Optimal. Leaf size=22 \[ 2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \]
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Rubi [A] time = 1.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 88, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6741, 6742, 2549} \begin {gather*} 2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2549
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 x \log (x)+8 \log \left (\frac {4}{\log ^2(x)}\right )-2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )+\left (2 x \log (x)+2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )\right ) \log \left (\frac {2}{x^2+x \log ^2\left (\frac {4}{\log ^2(x)}\right )}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )} \, dx\\ &=\int \left (-\frac {2 \left (2 x \log (x)-4 \log \left (\frac {4}{\log ^2(x)}\right )+\log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}+2 \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right )\right ) \, dx\\ &=-\left (2 \int \frac {2 x \log (x)-4 \log \left (\frac {4}{\log ^2(x)}\right )+\log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )} \, dx\right )+2 \int \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \, dx\\ &=2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right )-2 \int \left (1+\frac {x \log (x)-4 \log \left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \, dx-2 \int \frac {4 \log \left (\frac {4}{\log ^2(x)}\right )-\log (x) \left (2 x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )} \, dx\\ &=-2 x+2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right )-2 \int \frac {x \log (x)-4 \log \left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )} \, dx-2 \int \left (-1+\frac {-x \log (x)+4 \log \left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \, dx\\ &=2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right )-2 \int \frac {-x \log (x)+4 \log \left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )} \, dx-2 \int \left (\frac {x}{x+\log ^2\left (\frac {4}{\log ^2(x)}\right )}-\frac {4 \log \left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \, dx\\ &=2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right )-2 \int \frac {x}{x+\log ^2\left (\frac {4}{\log ^2(x)}\right )} \, dx-2 \int \left (-\frac {x}{x+\log ^2\left (\frac {4}{\log ^2(x)}\right )}+\frac {4 \log \left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \, dx+8 \int \frac {\log \left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )} \, dx\\ &=2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.10, size = 22, normalized size = 1.00 \begin {gather*} 2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 23, normalized size = 1.05 \begin {gather*} 2 \, x \log \left (\frac {2}{x \log \left (\frac {4}{\log \relax (x)^{2}}\right )^{2} + x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.65, size = 39, normalized size = 1.77 \begin {gather*} 2 \, x \log \relax (2) - 2 \, x \log \left (4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) \log \left (\log \relax (x)^{2}\right ) + \log \left (\log \relax (x)^{2}\right )^{2} + x\right ) - 2 \, x \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.84, size = 2139, normalized size = 97.23
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2139\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 37, normalized size = 1.68 \begin {gather*} 2 \, x \log \relax (2) - 2 \, x \log \left (4 \, \log \relax (2)^{2} - 8 \, \log \relax (2) \log \left (\log \relax (x)\right ) + 4 \, \log \left (\log \relax (x)\right )^{2} + x\right ) - 2 \, x \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.48, size = 24, normalized size = 1.09 \begin {gather*} 2\,x\,\left (\ln \left (\frac {1}{x^2+x\,{\ln \left (\frac {4}{{\ln \relax (x)}^2}\right )}^2}\right )+\ln \relax (2)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.88, size = 20, normalized size = 0.91 \begin {gather*} 2 x \log {\left (\frac {2}{x^{2} + x \log {\left (\frac {4}{\log {\relax (x )}^{2}} \right )}^{2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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