Optimal. Leaf size=24 \[ \log \left (x^4 \left (e^x+x\right ) \left (4+\frac {-2+\log (x)}{e^x+x}\right )\right ) \]
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Rubi [F] time = 0.79, 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 {-7+20 x+e^x (16+4 x)+4 \log (x)}{-2 x+4 e^x x+4 x^2+x \log (x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {4+x}{x}-\frac {-1-6 x+4 x^2+x \log (x)}{x \left (-2+4 e^x+4 x+\log (x)\right )}\right ) \, dx\\ &=\int \frac {4+x}{x} \, dx-\int \frac {-1-6 x+4 x^2+x \log (x)}{x \left (-2+4 e^x+4 x+\log (x)\right )} \, dx\\ &=\int \left (1+\frac {4}{x}\right ) \, dx-\int \left (-\frac {6}{-2+4 e^x+4 x+\log (x)}-\frac {1}{x \left (-2+4 e^x+4 x+\log (x)\right )}+\frac {4 x}{-2+4 e^x+4 x+\log (x)}+\frac {\log (x)}{-2+4 e^x+4 x+\log (x)}\right ) \, dx\\ &=x+4 \log (x)-4 \int \frac {x}{-2+4 e^x+4 x+\log (x)} \, dx+6 \int \frac {1}{-2+4 e^x+4 x+\log (x)} \, dx+\int \frac {1}{x \left (-2+4 e^x+4 x+\log (x)\right )} \, dx-\int \frac {\log (x)}{-2+4 e^x+4 x+\log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.19, size = 20, normalized size = 0.83 \begin {gather*} 4 \log (x)+\log \left (2-4 e^x-4 x-\log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 17, normalized size = 0.71 \begin {gather*} \log \left (4 \, x + 4 \, e^{x} + \log \relax (x) - 2\right ) + 4 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 17, normalized size = 0.71 \begin {gather*} \log \left (4 \, x + 4 \, e^{x} + \log \relax (x) - 2\right ) + 4 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 18, normalized size = 0.75
method | result | size |
norman | \(4 \ln \relax (x )+\ln \left (\ln \relax (x )+4 \,{\mathrm e}^{x}+4 x -2\right )\) | \(18\) |
risch | \(4 \ln \relax (x )+\ln \left (\ln \relax (x )+4 \,{\mathrm e}^{x}+4 x -2\right )\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 15, normalized size = 0.62 \begin {gather*} \log \left (x + e^{x} + \frac {1}{4} \, \log \relax (x) - \frac {1}{2}\right ) + 4 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 17, normalized size = 0.71 \begin {gather*} \ln \left (4\,x+4\,{\mathrm {e}}^x+\ln \relax (x)-2\right )+4\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 19, normalized size = 0.79 \begin {gather*} 4 \log {\relax (x )} + \log {\left (x + e^{x} + \frac {\log {\relax (x )}}{4} - \frac {1}{2} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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