Integrand size = 123, antiderivative size = 29 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=e^{-\frac {-1+e^{e^{-e^{-x} x+x^6}}}{x}+x}+x \]
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\[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=\int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (1+\frac {e^{\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{-e^{-x} x+x^6}} \left (e^x-e^{-e^{-x} x+x^6} x \left (-1+x+6 e^x x^5\right )\right )\right )}{x^2}\right ) \, dx \\ & = x+\int \frac {e^{\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{-e^{-x} x+x^6}} \left (e^x-e^{-e^{-x} x+x^6} x \left (-1+x+6 e^x x^5\right )\right )\right )}{x^2} \, dx \\ & = x+\int \left (\frac {e^{\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x} \left (-1+e^{e^{-e^{-x} x+x^6}}+x^2\right )}{x^2}-\frac {\exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}-e^{-x} x+x^6\right ) \left (-1+x+6 e^x x^5\right )}{x}\right ) \, dx \\ & = x+\int \frac {e^{\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x} \left (-1+e^{e^{-e^{-x} x+x^6}}+x^2\right )}{x^2} \, dx-\int \frac {\exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}-e^{-x} x+x^6\right ) \left (-1+x+6 e^x x^5\right )}{x} \, dx \\ & = x-\int \left (\frac {\exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}-e^{-x} x+x^6\right ) (-1+x)}{x}+6 \exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x-e^{-x} x+x^6\right ) x^4\right ) \, dx+\int \left (\frac {\exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x\right )}{x^2}+\frac {e^{\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x} \left (-1+x^2\right )}{x^2}\right ) \, dx \\ & = x-6 \int \exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x-e^{-x} x+x^6\right ) x^4 \, dx+\int \frac {\exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x\right )}{x^2} \, dx-\int \frac {\exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}-e^{-x} x+x^6\right ) (-1+x)}{x} \, dx+\int \frac {e^{\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x} \left (-1+x^2\right )}{x^2} \, dx \\ & = x-6 \int \exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x-e^{-x} x+x^6\right ) x^4 \, dx+\int \left (e^{\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x}-\frac {e^{\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x}}{x^2}\right ) \, dx-\int \left (\exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}-e^{-x} x+x^6\right )-\frac {\exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}-e^{-x} x+x^6\right )}{x}\right ) \, dx+\int \frac {\exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x\right )}{x^2} \, dx \\ & = x-6 \int \exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x-e^{-x} x+x^6\right ) x^4 \, dx+\int e^{\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x} \, dx-\int \exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}-e^{-x} x+x^6\right ) \, dx-\int \frac {e^{\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x}}{x^2} \, dx+\int \frac {\exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x\right )}{x^2} \, dx+\int \frac {\exp \left (e^{-e^{-x} x+x^6}+\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}-e^{-x} x+x^6\right )}{x} \, dx \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=e^{\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x}+x \]
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Time = 51.42 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07
method | result | size |
risch | \(x +{\mathrm e}^{\frac {-{\mathrm e}^{{\mathrm e}^{x \left (x^{5} {\mathrm e}^{x}-1\right ) {\mathrm e}^{-x}}}+x^{2}+1}{x}}\) | \(31\) |
parallelrisch | \(x +{\mathrm e}^{\frac {-{\mathrm e}^{{\mathrm e}^{\left (x^{6} {\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}}}+x^{2}+1}{x}}\) | \(32\) |
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Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=x + e^{\left (\frac {x^{2} - e^{\left (e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )} + 1}{x}\right )} \]
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Time = 9.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=x + e^{\frac {x^{2} - e^{e^{\left (x^{6} e^{x} - x\right ) e^{- x}}} + 1}{x}} \]
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Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=x + e^{\left (x - \frac {e^{\left (e^{\left (x^{6} - x e^{\left (-x\right )}\right )}\right )}}{x} + \frac {1}{x}\right )} \]
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\[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=\int { \frac {{\left (x^{2} e^{x} + {\left ({\left (x^{2} - 1\right )} e^{x} - {\left ({\left (6 \, x^{6} e^{x} + x^{2} - x\right )} e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )} - e^{x}\right )} e^{\left (e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )}\right )} e^{\left (\frac {x^{2} - e^{\left (e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )} + 1}{x}\right )}\right )} e^{\left (-x\right )}}{x^{2}} \,d x } \]
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Time = 9.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=x+{\mathrm {e}}^{-\frac {{\mathrm {e}}^{{\mathrm {e}}^{x^6}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{-x}}}}{x}}\,{\mathrm {e}}^{1/x}\,{\mathrm {e}}^x \]
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