Optimal. Leaf size=27 \[ 16-x+\frac {\log (x)}{-4+x}+\frac {\log ^2(3)}{2-\log (\log (x))} \]
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Rubi [A] time = 0.20, antiderivative size = 38, normalized size of antiderivative = 1.41, number of steps used = 9, number of rules used = 6, integrand size = 162, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {6688, 893, 2314, 31, 2302, 30} \begin {gather*} -x+\frac {\log ^2(3)}{2-\log (\log (x))}-\frac {x \log (x)}{4 (4-x)}-\frac {\log (x)}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 31
Rule 893
Rule 2302
Rule 2314
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1+4 x-x^2}{(-4+x) x}-\frac {\log (x)}{(-4+x)^2}+\frac {\log ^2(3)}{x \log (x) (-2+\log (\log (x)))^2}\right ) \, dx\\ &=\log ^2(3) \int \frac {1}{x \log (x) (-2+\log (\log (x)))^2} \, dx+\int \frac {1+4 x-x^2}{(-4+x) x} \, dx-\int \frac {\log (x)}{(-4+x)^2} \, dx\\ &=-\frac {x \log (x)}{4 (4-x)}-\frac {1}{4} \int \frac {1}{-4+x} \, dx+\log ^2(3) \operatorname {Subst}\left (\int \frac {1}{x (-2+\log (x))^2} \, dx,x,\log (x)\right )+\int \left (-1+\frac {1}{4 (-4+x)}-\frac {1}{4 x}\right ) \, dx\\ &=-x-\frac {\log (x)}{4}-\frac {x \log (x)}{4 (4-x)}+\log ^2(3) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,-2+\log (\log (x))\right )\\ &=-x-\frac {\log (x)}{4}-\frac {x \log (x)}{4 (4-x)}+\frac {\log ^2(3)}{2-\log (\log (x))}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 25, normalized size = 0.93 \begin {gather*} -x+\frac {\log (x)}{-4+x}-\frac {\log ^2(3)}{-2+\log (\log (x))} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 52, normalized size = 1.93 \begin {gather*} -\frac {{\left (x - 4\right )} \log \relax (3)^{2} - 2 \, x^{2} + {\left (x^{2} - 4 \, x - \log \relax (x)\right )} \log \left (\log \relax (x)\right ) + 8 \, x + 2 \, \log \relax (x)}{{\left (x - 4\right )} \log \left (\log \relax (x)\right ) - 2 \, x + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.99, size = 25, normalized size = 0.93 \begin {gather*} -x - \frac {\log \relax (3)^{2}}{\log \left (\log \relax (x)\right ) - 2} + \frac {\log \relax (x)}{x - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 1.19
method | result | size |
default | \(-\frac {\ln \relax (3)^{2}}{\ln \left (\ln \relax (x )\right )-2}+\frac {x \ln \relax (x )}{4 x -16}-x -\frac {\ln \relax (x )}{4}\) | \(32\) |
risch | \(-\frac {x^{2}-4 x -\ln \relax (x )}{x -4}-\frac {\ln \relax (3)^{2}}{\ln \left (\ln \relax (x )\right )-2}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.40, size = 55, normalized size = 2.04 \begin {gather*} -\frac {{\left (\log \relax (3)^{2} + 8\right )} x - 2 \, x^{2} - 4 \, \log \relax (3)^{2} + {\left (x^{2} - 4 \, x - \log \relax (x)\right )} \log \left (\log \relax (x)\right ) + 2 \, \log \relax (x)}{{\left (x - 4\right )} \log \left (\log \relax (x)\right ) - 2 \, x + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.25, size = 25, normalized size = 0.93 \begin {gather*} \frac {\ln \relax (x)}{x-4}-\frac {{\ln \relax (3)}^2}{\ln \left (\ln \relax (x)\right )-2}-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 19, normalized size = 0.70 \begin {gather*} - x - \frac {\log {\relax (3 )}^{2}}{\log {\left (\log {\relax (x )} \right )} - 2} + \frac {\log {\relax (x )}}{x - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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