Integrand size = 112, antiderivative size = 33 \[ \int \frac {-36 x+42 x^2-12 x^3+e^2 \left (-18+24 x-6 x^2\right )+\left (-72 x+90 x^2-24 x^3\right ) \log (x)}{e^4 \left (9 x^2-12 x^3+4 x^4\right )+e^2 \left (36 x^3-48 x^4+16 x^5\right ) \log (x)+\left (36 x^4-48 x^5+16 x^6\right ) \log ^2(x)} \, dx=\frac {\frac {1}{x}+\frac {3-x}{3 x-2 x^2}}{e^2+2 x \log (x)} \]
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\[ \int \frac {-36 x+42 x^2-12 x^3+e^2 \left (-18+24 x-6 x^2\right )+\left (-72 x+90 x^2-24 x^3\right ) \log (x)}{e^4 \left (9 x^2-12 x^3+4 x^4\right )+e^2 \left (36 x^3-48 x^4+16 x^5\right ) \log (x)+\left (36 x^4-48 x^5+16 x^6\right ) \log ^2(x)} \, dx=\int \frac {-36 x+42 x^2-12 x^3+e^2 \left (-18+24 x-6 x^2\right )+\left (-72 x+90 x^2-24 x^3\right ) \log (x)}{e^4 \left (9 x^2-12 x^3+4 x^4\right )+e^2 \left (36 x^3-48 x^4+16 x^5\right ) \log (x)+\left (36 x^4-48 x^5+16 x^6\right ) \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {6 \left (-e^2 \left (3-4 x+x^2\right )-x \left (6-7 x+2 x^2\right )-x \left (12-15 x+4 x^2\right ) \log (x)\right )}{(3-2 x)^2 x^2 \left (e^2+2 x \log (x)\right )^2} \, dx \\ & = 6 \int \frac {-e^2 \left (3-4 x+x^2\right )-x \left (6-7 x+2 x^2\right )-x \left (12-15 x+4 x^2\right ) \log (x)}{(3-2 x)^2 x^2 \left (e^2+2 x \log (x)\right )^2} \, dx \\ & = 6 \int \left (\frac {\left (e^2-2 x\right ) (-2+x)}{2 x^2 (-3+2 x) \left (e^2+2 x \log (x)\right )^2}+\frac {-12+15 x-4 x^2}{2 x^2 (-3+2 x)^2 \left (e^2+2 x \log (x)\right )}\right ) \, dx \\ & = 3 \int \frac {\left (e^2-2 x\right ) (-2+x)}{x^2 (-3+2 x) \left (e^2+2 x \log (x)\right )^2} \, dx+3 \int \frac {-12+15 x-4 x^2}{x^2 (-3+2 x)^2 \left (e^2+2 x \log (x)\right )} \, dx \\ & = 3 \int \left (\frac {2 e^2}{3 x^2 \left (e^2+2 x \log (x)\right )^2}+\frac {-12+e^2}{9 x \left (e^2+2 x \log (x)\right )^2}-\frac {2 \left (-3+e^2\right )}{9 (-3+2 x) \left (e^2+2 x \log (x)\right )^2}\right ) \, dx+3 \int \left (-\frac {4}{3 x^2 \left (e^2+2 x \log (x)\right )}-\frac {1}{9 x \left (e^2+2 x \log (x)\right )}+\frac {2}{3 (-3+2 x)^2 \left (e^2+2 x \log (x)\right )}+\frac {2}{9 (-3+2 x) \left (e^2+2 x \log (x)\right )}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {1}{x \left (e^2+2 x \log (x)\right )} \, dx\right )+\frac {2}{3} \int \frac {1}{(-3+2 x) \left (e^2+2 x \log (x)\right )} \, dx+2 \int \frac {1}{(-3+2 x)^2 \left (e^2+2 x \log (x)\right )} \, dx-4 \int \frac {1}{x^2 \left (e^2+2 x \log (x)\right )} \, dx+\left (2 e^2\right ) \int \frac {1}{x^2 \left (e^2+2 x \log (x)\right )^2} \, dx+\frac {1}{3} \left (2 \left (3-e^2\right )\right ) \int \frac {1}{(-3+2 x) \left (e^2+2 x \log (x)\right )^2} \, dx+\frac {1}{3} \left (-12+e^2\right ) \int \frac {1}{x \left (e^2+2 x \log (x)\right )^2} \, dx \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {-36 x+42 x^2-12 x^3+e^2 \left (-18+24 x-6 x^2\right )+\left (-72 x+90 x^2-24 x^3\right ) \log (x)}{e^4 \left (9 x^2-12 x^3+4 x^4\right )+e^2 \left (36 x^3-48 x^4+16 x^5\right ) \log (x)+\left (36 x^4-48 x^5+16 x^6\right ) \log ^2(x)} \, dx=-\frac {3 (2-x)}{x (-3+2 x) \left (e^2+2 x \log (x)\right )} \]
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Time = 3.68 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {-6+3 x}{x \left (-3+2 x \right ) \left (2 x \ln \left (x \right )+{\mathrm e}^{2}\right )}\) | \(26\) |
norman | \(\frac {-6+3 x}{x \left (-3+2 x \right ) \left (2 x \ln \left (x \right )+{\mathrm e}^{2}\right )}\) | \(27\) |
default | \(-\frac {6 \left (1-\frac {x}{2}\right )}{x \left (-3+2 x \right ) \left (2 x \ln \left (x \right )+{\mathrm e}^{2}\right )}\) | \(28\) |
parallelrisch | \(\frac {12 x -24}{4 x \left (4 x^{2} \ln \left (x \right )+2 \,{\mathrm e}^{2} x -6 x \ln \left (x \right )-3 \,{\mathrm e}^{2}\right )}\) | \(35\) |
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Time = 0.33 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {-36 x+42 x^2-12 x^3+e^2 \left (-18+24 x-6 x^2\right )+\left (-72 x+90 x^2-24 x^3\right ) \log (x)}{e^4 \left (9 x^2-12 x^3+4 x^4\right )+e^2 \left (36 x^3-48 x^4+16 x^5\right ) \log (x)+\left (36 x^4-48 x^5+16 x^6\right ) \log ^2(x)} \, dx=\frac {3 \, {\left (x - 2\right )}}{{\left (2 \, x^{2} - 3 \, x\right )} e^{2} + 2 \, {\left (2 \, x^{3} - 3 \, x^{2}\right )} \log \left (x\right )} \]
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Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {-36 x+42 x^2-12 x^3+e^2 \left (-18+24 x-6 x^2\right )+\left (-72 x+90 x^2-24 x^3\right ) \log (x)}{e^4 \left (9 x^2-12 x^3+4 x^4\right )+e^2 \left (36 x^3-48 x^4+16 x^5\right ) \log (x)+\left (36 x^4-48 x^5+16 x^6\right ) \log ^2(x)} \, dx=\frac {3 x - 6}{2 x^{2} e^{2} - 3 x e^{2} + \left (4 x^{3} - 6 x^{2}\right ) \log {\left (x \right )}} \]
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Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {-36 x+42 x^2-12 x^3+e^2 \left (-18+24 x-6 x^2\right )+\left (-72 x+90 x^2-24 x^3\right ) \log (x)}{e^4 \left (9 x^2-12 x^3+4 x^4\right )+e^2 \left (36 x^3-48 x^4+16 x^5\right ) \log (x)+\left (36 x^4-48 x^5+16 x^6\right ) \log ^2(x)} \, dx=\frac {3 \, {\left (x - 2\right )}}{2 \, x^{2} e^{2} - 3 \, x e^{2} + 2 \, {\left (2 \, x^{3} - 3 \, x^{2}\right )} \log \left (x\right )} \]
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Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {-36 x+42 x^2-12 x^3+e^2 \left (-18+24 x-6 x^2\right )+\left (-72 x+90 x^2-24 x^3\right ) \log (x)}{e^4 \left (9 x^2-12 x^3+4 x^4\right )+e^2 \left (36 x^3-48 x^4+16 x^5\right ) \log (x)+\left (36 x^4-48 x^5+16 x^6\right ) \log ^2(x)} \, dx=\frac {3 \, {\left (x - 2\right )}}{4 \, x^{3} \log \left (x\right ) + 2 \, x^{2} e^{2} - 6 \, x^{2} \log \left (x\right ) - 3 \, x e^{2}} \]
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Time = 17.96 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {-36 x+42 x^2-12 x^3+e^2 \left (-18+24 x-6 x^2\right )+\left (-72 x+90 x^2-24 x^3\right ) \log (x)}{e^4 \left (9 x^2-12 x^3+4 x^4\right )+e^2 \left (36 x^3-48 x^4+16 x^5\right ) \log (x)+\left (36 x^4-48 x^5+16 x^6\right ) \log ^2(x)} \, dx=\frac {3\,\left (x-2\right )}{x\,\left (2\,x-3\right )\,\left ({\mathrm {e}}^2+2\,x\,\ln \left (x\right )\right )} \]
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