Optimal. Leaf size=25 \[ \frac {\left (\log \left (\frac {1}{5 x}\right )+\log \left (\frac {\log (x)}{x}\right )\right )^4}{4 x^2} \]
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Rubi [F] time = 3.31, 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 {2 \log ^3\left (\frac {1}{5 x}\right )+\left (-4 \log ^3\left (\frac {1}{5 x}\right )-\log ^4\left (\frac {1}{5 x}\right )\right ) \log (x)+\left (6 \log ^2\left (\frac {1}{5 x}\right )+\left (-12 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )+\left (6 \log \left (\frac {1}{5 x}\right )+\left (-12 \log \left (\frac {1}{5 x}\right )-6 \log ^2\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (2+\left (-4-4 \log \left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )-\log (x) \log ^4\left (\frac {\log (x)}{x}\right )}{2 x^3 \log (x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {2 \log ^3\left (\frac {1}{5 x}\right )+\left (-4 \log ^3\left (\frac {1}{5 x}\right )-\log ^4\left (\frac {1}{5 x}\right )\right ) \log (x)+\left (6 \log ^2\left (\frac {1}{5 x}\right )+\left (-12 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )+\left (6 \log \left (\frac {1}{5 x}\right )+\left (-12 \log \left (\frac {1}{5 x}\right )-6 \log ^2\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (2+\left (-4-4 \log \left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )-\log (x) \log ^4\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx\\ &=\frac {1}{2} \int \frac {\left (\log \left (\frac {1}{5 x}\right )+\log \left (\frac {\log (x)}{x}\right )\right )^3 \left (2-\log (x) \left (4+\log \left (\frac {1}{5 x}\right )+\log \left (\frac {\log (x)}{x}\right )\right )\right )}{x^3 \log (x)} \, dx\\ &=\frac {1}{2} \int \left (-\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (-2+4 \log (x)+\log \left (\frac {1}{5 x}\right ) \log (x)\right )}{x^3 \log (x)}-\frac {2 \log ^2\left (\frac {1}{5 x}\right ) \left (-3+6 \log (x)+2 \log \left (\frac {1}{5 x}\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)}-\frac {6 \log \left (\frac {1}{5 x}\right ) \left (-1+2 \log (x)+\log \left (\frac {1}{5 x}\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)}-\frac {2 \left (-1+2 \log (x)+2 \log \left (\frac {1}{5 x}\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)}-\frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \left (-2+4 \log (x)+\log \left (\frac {1}{5 x}\right ) \log (x)\right )}{x^3 \log (x)} \, dx\right )-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-3 \int \frac {\log \left (\frac {1}{5 x}\right ) \left (-1+2 \log (x)+\log \left (\frac {1}{5 x}\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-\int \frac {\log ^2\left (\frac {1}{5 x}\right ) \left (-3+6 \log (x)+2 \log \left (\frac {1}{5 x}\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-\int \frac {\left (-1+2 \log (x)+2 \log \left (\frac {1}{5 x}\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{x^3}-\frac {2 \log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)}\right ) \, dx\right )-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-3 \int \left (\frac {2 \log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3}+\frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3}-\frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)}\right ) \, dx-\int \left (\frac {6 \log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3}+\frac {2 \log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)}\right ) \, dx-\int \left (\frac {2 \log ^3\left (\frac {\log (x)}{x}\right )}{x^3}+\frac {2 \log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3}-\frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{x^3} \, dx\right )-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx\\ &=-\frac {3 \left (4+\log \left (\frac {1}{5 x}\right )\right )}{16 x^2}+\frac {3 \log \left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}+\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{4 x^2}-\frac {1}{2} \int \frac {3-6 \log \left (\frac {1}{5 x}\right )+6 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )}{8 x^3} \, dx-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx\\ &=-\frac {3 \left (4+\log \left (\frac {1}{5 x}\right )\right )}{16 x^2}+\frac {3 \log \left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}+\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{4 x^2}-\frac {1}{16} \int \frac {3-6 \log \left (\frac {1}{5 x}\right )+6 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )}{x^3} \, dx-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx\\ &=-\frac {3 \left (4+\log \left (\frac {1}{5 x}\right )\right )}{16 x^2}+\frac {3 \log \left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}+\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{4 x^2}-\frac {1}{16} \int \left (\frac {3}{x^3}-\frac {6 \log \left (\frac {1}{5 x}\right )}{x^3}+\frac {6 \log ^2\left (\frac {1}{5 x}\right )}{x^3}-\frac {4 \log ^3\left (\frac {1}{5 x}\right )}{x^3}\right ) \, dx-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx\\ &=\frac {3}{32 x^2}-\frac {3 \left (4+\log \left (\frac {1}{5 x}\right )\right )}{16 x^2}+\frac {3 \log \left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}+\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{4 x^2}+\frac {1}{4} \int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3} \, dx+\frac {3}{8} \int \frac {\log \left (\frac {1}{5 x}\right )}{x^3} \, dx-\frac {3}{8} \int \frac {\log ^2\left (\frac {1}{5 x}\right )}{x^3} \, dx-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx\\ &=\frac {3}{16 x^2}-\frac {3 \log \left (\frac {1}{5 x}\right )}{16 x^2}+\frac {3 \log ^2\left (\frac {1}{5 x}\right )}{16 x^2}-\frac {\log ^3\left (\frac {1}{5 x}\right )}{8 x^2}-\frac {3 \left (4+\log \left (\frac {1}{5 x}\right )\right )}{16 x^2}+\frac {3 \log \left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}+\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{4 x^2}+\frac {3}{8} \int \frac {\log \left (\frac {1}{5 x}\right )}{x^3} \, dx-\frac {3}{8} \int \frac {\log ^2\left (\frac {1}{5 x}\right )}{x^3} \, dx-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx\\ &=\frac {9}{32 x^2}-\frac {3 \log \left (\frac {1}{5 x}\right )}{8 x^2}+\frac {3 \log ^2\left (\frac {1}{5 x}\right )}{8 x^2}-\frac {\log ^3\left (\frac {1}{5 x}\right )}{8 x^2}-\frac {3 \left (4+\log \left (\frac {1}{5 x}\right )\right )}{16 x^2}+\frac {3 \log \left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}+\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{4 x^2}+\frac {3}{8} \int \frac {\log \left (\frac {1}{5 x}\right )}{x^3} \, dx-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx\\ &=\frac {3}{8 x^2}-\frac {9 \log \left (\frac {1}{5 x}\right )}{16 x^2}+\frac {3 \log ^2\left (\frac {1}{5 x}\right )}{8 x^2}-\frac {\log ^3\left (\frac {1}{5 x}\right )}{8 x^2}-\frac {3 \left (4+\log \left (\frac {1}{5 x}\right )\right )}{16 x^2}+\frac {3 \log \left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}+\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{4 x^2}-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.27, size = 25, normalized size = 1.00 \begin {gather*} \frac {\left (\log \left (\frac {1}{5 x}\right )+\log \left (\frac {\log (x)}{x}\right )\right )^4}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 108, normalized size = 4.32 \begin {gather*} \frac {\log \left (-\frac {\log \relax (5) + \log \left (\frac {1}{5 \, x}\right )}{x}\right )^{4} + 4 \, \log \left (-\frac {\log \relax (5) + \log \left (\frac {1}{5 \, x}\right )}{x}\right )^{3} \log \left (\frac {1}{5 \, x}\right ) + 6 \, \log \left (-\frac {\log \relax (5) + \log \left (\frac {1}{5 \, x}\right )}{x}\right )^{2} \log \left (\frac {1}{5 \, x}\right )^{2} + 4 \, \log \left (-\frac {\log \relax (5) + \log \left (\frac {1}{5 \, x}\right )}{x}\right ) \log \left (\frac {1}{5 \, x}\right )^{3} + \log \left (\frac {1}{5 \, x}\right )^{4}}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 164, normalized size = 6.56 \begin {gather*} -{\left (\frac {\log \relax (5)}{x^{2}} + \frac {2 \, \log \relax (x)}{x^{2}}\right )} \log \left (\log \relax (x)\right )^{3} + \frac {3}{2} \, {\left (\frac {\log \relax (5)^{2}}{x^{2}} + \frac {4 \, \log \relax (5) \log \relax (x)}{x^{2}} + \frac {4 \, \log \relax (x)^{2}}{x^{2}}\right )} \log \left (\log \relax (x)\right )^{2} + \frac {\log \relax (5)^{4}}{4 \, x^{2}} + \frac {2 \, \log \relax (5)^{3} \log \relax (x)}{x^{2}} + \frac {6 \, \log \relax (5)^{2} \log \relax (x)^{2}}{x^{2}} + \frac {8 \, \log \relax (5) \log \relax (x)^{3}}{x^{2}} + \frac {4 \, \log \relax (x)^{4}}{x^{2}} - {\left (\frac {\log \relax (5)^{3}}{x^{2}} + \frac {6 \, \log \relax (5)^{2} \log \relax (x)}{x^{2}} + \frac {12 \, \log \relax (5) \log \relax (x)^{2}}{x^{2}} + \frac {8 \, \log \relax (x)^{3}}{x^{2}}\right )} \log \left (\log \relax (x)\right ) + \frac {\log \left (\log \relax (x)\right )^{4}}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.57, size = 4833, normalized size = 193.32
method | result | size |
risch | \(\text {Expression too large to display}\) | \(4833\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 214, normalized size = 8.56 \begin {gather*} \frac {\log \left (\frac {1}{5 \, x}\right )^{4}}{4 \, x^{2}} + \frac {\log \left (\frac {1}{5 \, x}\right )^{3}}{2 \, x^{2}} - \frac {3 \, \log \left (\frac {1}{5 \, x}\right )^{2}}{4 \, x^{2}} + \frac {4 \, {\left (14 \, \log \relax (5) + 1\right )} \log \relax (x)^{3} + 30 \, \log \relax (x)^{4} - 8 \, {\left (\log \relax (5) + 2 \, \log \relax (x)\right )} \log \left (\log \relax (x)\right )^{3} + 2 \, \log \left (\log \relax (x)\right )^{4} + 4 \, \log \relax (5)^{3} + 6 \, {\left (6 \, \log \relax (5)^{2} + 2 \, \log \relax (5) + 1\right )} \log \relax (x)^{2} + 12 \, {\left (\log \relax (5)^{2} + 4 \, \log \relax (5) \log \relax (x) + 4 \, \log \relax (x)^{2}\right )} \log \left (\log \relax (x)\right )^{2} + 6 \, \log \relax (5)^{2} + 2 \, {\left (4 \, \log \relax (5)^{3} + 6 \, \log \relax (5)^{2} + 6 \, \log \relax (5) + 3\right )} \log \relax (x) - 8 \, {\left (\log \relax (5)^{3} + 6 \, \log \relax (5)^{2} \log \relax (x) + 12 \, \log \relax (5) \log \relax (x)^{2} + 8 \, \log \relax (x)^{3}\right )} \log \left (\log \relax (x)\right ) + 6 \, \log \relax (5) + 3}{8 \, x^{2}} + \frac {3 \, \log \left (\frac {1}{5 \, x}\right )}{4 \, x^{2}} - \frac {3}{8 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 240, normalized size = 9.60 \begin {gather*} \frac {{\ln \relax (5)}^4}{4\,x^2}+\frac {{\ln \left (\frac {\ln \relax (x)}{x}\right )}^4}{4\,x^2}+\frac {{\ln \left (\frac {1}{x}\right )}^4}{4\,x^2}-\frac {\ln \left (\frac {1}{x}\right )\,{\ln \relax (5)}^3}{x^2}-\frac {{\ln \left (\frac {1}{x}\right )}^3\,\ln \relax (5)}{x^2}+\frac {\ln \left (\frac {1}{x}\right )\,{\ln \left (\frac {\ln \relax (x)}{x}\right )}^3}{x^2}+\frac {{\ln \left (\frac {1}{x}\right )}^3\,\ln \left (\frac {\ln \relax (x)}{x}\right )}{x^2}-\frac {\ln \relax (5)\,{\ln \left (\frac {\ln \relax (x)}{x}\right )}^3}{x^2}-\frac {{\ln \relax (5)}^3\,\ln \left (\frac {\ln \relax (x)}{x}\right )}{x^2}+\frac {3\,{\ln \left (\frac {1}{x}\right )}^2\,{\ln \relax (5)}^2}{2\,x^2}+\frac {3\,{\ln \left (\frac {1}{x}\right )}^2\,{\ln \left (\frac {\ln \relax (x)}{x}\right )}^2}{2\,x^2}+\frac {3\,{\ln \relax (5)}^2\,{\ln \left (\frac {\ln \relax (x)}{x}\right )}^2}{2\,x^2}-\frac {3\,\ln \left (\frac {1}{x}\right )\,\ln \relax (5)\,{\ln \left (\frac {\ln \relax (x)}{x}\right )}^2}{x^2}+\frac {3\,\ln \left (\frac {1}{x}\right )\,{\ln \relax (5)}^2\,\ln \left (\frac {\ln \relax (x)}{x}\right )}{x^2}-\frac {3\,{\ln \left (\frac {1}{x}\right )}^2\,\ln \relax (5)\,\ln \left (\frac {\ln \relax (x)}{x}\right )}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 20.07, size = 172, normalized size = 6.88 \begin {gather*} \frac {\left (- \log {\relax (x )} - \log {\relax (5 )}\right ) \log {\left (\frac {\log {\relax (x )}}{x} \right )}^{3}}{x^{2}} + \frac {\left (3 \log {\relax (x )}^{2} + 6 \log {\relax (5 )} \log {\relax (x )} + 3 \log {\relax (5 )}^{2}\right ) \log {\left (\frac {\log {\relax (x )}}{x} \right )}^{2}}{2 x^{2}} + \frac {\left (- \log {\relax (x )}^{3} - 3 \log {\relax (5 )} \log {\relax (x )}^{2} - 3 \log {\relax (5 )}^{2} \log {\relax (x )} - \log {\relax (5 )}^{3}\right ) \log {\left (\frac {\log {\relax (x )}}{x} \right )}}{x^{2}} + \frac {\log {\relax (x )}^{4}}{4 x^{2}} + \frac {\log {\relax (5 )} \log {\relax (x )}^{3}}{x^{2}} + \frac {3 \log {\relax (5 )}^{2} \log {\relax (x )}^{2}}{2 x^{2}} + \frac {\log {\relax (5 )}^{3} \log {\relax (x )}}{x^{2}} + \frac {\log {\left (\frac {\log {\relax (x )}}{x} \right )}^{4}}{4 x^{2}} + \frac {\log {\relax (5 )}^{4}}{4 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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