Integrand size = 172, antiderivative size = 33 \[ \int \frac {1-2 x+x^2+e^3 \left (-1+2 x-x^2\right )+\left (-x+x^2+e^3 \left (x-x^2\right )\right ) \log (x)+\left (-4+e^3 (4-2 x)+2 x+\left (-2+e^3 (2-2 x)+2 x\right ) \log (x)\right ) \log (2+\log (2))+\left (-x+e^3 x+\left (-2+2 e^3\right ) \log (2+\log (2))\right ) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )}{-x^2-x^2 \log (x)+(-2 x-2 x \log (x)) \log (2+\log (2))+(x+2 \log (2+\log (2))) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )} \, dx=\left (1-e^3\right ) \left (-x+\log \left (x+x \log (x)-\log \left (\frac {x}{2}+\log (2+\log (2))\right )\right )\right ) \]
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Time = 0.38 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6820, 12, 6874, 6816} \[ \int \frac {1-2 x+x^2+e^3 \left (-1+2 x-x^2\right )+\left (-x+x^2+e^3 \left (x-x^2\right )\right ) \log (x)+\left (-4+e^3 (4-2 x)+2 x+\left (-2+e^3 (2-2 x)+2 x\right ) \log (x)\right ) \log (2+\log (2))+\left (-x+e^3 x+\left (-2+2 e^3\right ) \log (2+\log (2))\right ) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )}{-x^2-x^2 \log (x)+(-2 x-2 x \log (x)) \log (2+\log (2))+(x+2 \log (2+\log (2))) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )} \, dx=\left (1-e^3\right ) \log \left (x+x \log (x)-\log \left (\frac {x}{2}+\log (2+\log (2))\right )\right )-\left (1-e^3\right ) x \]
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Rule 12
Rule 6816
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (1-e^3\right ) \left (-1-x^2+2 x (1-\log (2+\log (2)))+4 \log (2+\log (2))-(-1+x) \log (x) (x+2 \log (2+\log (2)))+(x+2 \log (2+\log (2))) \log \left (\frac {x}{2}+\log (2+\log (2))\right )\right )}{(x+2 \log (2+\log (2))) \left (x+x \log (x)-\log \left (\frac {x}{2}+\log (2+\log (2))\right )\right )} \, dx \\ & = \left (1-e^3\right ) \int \frac {-1-x^2+2 x (1-\log (2+\log (2)))+4 \log (2+\log (2))-(-1+x) \log (x) (x+2 \log (2+\log (2)))+(x+2 \log (2+\log (2))) \log \left (\frac {x}{2}+\log (2+\log (2))\right )}{(x+2 \log (2+\log (2))) \left (x+x \log (x)-\log \left (\frac {x}{2}+\log (2+\log (2))\right )\right )} \, dx \\ & = \left (1-e^3\right ) \int \left (-1+\frac {-1+2 x+x \log (x)+4 \log (2+\log (2))+2 \log (x) \log (2+\log (2))}{(x+2 \log (2+\log (2))) \left (x+x \log (x)-\log \left (\frac {x}{2}+\log (2+\log (2))\right )\right )}\right ) \, dx \\ & = -\left (\left (1-e^3\right ) x\right )+\left (1-e^3\right ) \int \frac {-1+2 x+x \log (x)+4 \log (2+\log (2))+2 \log (x) \log (2+\log (2))}{(x+2 \log (2+\log (2))) \left (x+x \log (x)-\log \left (\frac {x}{2}+\log (2+\log (2))\right )\right )} \, dx \\ & = -\left (\left (1-e^3\right ) x\right )+\left (1-e^3\right ) \log \left (x+x \log (x)-\log \left (\frac {x}{2}+\log (2+\log (2))\right )\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {1-2 x+x^2+e^3 \left (-1+2 x-x^2\right )+\left (-x+x^2+e^3 \left (x-x^2\right )\right ) \log (x)+\left (-4+e^3 (4-2 x)+2 x+\left (-2+e^3 (2-2 x)+2 x\right ) \log (x)\right ) \log (2+\log (2))+\left (-x+e^3 x+\left (-2+2 e^3\right ) \log (2+\log (2))\right ) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )}{-x^2-x^2 \log (x)+(-2 x-2 x \log (x)) \log (2+\log (2))+(x+2 \log (2+\log (2))) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )} \, dx=\left (-1+e^3\right ) \left (x-\log \left (x+x \log (x)-\log \left (\frac {x}{2}+\log (2+\log (2))\right )\right )\right ) \]
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Time = 3.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\left ({\mathrm e}^{3}-1\right ) x +\left (-{\mathrm e}^{3}+1\right ) \ln \left (x +x \ln \left (x \right )-\ln \left (\ln \left (\ln \left (2\right )+2\right )+\frac {x}{2}\right )\right )\) | \(34\) |
default | \(\left ({\mathrm e}^{3}-1\right ) x +\left (-{\mathrm e}^{3}+1\right ) \ln \left (x \ln \left (x \right )+\ln \left (2\right )+x -\ln \left (2 \ln \left (\ln \left (2\right )+2\right )+x \right )\right )\) | \(36\) |
risch | \(-\ln \left (\ln \left (\ln \left (\ln \left (2\right )+2\right )+\frac {x}{2}\right )-\left (\ln \left (x \right )+1\right ) x \right ) {\mathrm e}^{3}+x \,{\mathrm e}^{3}+\ln \left (\ln \left (\ln \left (\ln \left (2\right )+2\right )+\frac {x}{2}\right )-\left (\ln \left (x \right )+1\right ) x \right )-x\) | \(51\) |
parallelrisch | \(-{\mathrm e}^{3} \ln \left (x +x \ln \left (x \right )-\ln \left (\ln \left (\ln \left (2\right )+2\right )+\frac {x}{2}\right )\right )-4 \ln \left (\ln \left (2\right )+2\right ) {\mathrm e}^{3}+x \,{\mathrm e}^{3}+\ln \left (x +x \ln \left (x \right )-\ln \left (\ln \left (\ln \left (2\right )+2\right )+\frac {x}{2}\right )\right )+4 \ln \left (\ln \left (2\right )+2\right )-x\) | \(67\) |
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {1-2 x+x^2+e^3 \left (-1+2 x-x^2\right )+\left (-x+x^2+e^3 \left (x-x^2\right )\right ) \log (x)+\left (-4+e^3 (4-2 x)+2 x+\left (-2+e^3 (2-2 x)+2 x\right ) \log (x)\right ) \log (2+\log (2))+\left (-x+e^3 x+\left (-2+2 e^3\right ) \log (2+\log (2))\right ) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )}{-x^2-x^2 \log (x)+(-2 x-2 x \log (x)) \log (2+\log (2))+(x+2 \log (2+\log (2))) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )} \, dx=x e^{3} - {\left (e^{3} - 1\right )} \log \left (-x \log \left (x\right ) - x + \log \left (\frac {1}{2} \, x + \log \left (\log \left (2\right ) + 2\right )\right )\right ) - x \]
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Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {1-2 x+x^2+e^3 \left (-1+2 x-x^2\right )+\left (-x+x^2+e^3 \left (x-x^2\right )\right ) \log (x)+\left (-4+e^3 (4-2 x)+2 x+\left (-2+e^3 (2-2 x)+2 x\right ) \log (x)\right ) \log (2+\log (2))+\left (-x+e^3 x+\left (-2+2 e^3\right ) \log (2+\log (2))\right ) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )}{-x^2-x^2 \log (x)+(-2 x-2 x \log (x)) \log (2+\log (2))+(x+2 \log (2+\log (2))) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )} \, dx=x \left (-1 + e^{3}\right ) - \left (-1 + e\right ) \left (1 + e + e^{2}\right ) \log {\left (- x \log {\left (x \right )} - x + \log {\left (\frac {x}{2} + \log {\left (\log {\left (2 \right )} + 2 \right )} \right )} \right )} \]
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Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {1-2 x+x^2+e^3 \left (-1+2 x-x^2\right )+\left (-x+x^2+e^3 \left (x-x^2\right )\right ) \log (x)+\left (-4+e^3 (4-2 x)+2 x+\left (-2+e^3 (2-2 x)+2 x\right ) \log (x)\right ) \log (2+\log (2))+\left (-x+e^3 x+\left (-2+2 e^3\right ) \log (2+\log (2))\right ) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )}{-x^2-x^2 \log (x)+(-2 x-2 x \log (x)) \log (2+\log (2))+(x+2 \log (2+\log (2))) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )} \, dx=x {\left (e^{3} - 1\right )} - {\left (e^{3} - 1\right )} \log \left (-x \log \left (x\right ) - x - \log \left (2\right ) + \log \left (x + 2 \, \log \left (\log \left (2\right ) + 2\right )\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (28) = 56\).
Time = 0.32 (sec) , antiderivative size = 153, normalized size of antiderivative = 4.64 \[ \int \frac {1-2 x+x^2+e^3 \left (-1+2 x-x^2\right )+\left (-x+x^2+e^3 \left (x-x^2\right )\right ) \log (x)+\left (-4+e^3 (4-2 x)+2 x+\left (-2+e^3 (2-2 x)+2 x\right ) \log (x)\right ) \log (2+\log (2))+\left (-x+e^3 x+\left (-2+2 e^3\right ) \log (2+\log (2))\right ) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )}{-x^2-x^2 \log (x)+(-2 x-2 x \log (x)) \log (2+\log (2))+(x+2 \log (2+\log (2))) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )} \, dx={\left (x + 2 \, \log \left (\log \left (2\right ) + 2\right )\right )} e^{3} - e^{3} \log \left (-{\left (x + 2 \, \log \left (\log \left (2\right ) + 2\right )\right )} \log \left (2\right ) - {\left (x + 2 \, \log \left (\log \left (2\right ) + 2\right )\right )} \log \left (\frac {1}{2} \, x\right ) + 2 \, \log \left (2\right ) \log \left (\log \left (2\right ) + 2\right ) + 2 \, \log \left (\frac {1}{2} \, x\right ) \log \left (\log \left (2\right ) + 2\right ) - x + \log \left (\frac {1}{2} \, x + \log \left (\log \left (2\right ) + 2\right )\right )\right ) - x + \log \left (-{\left (x + 2 \, \log \left (\log \left (2\right ) + 2\right )\right )} \log \left (2\right ) - {\left (x + 2 \, \log \left (\log \left (2\right ) + 2\right )\right )} \log \left (\frac {1}{2} \, x\right ) + 2 \, \log \left (2\right ) \log \left (\log \left (2\right ) + 2\right ) + 2 \, \log \left (\frac {1}{2} \, x\right ) \log \left (\log \left (2\right ) + 2\right ) - x + \log \left (\frac {1}{2} \, x + \log \left (\log \left (2\right ) + 2\right )\right )\right ) - 2 \, \log \left (\log \left (2\right ) + 2\right ) \]
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Timed out. \[ \int \frac {1-2 x+x^2+e^3 \left (-1+2 x-x^2\right )+\left (-x+x^2+e^3 \left (x-x^2\right )\right ) \log (x)+\left (-4+e^3 (4-2 x)+2 x+\left (-2+e^3 (2-2 x)+2 x\right ) \log (x)\right ) \log (2+\log (2))+\left (-x+e^3 x+\left (-2+2 e^3\right ) \log (2+\log (2))\right ) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )}{-x^2-x^2 \log (x)+(-2 x-2 x \log (x)) \log (2+\log (2))+(x+2 \log (2+\log (2))) \log \left (\frac {1}{2} (x+2 \log (2+\log (2)))\right )} \, dx=\int -\frac {\ln \left (x\right )\,\left ({\mathrm {e}}^3\,\left (x-x^2\right )-x+x^2\right )-2\,x-\ln \left (\ln \left (2\right )+2\right )\,\left (\ln \left (x\right )\,\left ({\mathrm {e}}^3\,\left (2\,x-2\right )-2\,x+2\right )-2\,x+{\mathrm {e}}^3\,\left (2\,x-4\right )+4\right )+\ln \left (\frac {x}{2}+\ln \left (\ln \left (2\right )+2\right )\right )\,\left (x\,{\mathrm {e}}^3-x+\ln \left (\ln \left (2\right )+2\right )\,\left (2\,{\mathrm {e}}^3-2\right )\right )-{\mathrm {e}}^3\,\left (x^2-2\,x+1\right )+x^2+1}{x^2\,\ln \left (x\right )+\ln \left (\ln \left (2\right )+2\right )\,\left (2\,x+2\,x\,\ln \left (x\right )\right )-\ln \left (\frac {x}{2}+\ln \left (\ln \left (2\right )+2\right )\right )\,\left (x+2\,\ln \left (\ln \left (2\right )+2\right )\right )+x^2} \,d x \]
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