Integrand size = 158, antiderivative size = 25 \[ \int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx=(4+x) \left (2+2 x+\log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )\right ) \]
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\[ \int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx=\int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-16+76 x+52 x^2-22 x^4-8 x^5-\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )-\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{x \left (4+x-x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx \\ & = \int \left (-\frac {76}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {16}{x \left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}-\frac {52 x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {22 x^3}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {8 x^4}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {2 (5+2 x) \log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )}+\log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )\right ) \, dx \\ & = 2 \int \frac {(5+2 x) \log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+8 \int \frac {x^4}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+16 \int \frac {1}{x \left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+22 \int \frac {x^3}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-52 \int \frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-76 \int \frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+\int \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right ) \, dx \\ & = 2 \int \left (\frac {5 \log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )}+\frac {2 x \log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )}\right ) \, dx+8 \int \left (\frac {x}{2+\log \left (\frac {4+x-x^3}{x}\right )}+\frac {x (4+x)}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}\right ) \, dx+16 \int \left (-\frac {1}{4 x \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {-1+x^2}{4 \left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}\right ) \, dx+22 \int \left (\frac {1}{2+\log \left (\frac {4+x-x^3}{x}\right )}+\frac {4+x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}\right ) \, dx-52 \int \frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-76 \int \frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+\int \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right ) \, dx \\ & = -\left (4 \int \frac {1}{x \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx\right )+4 \int \frac {-1+x^2}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+4 \int \frac {x \log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+8 \int \frac {x}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+8 \int \frac {x (4+x)}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+10 \int \frac {\log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+22 \int \frac {1}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+22 \int \frac {4+x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-52 \int \frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-76 \int \frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+\int \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right ) \, dx \\ & = -\left (4 \int \frac {1}{x \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx\right )+4 \int \frac {x \log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+4 \int \left (-\frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {x^2}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}\right ) \, dx+8 \int \frac {x}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+8 \int \left (\frac {4 x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {x^2}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}\right ) \, dx+10 \int \frac {\log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+22 \int \frac {1}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+22 \int \left (\frac {4}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}+\frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}\right ) \, dx-52 \int \frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-76 \int \frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+\int \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right ) \, dx \\ & = -\left (4 \int \frac {1}{x \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx\right )-4 \int \frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+4 \int \frac {x^2}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+4 \int \frac {x \log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+8 \int \frac {x}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+8 \int \frac {x^2}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+10 \int \frac {\log \left (1+\frac {4}{x}-x^2\right )}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+22 \int \frac {1}{2+\log \left (\frac {4+x-x^3}{x}\right )} \, dx+22 \int \frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+32 \int \frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-52 \int \frac {x}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx-76 \int \frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+88 \int \frac {1}{\left (-4-x+x^3\right ) \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )} \, dx+\int \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right ) \, dx \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx=10 x+2 x^2+4 \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )+x \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right ) \]
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Time = 1.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96
method | result | size |
parallelrisch | \(-39+2 x^{2}+\ln \left (\ln \left (-\frac {x^{3}-x -4}{x}\right )+2\right ) x +4 \ln \left (\ln \left (-\frac {x^{3}-x -4}{x}\right )+2\right )+10 x\) | \(49\) |
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Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx=2 \, x^{2} + {\left (x + 4\right )} \log \left (\log \left (-\frac {x^{3} - x - 4}{x}\right ) + 2\right ) + 10 \, x \]
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Time = 0.40 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx=2 x^{2} + x \log {\left (\log {\left (\frac {- x^{3} + x + 4}{x} \right )} + 2 \right )} + 10 x + 4 \log {\left (\log {\left (\frac {- x^{3} + x + 4}{x} \right )} + 2 \right )} \]
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Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx=2 \, x^{2} + {\left (x + 4\right )} \log \left (\log \left (-x^{3} + x + 4\right ) - \log \left (x\right ) + 2\right ) + 10 \, x \]
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Time = 0.38 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx=2 \, x^{2} + x \log \left (\log \left (-\frac {x^{3} - x - 4}{x}\right ) + 2\right ) + 10 \, x + 4 \, \log \left (\log \left (-x^{3} + x + 4\right ) - \log \left (x\right ) + 2\right ) \]
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Time = 8.63 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx=10\,x+4\,\ln \left (\ln \left (\frac {-x^3+x+4}{x}\right )+2\right )+x\,\ln \left (\ln \left (\frac {-x^3+x+4}{x}\right )+2\right )+2\,x^2 \]
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