Integrand size = 34, antiderivative size = 12 \[ \int \frac {4}{(5-20 x+(-1+4 x) \log (5-20 x)) \log (-20+4 \log (5-20 x))} \, dx=\log (\log (4 (-5+\log (5-20 x)))) \]
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Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {12, 6816} \[ \int \frac {4}{(5-20 x+(-1+4 x) \log (5-20 x)) \log (-20+4 \log (5-20 x))} \, dx=\log (\log (4 \log (5-20 x)-20)) \]
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Rule 12
Rule 6816
Rubi steps \begin{align*} \text {integral}& = 4 \int \frac {1}{(5-20 x+(-1+4 x) \log (5-20 x)) \log (-20+4 \log (5-20 x))} \, dx \\ & = \log (\log (-20+4 \log (5-20 x))) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {4}{(5-20 x+(-1+4 x) \log (5-20 x)) \log (-20+4 \log (5-20 x))} \, dx=\log (\log (4 (-5+\log (5-20 x)))) \]
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Time = 12.62 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08
method | result | size |
norman | \(\ln \left (\ln \left (4 \ln \left (-20 x +5\right )-20\right )\right )\) | \(13\) |
risch | \(\ln \left (\ln \left (4 \ln \left (-20 x +5\right )-20\right )\right )\) | \(13\) |
parallelrisch | \(\ln \left (\ln \left (4 \ln \left (-20 x +5\right )-20\right )\right )\) | \(13\) |
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none
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {4}{(5-20 x+(-1+4 x) \log (5-20 x)) \log (-20+4 \log (5-20 x))} \, dx=\log \left (\log \left (4 \, \log \left (-20 \, x + 5\right ) - 20\right )\right ) \]
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Time = 0.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {4}{(5-20 x+(-1+4 x) \log (5-20 x)) \log (-20+4 \log (5-20 x))} \, dx=\log {\left (\log {\left (4 \log {\left (5 - 20 x \right )} - 20 \right )} \right )} \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.67 \[ \int \frac {4}{(5-20 x+(-1+4 x) \log (5-20 x)) \log (-20+4 \log (5-20 x))} \, dx=\log \left (2 \, \log \left (2\right ) + \log \left (i \, \pi + \log \left (5\right ) + \log \left (4 \, x - 1\right ) - 5\right )\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {4}{(5-20 x+(-1+4 x) \log (5-20 x)) \log (-20+4 \log (5-20 x))} \, dx=\log \left (\log \left (4 \, \log \left (-20 \, x + 5\right ) - 20\right )\right ) \]
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Time = 15.43 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {4}{(5-20 x+(-1+4 x) \log (5-20 x)) \log (-20+4 \log (5-20 x))} \, dx=\ln \left (\ln \left (4\,\ln \left (5-20\,x\right )-20\right )\right ) \]
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