Optimal. Leaf size=24 \[ \log \left (\frac {1}{3 \log \left (\frac {5 x}{4}\right ) \log ^2\left (\frac {\log ^2(x)}{x^2}\right )}\right ) \]
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Rubi [A]
time = 0.22, antiderivative size = 22, normalized size of antiderivative = 0.92, number of steps
used = 6, number of rules used = 5, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {6820, 14, 2339,
29, 6816} \begin {gather*} -2 \log \left (\log \left (\frac {\log ^2(x)}{x^2}\right )\right )-\log \left (\log \left (\frac {5 x}{4}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 29
Rule 2339
Rule 6816
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-\frac {1}{\log \left (\frac {5 x}{4}\right )}+\frac {4 (-1+\log (x))}{\log (x) \log \left (\frac {\log ^2(x)}{x^2}\right )}}{x} \, dx\\ &=\int \left (-\frac {1}{x \log \left (\frac {5 x}{4}\right )}+\frac {4 (-1+\log (x))}{x \log (x) \log \left (\frac {\log ^2(x)}{x^2}\right )}\right ) \, dx\\ &=4 \int \frac {-1+\log (x)}{x \log (x) \log \left (\frac {\log ^2(x)}{x^2}\right )} \, dx-\int \frac {1}{x \log \left (\frac {5 x}{4}\right )} \, dx\\ &=-2 \log \left (\log \left (\frac {\log ^2(x)}{x^2}\right )\right )-\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {5 x}{4}\right )\right )\\ &=-\log \left (\log \left (\frac {5 x}{4}\right )\right )-2 \log \left (\log \left (\frac {\log ^2(x)}{x^2}\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.10, size = 22, normalized size = 0.92 \begin {gather*} -\log \left (\log \left (\frac {5 x}{4}\right )\right )-2 \log \left (\log \left (\frac {\log ^2(x)}{x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.29, size = 26, normalized size = 1.08
method | result | size |
default | \(-\ln \left (\ln \left (5\right )-2 \ln \left (2\right )+\ln \left (x \right )\right )-2 \ln \left (\ln \left (\frac {\ln \left (x \right )^{2}}{x^{2}}\right )\right )\) | \(26\) |
risch | \(-\ln \left (\ln \left (x \right )+\frac {i \left (4 i \ln \left (2\right )-2 i \ln \left (5\right )\right )}{2}\right )-2 \ln \left (\ln \left (\ln \left (x \right )\right )-\frac {i \left (\pi \mathrm {csgn}\left (i \ln \left (x \right )\right )^{2} \mathrm {csgn}\left (i \ln \left (x \right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i \ln \left (x \right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \ln \left (x \right )^{2}}{x^{2}}\right )^{3}-\pi \mathrm {csgn}\left (\frac {i \ln \left (x \right )^{2}}{x^{2}}\right )^{2} \mathrm {csgn}\left (i \ln \left (x \right )^{2}\right )-\pi \mathrm {csgn}\left (\frac {i \ln \left (x \right )^{2}}{x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right )+\pi \,\mathrm {csgn}\left (\frac {i \ln \left (x \right )^{2}}{x^{2}}\right ) \mathrm {csgn}\left (i \ln \left (x \right )^{2}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right )-\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i \ln \left (x \right )^{2}\right )^{3}-4 i \ln \left (x \right )\right )}{4}\right )\) | \(223\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 19, normalized size = 0.79 \begin {gather*} -2 \, \log \left (-\log \left (x\right ) + \log \left (\log \left (x\right )\right )\right ) - \log \left (\log \left (\frac {5}{4} \, x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.52, size = 21, normalized size = 0.88 \begin {gather*} -\log \left (\log \left (\frac {5}{4}\right ) + \log \left (x\right )\right ) - 2 \, \log \left (\log \left (\frac {\log \left (x\right )^{2}}{x^{2}}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 27, normalized size = 1.12 \begin {gather*} - \log {\left (\log {\left (x \right )} - 2 \log {\left (2 \right )} + \log {\left (5 \right )} \right )} - 2 \log {\left (\log {\left (\frac {\log {\left (x \right )}^{2}}{x^{2}} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 32, normalized size = 1.33 \begin {gather*} -\log \left (-\log \left (5\right ) + 2 \, \log \left (2\right ) - \log \left (x\right )\right ) - 2 \, \log \left (-\log \left (\log \left (x\right )^{2}\right ) + 2 \, \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.92, size = 20, normalized size = 0.83 \begin {gather*} -2\,\ln \left (\ln \left (\frac {{\ln \left (x\right )}^2}{x^2}\right )\right )-\ln \left (\ln \left (\frac {5\,x}{4}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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