Optimal. Leaf size=15 \[ 1+\log \left (\frac {1}{x+\frac {4+x}{\log (x)}}\right ) \]
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Rubi [F]
time = 0.44, 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 {4+x-x \log (x)-x \log ^2(x)}{\left (4 x+x^2\right ) \log (x)+x^2 \log ^2(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 {4+x-x \log (x)-x \log ^2(x)}{x \log (x) (4+x+x \log (x))} \, dx\\ &=\int \left (-\frac {1}{x}+\frac {1}{x \log (x)}+\frac {4-x}{x (4+x+x \log (x))}\right ) \, dx\\ &=-\log (x)+\int \frac {1}{x \log (x)} \, dx+\int \frac {4-x}{x (4+x+x \log (x))} \, dx\\ &=-\log (x)+\int \left (\frac {1}{-4-x-x \log (x)}+\frac {4}{x (4+x+x \log (x))}\right ) \, dx+\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-\log (x)+\log (\log (x))+4 \int \frac {1}{x (4+x+x \log (x))} \, dx+\int \frac {1}{-4-x-x \log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 14, normalized size = 0.93 \begin {gather*} \log (\log (x))-\log (4+x+x \log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 15, normalized size = 1.00
method | result | size |
default | \(\ln \left (\ln \left (x \right )\right )-\ln \left (x \ln \left (x \right )+x +4\right )\) | \(15\) |
norman | \(\ln \left (\ln \left (x \right )\right )-\ln \left (x \ln \left (x \right )+x +4\right )\) | \(15\) |
risch | \(-\ln \left (x \right )+\ln \left (\ln \left (x \right )\right )-\ln \left (\ln \left (x \right )+\frac {4+x}{x}\right )\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 22, normalized size = 1.47 \begin {gather*} -\log \left (x\right ) - \log \left (\frac {x \log \left (x\right ) + x + 4}{x}\right ) + \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 22, normalized size = 1.47 \begin {gather*} -\log \left (x\right ) - \log \left (\frac {x \log \left (x\right ) + x + 4}{x}\right ) + \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 20, normalized size = 1.33 \begin {gather*} - \log {\left (x \right )} - \log {\left (\log {\left (x \right )} + \frac {2 x + 8}{2 x} \right )} + \log {\left (\log {\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 14, normalized size = 0.93 \begin {gather*} -\log \left (x \log \left (x\right ) + x + 4\right ) + \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.85, size = 14, normalized size = 0.93 \begin {gather*} \ln \left (\ln \left (x\right )\right )-\ln \left (x+x\,\ln \left (x\right )+4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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