Integrand size = 165, antiderivative size = 26 \[ \int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx=\frac {4}{5+\left (2+e^{12+x^2}\right )^2+\frac {5}{x^2}-\log (x)} \]
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\[ \int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx=\int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {4 x \left (10+x^2-4 e^{12+x^2} \left (2+e^{12+x^2}\right ) x^4\right )}{\left (5+\left (9+4 e^{12+x^2}+e^{24+2 x^2}\right ) x^2-x^2 \log (x)\right )^2} \, dx \\ & = 4 \int \frac {x \left (10+x^2-4 e^{12+x^2} \left (2+e^{12+x^2}\right ) x^4\right )}{\left (5+\left (9+4 e^{12+x^2}+e^{24+2 x^2}\right ) x^2-x^2 \log (x)\right )^2} \, dx \\ & = 4 \int \left (-\frac {4 x^3}{5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)}+\frac {x \left (10+21 x^2+36 x^4+8 e^{12+x^2} x^4-4 x^4 \log (x)\right )}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2}\right ) \, dx \\ & = 4 \int \frac {x \left (10+21 x^2+36 x^4+8 e^{12+x^2} x^4-4 x^4 \log (x)\right )}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2} \, dx-16 \int \frac {x^3}{5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)} \, dx \\ & = 4 \int \left (\frac {10 x}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2}+\frac {21 x^3}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2}+\frac {36 x^5}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2}+\frac {8 e^{12+x^2} x^5}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2}-\frac {4 x^5 \log (x)}{\left (-5-9 x^2-4 e^{12+x^2} x^2-e^{24+2 x^2} x^2+x^2 \log (x)\right )^2}\right ) \, dx-16 \int \frac {x^3}{5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)} \, dx \\ & = -\left (16 \int \frac {x^3}{5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)} \, dx\right )-16 \int \frac {x^5 \log (x)}{\left (-5-9 x^2-4 e^{12+x^2} x^2-e^{24+2 x^2} x^2+x^2 \log (x)\right )^2} \, dx+32 \int \frac {e^{12+x^2} x^5}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2} \, dx+40 \int \frac {x}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2} \, dx+84 \int \frac {x^3}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2} \, dx+144 \int \frac {x^5}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2} \, dx \\ \end{align*}
Time = 0.73 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx=\frac {4 x^2}{5+\left (9+4 e^{12+x^2}+e^{24+2 x^2}\right ) x^2-x^2 \log (x)} \]
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Time = 0.63 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73
method | result | size |
risch | \(\frac {4 x^{2}}{{\mathrm e}^{2 x^{2}+24} x^{2}+4 \,{\mathrm e}^{x^{2}+12} x^{2}-x^{2} \ln \left (x \right )+9 x^{2}+5}\) | \(45\) |
parallelrisch | \(-\frac {4 x^{2}}{-{\mathrm e}^{2 x^{2}+24} x^{2}+x^{2} \ln \left (x \right )-4 \,{\mathrm e}^{x^{2}+12} x^{2}-9 x^{2}-5}\) | \(45\) |
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Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx=\frac {4 \, x^{2}}{x^{2} e^{\left (2 \, x^{2} + 24\right )} + 4 \, x^{2} e^{\left (x^{2} + 12\right )} - x^{2} \log \left (x\right ) + 9 \, x^{2} + 5} \]
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx=\frac {4 x^{2}}{4 x^{2} e^{x^{2} + 12} + x^{2} e^{2 x^{2} + 24} - x^{2} \log {\left (x \right )} + 9 x^{2} + 5} \]
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Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx=\frac {4 \, x^{2}}{x^{2} e^{\left (2 \, x^{2} + 24\right )} + 4 \, x^{2} e^{\left (x^{2} + 12\right )} - x^{2} \log \left (x\right ) + 9 \, x^{2} + 5} \]
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Time = 0.89 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx=\frac {8 \, x^{2}}{x^{2} e^{\left (2 \, x^{2} + 24\right )} + 4 \, x^{2} e^{\left (x^{2} + 12\right )} - x^{2} \log \left (x\right ) + 9 \, x^{2} + 5} \]
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Timed out. \[ \int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx=\int \frac {40\,x-16\,x^5\,{\mathrm {e}}^{2\,x^2+24}-32\,x^5\,{\mathrm {e}}^{x^2+12}+4\,x^3}{{\mathrm {e}}^{2\,x^2+24}\,\left (34\,x^4+10\,x^2\right )-\ln \left (x\right )\,\left (2\,x^4\,{\mathrm {e}}^{2\,x^2+24}+8\,x^4\,{\mathrm {e}}^{x^2+12}+10\,x^2+18\,x^4\right )+8\,x^4\,{\mathrm {e}}^{3\,x^2+36}+x^4\,{\mathrm {e}}^{4\,x^2+48}+x^4\,{\ln \left (x\right )}^2+{\mathrm {e}}^{x^2+12}\,\left (72\,x^4+40\,x^2\right )+90\,x^2+81\,x^4+25} \,d x \]
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