Optimal. Leaf size=25 \[ \frac {16-\log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (4+\frac {\log (x)}{x^2}\right )^2} \]
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Rubi [F] time = 14.27, 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 {4 x^5+x^3 \log (x)+\left (-32 x^3 \log \left (\frac {4}{x}\right )+64 x^3 \log \left (\frac {4}{x}\right ) \log (x)\right ) \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )+\left (2 x^3 \log \left (\frac {4}{x}\right )-4 x^3 \log \left (\frac {4}{x}\right ) \log (x)\right ) \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (64 x^6 \log \left (\frac {4}{x}\right )+48 x^4 \log \left (\frac {4}{x}\right ) \log (x)+12 x^2 \log \left (\frac {4}{x}\right ) \log ^2(x)+\log \left (\frac {4}{x}\right ) \log ^3(x)\right ) \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^3 \left (4 x^2+\log (x)+2 \log \left (\frac {4}{x}\right ) \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right ) \left (-16+\log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )\right )-4 \log \left (\frac {4}{x}\right ) \log (x) \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right ) \left (-16+\log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )\right )\right )}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx\\ &=\int \left (-\frac {32 x^3}{\left (4 x^2+\log (x)\right )^3}+\frac {64 x^3 \log (x)}{\left (4 x^2+\log (x)\right )^3}+\frac {4 x^5}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )}+\frac {x^3 \log (x)}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )}-\frac {2 x^3 (-1+2 \log (x)) \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (4 x^2+\log (x)\right )^3}\right ) \, dx\\ &=-\left (2 \int \frac {x^3 (-1+2 \log (x)) \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (4 x^2+\log (x)\right )^3} \, dx\right )+4 \int \frac {x^5}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx-32 \int \frac {x^3}{\left (4 x^2+\log (x)\right )^3} \, dx+64 \int \frac {x^3 \log (x)}{\left (4 x^2+\log (x)\right )^3} \, dx+\int \frac {x^3 \log (x)}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx\\ &=-\left (2 \int \left (-\frac {x^3 \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (4 x^2+\log (x)\right )^3}+\frac {2 x^3 \log (x) \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (4 x^2+\log (x)\right )^3}\right ) \, dx\right )+4 \int \frac {x^5}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx-32 \int \frac {x^3}{\left (4 x^2+\log (x)\right )^3} \, dx+64 \int \left (-\frac {4 x^5}{\left (4 x^2+\log (x)\right )^3}+\frac {x^3}{\left (4 x^2+\log (x)\right )^2}\right ) \, dx+\int \frac {x^3 \log (x)}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx\\ &=2 \int \frac {x^3 \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (4 x^2+\log (x)\right )^3} \, dx+4 \int \frac {x^5}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx-4 \int \frac {x^3 \log (x) \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (4 x^2+\log (x)\right )^3} \, dx-32 \int \frac {x^3}{\left (4 x^2+\log (x)\right )^3} \, dx+64 \int \frac {x^3}{\left (4 x^2+\log (x)\right )^2} \, dx-256 \int \frac {x^5}{\left (4 x^2+\log (x)\right )^3} \, dx+\int \frac {x^3 \log (x)}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.13, size = 27, normalized size = 1.08 \begin {gather*} -\frac {x^4 \left (-16+\log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )\right )}{\left (4 x^2+\log (x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.95, size = 67, normalized size = 2.68 \begin {gather*} -\frac {x^{4} \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right ) - 16 \, x^{4}}{16 \, x^{4} + 16 \, x^{2} \log \relax (2) + 4 \, \log \relax (2)^{2} - 4 \, {\left (2 \, x^{2} + \log \relax (2)\right )} \log \left (\frac {4}{x}\right ) + \log \left (\frac {4}{x}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 116, normalized size = 4.64 \begin {gather*} -\frac {{\left (8 \, x^{6} + x^{4}\right )} \log \left (\log \left (\log \left (\log \left (2 \, \log \relax (2) - \log \relax (x)\right )\right )\right )\right )}{128 \, x^{6} + 64 \, x^{4} \log \relax (x) + 16 \, x^{4} + 8 \, x^{2} \log \relax (x)^{2} + 8 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} + \frac {16 \, {\left (8 \, x^{6} + x^{4}\right )}}{128 \, x^{6} + 64 \, x^{4} \log \relax (x) + 16 \, x^{4} + 8 \, x^{2} \log \relax (x)^{2} + 8 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 45, normalized size = 1.80
method | result | size |
risch | \(-\frac {x^{4} \ln \left (\ln \left (\ln \left (\ln \left (2 \ln \relax (2)-\ln \relax (x )\right )\right )\right )\right )}{\left (4 x^{2}+\ln \relax (x )\right )^{2}}+\frac {16 x^{4}}{\left (4 x^{2}+\ln \relax (x )\right )^{2}}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 44, normalized size = 1.76 \begin {gather*} -\frac {x^{4} \log \left (\log \left (\log \left (\log \left (2 \, \log \relax (2) - \log \relax (x)\right )\right )\right )\right ) - 16 \, x^{4}}{16 \, x^{4} + 8 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.94, size = 168, normalized size = 6.72 \begin {gather*} \frac {\frac {16\,x^4}{8\,x^2+1}-\frac {32\,x^4\,\ln \relax (x)}{8\,x^2+1}}{16\,x^4+8\,x^2\,\ln \relax (x)+{\ln \relax (x)}^2}-\frac {\frac {x^4}{8}+\frac {3\,x^2}{64}+\frac {1}{512}}{x^6+\frac {3\,x^4}{8}+\frac {3\,x^2}{64}+\frac {1}{512}}+\frac {\frac {32\,x^4}{{\left (8\,x^2+1\right )}^3}-\frac {128\,x^4\,\ln \relax (x)\,\left (4\,x^2+1\right )}{{\left (8\,x^2+1\right )}^3}}{\ln \relax (x)+4\,x^2}-\frac {x^5\,\ln \left (\ln \left (\ln \left (\ln \left (\ln \left (\frac {4}{x}\right )\right )\right )\right )\right )}{16\,x^5+8\,x^3\,\ln \relax (x)+x\,{\ln \relax (x)}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 23.31, size = 58, normalized size = 2.32 \begin {gather*} - \frac {x^{4} \log {\left (\log {\left (\log {\left (\log {\left (- \log {\relax (x )} + \log {\relax (4 )} \right )} \right )} \right )} \right )}}{16 x^{4} + 8 x^{2} \log {\relax (x )} + \log {\relax (x )}^{2}} + \frac {16 x^{4}}{16 x^{4} + 8 x^{2} \log {\relax (x )} + \log {\relax (x )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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