Integrand size = 75, antiderivative size = 17 \[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\frac {2}{3} x^2 \left (x+\frac {1}{\left (x^2+\log (x)\right )^2}\right ) \]
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\[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x \left (-2-2 x^2+3 x^7+\left (2+9 x^5\right ) \log (x)+9 x^3 \log ^2(x)+3 x \log ^3(x)\right )}{3 \left (x^2+\log (x)\right )^3} \, dx \\ & = \frac {2}{3} \int \frac {x \left (-2-2 x^2+3 x^7+\left (2+9 x^5\right ) \log (x)+9 x^3 \log ^2(x)+3 x \log ^3(x)\right )}{\left (x^2+\log (x)\right )^3} \, dx \\ & = \frac {2}{3} \int \left (3 x^2+\frac {2 x \left (-1-2 x^2\right )}{\left (x^2+\log (x)\right )^3}+\frac {2 x}{\left (x^2+\log (x)\right )^2}\right ) \, dx \\ & = \frac {2 x^3}{3}+\frac {4}{3} \int \frac {x \left (-1-2 x^2\right )}{\left (x^2+\log (x)\right )^3} \, dx+\frac {4}{3} \int \frac {x}{\left (x^2+\log (x)\right )^2} \, dx \\ & = \frac {2 x^3}{3}+\frac {4}{3} \int \frac {x}{\left (x^2+\log (x)\right )^2} \, dx+\frac {4}{3} \int \left (-\frac {x}{\left (x^2+\log (x)\right )^3}-\frac {2 x^3}{\left (x^2+\log (x)\right )^3}\right ) \, dx \\ & = \frac {2 x^3}{3}-\frac {4}{3} \int \frac {x}{\left (x^2+\log (x)\right )^3} \, dx+\frac {4}{3} \int \frac {x}{\left (x^2+\log (x)\right )^2} \, dx-\frac {8}{3} \int \frac {x^3}{\left (x^2+\log (x)\right )^3} \, dx \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\frac {2}{3} \left (x^3+\frac {x^2}{\left (x^2+\log (x)\right )^2}\right ) \]
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Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18
method | result | size |
default | \(\frac {2 x^{3}}{3}+\frac {2 x^{2}}{3 \left (\ln \left (x \right )+x^{2}\right )^{2}}\) | \(20\) |
risch | \(\frac {2 x^{3}}{3}+\frac {2 x^{2}}{3 \left (\ln \left (x \right )+x^{2}\right )^{2}}\) | \(20\) |
parallelrisch | \(\frac {2 x^{7}+4 x^{5} \ln \left (x \right )+2 x^{3} \ln \left (x \right )^{2}+2 x^{2}}{3 x^{4}+6 x^{2} \ln \left (x \right )+3 \ln \left (x \right )^{2}}\) | \(47\) |
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (15) = 30\).
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.41 \[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\frac {2 \, {\left (x^{7} + 2 \, x^{5} \log \left (x\right ) + x^{3} \log \left (x\right )^{2} + x^{2}\right )}}{3 \, {\left (x^{4} + 2 \, x^{2} \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\frac {2 x^{3}}{3} + \frac {2 x^{2}}{3 x^{4} + 6 x^{2} \log {\left (x \right )} + 3 \log {\left (x \right )}^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (15) = 30\).
Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.41 \[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\frac {2 \, {\left (x^{7} + 2 \, x^{5} \log \left (x\right ) + x^{3} \log \left (x\right )^{2} + x^{2}\right )}}{3 \, {\left (x^{4} + 2 \, x^{2} \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (15) = 30\).
Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 3.24 \[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\frac {2}{3} \, x^{3} + \frac {2 \, {\left (2 \, x^{4} + x^{2}\right )}}{3 \, {\left (2 \, x^{6} + 4 \, x^{4} \log \left (x\right ) + x^{4} + 2 \, x^{2} \log \left (x\right )^{2} + 2 \, x^{2} \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \]
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Time = 8.73 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\frac {2\,x^2}{3\,{\left (\ln \left (x\right )+x^2\right )}^2}+\frac {2\,x^3}{3} \]
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