Integrand size = 53, antiderivative size = 23 \[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\frac {1}{21} \left (25+e^{1+e^{-4 e^{-2 x} x} \log (x)}\right ) \]
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\[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\int \frac {\exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {1}{21} \int \frac {\exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{x} \, dx \\ & = \frac {1}{21} \int \left (\frac {\exp \left (1-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right )}{x}+4 \exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) (-1+2 x) \log (x)\right ) \, dx \\ & = \frac {1}{21} \int \frac {\exp \left (1-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right )}{x} \, dx+\frac {4}{21} \int \exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) (-1+2 x) \log (x) \, dx \\ & = \frac {1}{21} \int \frac {\exp \left (1-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right )}{x} \, dx+\frac {4}{21} \int \left (-\exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) \log (x)+2 \exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) x \log (x)\right ) \, dx \\ & = \frac {1}{21} \int \frac {\exp \left (1-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right )}{x} \, dx-\frac {4}{21} \int \exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) \log (x) \, dx+\frac {8}{21} \int \exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) x \log (x) \, dx \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\frac {1}{21} e x^{e^{-4 e^{-2 x} x}} \]
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Time = 1.52 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65
method | result | size |
risch | \(\frac {x^{{\mathrm e}^{-4 \,{\mathrm e}^{-2 x} x}} {\mathrm e}}{21}\) | \(15\) |
parallelrisch | \(\frac {{\mathrm e}^{\ln \left (x \right ) {\mathrm e}^{-4 \,{\mathrm e}^{-2 x} x}+1}}{21}\) | \(17\) |
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Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (18) = 36\).
Time = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.17 \[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\frac {1}{21} \, e^{\left (-{\left ({\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - e^{\left (-4 \, x e^{\left (-2 \, x\right )} + 2 \, x\right )} \log \left (x\right ) + 4 \, x\right )} e^{\left (-2 \, x\right )} + 4 \, x e^{\left (-2 \, x\right )} + 2 \, x\right )} \]
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Time = 0.84 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\frac {e^{1 + e^{- 4 x e^{- 2 x}} \log {\left (x \right )}}}{21} \]
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Time = 0.35 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\frac {1}{21} \, e^{\left (e^{\left (-4 \, x e^{\left (-2 \, x\right )}\right )} \log \left (x\right ) + 1\right )} \]
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Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\frac {1}{21} \, x^{e^{\left (-4 \, x e^{\left (-2 \, x\right )}\right )}} e \]
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Time = 11.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\frac {x^{{\mathrm {e}}^{-4\,x\,{\mathrm {e}}^{-2\,x}}}\,\mathrm {e}}{21} \]
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