Optimal. Leaf size=23 \[ x-\log \left (\frac {4 \log (x)}{\log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}\right ) \]
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Rubi [F] time = 1.22, 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 {(-8+2 x) \log (x)+x \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((-2+2 x \log (x)) \log \left (x^2\right )+\left (4-x+\left (-4 x+x^2\right ) \log (x)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}{\left (2 x \log (x) \log \left (x^2\right )+\left (-4 x+x^2\right ) \log (x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-\frac {1}{x \log (x)}+\frac {2 (-4+x)}{x \log \left (x^2\right ) \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}+\frac {\log \left (\log \left (x^2\right )\right )}{\left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}\right ) \, dx\\ &=x+2 \int \frac {-4+x}{x \log \left (x^2\right ) \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx-\int \frac {1}{x \log (x)} \, dx+\int \frac {\log \left (\log \left (x^2\right )\right )}{\left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx\\ &=x+2 \int \left (\frac {1}{\log \left (x^2\right ) \left (2-4 \log \left (\log \left (x^2\right )\right )+x \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}-\frac {4}{x \log \left (x^2\right ) \left (2-4 \log \left (\log \left (x^2\right )\right )+x \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )}\right ) \, dx+\int \frac {\log \left (\log \left (x^2\right )\right )}{\left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=x-\log (\log (x))+2 \int \frac {1}{\log \left (x^2\right ) \left (2-4 \log \left (\log \left (x^2\right )\right )+x \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx-8 \int \frac {1}{x \log \left (x^2\right ) \left (2-4 \log \left (\log \left (x^2\right )\right )+x \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx+\int \frac {\log \left (\log \left (x^2\right )\right )}{\left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx\\ &=x-\log (\log (x))+2 \int \frac {1}{\log \left (x^2\right ) \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx-8 \int \frac {1}{x \log \left (x^2\right ) \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx+\int \frac {\log \left (\log \left (x^2\right )\right )}{\left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right ) \log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.28, size = 20, normalized size = 0.87 \begin {gather*} x-\log (\log (x))+\log \left (\log \left (2+(-4+x) \log \left (\log \left (x^2\right )\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 22, normalized size = 0.96 \begin {gather*} x - \log \left (2 \, \log \relax (x)\right ) + \log \left (\log \left ({\left (x - 4\right )} \log \left (2 \, \log \relax (x)\right ) + 2\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 50, normalized size = 2.17
method | result | size |
risch | \(x -\ln \left (\ln \relax (x )\right )+\ln \left (\ln \left (\left (x -4\right ) \ln \left (2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )+2\right )\right )\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 26, normalized size = 1.13 \begin {gather*} x + \log \left (\log \left (x \log \relax (2) + {\left (x - 4\right )} \log \left (\log \relax (x)\right ) - 4 \, \log \relax (2) + 2\right )\right ) - \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.91, size = 20, normalized size = 0.87 \begin {gather*} x+\ln \left (\ln \left (\ln \left (\ln \left (x^2\right )\right )\,\left (x-4\right )+2\right )\right )-\ln \left (\ln \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.01, size = 20, normalized size = 0.87 \begin {gather*} x - \log {\left (\log {\relax (x )} \right )} + \log {\left (\log {\left (\left (x - 4\right ) \log {\left (2 \log {\relax (x )} \right )} + 2 \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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