Optimal. Leaf size=21 \[ \frac {2}{9 x^4 \log ^2\left (\frac {x^2}{4}+\log (\log (x))\right )} \]
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Rubi [F] time = 2.28, 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 {-16-8 x^2 \log (x)+\left (-8 x^2 \log (x)-32 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}{\left (9 x^7 \log (x)+36 x^5 \log (x) \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\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 \frac {-16-8 x^2 \log (x)+\left (-8 x^2 \log (x)-32 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}{9 x^5 \log (x) \left (x^2+4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx\\ &=\frac {1}{9} \int \frac {-16-8 x^2 \log (x)+\left (-8 x^2 \log (x)-32 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}{x^5 \log (x) \left (x^2+4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx\\ &=\frac {1}{9} \int \left (\frac {8 \left (-2-x^2 \log (x)\right )}{x^5 \log (x) \left (x^2+4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}-\frac {8}{x^5 \log ^2\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}\right ) \, dx\\ &=\frac {8}{9} \int \frac {-2-x^2 \log (x)}{x^5 \log (x) \left (x^2+4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx-\frac {8}{9} \int \frac {1}{x^5 \log ^2\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx\\ &=\frac {8}{9} \int \left (\frac {1}{x^3 \left (-x^2-4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}+\frac {2}{x^5 \log (x) \left (-x^2-4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}\right ) \, dx-\frac {8}{9} \int \frac {1}{x^5 \log ^2\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx\\ &=\frac {8}{9} \int \frac {1}{x^3 \left (-x^2-4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx-\frac {8}{9} \int \frac {1}{x^5 \log ^2\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx+\frac {16}{9} \int \frac {1}{x^5 \log (x) \left (-x^2-4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.56, size = 21, normalized size = 1.00 \begin {gather*} \frac {2}{9 x^4 \log ^2\left (\frac {x^2}{4}+\log (\log (x))\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 17, normalized size = 0.81 \begin {gather*} \frac {2}{9 \, x^{4} \log \left (\frac {1}{4} \, x^{2} + \log \left (\log \relax (x)\right )\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 104, normalized size = 4.95 \begin {gather*} \frac {2 \, {\left (x^{2} \log \relax (x) + 2\right )}}{9 \, {\left (4 \, x^{6} \log \relax (2)^{2} \log \relax (x) - 4 \, x^{6} \log \relax (2) \log \left (x^{2} + 4 \, \log \left (\log \relax (x)\right )\right ) \log \relax (x) + x^{6} \log \left (x^{2} + 4 \, \log \left (\log \relax (x)\right )\right )^{2} \log \relax (x) + 8 \, x^{4} \log \relax (2)^{2} - 8 \, x^{4} \log \relax (2) \log \left (x^{2} + 4 \, \log \left (\log \relax (x)\right )\right ) + 2 \, x^{4} \log \left (x^{2} + 4 \, \log \left (\log \relax (x)\right )\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 18, normalized size = 0.86
method | result | size |
risch | \(\frac {2}{9 x^{4} \ln \left (\ln \left (\ln \relax (x )\right )+\frac {x^{2}}{4}\right )^{2}}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 47, normalized size = 2.24 \begin {gather*} \frac {2}{9 \, {\left (4 \, x^{4} \log \relax (2)^{2} - 4 \, x^{4} \log \relax (2) \log \left (x^{2} + 4 \, \log \left (\log \relax (x)\right )\right ) + x^{4} \log \left (x^{2} + 4 \, \log \left (\log \relax (x)\right )\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.76, size = 519, normalized size = 24.71 \begin {gather*} \frac {4}{9\,x^4}-\frac {\frac {2\,\ln \relax (x)\,\left (4\,\ln \left (\ln \relax (x)\right )+x^2\right )}{9\,x^4\,\left (x^2\,\ln \relax (x)+2\right )}-\frac {2\,\ln \left (\ln \left (\ln \relax (x)\right )+\frac {x^2}{4}\right )\,\ln \relax (x)\,\left (4\,\ln \left (\ln \relax (x)\right )+x^2\right )\,\left (4\,\ln \left (\ln \relax (x)\right )-2\,x^4\,{\ln \relax (x)}^2-16\,\ln \left (\ln \relax (x)\right )\,\ln \relax (x)+x^2-12\,x^2\,\ln \left (\ln \relax (x)\right )\,{\ln \relax (x)}^2+4\right )}{9\,x^4\,{\left (x^2\,\ln \relax (x)+2\right )}^3}}{\ln \left (\ln \left (\ln \relax (x)\right )+\frac {x^2}{4}\right )}+\frac {\frac {2}{9\,x^4}+\frac {2\,\ln \left (\ln \left (\ln \relax (x)\right )+\frac {x^2}{4}\right )\,\ln \relax (x)\,\left (4\,\ln \left (\ln \relax (x)\right )+x^2\right )}{9\,x^4\,\left (x^2\,\ln \relax (x)+2\right )}}{{\ln \left (\ln \left (\ln \relax (x)\right )+\frac {x^2}{4}\right )}^2}-\frac {\ln \left (\ln \relax (x)\right )\,\left (\ln \relax (x)\,\left (x^2\,\left (\frac {16\,\left (x^2-20\right )}{9\,x^6}+\frac {416}{9\,x^6}\right )-\frac {64}{9\,x^4}\right )-\frac {40\,{\ln \relax (x)}^3}{9}-\frac {32\,\left (x^2-4\right )}{9\,x^6}+\frac {32\,\left (x^2-20\right )}{9\,x^6}-\frac {32\,{\ln \relax (x)}^2}{9\,x^2}+\frac {512}{9\,x^6}\right )}{x^6\,{\ln \relax (x)}^3+6\,x^4\,{\ln \relax (x)}^2+12\,x^2\,\ln \relax (x)+8}-\frac {{\ln \left (\ln \relax (x)\right )}^2\,\left (\ln \relax (x)\,\left (x^2\,\left (\frac {32\,\left (x^2-20\right )}{9\,x^8}+\frac {896}{9\,x^8}\right )-\frac {256}{9\,x^6}\right )-\frac {64\,\left (x^2-4\right )}{9\,x^8}+\frac {64\,\left (x^2-20\right )}{9\,x^8}-\frac {32\,{\ln \relax (x)}^3}{3\,x^2}-\frac {128\,{\ln \relax (x)}^2}{9\,x^4}+\frac {1024}{9\,x^8}\right )}{x^6\,{\ln \relax (x)}^3+6\,x^4\,{\ln \relax (x)}^2+12\,x^2\,\ln \relax (x)+8}+\frac {8\,\left (x^2-4\right )}{3\,x^3\,\left (4\,x-x^3\right )\,\left (x^2\,\ln \relax (x)+2\right )}-\frac {4\,\left (x^5-8\,x^3+16\,x\right )}{9\,x^4\,\left (4\,x-x^3\right )\,\left (x^6\,{\ln \relax (x)}^3+6\,x^4\,{\ln \relax (x)}^2+12\,x^2\,\ln \relax (x)+8\right )}+\frac {2\,\left (x^4-24\,x^2+80\right )}{9\,x^3\,\left (4\,x-x^3\right )\,\left (x^4\,{\ln \relax (x)}^2+4\,x^2\,\ln \relax (x)+4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 19, normalized size = 0.90 \begin {gather*} \frac {2}{9 x^{4} \log {\left (\frac {x^{2}}{4} + \log {\left (\log {\relax (x )} \right )} \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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