Integrand size = 103, antiderivative size = 20 \[ \int \frac {e^{-2 x \log \left (\frac {x}{3+x}\right )+x^2 \log (-3+x) \log \left (\frac {x}{3+x}\right )} \left (-9+18 x-5 x^2+\left (18 x+x^3+x^4\right ) \log \left (\frac {x}{3+x}\right )+\log (-3+x) \left (-9 x^2+3 x^3+\left (-18 x^2+2 x^4\right ) \log \left (\frac {x}{3+x}\right )\right )\right )}{-9+x^2} \, dx=x \left (\frac {x}{3+x}\right )^{x (-2+x \log (-3+x))} \]
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\[ \int \frac {e^{-2 x \log \left (\frac {x}{3+x}\right )+x^2 \log (-3+x) \log \left (\frac {x}{3+x}\right )} \left (-9+18 x-5 x^2+\left (18 x+x^3+x^4\right ) \log \left (\frac {x}{3+x}\right )+\log (-3+x) \left (-9 x^2+3 x^3+\left (-18 x^2+2 x^4\right ) \log \left (\frac {x}{3+x}\right )\right )\right )}{-9+x^2} \, dx=\int \frac {\exp \left (-2 x \log \left (\frac {x}{3+x}\right )+x^2 \log (-3+x) \log \left (\frac {x}{3+x}\right )\right ) \left (-9+18 x-5 x^2+\left (18 x+x^3+x^4\right ) \log \left (\frac {x}{3+x}\right )+\log (-3+x) \left (-9 x^2+3 x^3+\left (-18 x^2+2 x^4\right ) \log \left (\frac {x}{3+x}\right )\right )\right )}{-9+x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {9 \left (\frac {x}{3+x}\right )^{x (-2+x \log (-3+x))}}{-9+x^2}+\frac {18 x \left (\frac {x}{3+x}\right )^{x (-2+x \log (-3+x))}}{-9+x^2}-\frac {5 x^2 \left (\frac {x}{3+x}\right )^{x (-2+x \log (-3+x))}}{-9+x^2}-\frac {9 x^2 \left (\frac {x}{3+x}\right )^{x (-2+x \log (-3+x))} \log (-3+x)}{-9+x^2}+\frac {3 x^3 \left (\frac {x}{3+x}\right )^{x (-2+x \log (-3+x))} \log (-3+x)}{-9+x^2}+\frac {x \left (\frac {x}{3+x}\right )^{x (-2+x \log (-3+x))} \left (6-2 x+x^2-6 x \log (-3+x)+2 x^2 \log (-3+x)\right ) \log \left (\frac {x}{3+x}\right )}{-3+x}\right ) \, dx \\ & = 3 \int \frac {x^3 \left (\frac {x}{3+x}\right )^{x (-2+x \log (-3+x))} \log (-3+x)}{-9+x^2} \, dx-5 \int \frac {x^2 \left (\frac {x}{3+x}\right )^{x (-2+x \log (-3+x))}}{-9+x^2} \, dx-9 \int \frac {\left (\frac {x}{3+x}\right )^{x (-2+x \log (-3+x))}}{-9+x^2} \, dx-9 \int \frac {x^2 \left (\frac {x}{3+x}\right )^{x (-2+x \log (-3+x))} \log (-3+x)}{-9+x^2} \, dx+18 \int \frac {x \left (\frac {x}{3+x}\right )^{x (-2+x \log (-3+x))}}{-9+x^2} \, dx+\int \frac {x \left (\frac {x}{3+x}\right )^{x (-2+x \log (-3+x))} \left (6-2 x+x^2-6 x \log (-3+x)+2 x^2 \log (-3+x)\right ) \log \left (\frac {x}{3+x}\right )}{-3+x} \, dx \\ & = 3 \int \left (\frac {9 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)}{2 (-3+x)}+x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)+\frac {9 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)}{2 (3+x)}\right ) \, dx-5 \int \left (\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}+\frac {9 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{-9+x^2}\right ) \, dx-9 \int \left (-\frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{6 (3-x)}-\frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{6 (3+x)}\right ) \, dx-9 \int \left (\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)+\frac {9 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)}{-9+x^2}\right ) \, dx+18 \int \left (\frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{2 (-3+x)}+\frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{2 (3+x)}\right ) \, dx+\int \frac {x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \left (-6+2 x-x^2+6 x \log (-3+x)-2 x^2 \log (-3+x)\right ) \log \left (\frac {x}{3+x}\right )}{3-x} \, dx \\ & = \frac {3}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3-x} \, dx+\frac {3}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx+3 \int x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \, dx-5 \int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \, dx+9 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{-3+x} \, dx+9 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx-9 \int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \, dx+\frac {27}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)}{-3+x} \, dx+\frac {27}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)}{3+x} \, dx-45 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{-9+x^2} \, dx-81 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)}{-9+x^2} \, dx+\int \left (\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \left (6-2 x+x^2-6 x \log (-3+x)+2 x^2 \log (-3+x)\right ) \log \left (\frac {x}{3+x}\right )+\frac {3 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \left (6-2 x+x^2-6 x \log (-3+x)+2 x^2 \log (-3+x)\right ) \log \left (\frac {x}{3+x}\right )}{-3+x}\right ) \, dx \\ & = \frac {3}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3-x} \, dx+\frac {3}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx+3 \int x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \, dx+3 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \left (6-2 x+x^2-6 x \log (-3+x)+2 x^2 \log (-3+x)\right ) \log \left (\frac {x}{3+x}\right )}{-3+x} \, dx-5 \int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \, dx+9 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{-3+x} \, dx+9 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx-9 \int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \, dx+\frac {27}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)}{-3+x} \, dx+\frac {27}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)}{3+x} \, dx-45 \int \left (-\frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{6 (3-x)}-\frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{6 (3+x)}\right ) \, dx-81 \int \left (\frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)}{6 (-3+x)}-\frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)}{6 (3+x)}\right ) \, dx+\int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \left (6-2 x+x^2-6 x \log (-3+x)+2 x^2 \log (-3+x)\right ) \log \left (\frac {x}{3+x}\right ) \, dx \\ & = \frac {3}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3-x} \, dx+\frac {3}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx+3 \int x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \, dx+3 \int \left (\frac {6 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )}{-3+x}-\frac {2 x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )}{-3+x}+\frac {x^2 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )}{-3+x}-\frac {6 x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right )}{-3+x}+\frac {2 x^2 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right )}{-3+x}\right ) \, dx-5 \int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \, dx+\frac {15}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3-x} \, dx+\frac {15}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx+9 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{-3+x} \, dx+9 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx-9 \int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \, dx+2 \left (\frac {27}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)}{3+x} \, dx\right )+\int \left (6 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )-2 x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )+x^2 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )-6 x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right )+2 x^2 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right )\right ) \, dx \\ & = \frac {3}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3-x} \, dx+\frac {3}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx-2 \int x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right ) \, dx+2 \int x^2 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right ) \, dx+3 \int x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \, dx+3 \int \frac {x^2 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )}{-3+x} \, dx-5 \int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \, dx+6 \int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right ) \, dx-6 \int \frac {x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )}{-3+x} \, dx-6 \int x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right ) \, dx+6 \int \frac {x^2 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right )}{-3+x} \, dx+\frac {15}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3-x} \, dx+\frac {15}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx+9 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{-3+x} \, dx+9 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx-9 \int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \, dx+2 \left (\frac {27}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)}{3+x} \, dx\right )+18 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )}{-3+x} \, dx-18 \int \frac {x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right )}{-3+x} \, dx+\int x^2 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right ) \, dx \\ & = \frac {3}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3-x} \, dx+\frac {3}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx-2 \int x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right ) \, dx+2 \int x^2 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right ) \, dx+3 \int x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \, dx+3 \int \left (3 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )+\frac {9 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )}{-3+x}+x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )\right ) \, dx-5 \int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \, dx+6 \int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right ) \, dx-6 \int x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right ) \, dx-6 \int \left (\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )+\frac {3 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )}{-3+x}\right ) \, dx+6 \int \left (3 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right )+\frac {9 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right )}{-3+x}+x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right )\right ) \, dx+\frac {15}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3-x} \, dx+\frac {15}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx+9 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{-3+x} \, dx+9 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx-9 \int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \, dx+2 \left (\frac {27}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)}{3+x} \, dx\right )+18 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )}{-3+x} \, dx-18 \int \left (\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right )+\frac {3 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right )}{-3+x}\right ) \, dx+\int x^2 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right ) \, dx \\ & = \frac {3}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3-x} \, dx+\frac {3}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx-2 \int x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right ) \, dx+2 \int x^2 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \log \left (\frac {x}{3+x}\right ) \, dx+3 \int x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \, dx+3 \int x \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right ) \, dx-5 \int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \, dx+\frac {15}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3-x} \, dx+\frac {15}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx+9 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{-3+x} \, dx+9 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)}}{3+x} \, dx-9 \int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x) \, dx+9 \int \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right ) \, dx+2 \left (\frac {27}{2} \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log (-3+x)}{3+x} \, dx\right )+27 \int \frac {\left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right )}{-3+x} \, dx+\int x^2 \left (\frac {x}{3+x}\right )^{-2 x+x^2 \log (-3+x)} \log \left (\frac {x}{3+x}\right ) \, dx \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {e^{-2 x \log \left (\frac {x}{3+x}\right )+x^2 \log (-3+x) \log \left (\frac {x}{3+x}\right )} \left (-9+18 x-5 x^2+\left (18 x+x^3+x^4\right ) \log \left (\frac {x}{3+x}\right )+\log (-3+x) \left (-9 x^2+3 x^3+\left (-18 x^2+2 x^4\right ) \log \left (\frac {x}{3+x}\right )\right )\right )}{-9+x^2} \, dx=(-3+x)^{x^2 \log \left (\frac {x}{3+x}\right )} x \left (\frac {x}{3+x}\right )^{-2 x} \]
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Time = 5.96 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10
method | result | size |
parallelrisch | \({\mathrm e}^{x \left (\ln \left (-3+x \right ) x -2\right ) \ln \left (\frac {x}{3+x}\right )} x\) | \(22\) |
risch | \(x \left (-3+x \right )^{\frac {x^{2} \left (-i \pi \,\operatorname {csgn}\left (\frac {i x}{3+x}\right )+i \pi \,\operatorname {csgn}\left (i x \right )+i \pi \,\operatorname {csgn}\left (\frac {i}{3+x}\right )-i \pi \,\operatorname {csgn}\left (\frac {i x}{3+x}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{3+x}\right )+2 \ln \left (x \right )-2 \ln \left (3+x \right )\right )}{2}} x^{-2 x} \left (3+x \right )^{2 x} {\mathrm e}^{-i \pi \,\operatorname {csgn}\left (\frac {i x}{3+x}\right ) x \left (\operatorname {csgn}\left (\frac {i x}{3+x}\right )-\operatorname {csgn}\left (\frac {i}{3+x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i x}{3+x}\right )+\operatorname {csgn}\left (i x \right )\right )}\) | \(155\) |
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Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55 \[ \int \frac {e^{-2 x \log \left (\frac {x}{3+x}\right )+x^2 \log (-3+x) \log \left (\frac {x}{3+x}\right )} \left (-9+18 x-5 x^2+\left (18 x+x^3+x^4\right ) \log \left (\frac {x}{3+x}\right )+\log (-3+x) \left (-9 x^2+3 x^3+\left (-18 x^2+2 x^4\right ) \log \left (\frac {x}{3+x}\right )\right )\right )}{-9+x^2} \, dx=x e^{\left (x^{2} \log \left (x - 3\right ) \log \left (\frac {x}{x + 3}\right ) - 2 \, x \log \left (\frac {x}{x + 3}\right )\right )} \]
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Time = 7.38 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {e^{-2 x \log \left (\frac {x}{3+x}\right )+x^2 \log (-3+x) \log \left (\frac {x}{3+x}\right )} \left (-9+18 x-5 x^2+\left (18 x+x^3+x^4\right ) \log \left (\frac {x}{3+x}\right )+\log (-3+x) \left (-9 x^2+3 x^3+\left (-18 x^2+2 x^4\right ) \log \left (\frac {x}{3+x}\right )\right )\right )}{-9+x^2} \, dx=x e^{x^{2} \log {\left (\frac {x}{x + 3} \right )} \log {\left (x - 3 \right )} - 2 x \log {\left (\frac {x}{x + 3} \right )}} \]
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Time = 0.35 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95 \[ \int \frac {e^{-2 x \log \left (\frac {x}{3+x}\right )+x^2 \log (-3+x) \log \left (\frac {x}{3+x}\right )} \left (-9+18 x-5 x^2+\left (18 x+x^3+x^4\right ) \log \left (\frac {x}{3+x}\right )+\log (-3+x) \left (-9 x^2+3 x^3+\left (-18 x^2+2 x^4\right ) \log \left (\frac {x}{3+x}\right )\right )\right )}{-9+x^2} \, dx=x e^{\left (-x^{2} \log \left (x + 3\right ) \log \left (x - 3\right ) + x^{2} \log \left (x - 3\right ) \log \left (x\right ) + 2 \, x \log \left (x + 3\right ) - 2 \, x \log \left (x\right )\right )} \]
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\[ \int \frac {e^{-2 x \log \left (\frac {x}{3+x}\right )+x^2 \log (-3+x) \log \left (\frac {x}{3+x}\right )} \left (-9+18 x-5 x^2+\left (18 x+x^3+x^4\right ) \log \left (\frac {x}{3+x}\right )+\log (-3+x) \left (-9 x^2+3 x^3+\left (-18 x^2+2 x^4\right ) \log \left (\frac {x}{3+x}\right )\right )\right )}{-9+x^2} \, dx=\int { -\frac {{\left (5 \, x^{2} - {\left (3 \, x^{3} - 9 \, x^{2} + 2 \, {\left (x^{4} - 9 \, x^{2}\right )} \log \left (\frac {x}{x + 3}\right )\right )} \log \left (x - 3\right ) - {\left (x^{4} + x^{3} + 18 \, x\right )} \log \left (\frac {x}{x + 3}\right ) - 18 \, x + 9\right )} e^{\left (x^{2} \log \left (x - 3\right ) \log \left (\frac {x}{x + 3}\right ) - 2 \, x \log \left (\frac {x}{x + 3}\right )\right )}}{x^{2} - 9} \,d x } \]
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Time = 13.42 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55 \[ \int \frac {e^{-2 x \log \left (\frac {x}{3+x}\right )+x^2 \log (-3+x) \log \left (\frac {x}{3+x}\right )} \left (-9+18 x-5 x^2+\left (18 x+x^3+x^4\right ) \log \left (\frac {x}{3+x}\right )+\log (-3+x) \left (-9 x^2+3 x^3+\left (-18 x^2+2 x^4\right ) \log \left (\frac {x}{3+x}\right )\right )\right )}{-9+x^2} \, dx=\frac {x\,{\left (x-3\right )}^{x^2\,\ln \left (\frac {x}{x+3}\right )}}{{\left (\frac {x}{x+3}\right )}^{2\,x}} \]
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