Integrand size = 43, antiderivative size = 25 \[ \int \frac {-10 x^2+\left (10 \log (x)-10 \log ^2(x)\right ) \log \left (\frac {x^2}{16}\right )}{3 x^3 \log \left (\frac {x^2}{16}\right )} \, dx=\frac {5}{3} \left (15+\frac {\log ^2(x)}{x^2}-\log \left (\log \left (\frac {x^2}{16}\right )\right )\right ) \]
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Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {12, 6820, 14, 2341, 2413, 2340, 2339, 29} \[ \int \frac {-10 x^2+\left (10 \log (x)-10 \log ^2(x)\right ) \log \left (\frac {x^2}{16}\right )}{3 x^3 \log \left (\frac {x^2}{16}\right )} \, dx=-\frac {5 (1-\log (x)) \log (x)}{3 x^2}+\frac {5 \log (x)}{3 x^2}-\frac {5}{3} \log \left (\log \left (\frac {x^2}{16}\right )\right ) \]
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Rule 12
Rule 14
Rule 29
Rule 2339
Rule 2340
Rule 2341
Rule 2413
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {-10 x^2+\left (10 \log (x)-10 \log ^2(x)\right ) \log \left (\frac {x^2}{16}\right )}{x^3 \log \left (\frac {x^2}{16}\right )} \, dx \\ & = \frac {1}{3} \int \frac {10 \left (\log (x)-\log ^2(x)-\frac {x^2}{\log \left (\frac {x^2}{16}\right )}\right )}{x^3} \, dx \\ & = \frac {10}{3} \int \frac {\log (x)-\log ^2(x)-\frac {x^2}{\log \left (\frac {x^2}{16}\right )}}{x^3} \, dx \\ & = \frac {10}{3} \int \left (-\frac {(-1+\log (x)) \log (x)}{x^3}-\frac {1}{x \log \left (\frac {x^2}{16}\right )}\right ) \, dx \\ & = -\left (\frac {10}{3} \int \frac {(-1+\log (x)) \log (x)}{x^3} \, dx\right )-\frac {10}{3} \int \frac {1}{x \log \left (\frac {x^2}{16}\right )} \, dx \\ & = \frac {5 \log (x)}{6 x^2}-\frac {5 (1-\log (x)) \log (x)}{3 x^2}-\frac {5}{3} \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {x^2}{16}\right )\right )+\frac {10}{3} \int \frac {1-2 \log (x)}{4 x^3} \, dx \\ & = \frac {5 \log (x)}{6 x^2}-\frac {5 (1-\log (x)) \log (x)}{3 x^2}-\frac {5}{3} \log \left (\log \left (\frac {x^2}{16}\right )\right )+\frac {5}{6} \int \frac {1-2 \log (x)}{x^3} \, dx \\ & = \frac {5 \log (x)}{3 x^2}-\frac {5 (1-\log (x)) \log (x)}{3 x^2}-\frac {5}{3} \log \left (\log \left (\frac {x^2}{16}\right )\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-10 x^2+\left (10 \log (x)-10 \log ^2(x)\right ) \log \left (\frac {x^2}{16}\right )}{3 x^3 \log \left (\frac {x^2}{16}\right )} \, dx=\frac {5 \log ^2(x)}{3 x^2}-\frac {5}{3} \log \left (\log \left (\frac {x^2}{16}\right )\right ) \]
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Time = 0.16 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80
method | result | size |
parts | \(\frac {5 \ln \left (x \right )^{2}}{3 x^{2}}-\frac {5 \ln \left (\ln \left (\frac {x^{2}}{16}\right )\right )}{3}\) | \(20\) |
default | \(\frac {5 \ln \left (x \right )^{2}}{3 x^{2}}-\frac {5 \ln \left (-4 \ln \left (2\right )+\ln \left (x^{2}\right )\right )}{3}\) | \(23\) |
parallelrisch | \(-\frac {20 \ln \left (\ln \left (\frac {x^{2}}{16}\right )\right ) x^{2}-20 \ln \left (x \right )^{2}}{12 x^{2}}\) | \(25\) |
risch | \(\frac {5 \ln \left (x \right )^{2}}{3 x^{2}}-\frac {5 \ln \left (-\frac {i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{4}+\frac {i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}}{4}-2 \ln \left (2\right )+\ln \left (x \right )\right )}{3}\) | \(70\) |
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Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-10 x^2+\left (10 \log (x)-10 \log ^2(x)\right ) \log \left (\frac {x^2}{16}\right )}{3 x^3 \log \left (\frac {x^2}{16}\right )} \, dx=-\frac {5 \, {\left (x^{2} \log \left (-4 \, \log \left (2\right ) + 2 \, \log \left (x\right )\right ) - \log \left (x\right )^{2}\right )}}{3 \, x^{2}} \]
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Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-10 x^2+\left (10 \log (x)-10 \log ^2(x)\right ) \log \left (\frac {x^2}{16}\right )}{3 x^3 \log \left (\frac {x^2}{16}\right )} \, dx=- \frac {5 \log {\left (\log {\left (x \right )} - 2 \log {\left (2 \right )} \right )}}{3} + \frac {5 \log {\left (x \right )}^{2}}{3 x^{2}} \]
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Time = 0.18 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {-10 x^2+\left (10 \log (x)-10 \log ^2(x)\right ) \log \left (\frac {x^2}{16}\right )}{3 x^3 \log \left (\frac {x^2}{16}\right )} \, dx=\frac {5 \, {\left (2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )}}{6 \, x^{2}} - \frac {5 \, \log \left (x\right )}{3 \, x^{2}} - \frac {5}{6 \, x^{2}} - \frac {5}{3} \, \log \left (\log \left (\frac {1}{16} \, x^{2}\right )\right ) \]
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Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-10 x^2+\left (10 \log (x)-10 \log ^2(x)\right ) \log \left (\frac {x^2}{16}\right )}{3 x^3 \log \left (\frac {x^2}{16}\right )} \, dx=\frac {5 \, \log \left (x\right )^{2}}{3 \, x^{2}} - \frac {5}{3} \, \log \left (-4 \, \log \left (2\right ) + 2 \, \log \left (x\right )\right ) \]
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Time = 10.95 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {-10 x^2+\left (10 \log (x)-10 \log ^2(x)\right ) \log \left (\frac {x^2}{16}\right )}{3 x^3 \log \left (\frac {x^2}{16}\right )} \, dx=\frac {5\,{\ln \left (x\right )}^2}{3\,x^2}-\frac {5\,\ln \left (\ln \left (\frac {x^2}{16}\right )\right )}{3} \]
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