Integrand size = 69, antiderivative size = 26 \[ \int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx=\log \left (\frac {1}{50} \left (-1-e^{e^{-4-e^{x^4}+x}}+x\right )^2\right ) \]
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\[ \int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx=\int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 e^{-4-e^{x^4}} \left (e^{4+e^{x^4}}-e^{e^{-4-e^{x^4}+x}+x}\right )}{1+e^{e^{-4-e^{x^4}+x}}-x}-\frac {8 e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x+x^4} x^3}{1+e^{e^{-4-e^{x^4}+x}}-x}\right ) \, dx \\ & = -\left (2 \int \frac {e^{-4-e^{x^4}} \left (e^{4+e^{x^4}}-e^{e^{-4-e^{x^4}+x}+x}\right )}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx\right )-8 \int \frac {e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x+x^4} x^3}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx \\ & = -\left (2 \int \frac {1-e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x}}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx\right )-8 \int \frac {e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x+x^4} x^3}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx \\ & = -\left (2 \int \left (\frac {1}{1+e^{e^{-4-e^{x^4}+x}}-x}-\frac {e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x}}{1+e^{e^{-4-e^{x^4}+x}}-x}\right ) \, dx\right )-8 \int \frac {e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x+x^4} x^3}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx \\ & = -\left (2 \int \frac {1}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx\right )+2 \int \frac {e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x}}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx-8 \int \frac {e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x+x^4} x^3}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx \\ \end{align*}
Time = 1.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx=2 \log \left (1+e^{e^{-4-e^{x^4}+x}}-x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77
method | result | size |
risch | \(2 \ln \left ({\mathrm e}^{{\mathrm e}^{-{\mathrm e}^{x^{4}}+x -4}}-x +1\right )\) | \(20\) |
parallelrisch | \(2 \ln \left (x -{\mathrm e}^{{\mathrm e}^{-{\mathrm e}^{x^{4}}+x -4}}-1\right )\) | \(20\) |
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx=2 \, \log \left (-x + e^{\left (e^{\left (x - e^{\left (x^{4}\right )} - 4\right )}\right )} + 1\right ) \]
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Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx=2 \log {\left (- x + e^{e^{x - e^{x^{4}} - 4}} + 1 \right )} \]
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Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx=2 \, \log \left (-x + e^{\left (e^{\left (x - e^{\left (x^{4}\right )} - 4\right )}\right )} + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (21) = 42\).
Time = 0.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx=-2 \, x + 2 \, e^{\left (x^{4}\right )} + 2 \, \log \left (-x e^{\left (x - e^{\left (x^{4}\right )}\right )} + e^{\left (x - e^{\left (x^{4}\right )} + e^{\left (x - e^{\left (x^{4}\right )} - 4\right )}\right )} + e^{\left (x - e^{\left (x^{4}\right )}\right )}\right ) \]
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Time = 8.63 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx=2\,\ln \left ({\mathrm {e}}^{{\mathrm {e}}^{-{\mathrm {e}}^{x^4}}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^x}-x+1\right ) \]
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