Integrand size = 73, antiderivative size = 14 \[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=x^4 \left (1+\frac {x}{(x+\log (x))^2}\right ) \]
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\[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=\int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 \left (x \left (-2+3 x+4 x^2\right )+x (5+12 x) \log (x)+12 x \log ^2(x)+4 \log ^3(x)\right )}{(x+\log (x))^3} \, dx \\ & = \int \left (4 x^3-\frac {2 x^4 (1+x)}{(x+\log (x))^3}+\frac {5 x^4}{(x+\log (x))^2}\right ) \, dx \\ & = x^4-2 \int \frac {x^4 (1+x)}{(x+\log (x))^3} \, dx+5 \int \frac {x^4}{(x+\log (x))^2} \, dx \\ & = x^4-2 \int \left (\frac {x^4}{(x+\log (x))^3}+\frac {x^5}{(x+\log (x))^3}\right ) \, dx+5 \int \frac {x^4}{(x+\log (x))^2} \, dx \\ & = x^4-2 \int \frac {x^4}{(x+\log (x))^3} \, dx-2 \int \frac {x^5}{(x+\log (x))^3} \, dx+5 \int \frac {x^4}{(x+\log (x))^2} \, dx \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=x^4+\frac {x^5}{(x+\log (x))^2} \]
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Time = 0.45 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07
method | result | size |
risch | \(x^{4}+\frac {x^{5}}{\left (x +\ln \left (x \right )\right )^{2}}\) | \(15\) |
default | \(-\frac {-x^{5}-x^{6}-2 x^{5} \ln \left (x \right )-x^{4} \ln \left (x \right )^{2}}{\left (x +\ln \left (x \right )\right )^{2}}\) | \(36\) |
parallelrisch | \(\frac {x^{6}+2 x^{5} \ln \left (x \right )+x^{4} \ln \left (x \right )^{2}+x^{5}}{\ln \left (x \right )^{2}+2 x \ln \left (x \right )+x^{2}}\) | \(39\) |
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Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (14) = 28\).
Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.71 \[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=\frac {x^{6} + 2 \, x^{5} \log \left (x\right ) + x^{4} \log \left (x\right )^{2} + x^{5}}{x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} \]
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Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=\frac {x^{5}}{x^{2} + 2 x \log {\left (x \right )} + \log {\left (x \right )}^{2}} + x^{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (14) = 28\).
Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.71 \[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=\frac {x^{6} + 2 \, x^{5} \log \left (x\right ) + x^{4} \log \left (x\right )^{2} + x^{5}}{x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (14) = 28\).
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 3.07 \[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=x^{4} + \frac {x^{6} + x^{5}}{x^{3} + 2 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} + x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} \]
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Time = 12.92 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86 \[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=x^4-\frac {x^6-x^4\,\left (x^2+x\right )}{{\left (x+\ln \left (x\right )\right )}^2} \]
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