Optimal. Leaf size=21 \[ -1+\log (-2 (1+x)+4 x \log (e x)+\log (x \log (2))) \]
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Rubi [F] time = 0.28, 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 {1+2 x+4 x \log (e x)}{-2 x-2 x^2+4 x^2 \log (e x)+x \log (x \log (2))} \, 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 {2}{-2+2 x+4 x \log (x)+\log (x \log (2))}+\frac {1}{x (-2+2 x+4 x \log (x)+\log (x \log (2)))}+\frac {4 (1+\log (x))}{-2+2 x+4 x \log (x)+\log (x \log (2))}\right ) \, dx\\ &=2 \int \frac {1}{-2+2 x+4 x \log (x)+\log (x \log (2))} \, dx+4 \int \frac {1+\log (x)}{-2+2 x+4 x \log (x)+\log (x \log (2))} \, dx+\int \frac {1}{x (-2+2 x+4 x \log (x)+\log (x \log (2)))} \, dx\\ &=2 \int \frac {1}{-2+2 x+4 x \log (x)+\log (x \log (2))} \, dx+4 \int \left (\frac {1}{-2+2 x+4 x \log (x)+\log (x \log (2))}+\frac {\log (x)}{-2+2 x+4 x \log (x)+\log (x \log (2))}\right ) \, dx+\int \frac {1}{x (-2+2 x+4 x \log (x)+\log (x \log (2)))} \, dx\\ &=2 \int \frac {1}{-2+2 x+4 x \log (x)+\log (x \log (2))} \, dx+4 \int \frac {1}{-2+2 x+4 x \log (x)+\log (x \log (2))} \, dx+4 \int \frac {\log (x)}{-2+2 x+4 x \log (x)+\log (x \log (2))} \, dx+\int \frac {1}{x (-2+2 x+4 x \log (x)+\log (x \log (2)))} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.15, size = 16, normalized size = 0.76 \begin {gather*} \log (-2+2 x+4 x \log (x)+\log (x \log (2))) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 38, normalized size = 1.81 \begin {gather*} \log \left (4 \, x + 1\right ) + \log \left (\frac {{\left (4 \, x + 1\right )} \log \left (x e\right ) - 2 \, x + \log \left (e^{\left (-1\right )} \log \relax (2)\right ) - 2}{4 \, x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 16, normalized size = 0.76 \begin {gather*} \log \left (4 \, x \log \relax (x) + 2 \, x + \log \relax (x) + \log \left (\log \relax (2)\right ) - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 20, normalized size = 0.95
method | result | size |
norman | \(\ln \left (\ln \left (x \ln \relax (2)\right )+4 x \ln \left (x \,{\mathrm e}\right )-2 x -2\right )\) | \(20\) |
risch | \(\ln \left (4 x +1\right )+\ln \left (\ln \relax (x )+\frac {i \left (-2 i \ln \left (\ln \relax (2)\right )-4 i x +4 i\right )}{8 x +2}\right )\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 32, normalized size = 1.52 \begin {gather*} \log \left (4 \, x + 1\right ) + \log \left (\frac {{\left (4 \, x + 1\right )} \log \relax (x) + 2 \, x + \log \left (\log \relax (2)\right ) - 2}{4 \, x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.18, size = 16, normalized size = 0.76 \begin {gather*} \ln \left (2\,x+\ln \left (\ln \relax (2)\right )+\ln \relax (x)+4\,x\,\ln \relax (x)-2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.60, size = 29, normalized size = 1.38 \begin {gather*} \log {\left (4 x + 1 \right )} + \log {\left (\log {\left (e x \right )} + \frac {- 2 x - 3 + \log {\left (\log {\relax (2 )} \right )}}{4 x + 1} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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