Integrand size = 37, antiderivative size = 25 \[ \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx=-\frac {e^{1-e^{x^2} x}}{-1+e^{x^2} x} \]
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\[ \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx=\int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{1-e^{x^2} x+2 x^2} x \left (1+2 x^2\right )}{\left (1-e^{x^2} x\right )^2} \, dx \\ & = \int \left (\frac {e^{1-e^{x^2} x+2 x^2} x}{\left (-1+e^{x^2} x\right )^2}+\frac {2 e^{1-e^{x^2} x+2 x^2} x^3}{\left (-1+e^{x^2} x\right )^2}\right ) \, dx \\ & = 2 \int \frac {e^{1-e^{x^2} x+2 x^2} x^3}{\left (-1+e^{x^2} x\right )^2} \, dx+\int \frac {e^{1-e^{x^2} x+2 x^2} x}{\left (-1+e^{x^2} x\right )^2} \, dx \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx=-\frac {e^{1-e^{x^2} x}}{-1+e^{x^2} x} \]
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Time = 0.14 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
method | result | size |
risch | \(-\frac {{\mathrm e}^{1-{\mathrm e}^{x^{2}} x}}{-1+{\mathrm e}^{x^{2}} x}\) | \(23\) |
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Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx=-\frac {e^{\left (2 \, x^{2} - x e^{\left (x^{2}\right )} + 1\right )}}{x e^{\left (3 \, x^{2}\right )} - e^{\left (2 \, x^{2}\right )}} \]
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Time = 0.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx=- \frac {e^{2 x^{2} - x e^{x^{2}} + 1}}{x e^{3 x^{2}} - e^{2 x^{2}}} \]
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Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx=-\frac {e^{\left (-x e^{\left (x^{2}\right )} + 1\right )}}{x e^{\left (x^{2}\right )} - 1} \]
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Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx=-\frac {e^{\left (2 \, x^{2} - x e^{\left (x^{2}\right )} + 1\right )}}{x e^{\left (3 \, x^{2}\right )} - e^{\left (2 \, x^{2}\right )}} \]
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Timed out. \[ \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx=\int \frac {{\mathrm {e}}^{2\,x^2-x\,{\mathrm {e}}^{x^2}+1}\,\left (2\,x^3+x\right )}{{\left (x\,{\mathrm {e}}^{x^2}-1\right )}^2} \,d x \]
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