Integrand size = 151, antiderivative size = 16 \[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx=(5+\log (x)) \log \left (\log \left (x-(2+x)^4\right )\right ) \]
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\[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx=\int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{x \left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx \\ & = \int \left (\frac {155}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {240 x}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {120 x^2}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {20 x^3}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {\left (31+48 x+24 x^2+4 x^3\right ) \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {\log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{x}\right ) \, dx \\ & = 20 \int \frac {x^3}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+120 \int \frac {x^2}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+155 \int \frac {1}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+240 \int \frac {x}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+\int \frac {\left (31+48 x+24 x^2+4 x^3\right ) \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+\int \frac {\log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{x} \, dx \\ & = 20 \int \frac {x^3}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+120 \int \frac {x^2}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+155 \int \frac {1}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+240 \int \frac {x}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+\int \left (\frac {31 \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {48 x \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {24 x^2 \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {4 x^3 \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}\right ) \, dx+\int \frac {\log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{x} \, dx \\ & = 4 \int \frac {x^3 \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+20 \int \frac {x^3}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+24 \int \frac {x^2 \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+31 \int \frac {\log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+48 \int \frac {x \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+120 \int \frac {x^2}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+155 \int \frac {1}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+240 \int \frac {x}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+\int \frac {\log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{x} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(50\) vs. \(2(16)=32\).
Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 3.12 \[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx=5 \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )+\log (x) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(16)=32\).
Time = 15.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 3.19
method | result | size |
risch | \(\ln \left (x \right ) \ln \left (\ln \left (-x^{4}-8 x^{3}-24 x^{2}-31 x -16\right )\right )+5 \ln \left (\ln \left (-x^{4}-8 x^{3}-24 x^{2}-31 x -16\right )\right )\) | \(51\) |
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx={\left (\log \left (x\right ) + 5\right )} \log \left (\log \left (-x^{4} - 8 \, x^{3} - 24 \, x^{2} - 31 \, x - 16\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (14) = 28\).
Time = 0.62 (sec) , antiderivative size = 51, normalized size of antiderivative = 3.19 \[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx=\log {\left (x \right )} \log {\left (\log {\left (- x^{4} - 8 x^{3} - 24 x^{2} - 31 x - 16 \right )} \right )} + 5 \log {\left (\log {\left (- x^{4} - 8 x^{3} - 24 x^{2} - 31 x - 16 \right )} \right )} \]
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Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx={\left (\log \left (x\right ) + 5\right )} \log \left (\log \left (-x^{4} - 8 \, x^{3} - 24 \, x^{2} - 31 \, x - 16\right )\right ) \]
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Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 52, normalized size of antiderivative = 3.25 \[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx=\log \left (x\right ) \log \left (\log \left (-x^{4} - 8 \, x^{3} - 24 \, x^{2} - 31 \, x - 16\right )\right ) + 5 \, \log \left (\pi - i \, \log \left (x^{4} + 8 \, x^{3} + 24 \, x^{2} + 31 \, x + 16\right )\right ) \]
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Time = 9.71 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx=\ln \left (\ln \left (-x^4-8\,x^3-24\,x^2-31\,x-16\right )\right )\,\left (\ln \left (x\right )+5\right ) \]
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