Integrand size = 166, antiderivative size = 35 \[ \int \frac {e^{-2 x} \left (e^{2 x} \left (-2 x+x^2\right ) \log ^2(2-x)+e^{\frac {9+3 x}{\log (2-x)}} \left (-18-6 x+(-12+6 x) \log (2-x)+(8-4 x) \log ^2(2-x)\right )+e^{\frac {9+3 x}{2 \log (2-x)}} \left (e^x \left (9 x+3 x^2\right )+e^x \left (6 x-3 x^2\right ) \log (2-x)+e^x \left (4-6 x+2 x^2\right ) \log ^2(2-x)\right )\right )}{(-4+2 x) \log ^2(2-x)} \, dx=\left (-e^{-x+\frac {9 \left (2+\frac {2 x}{3}\right )}{4 \log (2-x)}}+\frac {x}{2}\right )^2 \]
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\[ \int \frac {e^{-2 x} \left (e^{2 x} \left (-2 x+x^2\right ) \log ^2(2-x)+e^{\frac {9+3 x}{\log (2-x)}} \left (-18-6 x+(-12+6 x) \log (2-x)+(8-4 x) \log ^2(2-x)\right )+e^{\frac {9+3 x}{2 \log (2-x)}} \left (e^x \left (9 x+3 x^2\right )+e^x \left (6 x-3 x^2\right ) \log (2-x)+e^x \left (4-6 x+2 x^2\right ) \log ^2(2-x)\right )\right )}{(-4+2 x) \log ^2(2-x)} \, dx=\int \frac {e^{-2 x} \left (e^{2 x} \left (-2 x+x^2\right ) \log ^2(2-x)+e^{\frac {9+3 x}{\log (2-x)}} \left (-18-6 x+(-12+6 x) \log (2-x)+(8-4 x) \log ^2(2-x)\right )+e^{\frac {9+3 x}{2 \log (2-x)}} \left (e^x \left (9 x+3 x^2\right )+e^x \left (6 x-3 x^2\right ) \log (2-x)+e^x \left (4-6 x+2 x^2\right ) \log ^2(2-x)\right )\right )}{(-4+2 x) \log ^2(2-x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x}{2}+\frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}} \left (9+3 x+6 \log (2-x)-3 x \log (2-x)-4 \log ^2(2-x)+2 x \log ^2(2-x)\right )}{(2-x) \log ^2(2-x)}+\frac {\exp \left (-2 x+\frac {9+3 x+2 x \log (2-x)}{2 \log (2-x)}\right ) \left (-9 x-3 x^2-6 x \log (2-x)+3 x^2 \log (2-x)-4 \log ^2(2-x)+6 x \log ^2(2-x)-2 x^2 \log ^2(2-x)\right )}{2 (2-x) \log ^2(2-x)}\right ) \, dx \\ & = \frac {x^2}{4}+\frac {1}{2} \int \frac {\exp \left (-2 x+\frac {9+3 x+2 x \log (2-x)}{2 \log (2-x)}\right ) \left (-9 x-3 x^2-6 x \log (2-x)+3 x^2 \log (2-x)-4 \log ^2(2-x)+6 x \log ^2(2-x)-2 x^2 \log ^2(2-x)\right )}{(2-x) \log ^2(2-x)} \, dx+\int \frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}} \left (9+3 x+6 \log (2-x)-3 x \log (2-x)-4 \log ^2(2-x)+2 x \log ^2(2-x)\right )}{(2-x) \log ^2(2-x)} \, dx \\ & = \frac {x^2}{4}+\frac {1}{2} \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} \left (-3 x (3+x)+3 (-2+x) x \log (2-x)-2 \left (2-3 x+x^2\right ) \log ^2(2-x)\right )}{(2-x) \log ^2(2-x)} \, dx+\int \frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}} \left (3 (3+x)-3 (-2+x) \log (2-x)+2 (-2+x) \log ^2(2-x)\right )}{(2-x) \log ^2(2-x)} \, dx \\ & = \frac {x^2}{4}+\frac {1}{2} \int \left (2 e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} (-1+x)+\frac {3 e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} x (3+x)}{(-2+x) \log ^2(2-x)}-\frac {3 e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} x}{\log (2-x)}\right ) \, dx+\int \left (-2 e^{-2 x+\frac {3 (3+x)}{\log (2-x)}}-\frac {3 e^{-2 x+\frac {3 (3+x)}{\log (2-x)}} (3+x)}{(-2+x) \log ^2(2-x)}+\frac {3 e^{-2 x+\frac {3 (3+x)}{\log (2-x)}}}{\log (2-x)}\right ) \, dx \\ & = \frac {x^2}{4}+\frac {3}{2} \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} x (3+x)}{(-2+x) \log ^2(2-x)} \, dx-\frac {3}{2} \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} x}{\log (2-x)} \, dx-2 \int e^{-2 x+\frac {3 (3+x)}{\log (2-x)}} \, dx-3 \int \frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}} (3+x)}{(-2+x) \log ^2(2-x)} \, dx+3 \int \frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}}}{\log (2-x)} \, dx+\int e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} (-1+x) \, dx \\ & = \frac {x^2}{4}+\frac {3}{2} \int \left (\frac {5 e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}}}{\log ^2(2-x)}+\frac {10 e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}}}{(-2+x) \log ^2(2-x)}+\frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} x}{\log ^2(2-x)}\right ) \, dx-\frac {3}{2} \int \left (\frac {2 e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}}}{\log (2-x)}-\frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} (2-x)}{\log (2-x)}\right ) \, dx-2 \int e^{-2 x+\frac {3 (3+x)}{\log (2-x)}} \, dx-3 \int \left (\frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}}}{\log ^2(2-x)}+\frac {5 e^{-2 x+\frac {3 (3+x)}{\log (2-x)}}}{(-2+x) \log ^2(2-x)}\right ) \, dx+3 \int \frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}}}{\log (2-x)} \, dx+\int \left (-e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}}+e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} x\right ) \, dx \\ & = \frac {x^2}{4}+\frac {3}{2} \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} x}{\log ^2(2-x)} \, dx+\frac {3}{2} \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} (2-x)}{\log (2-x)} \, dx-2 \int e^{-2 x+\frac {3 (3+x)}{\log (2-x)}} \, dx-3 \int \frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}}}{\log ^2(2-x)} \, dx+3 \int \frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}}}{\log (2-x)} \, dx-3 \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}}}{\log (2-x)} \, dx+\frac {15}{2} \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}}}{\log ^2(2-x)} \, dx-15 \int \frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}}}{(-2+x) \log ^2(2-x)} \, dx+15 \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}}}{(-2+x) \log ^2(2-x)} \, dx-\int e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} \, dx+\int e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} x \, dx \\ & = \frac {x^2}{4}+\frac {3}{2} \int \left (\frac {2 e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}}}{\log ^2(2-x)}-\frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} (2-x)}{\log ^2(2-x)}\right ) \, dx+\frac {3}{2} \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} (2-x)}{\log (2-x)} \, dx-2 \int e^{-2 x+\frac {3 (3+x)}{\log (2-x)}} \, dx-3 \int \frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}}}{\log ^2(2-x)} \, dx+3 \int \frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}}}{\log (2-x)} \, dx-3 \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}}}{\log (2-x)} \, dx+\frac {15}{2} \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}}}{\log ^2(2-x)} \, dx-15 \int \frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}}}{(-2+x) \log ^2(2-x)} \, dx+15 \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}}}{(-2+x) \log ^2(2-x)} \, dx-\int e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} \, dx+\int e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} x \, dx \\ & = \frac {x^2}{4}-\frac {3}{2} \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} (2-x)}{\log ^2(2-x)} \, dx+\frac {3}{2} \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} (2-x)}{\log (2-x)} \, dx-2 \int e^{-2 x+\frac {3 (3+x)}{\log (2-x)}} \, dx-3 \int \frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}}}{\log ^2(2-x)} \, dx+3 \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}}}{\log ^2(2-x)} \, dx+3 \int \frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}}}{\log (2-x)} \, dx-3 \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}}}{\log (2-x)} \, dx+\frac {15}{2} \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}}}{\log ^2(2-x)} \, dx-15 \int \frac {e^{-2 x+\frac {3 (3+x)}{\log (2-x)}}}{(-2+x) \log ^2(2-x)} \, dx+15 \int \frac {e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}}}{(-2+x) \log ^2(2-x)} \, dx-\int e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} \, dx+\int e^{\frac {9+3 x-2 x \log (2-x)}{2 \log (2-x)}} x \, dx \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-2 x} \left (e^{2 x} \left (-2 x+x^2\right ) \log ^2(2-x)+e^{\frac {9+3 x}{\log (2-x)}} \left (-18-6 x+(-12+6 x) \log (2-x)+(8-4 x) \log ^2(2-x)\right )+e^{\frac {9+3 x}{2 \log (2-x)}} \left (e^x \left (9 x+3 x^2\right )+e^x \left (6 x-3 x^2\right ) \log (2-x)+e^x \left (4-6 x+2 x^2\right ) \log ^2(2-x)\right )\right )}{(-4+2 x) \log ^2(2-x)} \, dx=\frac {1}{4} e^{-2 x} \left (-2 e^{\frac {3 (3+x)}{2 \log (2-x)}}+e^x x\right )^2 \]
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Time = 2.79 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54
method | result | size |
parallelrisch | \(\frac {\left (2 \,{\mathrm e}^{2 x} x^{2}-8 \,{\mathrm e}^{x} {\mathrm e}^{\frac {\frac {9}{2}+\frac {3 x}{2}}{\ln \left (2-x \right )}} x +8 \,{\mathrm e}^{\frac {3 x +9}{\ln \left (2-x \right )}}\right ) {\mathrm e}^{-2 x}}{8}\) | \(54\) |
risch | \(\frac {x^{2}}{4}+{\mathrm e}^{-\frac {2 x \ln \left (2-x \right )-3 x -9}{\ln \left (2-x \right )}}-x \,{\mathrm e}^{-\frac {2 x \ln \left (2-x \right )-3 x -9}{2 \ln \left (2-x \right )}}\) | \(60\) |
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Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.43 \[ \int \frac {e^{-2 x} \left (e^{2 x} \left (-2 x+x^2\right ) \log ^2(2-x)+e^{\frac {9+3 x}{\log (2-x)}} \left (-18-6 x+(-12+6 x) \log (2-x)+(8-4 x) \log ^2(2-x)\right )+e^{\frac {9+3 x}{2 \log (2-x)}} \left (e^x \left (9 x+3 x^2\right )+e^x \left (6 x-3 x^2\right ) \log (2-x)+e^x \left (4-6 x+2 x^2\right ) \log ^2(2-x)\right )\right )}{(-4+2 x) \log ^2(2-x)} \, dx=\frac {1}{4} \, {\left (x^{2} e^{\left (2 \, x\right )} - 4 \, x e^{\left (x + \frac {3 \, {\left (x + 3\right )}}{2 \, \log \left (-x + 2\right )}\right )} + 4 \, e^{\left (\frac {3 \, {\left (x + 3\right )}}{\log \left (-x + 2\right )}\right )}\right )} e^{\left (-2 \, x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
Time = 0.44 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31 \[ \int \frac {e^{-2 x} \left (e^{2 x} \left (-2 x+x^2\right ) \log ^2(2-x)+e^{\frac {9+3 x}{\log (2-x)}} \left (-18-6 x+(-12+6 x) \log (2-x)+(8-4 x) \log ^2(2-x)\right )+e^{\frac {9+3 x}{2 \log (2-x)}} \left (e^x \left (9 x+3 x^2\right )+e^x \left (6 x-3 x^2\right ) \log (2-x)+e^x \left (4-6 x+2 x^2\right ) \log ^2(2-x)\right )\right )}{(-4+2 x) \log ^2(2-x)} \, dx=\frac {x^{2}}{4} + \left (- x e^{2 x} e^{\frac {\frac {3 x}{2} + \frac {9}{2}}{\log {\left (2 - x \right )}}} + e^{x} e^{\frac {2 \cdot \left (\frac {3 x}{2} + \frac {9}{2}\right )}{\log {\left (2 - x \right )}}}\right ) e^{- 3 x} \]
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Exception generated. \[ \int \frac {e^{-2 x} \left (e^{2 x} \left (-2 x+x^2\right ) \log ^2(2-x)+e^{\frac {9+3 x}{\log (2-x)}} \left (-18-6 x+(-12+6 x) \log (2-x)+(8-4 x) \log ^2(2-x)\right )+e^{\frac {9+3 x}{2 \log (2-x)}} \left (e^x \left (9 x+3 x^2\right )+e^x \left (6 x-3 x^2\right ) \log (2-x)+e^x \left (4-6 x+2 x^2\right ) \log ^2(2-x)\right )\right )}{(-4+2 x) \log ^2(2-x)} \, dx=\text {Exception raised: RuntimeError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (26) = 52\).
Time = 0.33 (sec) , antiderivative size = 122, normalized size of antiderivative = 3.49 \[ \int \frac {e^{-2 x} \left (e^{2 x} \left (-2 x+x^2\right ) \log ^2(2-x)+e^{\frac {9+3 x}{\log (2-x)}} \left (-18-6 x+(-12+6 x) \log (2-x)+(8-4 x) \log ^2(2-x)\right )+e^{\frac {9+3 x}{2 \log (2-x)}} \left (e^x \left (9 x+3 x^2\right )+e^x \left (6 x-3 x^2\right ) \log (2-x)+e^x \left (4-6 x+2 x^2\right ) \log ^2(2-x)\right )\right )}{(-4+2 x) \log ^2(2-x)} \, dx=\frac {1}{4} \, {\left (x - 2\right )}^{2} - {\left (x - 2\right )} e^{\left (-\frac {2 \, {\left (x - 2\right )} \log \left (-x + 2\right ) - 3 \, x + 4 \, \log \left (-x + 2\right ) - 9}{2 \, \log \left (-x + 2\right )}\right )} + x - 2 \, e^{\left (-\frac {2 \, {\left (x - 2\right )} \log \left (-x + 2\right ) - 3 \, x + 4 \, \log \left (-x + 2\right ) - 9}{2 \, \log \left (-x + 2\right )}\right )} + e^{\left (-\frac {2 \, {\left (x - 2\right )} \log \left (-x + 2\right ) - 3 \, x + 4 \, \log \left (-x + 2\right ) - 9}{\log \left (-x + 2\right )}\right )} - 2 \]
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Timed out. \[ \int \frac {e^{-2 x} \left (e^{2 x} \left (-2 x+x^2\right ) \log ^2(2-x)+e^{\frac {9+3 x}{\log (2-x)}} \left (-18-6 x+(-12+6 x) \log (2-x)+(8-4 x) \log ^2(2-x)\right )+e^{\frac {9+3 x}{2 \log (2-x)}} \left (e^x \left (9 x+3 x^2\right )+e^x \left (6 x-3 x^2\right ) \log (2-x)+e^x \left (4-6 x+2 x^2\right ) \log ^2(2-x)\right )\right )}{(-4+2 x) \log ^2(2-x)} \, dx=-\int \frac {{\mathrm {e}}^{-2\,x}\,\left ({\mathrm {e}}^{\frac {2\,\left (\frac {3\,x}{2}+\frac {9}{2}\right )}{\ln \left (2-x\right )}}\,\left (\left (4\,x-8\right )\,{\ln \left (2-x\right )}^2+\left (12-6\,x\right )\,\ln \left (2-x\right )+6\,x+18\right )-{\mathrm {e}}^{\frac {\frac {3\,x}{2}+\frac {9}{2}}{\ln \left (2-x\right )}}\,\left ({\mathrm {e}}^x\,\left (2\,x^2-6\,x+4\right )\,{\ln \left (2-x\right )}^2+{\mathrm {e}}^x\,\left (6\,x-3\,x^2\right )\,\ln \left (2-x\right )+{\mathrm {e}}^x\,\left (3\,x^2+9\,x\right )\right )+{\mathrm {e}}^{2\,x}\,{\ln \left (2-x\right )}^2\,\left (2\,x-x^2\right )\right )}{{\ln \left (2-x\right )}^2\,\left (2\,x-4\right )} \,d x \]
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