Optimal. Leaf size=19 \[ 61-x \left (-4+\frac {2 \log (2)}{1+x+\log (\log (x))}\right ) \]
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Rubi [F]
time = 0.45, 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 {2 \log (2)+\left (4+8 x+4 x^2-2 \log (2)\right ) \log (x)+(8+8 x-2 \log (2)) \log (x) \log (\log (x))+4 \log (x) \log ^2(\log (x))}{\left (1+2 x+x^2\right ) \log (x)+(2+2 x) \log (x) \log (\log (x))+\log (x) \log ^2(\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 \frac {2 \left (\log (2)+\log (x) \left (2+4 x+2 x^2-\log (2)+(4+4 x-\log (2)) \log (\log (x))+2 \log ^2(\log (x))\right )\right )}{\log (x) (1+x+\log (\log (x)))^2} \, dx\\ &=2 \int \frac {\log (2)+\log (x) \left (2+4 x+2 x^2-\log (2)+(4+4 x-\log (2)) \log (\log (x))+2 \log ^2(\log (x))\right )}{\log (x) (1+x+\log (\log (x)))^2} \, dx\\ &=2 \int \left (2+\frac {\log (2) (1+x \log (x))}{\log (x) (1+x+\log (\log (x)))^2}-\frac {\log (2)}{1+x+\log (\log (x))}\right ) \, dx\\ &=4 x+(2 \log (2)) \int \frac {1+x \log (x)}{\log (x) (1+x+\log (\log (x)))^2} \, dx-(2 \log (2)) \int \frac {1}{1+x+\log (\log (x))} \, dx\\ &=4 x-(2 \log (2)) \int \frac {1}{1+x+\log (\log (x))} \, dx+(2 \log (2)) \int \left (\frac {x}{(1+x+\log (\log (x)))^2}+\frac {1}{\log (x) (1+x+\log (\log (x)))^2}\right ) \, dx\\ &=4 x+(2 \log (2)) \int \frac {x}{(1+x+\log (\log (x)))^2} \, dx+(2 \log (2)) \int \frac {1}{\log (x) (1+x+\log (\log (x)))^2} \, dx-(2 \log (2)) \int \frac {1}{1+x+\log (\log (x))} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 19, normalized size = 1.00 \begin {gather*} 2 \left (2 x-\frac {x \log (2)}{1+x+\log (\log (x))}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.41, size = 18, normalized size = 0.95
method | result | size |
risch | \(4 x -\frac {2 x \ln \left (2\right )}{1+\ln \left (\ln \left (x \right )\right )+x}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 29, normalized size = 1.53 \begin {gather*} \frac {2 \, {\left (2 \, x^{2} - x {\left (\log \left (2\right ) - 2\right )} + 2 \, x \log \left (\log \left (x\right )\right )\right )}}{x + \log \left (\log \left (x\right )\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 30, normalized size = 1.58 \begin {gather*} \frac {2 \, {\left (2 \, x^{2} - x \log \left (2\right ) + 2 \, x \log \left (\log \left (x\right )\right ) + 2 \, x\right )}}{x + \log \left (\log \left (x\right )\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 17, normalized size = 0.89 \begin {gather*} 4 x - \frac {2 x \log {\left (2 \right )}}{x + \log {\left (\log {\left (x \right )} \right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 17, normalized size = 0.89 \begin {gather*} 4 \, x - \frac {2 \, x \log \left (2\right )}{x + \log \left (\log \left (x\right )\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.43, size = 32, normalized size = 1.68 \begin {gather*} \frac {4\,x+\ln \left (4\right )+\ln \left (\ln \left (x\right )\right )\,\ln \left (4\right )+4\,x\,\ln \left (\ln \left (x\right )\right )+4\,x^2}{x+\ln \left (\ln \left (x\right )\right )+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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