Integrand size = 53, antiderivative size = 27 \[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=-2-\frac {4 x \left (\frac {\log (x)}{x}+x^2 \log ^2(x)\right )}{3-x} \]
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Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89, number of steps used = 24, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.283, Rules used = {1608, 27, 6874, 36, 31, 29, 2404, 2332, 2351, 2353, 2352, 2341, 2333, 2355, 2342} \[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=4 x^2 \log ^2(x)-\frac {36 x \log ^2(x)}{3-x}+12 x \log ^2(x)-\frac {4 x \log (x)}{3 (3-x)}-\frac {4 \log (x)}{3} \]
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Rule 27
Rule 29
Rule 31
Rule 36
Rule 1608
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2351
Rule 2352
Rule 2353
Rule 2355
Rule 2404
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{x \left (9-6 x+x^2\right )} \, dx \\ & = \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{(-3+x)^2 x} \, dx \\ & = \int \left (\frac {4}{(-3+x) x}+\frac {4 \left (-1-6 x^2+2 x^3\right ) \log (x)}{(-3+x)^2}+\frac {4 x^2 (-9+2 x) \log ^2(x)}{(-3+x)^2}\right ) \, dx \\ & = 4 \int \frac {1}{(-3+x) x} \, dx+4 \int \frac {\left (-1-6 x^2+2 x^3\right ) \log (x)}{(-3+x)^2} \, dx+4 \int \frac {x^2 (-9+2 x) \log ^2(x)}{(-3+x)^2} \, dx \\ & = \frac {4}{3} \int \frac {1}{-3+x} \, dx-\frac {4}{3} \int \frac {1}{x} \, dx+4 \int \left (6 \log (x)-\frac {\log (x)}{(-3+x)^2}+\frac {18 \log (x)}{-3+x}+2 x \log (x)\right ) \, dx+4 \int \left (3 \log ^2(x)-\frac {27 \log ^2(x)}{(-3+x)^2}+2 x \log ^2(x)\right ) \, dx \\ & = \frac {4}{3} \log (3-x)-\frac {4 \log (x)}{3}-4 \int \frac {\log (x)}{(-3+x)^2} \, dx+8 \int x \log (x) \, dx+8 \int x \log ^2(x) \, dx+12 \int \log ^2(x) \, dx+24 \int \log (x) \, dx+72 \int \frac {\log (x)}{-3+x} \, dx-108 \int \frac {\log ^2(x)}{(-3+x)^2} \, dx \\ & = -24 x-2 x^2+\frac {4}{3} \log (3-x)+72 \log (3) \log (-3+x)-\frac {4 \log (x)}{3}+24 x \log (x)-\frac {4 x \log (x)}{3 (3-x)}+4 x^2 \log (x)+12 x \log ^2(x)-\frac {36 x \log ^2(x)}{3-x}+4 x^2 \log ^2(x)-\frac {4}{3} \int \frac {1}{-3+x} \, dx-8 \int x \log (x) \, dx-24 \int \log (x) \, dx+72 \int \frac {\log \left (\frac {x}{3}\right )}{-3+x} \, dx-72 \int \frac {\log (x)}{-3+x} \, dx \\ & = -\frac {4 \log (x)}{3}-\frac {4 x \log (x)}{3 (3-x)}+12 x \log ^2(x)-\frac {36 x \log ^2(x)}{3-x}+4 x^2 \log ^2(x)-72 \operatorname {PolyLog}\left (2,1-\frac {x}{3}\right )-72 \int \frac {\log \left (\frac {x}{3}\right )}{-3+x} \, dx \\ & = -\frac {4 \log (x)}{3}-\frac {4 x \log (x)}{3 (3-x)}+12 x \log ^2(x)-\frac {36 x \log ^2(x)}{3-x}+4 x^2 \log ^2(x) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=\frac {4}{3} \left (-\log (3-x)+\log (-3+x)+\frac {3 \log (x) \left (1+x^3 \log (x)\right )}{-3+x}\right ) \]
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Time = 0.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78
method | result | size |
norman | \(\frac {4 \ln \left (x \right )+4 x^{3} \ln \left (x \right )^{2}}{-3+x}\) | \(21\) |
parallelrisch | \(\frac {4 \ln \left (x \right )+4 x^{3} \ln \left (x \right )^{2}}{-3+x}\) | \(21\) |
risch | \(\frac {4 x^{3} \ln \left (x \right )^{2}}{-3+x}+\frac {4 \ln \left (x \right )}{-3+x}\) | \(25\) |
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Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=\frac {4 \, {\left (x^{3} \log \left (x\right )^{2} + \log \left (x\right )\right )}}{x - 3} \]
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Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=\frac {4 x^{3} \log {\left (x \right )}^{2}}{x - 3} + \frac {4 \log {\left (x \right )}}{x - 3} \]
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Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=\frac {4 \, x^{3} \log \left (x\right )^{2}}{x - 3} + \frac {4 \, \log \left (x\right )}{x - 3} \]
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\[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=\int { \frac {4 \, {\left ({\left (2 \, x^{4} - 9 \, x^{3}\right )} \log \left (x\right )^{2} + {\left (2 \, x^{4} - 6 \, x^{3} - x\right )} \log \left (x\right ) + x - 3\right )}}{x^{3} - 6 \, x^{2} + 9 \, x} \,d x } \]
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Time = 12.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=\frac {4\,\ln \left (x\right )\,\left (x^3\,\ln \left (x\right )+1\right )}{x-3} \]
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