Optimal. Leaf size=30 \[ \log \left (e^{-2 x} \left (-1+\log \left (-1+x+x^2 (2+x)^2+x \log (4 x)\right )\right )^2\right ) \]
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Rubi [A] time = 0.81, antiderivative size = 34, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 3, integrand size = 157, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6741, 6742, 6684} \begin {gather*} 2 \log \left (1-\log \left (x^4+4 x^3+4 x^2+x+x \log (4 x)-1\right )\right )-2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 6684
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2+18 x+32 x^2+16 x^3+2 x^4+(2+2 x) \log (4 x)+\left (2-2 x-8 x^2-8 x^3-2 x^4-2 x \log (4 x)\right ) \log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )}{\left (1-x-4 x^2-4 x^3-x^4-x \log (4 x)\right ) \left (1-\log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )\right )} \, dx\\ &=\int \left (-2+\frac {2 \left (2+8 x+12 x^2+4 x^3+\log (4 x)\right )}{\left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right ) \left (-1+\log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )\right )}\right ) \, dx\\ &=-2 x+2 \int \frac {2+8 x+12 x^2+4 x^3+\log (4 x)}{\left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right ) \left (-1+\log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )\right )} \, dx\\ &=-2 x+2 \log \left (1-\log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 34, normalized size = 1.13 \begin {gather*} 2 \left (-x+\log \left (1-\log \left (-1+x+4 x^2+4 x^3+x^4+x \log (4 x)\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 32, normalized size = 1.07 \begin {gather*} -2 \, x + 2 \, \log \left (\log \left (x^{4} + 4 \, x^{3} + 4 \, x^{2} + x \log \left (4 \, x\right ) + x - 1\right ) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.56, size = 32, normalized size = 1.07 \begin {gather*} -2 \, x + 2 \, \log \left (\log \left (x^{4} + 4 \, x^{3} + 4 \, x^{2} + x \log \left (4 \, x\right ) + x - 1\right ) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 33, normalized size = 1.10
method | result | size |
risch | \(-2 x +2 \ln \left (\ln \left (x \ln \left (4 x \right )+x^{4}+4 x^{3}+4 x^{2}+x -1\right )-1\right )\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 37, normalized size = 1.23 \begin {gather*} -2 \, x + 2 \, \log \left (\log \left (x^{4} + 4 \, x^{3} + 4 \, x^{2} + x {\left (2 \, \log \relax (2) + 1\right )} + x \log \relax (x) - 1\right ) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.82, size = 32, normalized size = 1.07 \begin {gather*} 2\,\ln \left (\ln \left (x+x\,\ln \left (4\,x\right )+4\,x^2+4\,x^3+x^4-1\right )-1\right )-2\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.73, size = 32, normalized size = 1.07 \begin {gather*} - 2 x + 2 \log {\left (\log {\left (x^{4} + 4 x^{3} + 4 x^{2} + x \log {\left (4 x \right )} + x - 1 \right )} - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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