Optimal. Leaf size=27 \[ \log \left (\frac {8 \left (e^4+\left (-x+\frac {\log \left (\log ^2(x)\right )}{x}\right )^2\right )^4}{x^2}\right ) \]
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Rubi [A] time = 0.80, antiderivative size = 36, normalized size of antiderivative = 1.33, number of steps used = 4, number of rules used = 3, integrand size = 90, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6741, 6742, 6684} \begin {gather*} 4 \log \left (x^4+e^4 x^2-2 x^2 \log \left (\log ^2(x)\right )+\log ^2\left (\log ^2(x)\right )\right )-10 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 6684
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-16 x^2+\left (-2 e^4 x^2+6 x^4\right ) \log (x)+\left (16+4 x^2 \log (x)\right ) \log \left (\log ^2(x)\right )-10 \log (x) \log ^2\left (\log ^2(x)\right )}{x \log (x) \left (e^4 x^2+x^4-2 x^2 \log \left (\log ^2(x)\right )+\log ^2\left (\log ^2(x)\right )\right )} \, dx\\ &=\int \left (-\frac {10}{x}+\frac {8 \left (-2 x^2+e^4 x^2 \log (x)+2 x^4 \log (x)+2 \log \left (\log ^2(x)\right )-2 x^2 \log (x) \log \left (\log ^2(x)\right )\right )}{x \log (x) \left (e^4 x^2+x^4-2 x^2 \log \left (\log ^2(x)\right )+\log ^2\left (\log ^2(x)\right )\right )}\right ) \, dx\\ &=-10 \log (x)+8 \int \frac {-2 x^2+e^4 x^2 \log (x)+2 x^4 \log (x)+2 \log \left (\log ^2(x)\right )-2 x^2 \log (x) \log \left (\log ^2(x)\right )}{x \log (x) \left (e^4 x^2+x^4-2 x^2 \log \left (\log ^2(x)\right )+\log ^2\left (\log ^2(x)\right )\right )} \, dx\\ &=-10 \log (x)+4 \log \left (e^4 x^2+x^4-2 x^2 \log \left (\log ^2(x)\right )+\log ^2\left (\log ^2(x)\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 38, normalized size = 1.41 \begin {gather*} -2 \left (5 \log (x)-2 \log \left (e^4 x^2+x^4-2 x^2 \log \left (\log ^2(x)\right )+\log ^2\left (\log ^2(x)\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 35, normalized size = 1.30 \begin {gather*} 4 \, \log \left (x^{4} + x^{2} e^{4} - 2 \, x^{2} \log \left (\log \relax (x)^{2}\right ) + \log \left (\log \relax (x)^{2}\right )^{2}\right ) - 10 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 35, normalized size = 1.30 \begin {gather*} 4 \, \log \left (x^{4} + x^{2} e^{4} - 2 \, x^{2} \log \left (\log \relax (x)^{2}\right ) + \log \left (\log \relax (x)^{2}\right )^{2}\right ) - 10 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 259, normalized size = 9.59
method | result | size |
risch | \(-10 \ln \relax (x )+4 \ln \left (-\frac {\pi ^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{4} \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{2}}{16}+\frac {\pi ^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{3} \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{3}}{4}-\frac {3 \pi ^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{4}}{8}+\frac {\pi ^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{5}}{4}-\frac {\pi ^{2} \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{6}}{16}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )}{4}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{2}}{2}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{3}}{4}+\frac {x^{4}}{4}+\frac {x^{2} {\mathrm e}^{4}}{4}+\ln \left (\ln \relax (x )\right )^{2}+\left (-\frac {i \pi \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )}{2}+i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{2}-\frac {i \pi \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{3}}{2}-x^{2}\right ) \ln \left (\ln \relax (x )\right )\right )\) | \(259\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 34, normalized size = 1.26 \begin {gather*} 4 \, \log \left (\frac {1}{4} \, x^{4} + \frac {1}{4} \, x^{2} e^{4} - x^{2} \log \left (\log \relax (x)\right ) + \log \left (\log \relax (x)\right )^{2}\right ) - 10 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {10\,{\ln \left ({\ln \relax (x)}^2\right )}^2\,\ln \relax (x)+\ln \relax (x)\,\left (2\,x^2\,{\mathrm {e}}^4-6\,x^4\right )-\ln \left ({\ln \relax (x)}^2\right )\,\left (4\,x^2\,\ln \relax (x)+16\right )+16\,x^2}{\ln \relax (x)\,\left (x^5+{\mathrm {e}}^4\,x^3\right )+x\,{\ln \left ({\ln \relax (x)}^2\right )}^2\,\ln \relax (x)-2\,x^3\,\ln \left ({\ln \relax (x)}^2\right )\,\ln \relax (x)} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 37, normalized size = 1.37 \begin {gather*} - 10 \log {\relax (x )} + 4 \log {\left (x^{4} - 2 x^{2} \log {\left (\log {\relax (x )}^{2} \right )} + x^{2} e^{4} + \log {\left (\log {\relax (x )}^{2} \right )}^{2} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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