Integrand size = 63, antiderivative size = 25 \[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=-4+2 x-\frac {2 x^5}{9 (2-x) \log ^2(2 x)} \]
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\[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=\int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{9 (-2+x)^2 \log ^3(2 x)} \, dx \\ & = \frac {1}{9} \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{(-2+x)^2 \log ^3(2 x)} \, dx \\ & = \frac {1}{9} \int \left (18-\frac {4 x^4}{(-2+x) \log ^3(2 x)}+\frac {4 x^4 (-5+2 x)}{(-2+x)^2 \log ^2(2 x)}\right ) \, dx \\ & = 2 x-\frac {4}{9} \int \frac {x^4}{(-2+x) \log ^3(2 x)} \, dx+\frac {4}{9} \int \frac {x^4 (-5+2 x)}{(-2+x)^2 \log ^2(2 x)} \, dx \\ \end{align*}
Time = 2.79 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=2 x+\frac {2 x^5}{9 (-2+x) \log ^2(2 x)} \]
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Time = 491.87 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84
method | result | size |
risch | \(2 x +\frac {2 x^{5}}{9 \left (-2+x \right ) \ln \left (2 x \right )^{2}}\) | \(21\) |
norman | \(\frac {-8 \ln \left (2 x \right )^{2}+\frac {2 x^{5}}{9}+2 x^{2} \ln \left (2 x \right )^{2}}{\left (-2+x \right ) \ln \left (2 x \right )^{2}}\) | \(38\) |
parallelrisch | \(\frac {2 x^{5}+18 x^{2} \ln \left (2 x \right )^{2}-72 \ln \left (2 x \right )^{2}}{9 \ln \left (2 x \right )^{2} \left (-2+x \right )}\) | \(39\) |
derivativedivides | \(2 x +\frac {32}{9 \ln \left (2 x \right )^{2}}+\frac {16 x}{9 \ln \left (2 x \right )^{2}}+\frac {4 x^{3}}{9 \ln \left (2 x \right )^{2}}+\frac {8 x^{2}}{9 \ln \left (2 x \right )^{2}}+\frac {2 x^{4}}{9 \ln \left (2 x \right )^{2}}+\frac {128}{9 \left (2 x -4\right ) \ln \left (2 x \right )^{2}}\) | \(70\) |
default | \(2 x +\frac {32}{9 \ln \left (2 x \right )^{2}}+\frac {16 x}{9 \ln \left (2 x \right )^{2}}+\frac {4 x^{3}}{9 \ln \left (2 x \right )^{2}}+\frac {8 x^{2}}{9 \ln \left (2 x \right )^{2}}+\frac {2 x^{4}}{9 \ln \left (2 x \right )^{2}}+\frac {128}{9 \left (2 x -4\right ) \ln \left (2 x \right )^{2}}\) | \(70\) |
parts | \(2 x +\frac {32}{9 \left (\ln \left (2\right )+\ln \left (x \right )\right )^{2}}+\frac {64}{9 \left (-2+x \right ) \left (\ln \left (2\right )+\ln \left (x \right )\right )^{2}}+\frac {2 x^{4}}{9 \left (\ln \left (2\right )+\ln \left (x \right )\right )^{2}}+\frac {16 x}{9 \left (\ln \left (2\right )+\ln \left (x \right )\right )^{2}}+\frac {8 x^{2}}{9 \left (\ln \left (2\right )+\ln \left (x \right )\right )^{2}}+\frac {4 x^{3}}{9 \left (\ln \left (2\right )+\ln \left (x \right )\right )^{2}}\) | \(74\) |
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Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=\frac {2 \, {\left (x^{5} + 9 \, {\left (x^{2} - 2 \, x\right )} \log \left (2 \, x\right )^{2}\right )}}{9 \, {\left (x - 2\right )} \log \left (2 \, x\right )^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=\frac {2 x^{5}}{\left (9 x - 18\right ) \log {\left (2 x \right )}^{2}} + 2 x \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (21) = 42\).
Time = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.48 \[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=\frac {2 \, {\left (x^{5} + 9 \, x^{2} \log \left (2\right )^{2} - 18 \, x \log \left (2\right )^{2} + 9 \, {\left (x^{2} - 2 \, x\right )} \log \left (x\right )^{2} + 18 \, {\left (x^{2} \log \left (2\right ) - 2 \, x \log \left (2\right )\right )} \log \left (x\right )\right )}}{9 \, {\left (x \log \left (2\right )^{2} + {\left (x - 2\right )} \log \left (x\right )^{2} - 2 \, \log \left (2\right )^{2} + 2 \, {\left (x \log \left (2\right ) - 2 \, \log \left (2\right )\right )} \log \left (x\right )\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=\frac {2 \, x^{5}}{9 \, {\left (x \log \left (2 \, x\right )^{2} - 2 \, \log \left (2 \, x\right )^{2}\right )}} + 2 \, x \]
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Time = 9.65 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=\frac {2\,x^5}{9\,{\ln \left (2\,x\right )}^2\,\left (x-2\right )}+\frac {2\,x\,\left (9\,x-18\right )}{9\,\left (x-2\right )} \]
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