Integrand size = 54, antiderivative size = 28 \[ \int \frac {4-4 \log (x)+(2-2 \log (x)) \log \left (\frac {4 x \log (4)-\log (4) \log (x)}{3 x}\right )}{-4 x^2 \log ^8(5)+x \log ^8(5) \log (x)} \, dx=\frac {\left (2+\log \left (\frac {\log (4) \left (x+\frac {1}{3} (x-\log (x))\right )}{x}\right )\right )^2}{\log ^8(5)} \]
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Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2641, 6873, 12, 6818} \[ \int \frac {4-4 \log (x)+(2-2 \log (x)) \log \left (\frac {4 x \log (4)-\log (4) \log (x)}{3 x}\right )}{-4 x^2 \log ^8(5)+x \log ^8(5) \log (x)} \, dx=\frac {\left (\log \left (\frac {\log (4) (4 x-\log (x))}{3 x}\right )+2\right )^2}{\log ^8(5)} \]
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Rule 12
Rule 2641
Rule 6818
Rule 6873
Rubi steps \begin{align*} \text {integral}& = \int \frac {4-4 \log (x)+(2-2 \log (x)) \log \left (\frac {4 x \log (4)-\log (4) \log (x)}{3 x}\right )}{x \left (-4 x \log ^8(5)+\log ^8(5) \log (x)\right )} \, dx \\ & = \int \frac {2 (1-\log (x)) \left (-2-\log \left (\frac {\log (4) (4 x-\log (x))}{3 x}\right )\right )}{x \log ^8(5) (4 x-\log (x))} \, dx \\ & = \frac {2 \int \frac {(1-\log (x)) \left (-2-\log \left (\frac {\log (4) (4 x-\log (x))}{3 x}\right )\right )}{x (4 x-\log (x))} \, dx}{\log ^8(5)} \\ & = \frac {\left (2+\log \left (\frac {\log (4) (4 x-\log (x))}{3 x}\right )\right )^2}{\log ^8(5)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {4-4 \log (x)+(2-2 \log (x)) \log \left (\frac {4 x \log (4)-\log (4) \log (x)}{3 x}\right )}{-4 x^2 \log ^8(5)+x \log ^8(5) \log (x)} \, dx=\frac {\left (2+\log \left (\frac {\log (4) (4 x-\log (x))}{3 x}\right )\right )^2}{\log ^8(5)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(25)=50\).
Time = 0.80 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00
method | result | size |
norman | \(\frac {\frac {4 \ln \left (\frac {-2 \ln \left (2\right ) \ln \left (x \right )+8 x \ln \left (2\right )}{3 x}\right )}{\ln \left (5\right )}+\frac {\ln \left (\frac {-2 \ln \left (2\right ) \ln \left (x \right )+8 x \ln \left (2\right )}{3 x}\right )^{2}}{\ln \left (5\right )}}{\ln \left (5\right )^{7}}\) | \(56\) |
default | \(\frac {-4 \ln \left (x \right )+4 \ln \left (-4 x +\ln \left (x \right )\right )+\ln \left (\frac {4 x -\ln \left (x \right )}{x}\right )^{2}-2 \ln \left (\ln \left (2\right )\right ) \left (\ln \left (x \right )-\ln \left (-4 x +\ln \left (x \right )\right )\right )}{\ln \left (5\right )^{8}}-\frac {2 \ln \left (2\right ) \left (\ln \left (x \right )-\ln \left (-4 x +\ln \left (x \right )\right )\right )}{\ln \left (5\right )^{8}}+\frac {2 \ln \left (3\right ) \left (\ln \left (x \right )-\ln \left (-4 x +\ln \left (x \right )\right )\right )}{\ln \left (5\right )^{8}}\) | \(96\) |
risch | \(\frac {\ln \left (x -\frac {\ln \left (x \right )}{4}\right )^{2}}{\ln \left (5\right )^{8}}-\frac {2 \ln \left (x \right ) \ln \left (x -\frac {\ln \left (x \right )}{4}\right )}{\ln \left (5\right )^{8}}+\frac {i \pi \ln \left (-4 x +\ln \left (x \right )\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (-x +\frac {\ln \left (x \right )}{4}\right )}{x}\right )^{2}}{\ln \left (5\right )^{8}}+\frac {i \pi \ln \left (-4 x +\ln \left (x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-x +\frac {\ln \left (x \right )}{4}\right )}{x}\right )^{3}}{\ln \left (5\right )^{8}}-\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (-x +\frac {\ln \left (x \right )}{4}\right )}{x}\right )^{2}}{\ln \left (5\right )^{8}}+\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (-x +\frac {\ln \left (x \right )}{4}\right )\right ) \operatorname {csgn}\left (\frac {i \left (-x +\frac {\ln \left (x \right )}{4}\right )}{x}\right )}{\ln \left (5\right )^{8}}-\frac {i \pi \ln \left (-4 x +\ln \left (x \right )\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (-x +\frac {\ln \left (x \right )}{4}\right )\right ) \operatorname {csgn}\left (\frac {i \left (-x +\frac {\ln \left (x \right )}{4}\right )}{x}\right )}{\ln \left (5\right )^{8}}+\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (i \left (-x +\frac {\ln \left (x \right )}{4}\right )\right ) \operatorname {csgn}\left (\frac {i \left (-x +\frac {\ln \left (x \right )}{4}\right )}{x}\right )^{2}}{\ln \left (5\right )^{8}}-\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i \left (-x +\frac {\ln \left (x \right )}{4}\right )}{x}\right )^{3}}{\ln \left (5\right )^{8}}-\frac {i \pi \ln \left (-4 x +\ln \left (x \right )\right ) \operatorname {csgn}\left (i \left (-x +\frac {\ln \left (x \right )}{4}\right )\right ) \operatorname {csgn}\left (\frac {i \left (-x +\frac {\ln \left (x \right )}{4}\right )}{x}\right )^{2}}{\ln \left (5\right )^{8}}-\frac {2 \ln \left (x \right ) \ln \left (\ln \left (2\right )\right )}{\ln \left (5\right )^{8}}+\frac {2 \ln \left (\ln \left (2\right )\right ) \ln \left (-4 x +\ln \left (x \right )\right )}{\ln \left (5\right )^{8}}+\frac {\ln \left (x \right )^{2}}{\ln \left (5\right )^{8}}-\frac {6 \ln \left (2\right ) \ln \left (x \right )}{\ln \left (5\right )^{8}}+\frac {2 \ln \left (3\right ) \ln \left (x \right )}{\ln \left (5\right )^{8}}+\frac {6 \ln \left (-4 x +\ln \left (x \right )\right ) \ln \left (2\right )}{\ln \left (5\right )^{8}}-\frac {2 \ln \left (-4 x +\ln \left (x \right )\right ) \ln \left (3\right )}{\ln \left (5\right )^{8}}-\frac {4 \ln \left (x \right )}{\ln \left (5\right )^{8}}+\frac {4 \ln \left (-4 x +\ln \left (x \right )\right )}{\ln \left (5\right )^{8}}\) | \(446\) |
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none
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {4-4 \log (x)+(2-2 \log (x)) \log \left (\frac {4 x \log (4)-\log (4) \log (x)}{3 x}\right )}{-4 x^2 \log ^8(5)+x \log ^8(5) \log (x)} \, dx=\frac {\log \left (\frac {2 \, {\left (4 \, x \log \left (2\right ) - \log \left (2\right ) \log \left (x\right )\right )}}{3 \, x}\right )^{2} + 4 \, \log \left (\frac {2 \, {\left (4 \, x \log \left (2\right ) - \log \left (2\right ) \log \left (x\right )\right )}}{3 \, x}\right )}{\log \left (5\right )^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).
Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {4-4 \log (x)+(2-2 \log (x)) \log \left (\frac {4 x \log (4)-\log (4) \log (x)}{3 x}\right )}{-4 x^2 \log ^8(5)+x \log ^8(5) \log (x)} \, dx=- \frac {4 \log {\left (x \right )}}{\log {\left (5 \right )}^{8}} + \frac {\log {\left (\frac {\frac {8 x \log {\left (2 \right )}}{3} - \frac {2 \log {\left (2 \right )} \log {\left (x \right )}}{3}}{x} \right )}^{2}}{\log {\left (5 \right )}^{8}} + \frac {4 \log {\left (- 4 x + \log {\left (x \right )} \right )}}{\log {\left (5 \right )}^{8}} \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.46 \[ \int \frac {4-4 \log (x)+(2-2 \log (x)) \log \left (\frac {4 x \log (4)-\log (4) \log (x)}{3 x}\right )}{-4 x^2 \log ^8(5)+x \log ^8(5) \log (x)} \, dx=-\frac {2 \, {\left (i \, \pi - \log \left (3\right ) + \log \left (2\right ) + \log \left (\log \left (2\right )\right ) + 2\right )} \log \left (x\right ) - \log \left (x\right )^{2} + 2 \, {\left (-i \, \pi + \log \left (3\right ) - \log \left (2\right ) + \log \left (x\right ) - \log \left (\log \left (2\right )\right ) - 2\right )} \log \left (-4 \, x + \log \left (x\right )\right ) - \log \left (-4 \, x + \log \left (x\right )\right )^{2}}{\log \left (5\right )^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.14 \[ \int \frac {4-4 \log (x)+(2-2 \log (x)) \log \left (\frac {4 x \log (4)-\log (4) \log (x)}{3 x}\right )}{-4 x^2 \log ^8(5)+x \log ^8(5) \log (x)} \, dx=\frac {\log \left (4 \, x \log \left (2\right ) - \log \left (2\right ) \log \left (x\right )\right )^{2}}{\log \left (5\right )^{8}} + \frac {2 \, {\left (\log \left (3\right ) - \log \left (2\right ) - 2\right )} \log \left (x\right )}{\log \left (5\right )^{8}} - \frac {2 \, \log \left (4 \, x \log \left (2\right ) - \log \left (2\right ) \log \left (x\right )\right ) \log \left (x\right )}{\log \left (5\right )^{8}} + \frac {\log \left (x\right )^{2}}{\log \left (5\right )^{8}} - \frac {2 \, {\left (\log \left (3\right ) - \log \left (2\right ) - 2\right )} \log \left (-4 \, x + \log \left (x\right )\right )}{\log \left (5\right )^{8}} \]
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Time = 11.44 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {4-4 \log (x)+(2-2 \log (x)) \log \left (\frac {4 x \log (4)-\log (4) \log (x)}{3 x}\right )}{-4 x^2 \log ^8(5)+x \log ^8(5) \log (x)} \, dx=\frac {{\ln \left (\frac {\frac {8\,x\,\ln \left (2\right )}{3}-\frac {2\,\ln \left (2\right )\,\ln \left (x\right )}{3}}{x}\right )}^2+4\,\ln \left (\ln \left (x\right )-4\,x\right )-4\,\ln \left (x\right )}{{\ln \left (5\right )}^8} \]
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