Integrand size = 120, antiderivative size = 27 \[ \int \frac {18+9 x+8 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6+6 x-12 x^2+4 x^3+6 x^4\right )}{9 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6 x^3+2 x^4\right )} \, dx=x+\frac {-\frac {3}{x}+x}{x \left (3+e^{(2-2 x) x}+x\right )} \]
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\[ \int \frac {18+9 x+8 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6+6 x-12 x^2+4 x^3+6 x^4\right )}{9 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6 x^3+2 x^4\right )} \, dx=\int \frac {18+9 x+8 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6+6 x-12 x^2+4 x^3+6 x^4\right )}{9 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6 x^3+2 x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{4 x} x^3+2 e^{2 x (1+x)} \left (3+3 x-6 x^2+2 x^3+3 x^4\right )+e^{4 x^2} \left (18+9 x+8 x^3+6 x^4+x^5\right )}{x^3 \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx \\ & = \int \left (\frac {2 e^{2 x (1+x)} \left (3+3 x-6 x^2+2 x^3+3 x^4\right )}{x^3 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2}+\frac {18 e^{4 x^2}+9 e^{4 x^2} x+e^{4 x} x^3+8 e^{4 x^2} x^3+6 e^{4 x^2} x^4+e^{4 x^2} x^5}{x^3 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2}\right ) \, dx \\ & = 2 \int \frac {e^{2 x (1+x)} \left (3+3 x-6 x^2+2 x^3+3 x^4\right )}{x^3 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2} \, dx+\int \frac {18 e^{4 x^2}+9 e^{4 x^2} x+e^{4 x} x^3+8 e^{4 x^2} x^3+6 e^{4 x^2} x^4+e^{4 x^2} x^5}{x^3 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2} \, dx \\ & = 2 \int \frac {e^{2 x (1+x)} \left (3+3 x-6 x^2+2 x^3+3 x^4\right )}{x^3 \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+\int \frac {e^{4 x} x^3+e^{4 x^2} \left (18+9 x+8 x^3+6 x^4+x^5\right )}{x^3 \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx \\ & = 2 \int \left (\frac {2 e^{2 x (1+x)}}{\left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2}+\frac {3 e^{2 x (1+x)}}{x^3 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2}+\frac {3 e^{2 x (1+x)}}{x^2 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2}-\frac {6 e^{2 x (1+x)}}{x \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2}+\frac {3 e^{2 x (1+x)} x}{\left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2}\right ) \, dx+\int \left (\frac {(2+x) \left (9+4 x^3+x^4\right )}{x^3 (3+x)^2}-\frac {2 e^{2 x} \left (18+9 x+8 x^3+6 x^4+x^5\right )}{x^3 (3+x)^2 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )}+\frac {e^{4 x} \left (18+9 x+17 x^3+12 x^4+2 x^5\right )}{x^3 (3+x)^2 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2}\right ) \, dx \\ & = -\left (2 \int \frac {e^{2 x} \left (18+9 x+8 x^3+6 x^4+x^5\right )}{x^3 (3+x)^2 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )} \, dx\right )+4 \int \frac {e^{2 x (1+x)}}{\left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2} \, dx+6 \int \frac {e^{2 x (1+x)}}{x^3 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2} \, dx+6 \int \frac {e^{2 x (1+x)}}{x^2 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2} \, dx+6 \int \frac {e^{2 x (1+x)} x}{\left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2} \, dx-12 \int \frac {e^{2 x (1+x)}}{x \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2} \, dx+\int \frac {(2+x) \left (9+4 x^3+x^4\right )}{x^3 (3+x)^2} \, dx+\int \frac {e^{4 x} \left (18+9 x+17 x^3+12 x^4+2 x^5\right )}{x^3 (3+x)^2 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2} \, dx \\ & = -\left (2 \int \frac {e^{2 x} \left (18+9 x+8 x^3+6 x^4+x^5\right )}{x^3 (3+x)^2 \left (e^{2 x}+e^{2 x^2} (3+x)\right )} \, dx\right )+4 \int \frac {e^{2 x (1+x)}}{\left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+6 \int \frac {e^{2 x (1+x)}}{x^3 \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+6 \int \frac {e^{2 x (1+x)}}{x^2 \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+6 \int \frac {e^{2 x (1+x)} x}{\left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx-12 \int \frac {e^{2 x (1+x)}}{x \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+\int \left (1+\frac {2}{x^3}-\frac {1}{3 x^2}-\frac {2}{3 (3+x)^2}\right ) \, dx+\int \frac {e^{4 x} \left (18+9 x+17 x^3+12 x^4+2 x^5\right )}{x^3 (3+x)^2 \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx \\ & = -\frac {1}{x^2}+\frac {1}{3 x}+x+\frac {2}{3 (3+x)}-2 \int \left (\frac {e^{2 x}}{e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x}+\frac {2 e^{2 x}}{x^3 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )}-\frac {e^{2 x}}{3 x^2 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )}-\frac {2 e^{2 x}}{3 (3+x)^2 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )}\right ) \, dx+4 \int \frac {e^{2 x (1+x)}}{\left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+6 \int \frac {e^{2 x (1+x)}}{x^3 \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+6 \int \frac {e^{2 x (1+x)}}{x^2 \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+6 \int \frac {e^{2 x (1+x)} x}{\left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx-12 \int \frac {e^{2 x (1+x)}}{x \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+\int \left (\frac {2 e^{4 x}}{\left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2}+\frac {2 e^{4 x}}{x^3 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2}-\frac {e^{4 x}}{3 x^2 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2}-\frac {2 e^{4 x}}{3 (3+x)^2 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2}\right ) \, dx \\ & = -\frac {1}{x^2}+\frac {1}{3 x}+x+\frac {2}{3 (3+x)}-\frac {1}{3} \int \frac {e^{4 x}}{x^2 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2} \, dx-\frac {2}{3} \int \frac {e^{4 x}}{(3+x)^2 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2} \, dx+\frac {2}{3} \int \frac {e^{2 x}}{x^2 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )} \, dx+\frac {4}{3} \int \frac {e^{2 x}}{(3+x)^2 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )} \, dx+2 \int \frac {e^{4 x}}{\left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2} \, dx+2 \int \frac {e^{4 x}}{x^3 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )^2} \, dx-2 \int \frac {e^{2 x}}{e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x} \, dx-4 \int \frac {e^{2 x}}{x^3 \left (e^{2 x}+3 e^{2 x^2}+e^{2 x^2} x\right )} \, dx+4 \int \frac {e^{2 x (1+x)}}{\left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+6 \int \frac {e^{2 x (1+x)}}{x^3 \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+6 \int \frac {e^{2 x (1+x)}}{x^2 \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+6 \int \frac {e^{2 x (1+x)} x}{\left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx-12 \int \frac {e^{2 x (1+x)}}{x \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx \\ & = -\frac {1}{x^2}+\frac {1}{3 x}+x+\frac {2}{3 (3+x)}-\frac {1}{3} \int \frac {e^{4 x}}{x^2 \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx-\frac {2}{3} \int \frac {e^{4 x}}{(3+x)^2 \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+\frac {2}{3} \int \frac {e^{2 x}}{x^2 \left (e^{2 x}+e^{2 x^2} (3+x)\right )} \, dx+\frac {4}{3} \int \frac {e^{2 x}}{(3+x)^2 \left (e^{2 x}+e^{2 x^2} (3+x)\right )} \, dx+2 \int \frac {e^{4 x}}{\left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+2 \int \frac {e^{4 x}}{x^3 \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx-2 \int \frac {e^{2 x}}{e^{2 x}+e^{2 x^2} (3+x)} \, dx+4 \int \frac {e^{2 x (1+x)}}{\left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx-4 \int \frac {e^{2 x}}{x^3 \left (e^{2 x}+e^{2 x^2} (3+x)\right )} \, dx+6 \int \frac {e^{2 x (1+x)}}{x^3 \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+6 \int \frac {e^{2 x (1+x)}}{x^2 \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx+6 \int \frac {e^{2 x (1+x)} x}{\left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx-12 \int \frac {e^{2 x (1+x)}}{x \left (e^{2 x}+e^{2 x^2} (3+x)\right )^2} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(197\) vs. \(2(27)=54\).
Time = 14.58 (sec) , antiderivative size = 197, normalized size of antiderivative = 7.30 \[ \int \frac {18+9 x+8 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6+6 x-12 x^2+4 x^3+6 x^4\right )}{9 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6 x^3+2 x^4\right )} \, dx=\frac {1}{450} \left (\frac {540}{x^3}+\frac {3564}{x^2}+\frac {19440}{x}+\frac {45 (2263+760 x)}{\left (-5+10 x+4 x^2\right )^2}+\frac {4 (838+17 x)}{-5+10 x+4 x^2}\right )+\frac {e^{2 x} \left (-13500-35100 x-162000 x^2+1601885 x^3-1516290 x^4-1671312 x^5-284312 x^6+36000 x^7+7200 x^8\right )+e^{2 x^2} \left (-40500-152550 x-386100 x^2+4573905 x^3-3099985 x^4-6524826 x^5-2488248 x^6-169112 x^7+57600 x^8+7200 x^9\right )}{450 x^3 \left (-5+10 x+4 x^2\right )^2 \left (e^{2 x}+e^{2 x^2} (3+x)\right )} \]
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Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
risch | \(x +\frac {x^{2}-3}{x^{2} \left (3+{\mathrm e}^{-2 x \left (-1+x \right )}+x \right )}\) | \(24\) |
parallelrisch | \(\frac {x^{4}+{\mathrm e}^{-2 x^{2}+2 x} x^{3}-3+3 x^{3}+x^{2}}{x^{2} \left (3+{\mathrm e}^{-2 x^{2}+2 x}+x \right )}\) | \(47\) |
norman | \(\frac {-3+x^{4}-8 x^{2}+{\mathrm e}^{-2 x^{2}+2 x} x^{3}-3 \,{\mathrm e}^{-2 x^{2}+2 x} x^{2}}{x^{2} \left (3+{\mathrm e}^{-2 x^{2}+2 x}+x \right )}\) | \(59\) |
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {18+9 x+8 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6+6 x-12 x^2+4 x^3+6 x^4\right )}{9 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6 x^3+2 x^4\right )} \, dx=\frac {x^{4} + x^{3} e^{\left (-2 \, x^{2} + 2 \, x\right )} + 3 \, x^{3} + x^{2} - 3}{x^{3} + x^{2} e^{\left (-2 \, x^{2} + 2 \, x\right )} + 3 \, x^{2}} \]
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Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {18+9 x+8 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6+6 x-12 x^2+4 x^3+6 x^4\right )}{9 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6 x^3+2 x^4\right )} \, dx=x + \frac {x^{2} - 3}{x^{3} + x^{2} e^{- 2 x^{2} + 2 x} + 3 x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (25) = 50\).
Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.11 \[ \int \frac {18+9 x+8 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6+6 x-12 x^2+4 x^3+6 x^4\right )}{9 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6 x^3+2 x^4\right )} \, dx=\frac {x^{3} e^{\left (2 \, x\right )} + {\left (x^{4} + 3 \, x^{3} + x^{2} - 3\right )} e^{\left (2 \, x^{2}\right )}}{x^{2} e^{\left (2 \, x\right )} + {\left (x^{3} + 3 \, x^{2}\right )} e^{\left (2 \, x^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (25) = 50\).
Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {18+9 x+8 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6+6 x-12 x^2+4 x^3+6 x^4\right )}{9 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6 x^3+2 x^4\right )} \, dx=\frac {x^{4} + x^{3} e^{\left (-2 \, x^{2} + 2 \, x\right )} + 3 \, x^{3} + x^{2} - 3}{x^{3} + x^{2} e^{\left (-2 \, x^{2} + 2 \, x\right )} + 3 \, x^{2}} \]
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Time = 0.60 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {18+9 x+8 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6+6 x-12 x^2+4 x^3+6 x^4\right )}{9 x^3+e^{4 x-4 x^2} x^3+6 x^4+x^5+e^{2 x-2 x^2} \left (6 x^3+2 x^4\right )} \, dx=x+\frac {x^2-3}{x^2\,{\mathrm {e}}^{2\,x-2\,x^2}+3\,x^2+x^3} \]
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