Integrand size = 147, antiderivative size = 26 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{e^5 x \left (5+x+\frac {1+x}{e^{4 x}+x+x^2}\right )} \]
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\[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=\int \frac {\exp \left (\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}\right ) \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{\left (e^{4 x}+x+x^2\right )^2} \, dx \\ & = \int \left (\exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) (5+2 x)-\frac {\exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) \left (-1+2 x+4 x^2\right )}{e^{4 x}+x+x^2}+\frac {\exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x \left (-1+x+6 x^2+4 x^3\right )}{\left (e^{4 x}+x+x^2\right )^2}\right ) \, dx \\ & = \int \exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) (5+2 x) \, dx-\int \frac {\exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) \left (-1+2 x+4 x^2\right )}{e^{4 x}+x+x^2} \, dx+\int \frac {\exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x \left (-1+x+6 x^2+4 x^3\right )}{\left (e^{4 x}+x+x^2\right )^2} \, dx \\ & = \int \left (5 \exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right )+2 \exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x\right ) \, dx+\int \left (-\frac {\exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x}{\left (e^{4 x}+x+x^2\right )^2}+\frac {\exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x^2}{\left (e^{4 x}+x+x^2\right )^2}+\frac {6 \exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x^3}{\left (e^{4 x}+x+x^2\right )^2}+\frac {4 \exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x^4}{\left (e^{4 x}+x+x^2\right )^2}\right ) \, dx-\int \left (-\frac {\exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right )}{e^{4 x}+x+x^2}+\frac {2 \exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x}{e^{4 x}+x+x^2}+\frac {4 \exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x^2}{e^{4 x}+x+x^2}\right ) \, dx \\ & = 2 \int \exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x \, dx-2 \int \frac {\exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x}{e^{4 x}+x+x^2} \, dx+4 \int \frac {\exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x^4}{\left (e^{4 x}+x+x^2\right )^2} \, dx-4 \int \frac {\exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x^2}{e^{4 x}+x+x^2} \, dx+5 \int \exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) \, dx+6 \int \frac {\exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x^3}{\left (e^{4 x}+x+x^2\right )^2} \, dx-\int \frac {\exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x}{\left (e^{4 x}+x+x^2\right )^2} \, dx+\int \frac {\exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right ) x^2}{\left (e^{4 x}+x+x^2\right )^2} \, dx+\int \frac {\exp \left (5+\frac {e^5 x \left (1+5 e^{4 x}+6 x+e^{4 x} x+6 x^2+x^3\right )}{e^{4 x}+x+x^2}\right )}{e^{4 x}+x+x^2} \, dx \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{5 e^5 x+e^5 x^2+\frac {e^5 x+e^5 x^2}{e^{4 x}+x+x^2}} \]
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Time = 5.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {{\mathrm e}^{5} x \left (x^{3}+x \,{\mathrm e}^{4 x}+6 x^{2}+5 \,{\mathrm e}^{4 x}+6 x +1\right )}{{\mathrm e}^{4 x}+x^{2}+x}}\) | \(42\) |
risch | \({\mathrm e}^{\frac {x \left (x^{3} {\mathrm e}^{5}+6 x^{2} {\mathrm e}^{5}+{\mathrm e}^{5+4 x} x +6 x \,{\mathrm e}^{5}+5 \,{\mathrm e}^{5+4 x}+{\mathrm e}^{5}\right )}{{\mathrm e}^{4 x}+x^{2}+x}}\) | \(52\) |
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Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (23) = 46\).
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.00 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{\left (\frac {{\left (x^{4} + 6 \, x^{3} + 6 \, x^{2} + x\right )} e^{10} + {\left (x^{2} + 5 \, x\right )} e^{\left (4 \, x + 10\right )}}{{\left (x^{2} + x\right )} e^{5} + e^{\left (4 \, x + 5\right )}}\right )} \]
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Time = 0.34 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{\frac {\left (x^{2} + 5 x\right ) e^{5} e^{4 x} + \left (x^{4} + 6 x^{3} + 6 x^{2} + x\right ) e^{5}}{x^{2} + x + e^{4 x}}} \]
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Time = 0.51 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{\left (x^{2} e^{5} + 5 \, x e^{5} - \frac {e^{\left (4 \, x + 5\right )}}{x^{2} + x + e^{\left (4 \, x\right )}} + e^{5}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (23) = 46\).
Time = 0.40 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.81 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{\left (\frac {x^{4} e^{5} + 6 \, x^{3} e^{5} + 6 \, x^{2} e^{5} + x^{2} e^{\left (4 \, x + 5\right )} + 5 \, x^{2} + x e^{5} + 5 \, x e^{\left (4 \, x + 5\right )} + 5 \, x + 5 \, e^{\left (4 \, x\right )}}{x^{2} + x + e^{\left (4 \, x\right )}} - 5\right )} \]
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Time = 11.67 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.46 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx={\mathrm {e}}^{\frac {5\,x\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}}\,{\mathrm {e}}^{\frac {x^4\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}}\,{\mathrm {e}}^{\frac {6\,x^2\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}}\,{\mathrm {e}}^{\frac {6\,x^3\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}}\,{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}} \]
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