Integrand size = 113, antiderivative size = 21 \[ \int \frac {x+e^{2 x} x+e^{-8 \log ^2(x)} x+x^3+e^x \left (-x-2 x^2\right )+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2-8 \log (x)\right )}{e^{2 x} x+e^{-8 \log ^2(x)} x-2 e^x x^2+x^3+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2\right )} \, dx=\frac {1}{e^x-e^{-4 \log ^2(x)}-x}+x \]
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\[ \int \frac {x+e^{2 x} x+e^{-8 \log ^2(x)} x+x^3+e^x \left (-x-2 x^2\right )+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2-8 \log (x)\right )}{e^{2 x} x+e^{-8 \log ^2(x)} x-2 e^x x^2+x^3+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2\right )} \, dx=\int \frac {x+e^{2 x} x+e^{-8 \log ^2(x)} x+x^3+e^x \left (-x-2 x^2\right )+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2-8 \log (x)\right )}{e^{2 x} x+e^{-8 \log ^2(x)} x-2 e^x x^2+x^3+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{8 \log ^2(x)} \left (x+e^{2 x} x+e^{-8 \log ^2(x)} x+x^3+e^x \left (-x-2 x^2\right )+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2-8 \log (x)\right )\right )}{x \left (1-e^{x+4 \log ^2(x)}+e^{4 \log ^2(x)} x\right )^2} \, dx \\ & = \int \left (1-\frac {8 e^{4 \log ^2(x)} \log (x)}{x}+\frac {8 e^{8 \log ^2(x)} \left (-e^x+x\right ) \log (x)}{x \left (1-e^{x+4 \log ^2(x)}+e^{4 \log ^2(x)} x\right )}-\frac {e^{8 \log ^2(x)} \left (-x+e^x x+8 e^x \log (x)-8 x \log (x)\right )}{x \left (1-e^{x+4 \log ^2(x)}+e^{4 \log ^2(x)} x\right )^2}\right ) \, dx \\ & = x-8 \int \frac {e^{4 \log ^2(x)} \log (x)}{x} \, dx+8 \int \frac {e^{8 \log ^2(x)} \left (-e^x+x\right ) \log (x)}{x \left (1-e^{x+4 \log ^2(x)}+e^{4 \log ^2(x)} x\right )} \, dx-\int \frac {e^{8 \log ^2(x)} \left (-x+e^x x+8 e^x \log (x)-8 x \log (x)\right )}{x \left (1-e^{x+4 \log ^2(x)}+e^{4 \log ^2(x)} x\right )^2} \, dx \\ & = x+8 \int \left (-\frac {e^{8 \log ^2(x)} \log (x)}{-1+e^{x+4 \log ^2(x)}-e^{4 \log ^2(x)} x}+\frac {e^{x+8 \log ^2(x)} \log (x)}{x \left (-1+e^{x+4 \log ^2(x)}-e^{4 \log ^2(x)} x\right )}\right ) \, dx-8 \text {Subst}\left (\int e^{4 x^2} x \, dx,x,\log (x)\right )-\int \left (-\frac {e^{8 \log ^2(x)}}{\left (-1+e^{x+4 \log ^2(x)}-e^{4 \log ^2(x)} x\right )^2}+\frac {e^{x+8 \log ^2(x)}}{\left (-1+e^{x+4 \log ^2(x)}-e^{4 \log ^2(x)} x\right )^2}-\frac {8 e^{8 \log ^2(x)} \log (x)}{\left (-1+e^{x+4 \log ^2(x)}-e^{4 \log ^2(x)} x\right )^2}+\frac {8 e^{x+8 \log ^2(x)} \log (x)}{x \left (-1+e^{x+4 \log ^2(x)}-e^{4 \log ^2(x)} x\right )^2}\right ) \, dx \\ & = -e^{4 \log ^2(x)}+x+8 \int \frac {e^{8 \log ^2(x)} \log (x)}{\left (-1+e^{x+4 \log ^2(x)}-e^{4 \log ^2(x)} x\right )^2} \, dx-8 \int \frac {e^{x+8 \log ^2(x)} \log (x)}{x \left (-1+e^{x+4 \log ^2(x)}-e^{4 \log ^2(x)} x\right )^2} \, dx-8 \int \frac {e^{8 \log ^2(x)} \log (x)}{-1+e^{x+4 \log ^2(x)}-e^{4 \log ^2(x)} x} \, dx+8 \int \frac {e^{x+8 \log ^2(x)} \log (x)}{x \left (-1+e^{x+4 \log ^2(x)}-e^{4 \log ^2(x)} x\right )} \, dx+\int \frac {e^{8 \log ^2(x)}}{\left (-1+e^{x+4 \log ^2(x)}-e^{4 \log ^2(x)} x\right )^2} \, dx-\int \frac {e^{x+8 \log ^2(x)}}{\left (-1+e^{x+4 \log ^2(x)}-e^{4 \log ^2(x)} x\right )^2} \, dx \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {x+e^{2 x} x+e^{-8 \log ^2(x)} x+x^3+e^x \left (-x-2 x^2\right )+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2-8 \log (x)\right )}{e^{2 x} x+e^{-8 \log ^2(x)} x-2 e^x x^2+x^3+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2\right )} \, dx=x+\frac {e^{4 \log ^2(x)}}{-1+e^{x+4 \log ^2(x)}-e^{4 \log ^2(x)} x} \]
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Time = 9.42 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95
method | result | size |
risch | \(x -\frac {1}{{\mathrm e}^{-4 \ln \left (x \right )^{2}}+x -{\mathrm e}^{x}}\) | \(20\) |
parallelrisch | \(\frac {{\mathrm e}^{-4 \ln \left (x \right )^{2}} x -1+x^{2}-{\mathrm e}^{x} x}{{\mathrm e}^{-4 \ln \left (x \right )^{2}}+x -{\mathrm e}^{x}}\) | \(40\) |
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67 \[ \int \frac {x+e^{2 x} x+e^{-8 \log ^2(x)} x+x^3+e^x \left (-x-2 x^2\right )+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2-8 \log (x)\right )}{e^{2 x} x+e^{-8 \log ^2(x)} x-2 e^x x^2+x^3+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2\right )} \, dx=\frac {x^{2} + x e^{\left (-4 \, \log \left (x\right )^{2}\right )} - x e^{x} - 1}{x + e^{\left (-4 \, \log \left (x\right )^{2}\right )} - e^{x}} \]
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Time = 0.12 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {x+e^{2 x} x+e^{-8 \log ^2(x)} x+x^3+e^x \left (-x-2 x^2\right )+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2-8 \log (x)\right )}{e^{2 x} x+e^{-8 \log ^2(x)} x-2 e^x x^2+x^3+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2\right )} \, dx=x + \frac {1}{- x + e^{x} - e^{- 4 \log {\left (x \right )}^{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86 \[ \int \frac {x+e^{2 x} x+e^{-8 \log ^2(x)} x+x^3+e^x \left (-x-2 x^2\right )+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2-8 \log (x)\right )}{e^{2 x} x+e^{-8 \log ^2(x)} x-2 e^x x^2+x^3+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2\right )} \, dx=\frac {{\left (x^{2} - x e^{x} - 1\right )} e^{\left (4 \, \log \left (x\right )^{2}\right )} + x}{{\left (x - e^{x}\right )} e^{\left (4 \, \log \left (x\right )^{2}\right )} + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 607 vs. \(2 (19) = 38\).
Time = 0.33 (sec) , antiderivative size = 607, normalized size of antiderivative = 28.90 \[ \int \frac {x+e^{2 x} x+e^{-8 \log ^2(x)} x+x^3+e^x \left (-x-2 x^2\right )+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2-8 \log (x)\right )}{e^{2 x} x+e^{-8 \log ^2(x)} x-2 e^x x^2+x^3+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2\right )} \, dx=\frac {8 \, x^{4} e^{\left (4 \, \log \left (x\right )^{2}\right )} \log \left (x\right ) + x^{4} e^{\left (4 \, \log \left (x\right )^{2}\right )} - x^{4} e^{\left (4 \, \log \left (x\right )^{2} + x\right )} - 24 \, x^{3} e^{\left (4 \, \log \left (x\right )^{2} + x\right )} \log \left (x\right ) + 2 \, x^{3} e^{\left (4 \, \log \left (x\right )^{2} + 2 \, x\right )} - 2 \, x^{3} e^{\left (4 \, \log \left (x\right )^{2} + x\right )} - 2 \, x^{3} e^{x} + 16 \, x^{3} \log \left (x\right ) - 8 \, x^{2} e^{\left (4 \, \log \left (x\right )^{2}\right )} \log \left (x\right ) + 8 \, x^{2} e^{\left (-4 \, \log \left (x\right )^{2}\right )} \log \left (x\right ) + 24 \, x^{2} e^{\left (4 \, \log \left (x\right )^{2} + 2 \, x\right )} \log \left (x\right ) - 32 \, x^{2} e^{x} \log \left (x\right ) + 2 \, x^{3} - x^{2} e^{\left (4 \, \log \left (x\right )^{2}\right )} + x^{2} e^{\left (-4 \, \log \left (x\right )^{2}\right )} - x^{2} e^{\left (4 \, \log \left (x\right )^{2} + 3 \, x\right )} + x^{2} e^{\left (4 \, \log \left (x\right )^{2} + 2 \, x\right )} + x^{2} e^{\left (4 \, \log \left (x\right )^{2} + x\right )} - x^{2} e^{\left (-4 \, \log \left (x\right )^{2} + x\right )} + 2 \, x^{2} e^{\left (2 \, x\right )} - 2 \, x^{2} e^{x} - 8 \, x e^{\left (4 \, \log \left (x\right )^{2} + 3 \, x\right )} \log \left (x\right ) + 16 \, x e^{\left (4 \, \log \left (x\right )^{2} + x\right )} \log \left (x\right ) - 8 \, x e^{\left (-4 \, \log \left (x\right )^{2} + x\right )} \log \left (x\right ) + 16 \, x e^{\left (2 \, x\right )} \log \left (x\right ) - x e^{\left (4 \, \log \left (x\right )^{2} + 2 \, x\right )} + x e^{\left (4 \, \log \left (x\right )^{2} + x\right )} + x e^{x} - 8 \, x \log \left (x\right ) - 8 \, e^{\left (4 \, \log \left (x\right )^{2} + 2 \, x\right )} \log \left (x\right ) + 8 \, e^{x} \log \left (x\right ) - x}{8 \, x^{3} e^{\left (4 \, \log \left (x\right )^{2}\right )} \log \left (x\right ) + x^{3} e^{\left (4 \, \log \left (x\right )^{2}\right )} - x^{3} e^{\left (4 \, \log \left (x\right )^{2} + x\right )} - 24 \, x^{2} e^{\left (4 \, \log \left (x\right )^{2} + x\right )} \log \left (x\right ) + 2 \, x^{2} e^{\left (4 \, \log \left (x\right )^{2} + 2 \, x\right )} - 2 \, x^{2} e^{\left (4 \, \log \left (x\right )^{2} + x\right )} - 2 \, x^{2} e^{x} + 16 \, x^{2} \log \left (x\right ) + 8 \, x e^{\left (-4 \, \log \left (x\right )^{2}\right )} \log \left (x\right ) + 24 \, x e^{\left (4 \, \log \left (x\right )^{2} + 2 \, x\right )} \log \left (x\right ) - 32 \, x e^{x} \log \left (x\right ) + 2 \, x^{2} + x e^{\left (-4 \, \log \left (x\right )^{2}\right )} - x e^{\left (4 \, \log \left (x\right )^{2} + 3 \, x\right )} + x e^{\left (4 \, \log \left (x\right )^{2} + 2 \, x\right )} - x e^{\left (-4 \, \log \left (x\right )^{2} + x\right )} + 2 \, x e^{\left (2 \, x\right )} - 2 \, x e^{x} - 8 \, e^{\left (4 \, \log \left (x\right )^{2} + 3 \, x\right )} \log \left (x\right ) - 8 \, e^{\left (-4 \, \log \left (x\right )^{2} + x\right )} \log \left (x\right ) + 16 \, e^{\left (2 \, x\right )} \log \left (x\right )} \]
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Time = 9.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {x+e^{2 x} x+e^{-8 \log ^2(x)} x+x^3+e^x \left (-x-2 x^2\right )+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2-8 \log (x)\right )}{e^{2 x} x+e^{-8 \log ^2(x)} x-2 e^x x^2+x^3+e^{-4 \log ^2(x)} \left (-2 e^x x+2 x^2\right )} \, dx=x-\frac {1}{x+{\mathrm {e}}^{-4\,{\ln \left (x\right )}^2}-{\mathrm {e}}^x} \]
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