Integrand size = 246, antiderivative size = 21 \[ \int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx=\left (-1+\frac {x}{x+\left (e^{x (4+x)}+x\right )^2}\right )^2 \]
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\[ \int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx=\int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (e^{x (4+x)}+x\right )^3 \left (x+e^{x (4+x)} \left (-1+8 x+4 x^2\right )\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx \\ & = 2 \int \frac {\left (e^{x (4+x)}+x\right )^3 \left (x+e^{x (4+x)} \left (-1+8 x+4 x^2\right )\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx \\ & = 2 \int \left (\frac {-1+8 x+4 x^2}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2}-\frac {2 x \left (-1-e^{x (4+x)}+7 x+4 e^{x (4+x)} x+8 x^2+2 e^{x (4+x)} x^2+2 x^3\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}+\frac {x^2 \left (-1-2 e^{x (4+x)}+6 x+8 e^{x (4+x)} x+12 x^2+4 e^{x (4+x)} x^2+4 x^3\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3}\right ) \, dx \\ & = 2 \int \frac {-1+8 x+4 x^2}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2} \, dx+2 \int \frac {x^2 \left (-1-2 e^{x (4+x)}+6 x+8 e^{x (4+x)} x+12 x^2+4 e^{x (4+x)} x^2+4 x^3\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx-4 \int \frac {x \left (-1-e^{x (4+x)}+7 x+4 e^{x (4+x)} x+8 x^2+2 e^{x (4+x)} x^2+2 x^3\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx \\ & = 2 \int \left (-\frac {1}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2}+\frac {8 x}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2}+\frac {4 x^2}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2}\right ) \, dx+2 \int \frac {x^2 \left (-1+6 x+12 x^2+4 x^3+e^{x (4+x)} \left (-2+8 x+4 x^2\right )\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx-4 \int \frac {x \left (-1+7 x+8 x^2+2 x^3+e^{x (4+x)} \left (-1+4 x+2 x^2\right )\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx \\ & = -\left (2 \int \frac {1}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2} \, dx\right )+2 \int \left (\frac {2 e^{4 x+x^2} x^2}{\left (-e^{2 x (4+x)}-x-2 e^{x (4+x)} x-x^2\right )^3}-\frac {x^2}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3}+\frac {6 x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3}+\frac {8 e^{4 x+x^2} x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3}+\frac {12 x^4}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3}+\frac {4 e^{4 x+x^2} x^4}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3}+\frac {4 x^5}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3}\right ) \, dx-4 \int \left (-\frac {x}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}-\frac {e^{4 x+x^2} x}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}+\frac {7 x^2}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}+\frac {4 e^{4 x+x^2} x^2}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}+\frac {8 x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}+\frac {2 e^{4 x+x^2} x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}+\frac {2 x^4}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}\right ) \, dx+8 \int \frac {x^2}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2} \, dx+16 \int \frac {x}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2} \, dx \\ & = -\left (2 \int \frac {x^2}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx\right )-2 \int \frac {1}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2} \, dx+4 \int \frac {e^{4 x+x^2} x^2}{\left (-e^{2 x (4+x)}-x-2 e^{x (4+x)} x-x^2\right )^3} \, dx+4 \int \frac {x}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx+4 \int \frac {e^{4 x+x^2} x}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx+8 \int \frac {e^{4 x+x^2} x^4}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx+8 \int \frac {x^5}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx-8 \int \frac {e^{4 x+x^2} x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx-8 \int \frac {x^4}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx+8 \int \frac {x^2}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2} \, dx+12 \int \frac {x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx+16 \int \frac {e^{4 x+x^2} x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx-16 \int \frac {e^{4 x+x^2} x^2}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx+16 \int \frac {x}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2} \, dx+24 \int \frac {x^4}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx-28 \int \frac {x^2}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx-32 \int \frac {x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(21)=42\).
Time = 3.76 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.62 \[ \int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx=-\frac {x \left (2 e^{2 x (4+x)}+x+4 e^{x (4+x)} x+2 x^2\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(20)=40\).
Time = 0.16 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.48
method | result | size |
risch | \(-\frac {\left (2 x^{2}+4 x \,{\mathrm e}^{\left (4+x \right ) x}+2 \,{\mathrm e}^{2 \left (4+x \right ) x}+x \right ) x}{\left ({\mathrm e}^{2 \left (4+x \right ) x}+2 x \,{\mathrm e}^{\left (4+x \right ) x}+x^{2}+x \right )^{2}}\) | \(52\) |
parallelrisch | \(\frac {-2 x^{3}-4 x^{2} {\mathrm e}^{x^{2}+4 x}-2 \,{\mathrm e}^{2 x^{2}+8 x} x -x^{2}}{x^{4}+4 \,{\mathrm e}^{x^{2}+4 x} x^{3}+6 \,{\mathrm e}^{2 x^{2}+8 x} x^{2}+4 \,{\mathrm e}^{3 x^{2}+12 x} x +{\mathrm e}^{4 x^{2}+16 x}+2 x^{3}+4 x^{2} {\mathrm e}^{x^{2}+4 x}+2 \,{\mathrm e}^{2 x^{2}+8 x} x +x^{2}}\) | \(130\) |
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Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 110, normalized size of antiderivative = 5.24 \[ \int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx=-\frac {2 \, x^{3} + 4 \, x^{2} e^{\left (x^{2} + 4 \, x\right )} + x^{2} + 2 \, x e^{\left (2 \, x^{2} + 8 \, x\right )}}{x^{4} + 2 \, x^{3} + x^{2} + 4 \, x e^{\left (3 \, x^{2} + 12 \, x\right )} + 2 \, {\left (3 \, x^{2} + x\right )} e^{\left (2 \, x^{2} + 8 \, x\right )} + 4 \, {\left (x^{3} + x^{2}\right )} e^{\left (x^{2} + 4 \, x\right )} + e^{\left (4 \, x^{2} + 16 \, x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (15) = 30\).
Time = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 5.19 \[ \int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx=\frac {- 2 x^{3} - 4 x^{2} e^{x^{2} + 4 x} - x^{2} - 2 x e^{2 x^{2} + 8 x}}{x^{4} + 2 x^{3} + x^{2} + 4 x e^{3 x^{2} + 12 x} + \left (6 x^{2} + 2 x\right ) e^{2 x^{2} + 8 x} + \left (4 x^{3} + 4 x^{2}\right ) e^{x^{2} + 4 x} + e^{4 x^{2} + 16 x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (20) = 40\).
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 5.24 \[ \int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx=-\frac {2 \, x^{3} + 4 \, x^{2} e^{\left (x^{2} + 4 \, x\right )} + x^{2} + 2 \, x e^{\left (2 \, x^{2} + 8 \, x\right )}}{x^{4} + 2 \, x^{3} + x^{2} + 4 \, x e^{\left (3 \, x^{2} + 12 \, x\right )} + 2 \, {\left (3 \, x^{2} + x\right )} e^{\left (2 \, x^{2} + 8 \, x\right )} + 4 \, {\left (x^{3} + x^{2}\right )} e^{\left (x^{2} + 4 \, x\right )} + e^{\left (4 \, x^{2} + 16 \, x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (20) = 40\).
Time = 0.66 (sec) , antiderivative size = 128, normalized size of antiderivative = 6.10 \[ \int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx=-\frac {2 \, {\left (2 \, x^{3} + 4 \, x^{2} e^{\left (x^{2} + 4 \, x\right )} + x^{2} + 2 \, x e^{\left (2 \, x^{2} + 8 \, x\right )}\right )}}{x^{4} + 4 \, x^{3} e^{\left (x^{2} + 4 \, x\right )} + 2 \, x^{3} + 6 \, x^{2} e^{\left (2 \, x^{2} + 8 \, x\right )} + 4 \, x^{2} e^{\left (x^{2} + 4 \, x\right )} + x^{2} + 4 \, x e^{\left (3 \, x^{2} + 12 \, x\right )} + 2 \, x e^{\left (2 \, x^{2} + 8 \, x\right )} + e^{\left (4 \, x^{2} + 16 \, x\right )}} \]
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Timed out. \[ \int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx=\int \frac {{\mathrm {e}}^{4\,x^2+16\,x}\,\left (8\,x^2+16\,x-2\right )+{\mathrm {e}}^{3\,x^2+12\,x}\,\left (24\,x^3+48\,x^2-4\,x\right )+{\mathrm {e}}^{x^2+4\,x}\,\left (8\,x^5+16\,x^4+4\,x^3\right )+2\,x^4+{\mathrm {e}}^{2\,x^2+8\,x}\,\left (24\,x^4+48\,x^3\right )}{{\mathrm {e}}^{6\,x^2+24\,x}+{\mathrm {e}}^{2\,x^2+8\,x}\,\left (15\,x^4+18\,x^3+3\,x^2\right )+{\mathrm {e}}^{4\,x^2+16\,x}\,\left (15\,x^2+3\,x\right )+6\,x\,{\mathrm {e}}^{5\,x^2+20\,x}+{\mathrm {e}}^{x^2+4\,x}\,\left (6\,x^5+12\,x^4+6\,x^3\right )+x^3+3\,x^4+3\,x^5+x^6+{\mathrm {e}}^{3\,x^2+12\,x}\,\left (20\,x^3+12\,x^2\right )} \,d x \]
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