{4}+\frac {x^{2}}{4}-2 x^{2} \ln \relax (3)-2 x \ln \relax (3)+\frac {{\mathrm e}^{x} x}{4}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 23, normalized size = 1.10 \begin {gather*} \frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{x} - 2 \, {\left (x^{2} + x\right )} \log \relax (3) + \frac {1}{4} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.23, size = 26, normalized size = 1.24 \begin {gather*} \frac {x\,{\mathrm {e}}^x}{4}-x^2\,\left (2\,\ln \relax (3)-\frac {1}{4}\right )-x\,\left (2\,\ln \relax (3)-\frac {1}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 27, normalized size = 1.29 \begin {gather*} x^{2} \left (\frac {1}{4} - 2 \log {\relax (3 )}\right ) + \frac {x e^{x}}{4} + x \left (\frac {1}{4} - 2 \log {\relax (3 )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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