Integrand size = 74, antiderivative size = 22 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^x}{4 \left (e^{x^2}+(-142-x)^2\right )} \]
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\[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \left (19880+e^{x^2} (1-2 x)+282 x+x^2\right )}{4 \left (e^{x^2}+(142+x)^2\right )^2} \, dx \\ & = \frac {1}{4} \int \frac {e^x \left (19880+e^{x^2} (1-2 x)+282 x+x^2\right )}{\left (e^{x^2}+(142+x)^2\right )^2} \, dx \\ & = \frac {1}{4} \int \left (-\frac {e^x (-1+2 x)}{20164+e^{x^2}+284 x+x^2}+\frac {2 e^x \left (-142+20163 x+284 x^2+x^3\right )}{\left (20164+e^{x^2}+284 x+x^2\right )^2}\right ) \, dx \\ & = -\left (\frac {1}{4} \int \frac {e^x (-1+2 x)}{20164+e^{x^2}+284 x+x^2} \, dx\right )+\frac {1}{2} \int \frac {e^x \left (-142+20163 x+284 x^2+x^3\right )}{\left (20164+e^{x^2}+284 x+x^2\right )^2} \, dx \\ & = -\left (\frac {1}{4} \int \left (-\frac {e^x}{20164+e^{x^2}+284 x+x^2}+\frac {2 e^x x}{20164+e^{x^2}+284 x+x^2}\right ) \, dx\right )+\frac {1}{2} \int \left (-\frac {142 e^x}{\left (20164+e^{x^2}+284 x+x^2\right )^2}+\frac {20163 e^x x}{\left (20164+e^{x^2}+284 x+x^2\right )^2}+\frac {284 e^x x^2}{\left (20164+e^{x^2}+284 x+x^2\right )^2}+\frac {e^x x^3}{\left (20164+e^{x^2}+284 x+x^2\right )^2}\right ) \, dx \\ & = \frac {1}{4} \int \frac {e^x}{20164+e^{x^2}+284 x+x^2} \, dx+\frac {1}{2} \int \frac {e^x x^3}{\left (20164+e^{x^2}+284 x+x^2\right )^2} \, dx-\frac {1}{2} \int \frac {e^x x}{20164+e^{x^2}+284 x+x^2} \, dx-71 \int \frac {e^x}{\left (20164+e^{x^2}+284 x+x^2\right )^2} \, dx+142 \int \frac {e^x x^2}{\left (20164+e^{x^2}+284 x+x^2\right )^2} \, dx+\frac {20163}{2} \int \frac {e^x x}{\left (20164+e^{x^2}+284 x+x^2\right )^2} \, dx \\ \end{align*}
Time = 2.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^x}{4 \left (20164+e^{x^2}+284 x+x^2\right )} \]
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Time = 0.14 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
norman | \(\frac {{\mathrm e}^{x}}{4 x^{2}+4 \,{\mathrm e}^{x^{2}}+1136 x +80656}\) | \(19\) |
risch | \(\frac {{\mathrm e}^{x}}{4 x^{2}+4 \,{\mathrm e}^{x^{2}}+1136 x +80656}\) | \(19\) |
parallelrisch | \(\frac {{\mathrm e}^{x}}{4 x^{2}+4 \,{\mathrm e}^{x^{2}}+1136 x +80656}\) | \(19\) |
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^{\left (x^{2} + x\right )}}{4 \, {\left ({\left (x^{2} + 284 \, x + 20164\right )} e^{\left (x^{2}\right )} + e^{\left (2 \, x^{2}\right )}\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^{x}}{4 x^{2} + 1136 x + 4 e^{x^{2}} + 80656} \]
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Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^{x}}{4 \, {\left (x^{2} + 284 \, x + e^{\left (x^{2}\right )} + 20164\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\frac {e^{x}}{4 \, {\left (x^{2} + 284 \, x + e^{\left (x^{2}\right )} + 20164\right )}} \]
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Timed out. \[ \int \frac {e^{x+x^2} (1-2 x)+e^x \left (19880+282 x+x^2\right )}{1626347584+4 e^{2 x^2}+45812608 x+483936 x^2+2272 x^3+4 x^4+e^{x^2} \left (161312+2272 x+8 x^2\right )} \, dx=\int \frac {{\mathrm {e}}^x\,\left (x^2+282\,x+19880\right )-{\mathrm {e}}^{x^2+x}\,\left (2\,x-1\right )}{45812608\,x+4\,{\mathrm {e}}^{2\,x^2}+{\mathrm {e}}^{x^2}\,\left (8\,x^2+2272\,x+161312\right )+483936\,x^2+2272\,x^3+4\,x^4+1626347584} \,d x \]
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