Integrand size = 130, antiderivative size = 29 \[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=\log \left (\frac {x}{1+\left (-e^{5-x^2}-x+x^2\right )^2+\log (x)}\right ) \]
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\[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=\int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-x^2+4 x^3-3 x^4+\log (x)}{x \left (1+x^2-2 x^3+x^4+\log (x)\right )}+\frac {e^5 \left (e^5+6 e^5 x^2+2 e^{x^2} x^2-6 e^5 x^3+6 e^{x^2} x^3+8 e^5 x^4-12 e^{x^2} x^4-8 e^5 x^5+14 e^{x^2} x^5+4 e^5 x^6-16 e^{x^2} x^6+12 e^{x^2} x^7-4 e^{x^2} x^8-2 e^{x^2} x \log (x)+4 e^5 x^2 \log (x)+4 e^{x^2} x^2 \log (x)+4 e^{x^2} x^3 \log (x)-4 e^{x^2} x^4 \log (x)\right )}{x \left (1+x^2-2 x^3+x^4+\log (x)\right ) \left (e^{10}+e^{2 x^2}+2 e^{5+x^2} x+e^{2 x^2} x^2-2 e^{5+x^2} x^2-2 e^{2 x^2} x^3+e^{2 x^2} x^4+e^{2 x^2} \log (x)\right )}\right ) \, dx \\ & = e^5 \int \frac {e^5+6 e^5 x^2+2 e^{x^2} x^2-6 e^5 x^3+6 e^{x^2} x^3+8 e^5 x^4-12 e^{x^2} x^4-8 e^5 x^5+14 e^{x^2} x^5+4 e^5 x^6-16 e^{x^2} x^6+12 e^{x^2} x^7-4 e^{x^2} x^8-2 e^{x^2} x \log (x)+4 e^5 x^2 \log (x)+4 e^{x^2} x^2 \log (x)+4 e^{x^2} x^3 \log (x)-4 e^{x^2} x^4 \log (x)}{x \left (1+x^2-2 x^3+x^4+\log (x)\right ) \left (e^{10}+e^{2 x^2}+2 e^{5+x^2} x+e^{2 x^2} x^2-2 e^{5+x^2} x^2-2 e^{2 x^2} x^3+e^{2 x^2} x^4+e^{2 x^2} \log (x)\right )} \, dx+\int \frac {-x^2+4 x^3-3 x^4+\log (x)}{x \left (1+x^2-2 x^3+x^4+\log (x)\right )} \, dx \\ & = e^5 \int \frac {-2 e^{x^2} x^2 \left (-1-3 x+6 x^2-7 x^3+8 x^4-6 x^5+2 x^6\right )+e^5 \left (1+6 x^2-6 x^3+8 x^4-8 x^5+4 x^6\right )-2 x \left (-2 e^5 x+e^{x^2} \left (1-2 x-2 x^2+2 x^3\right )\right ) \log (x)}{x \left (1+x^2-2 x^3+x^4+\log (x)\right ) \left (e^{10}-2 e^{5+x^2} (-1+x) x+e^{2 x^2} \left (1+x^2-2 x^3+x^4\right )+e^{2 x^2} \log (x)\right )} \, dx+\int \left (\frac {1}{x}+\frac {-1-2 x^2+6 x^3-4 x^4}{x \left (1+x^2-2 x^3+x^4+\log (x)\right )}\right ) \, dx \\ & = \text {Too large to display} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(88\) vs. \(2(29)=58\).
Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.03 \[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=2 x^2+\log (x)-\log \left (e^{10}+e^{2 x^2}+2 e^{5+x^2} x+e^{2 x^2} x^2-2 e^{5+x^2} x^2-2 e^{2 x^2} x^3+e^{2 x^2} x^4+e^{2 x^2} \log (x)\right ) \]
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Time = 1.48 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86
method | result | size |
risch | \(\ln \left (x \right )-\ln \left (x^{4}-2 x^{3}+x^{2}-2 x^{2} {\mathrm e}^{-x^{2}+5}+2 x \,{\mathrm e}^{-x^{2}+5}+1+{\mathrm e}^{-2 x^{2}+10}+\ln \left (x \right )\right )\) | \(54\) |
parallelrisch | \(2 x^{2}-\ln \left ({\mathrm e}^{2 x^{2}} x^{4}-2 \,{\mathrm e}^{2 x^{2}} x^{3}-2 x^{2} {\mathrm e}^{5} {\mathrm e}^{x^{2}}+{\mathrm e}^{2 x^{2}} x^{2}+2 \,{\mathrm e}^{5} {\mathrm e}^{x^{2}} x +{\mathrm e}^{2 x^{2}} \ln \left (x \right )+{\mathrm e}^{10}+{\mathrm e}^{2 x^{2}}\right )+\ln \left (x \right )\) | \(83\) |
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (28) = 56\).
Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.28 \[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=-\log \left ({\left ({\left (x^{4} - 2 \, x^{3} + x^{2} + 1\right )} e^{\left (2 \, x^{2} + 10\right )} - 2 \, {\left (x^{2} - x\right )} e^{\left (x^{2} + 15\right )} + e^{\left (2 \, x^{2} + 10\right )} \log \left (x\right ) + e^{20}\right )} e^{\left (-2 \, x^{2} - 10\right )}\right ) + \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (22) = 44\).
Time = 4.37 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.17 \[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=2 x^{2} + \log {\left (x \right )} - \log {\left (\frac {\left (- 2 x^{2} e^{5} + 2 x e^{5}\right ) e^{x^{2}}}{x^{4} - 2 x^{3} + x^{2} + \log {\left (x \right )} + 1} + e^{2 x^{2}} + \frac {e^{10}}{x^{4} - 2 x^{3} + x^{2} + \log {\left (x \right )} + 1} \right )} - \log {\left (x^{4} - 2 x^{3} + x^{2} + \log {\left (x \right )} + 1 \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (28) = 56\).
Time = 0.23 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.10 \[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=2 \, x^{2} - \log \left (x^{4} - 2 \, x^{3} + x^{2} + \log \left (x\right ) + 1\right ) + \log \left (x\right ) - \log \left (\frac {{\left (x^{4} - 2 \, x^{3} + x^{2} + \log \left (x\right ) + 1\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{2} e^{5} - x e^{5}\right )} e^{\left (x^{2}\right )} + e^{10}}{x^{4} - 2 \, x^{3} + x^{2} + \log \left (x\right ) + 1}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (28) = 56\).
Time = 0.35 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.14 \[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=2 \, x^{2} - \log \left (x^{4} e^{\left (2 \, x^{2}\right )} - 2 \, x^{3} e^{\left (2 \, x^{2}\right )} + x^{2} e^{\left (2 \, x^{2}\right )} - 2 \, x^{2} e^{\left (x^{2} + 5\right )} + 2 \, x e^{\left (x^{2} + 5\right )} + e^{\left (2 \, x^{2}\right )} \log \left (x\right ) + e^{10} + e^{\left (2 \, x^{2}\right )}\right ) + \log \left (x^{4} - 2 \, x^{3} + x^{2} + \log \left (x\right ) + 1\right ) - \log \left (-x^{4} + 2 \, x^{3} - x^{2} - \log \left (x\right ) - 1\right ) + \log \left (x\right ) \]
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Timed out. \[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=\int \frac {{\mathrm {e}}^{10}\,\left (4\,x^2+1\right )-{\mathrm {e}}^{2\,x^2}\,\left (3\,x^4-4\,x^3+x^2\right )+{\mathrm {e}}^{2\,x^2}\,\ln \left (x\right )+{\mathrm {e}}^{x^2+5}\,\left (-4\,x^4+4\,x^3+2\,x^2\right )}{x\,{\mathrm {e}}^{10}+{\mathrm {e}}^{2\,x^2}\,\left (x^5-2\,x^4+x^3+x\right )+{\mathrm {e}}^{x^2+5}\,\left (2\,x^2-2\,x^3\right )+x\,{\mathrm {e}}^{2\,x^2}\,\ln \left (x\right )} \,d x \]
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