Integrand size = 200, antiderivative size = 33 \[ \int \frac {e^{-x} \left (e^x \left (2 x^2-8 x^3+4 x^4+8 x^5+2 x^6+e^6 \left (2-4 x^2+2 x^4\right )+e^3 \left (-4 x+8 x^2+8 x^3-8 x^4-4 x^5\right )\right )+\left (x^2-x^3+3 x^4+x^5+e^6 \left (1-x+x^2+x^3\right )+e^3 \left (-2 x-4 x^3-2 x^4\right )\right ) \log (4)\right )}{x^2-4 x^3+2 x^4+4 x^5+x^6+e^6 \left (1-2 x^2+x^4\right )+e^3 \left (-2 x+4 x^2+4 x^3-4 x^4-2 x^5\right )} \, dx=2 x+\frac {e^{-x} \log (4)}{\frac {1}{x}-x+\frac {2 x}{e^3-x}} \]
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\[ \int \frac {e^{-x} \left (e^x \left (2 x^2-8 x^3+4 x^4+8 x^5+2 x^6+e^6 \left (2-4 x^2+2 x^4\right )+e^3 \left (-4 x+8 x^2+8 x^3-8 x^4-4 x^5\right )\right )+\left (x^2-x^3+3 x^4+x^5+e^6 \left (1-x+x^2+x^3\right )+e^3 \left (-2 x-4 x^3-2 x^4\right )\right ) \log (4)\right )}{x^2-4 x^3+2 x^4+4 x^5+x^6+e^6 \left (1-2 x^2+x^4\right )+e^3 \left (-2 x+4 x^2+4 x^3-4 x^4-2 x^5\right )} \, dx=\int \frac {e^{-x} \left (e^x \left (2 x^2-8 x^3+4 x^4+8 x^5+2 x^6+e^6 \left (2-4 x^2+2 x^4\right )+e^3 \left (-4 x+8 x^2+8 x^3-8 x^4-4 x^5\right )\right )+\left (x^2-x^3+3 x^4+x^5+e^6 \left (1-x+x^2+x^3\right )+e^3 \left (-2 x-4 x^3-2 x^4\right )\right ) \log (4)\right )}{x^2-4 x^3+2 x^4+4 x^5+x^6+e^6 \left (1-2 x^2+x^4\right )+e^3 \left (-2 x+4 x^2+4 x^3-4 x^4-2 x^5\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x} \left (2 e^{6+x} \left (-1+x^2\right )^2+2 e^x x^2 \left (-1+2 x+x^2\right )^2-4 e^{3+x} x \left (1-2 x-2 x^2+2 x^3+x^4\right )+e^6 \left (1-x+x^2+x^3\right ) \log (4)-2 e^3 x \left (1+2 x^2+x^3\right ) \log (4)+x^2 \left (1-x+3 x^2+x^3\right ) \log (4)\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx \\ & = \int \left (2+\frac {2 e^{3-x} x \left (-1-2 x^2-x^3\right ) \log (4)}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}+\frac {e^{6-x} \left (1-x+x^2+x^3\right ) \log (4)}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}+\frac {e^{-x} x^2 \left (1-x+3 x^2+x^3\right ) \log (4)}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}\right ) \, dx \\ & = 2 x+\log (4) \int \frac {e^{6-x} \left (1-x+x^2+x^3\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+\log (4) \int \frac {e^{-x} x^2 \left (1-x+3 x^2+x^3\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+(2 \log (4)) \int \frac {e^{3-x} x \left (-1-2 x^2-x^3\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx \\ & = 2 x+\log (4) \int \left (\frac {e^{6-x} \left (1-e^3\right ) \left (1-x^2\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}+\frac {e^{6-x}}{e^3-x+\left (2-e^3\right ) x^2+x^3}\right ) \, dx+\log (4) \int \left (\frac {e^{-x} \left (e^3 \left (2+e^3-e^6\right )-2 \left (1+e^3\right ) x+\left (6-3 e^6+e^9\right ) x^2\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}+\frac {e^{-x} \left (-2-e^3+e^6+\left (1+e^3\right ) x+x^2\right )}{e^3-x+\left (2-e^3\right ) x^2+x^3}\right ) \, dx+(2 \log (4)) \int \left (\frac {e^{3-x} \left (e^6-x-\left (1-e^3\right )^2 x^2\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}+\frac {e^{3-x} \left (-e^3-x\right )}{e^3-x+\left (2-e^3\right ) x^2+x^3}\right ) \, dx \\ & = 2 x+\log (4) \int \frac {e^{-x} \left (e^3 \left (2+e^3-e^6\right )-2 \left (1+e^3\right ) x+\left (6-3 e^6+e^9\right ) x^2\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+\log (4) \int \frac {e^{6-x}}{e^3-x+\left (2-e^3\right ) x^2+x^3} \, dx+\log (4) \int \frac {e^{-x} \left (-2-e^3+e^6+\left (1+e^3\right ) x+x^2\right )}{e^3-x+\left (2-e^3\right ) x^2+x^3} \, dx+(2 \log (4)) \int \frac {e^{3-x} \left (e^6-x-\left (1-e^3\right )^2 x^2\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+(2 \log (4)) \int \frac {e^{3-x} \left (-e^3-x\right )}{e^3-x+\left (2-e^3\right ) x^2+x^3} \, dx+\left (\left (1-e^3\right ) \log (4)\right ) \int \frac {e^{6-x} \left (1-x^2\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx \\ & = 2 x+\log (4) \int \frac {e^{6-x}}{e^3-x+\left (2-e^3\right ) x^2+x^3} \, dx+\log (4) \int \left (\frac {e^{3-x} \left (2+e^3-e^6\right )}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}+\frac {2 e^{-x} \left (-1-e^3\right ) x}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}+\frac {e^{-x} \left (6-3 e^6+e^9\right ) x^2}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}\right ) \, dx+\log (4) \int \left (\frac {2 e^{-x} \left (1-\frac {1}{2} e^3 \left (-1+e^3\right )\right )}{-e^3+x-\left (2-e^3\right ) x^2-x^3}+\frac {e^{-x} \left (1+e^3\right ) x}{e^3-x+\left (2-e^3\right ) x^2+x^3}+\frac {e^{-x} x^2}{e^3-x+\left (2-e^3\right ) x^2+x^3}\right ) \, dx+(2 \log (4)) \int \left (\frac {e^{6-x}}{-e^3+x-\left (2-e^3\right ) x^2-x^3}+\frac {e^{3-x} x}{-e^3+x-\left (2-e^3\right ) x^2-x^3}\right ) \, dx+(2 \log (4)) \int \left (\frac {e^{9-x}}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}-\frac {e^{3-x} x}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}-\frac {e^{3-x} \left (1-e^3\right )^2 x^2}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}\right ) \, dx+\left (\left (1-e^3\right ) \log (4)\right ) \int \left (\frac {e^{6-x}}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}-\frac {e^{6-x} x^2}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2}\right ) \, dx \\ & = 2 x+\log (4) \int \frac {e^{6-x}}{e^3-x+\left (2-e^3\right ) x^2+x^3} \, dx+\log (4) \int \frac {e^{-x} x^2}{e^3-x+\left (2-e^3\right ) x^2+x^3} \, dx+(2 \log (4)) \int \frac {e^{6-x}}{-e^3+x-\left (2-e^3\right ) x^2-x^3} \, dx+(2 \log (4)) \int \frac {e^{3-x} x}{-e^3+x-\left (2-e^3\right ) x^2-x^3} \, dx+(2 \log (4)) \int \frac {e^{9-x}}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx-(2 \log (4)) \int \frac {e^{3-x} x}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+\left (\left (1-e^3\right ) \log (4)\right ) \int \frac {e^{6-x}}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx-\left (\left (1-e^3\right ) \log (4)\right ) \int \frac {e^{6-x} x^2}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx-\left (2 \left (1-e^3\right )^2 \log (4)\right ) \int \frac {e^{3-x} x^2}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+\left (\left (1+e^3\right ) \log (4)\right ) \int \frac {e^{-x} x}{e^3-x+\left (2-e^3\right ) x^2+x^3} \, dx-\left (2 \left (1+e^3\right ) \log (4)\right ) \int \frac {e^{-x} x}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+\left (\left (2+e^3-e^6\right ) \log (4)\right ) \int \frac {e^{-x}}{-e^3+x-\left (2-e^3\right ) x^2-x^3} \, dx+\left (\left (2+e^3-e^6\right ) \log (4)\right ) \int \frac {e^{3-x}}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx+\left (\left (6-3 e^6+e^9\right ) \log (4)\right ) \int \frac {e^{-x} x^2}{\left (e^3-x+\left (2-e^3\right ) x^2+x^3\right )^2} \, dx \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \[ \int \frac {e^{-x} \left (e^x \left (2 x^2-8 x^3+4 x^4+8 x^5+2 x^6+e^6 \left (2-4 x^2+2 x^4\right )+e^3 \left (-4 x+8 x^2+8 x^3-8 x^4-4 x^5\right )\right )+\left (x^2-x^3+3 x^4+x^5+e^6 \left (1-x+x^2+x^3\right )+e^3 \left (-2 x-4 x^3-2 x^4\right )\right ) \log (4)\right )}{x^2-4 x^3+2 x^4+4 x^5+x^6+e^6 \left (1-2 x^2+x^4\right )+e^3 \left (-2 x+4 x^2+4 x^3-4 x^4-2 x^5\right )} \, dx=x \left (2-\frac {e^{-x} \left (e^3-x\right ) \log (4)}{e^3 \left (-1+x^2\right )-x \left (-1+2 x+x^2\right )}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(88\) vs. \(2(32)=64\).
Time = 0.60 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.70
method | result | size |
norman | \(\frac {\left (\left (4 \,{\mathrm e}^{3}-2 \,{\mathrm e}^{6}\right ) {\mathrm e}^{x}-4 \,{\mathrm e}^{x} x +\left (2 \,{\mathrm e}^{6}-8 \,{\mathrm e}^{3}+10\right ) x^{2} {\mathrm e}^{x}+2 x^{2} \ln \left (2\right )-2 \,{\mathrm e}^{x} x^{4}-2 \,{\mathrm e}^{3} \ln \left (2\right ) x \right ) {\mathrm e}^{-x}}{x^{2} {\mathrm e}^{3}-x^{3}-2 x^{2}-{\mathrm e}^{3}+x}\) | \(89\) |
parallelrisch | \(-\frac {\left (-2 x^{2} {\mathrm e}^{6} {\mathrm e}^{x}-4 \,{\mathrm e}^{x} {\mathrm e}^{3}+2 \,{\mathrm e}^{x} x^{4}-2 x^{2} \ln \left (2\right )-10 \,{\mathrm e}^{x} x^{2}+4 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{6} {\mathrm e}^{x}+8 x^{2} {\mathrm e}^{3} {\mathrm e}^{x}+2 \,{\mathrm e}^{3} \ln \left (2\right ) x \right ) {\mathrm e}^{-x}}{x^{2} {\mathrm e}^{3}-x^{3}-2 x^{2}-{\mathrm e}^{3}+x}\) | \(99\) |
parts | \(\text {Expression too large to display}\) | \(2091\) |
default | \(\text {Expression too large to display}\) | \(4668\) |
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (32) = 64\).
Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.15 \[ \int \frac {e^{-x} \left (e^x \left (2 x^2-8 x^3+4 x^4+8 x^5+2 x^6+e^6 \left (2-4 x^2+2 x^4\right )+e^3 \left (-4 x+8 x^2+8 x^3-8 x^4-4 x^5\right )\right )+\left (x^2-x^3+3 x^4+x^5+e^6 \left (1-x+x^2+x^3\right )+e^3 \left (-2 x-4 x^3-2 x^4\right )\right ) \log (4)\right )}{x^2-4 x^3+2 x^4+4 x^5+x^6+e^6 \left (1-2 x^2+x^4\right )+e^3 \left (-2 x+4 x^2+4 x^3-4 x^4-2 x^5\right )} \, dx=\frac {2 \, {\left ({\left (x^{4} + 2 \, x^{3} - x^{2} - {\left (x^{3} - x\right )} e^{3}\right )} e^{x} - {\left (x^{2} - x e^{3}\right )} \log \left (2\right )\right )} e^{\left (-x\right )}}{x^{3} + 2 \, x^{2} - {\left (x^{2} - 1\right )} e^{3} - x} \]
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Time = 0.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-x} \left (e^x \left (2 x^2-8 x^3+4 x^4+8 x^5+2 x^6+e^6 \left (2-4 x^2+2 x^4\right )+e^3 \left (-4 x+8 x^2+8 x^3-8 x^4-4 x^5\right )\right )+\left (x^2-x^3+3 x^4+x^5+e^6 \left (1-x+x^2+x^3\right )+e^3 \left (-2 x-4 x^3-2 x^4\right )\right ) \log (4)\right )}{x^2-4 x^3+2 x^4+4 x^5+x^6+e^6 \left (1-2 x^2+x^4\right )+e^3 \left (-2 x+4 x^2+4 x^3-4 x^4-2 x^5\right )} \, dx=2 x + \frac {\left (- 2 x^{2} \log {\left (2 \right )} + 2 x e^{3} \log {\left (2 \right )}\right ) e^{- x}}{x^{3} - x^{2} e^{3} + 2 x^{2} - x + e^{3}} \]
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Time = 0.41 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.94 \[ \int \frac {e^{-x} \left (e^x \left (2 x^2-8 x^3+4 x^4+8 x^5+2 x^6+e^6 \left (2-4 x^2+2 x^4\right )+e^3 \left (-4 x+8 x^2+8 x^3-8 x^4-4 x^5\right )\right )+\left (x^2-x^3+3 x^4+x^5+e^6 \left (1-x+x^2+x^3\right )+e^3 \left (-2 x-4 x^3-2 x^4\right )\right ) \log (4)\right )}{x^2-4 x^3+2 x^4+4 x^5+x^6+e^6 \left (1-2 x^2+x^4\right )+e^3 \left (-2 x+4 x^2+4 x^3-4 x^4-2 x^5\right )} \, dx=\frac {2 \, {\left (x^{4} - x^{3} {\left (e^{3} - 2\right )} - x^{2} + x e^{3} - {\left (x^{2} \log \left (2\right ) - x e^{3} \log \left (2\right )\right )} e^{\left (-x\right )}\right )}}{x^{3} - x^{2} {\left (e^{3} - 2\right )} - x + e^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (32) = 64\).
Time = 0.41 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18 \[ \int \frac {e^{-x} \left (e^x \left (2 x^2-8 x^3+4 x^4+8 x^5+2 x^6+e^6 \left (2-4 x^2+2 x^4\right )+e^3 \left (-4 x+8 x^2+8 x^3-8 x^4-4 x^5\right )\right )+\left (x^2-x^3+3 x^4+x^5+e^6 \left (1-x+x^2+x^3\right )+e^3 \left (-2 x-4 x^3-2 x^4\right )\right ) \log (4)\right )}{x^2-4 x^3+2 x^4+4 x^5+x^6+e^6 \left (1-2 x^2+x^4\right )+e^3 \left (-2 x+4 x^2+4 x^3-4 x^4-2 x^5\right )} \, dx=\frac {2 \, {\left (x^{4} - x^{3} e^{3} - 2 \, x^{2} e^{\left (-x\right )} \log \left (2\right ) + 2 \, x^{3} + 2 \, x e^{\left (-x + 3\right )} \log \left (2\right ) - x^{2} + x e^{3}\right )}}{x^{3} - x^{2} e^{3} + 2 \, x^{2} - x + e^{3}} \]
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Time = 13.60 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-x} \left (e^x \left (2 x^2-8 x^3+4 x^4+8 x^5+2 x^6+e^6 \left (2-4 x^2+2 x^4\right )+e^3 \left (-4 x+8 x^2+8 x^3-8 x^4-4 x^5\right )\right )+\left (x^2-x^3+3 x^4+x^5+e^6 \left (1-x+x^2+x^3\right )+e^3 \left (-2 x-4 x^3-2 x^4\right )\right ) \log (4)\right )}{x^2-4 x^3+2 x^4+4 x^5+x^6+e^6 \left (1-2 x^2+x^4\right )+e^3 \left (-2 x+4 x^2+4 x^3-4 x^4-2 x^5\right )} \, dx=2\,x+\frac {{\mathrm {e}}^{-x}\,\left (2\,x^2\,\ln \left (2\right )-2\,x\,{\mathrm {e}}^3\,\ln \left (2\right )\right )}{-x^3+\left ({\mathrm {e}}^3-2\right )\,x^2+x-{\mathrm {e}}^3} \]
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