Integrand size = 69, antiderivative size = 30 \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=2 x+\left (1+2 x^2\right ) \left (3-\frac {e^{e^8}}{-x+x^4}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.072, Rules used = {1608, 28, 1843, 1600, 14} \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=\frac {e^{e^8} (x+2) x}{1-x^3}+6 x^2+2 x+\frac {e^{e^8}}{x} \]
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Rule 14
Rule 28
Rule 1600
Rule 1608
Rule 1843
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2 \left (1-2 x^3+x^6\right )} \, dx \\ & = \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2 \left (-1+x^3\right )^2} \, dx \\ & = \frac {e^{e^8} x (2+x)}{1-x^3}+\frac {1}{3} \int \frac {3 e^{e^8}-6 x^2-3 \left (12+e^{e^8}\right ) x^3+6 x^5+36 x^6}{x^2 \left (-1+x^3\right )} \, dx \\ & = \frac {e^{e^8} x (2+x)}{1-x^3}+\frac {1}{3} \int \frac {-3 e^{e^8}+6 x^2+36 x^3}{x^2} \, dx \\ & = \frac {e^{e^8} x (2+x)}{1-x^3}+\frac {1}{3} \int \left (6-\frac {3 e^{e^8}}{x^2}+36 x\right ) \, dx \\ & = \frac {e^{e^8}}{x}+2 x+6 x^2+\frac {e^{e^8} x (2+x)}{1-x^3} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=\frac {e^{e^8}}{x}+2 x+6 x^2+\frac {e^{e^8} \left (-2 x-x^2\right )}{-1+x^3} \]
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Time = 0.07 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17
method | result | size |
risch | \(6 x^{2}+2 x +\frac {-2 \,{\mathrm e}^{{\mathrm e}^{8}} x^{2}-{\mathrm e}^{{\mathrm e}^{8}}}{x \left (x^{3}-1\right )}\) | \(35\) |
default | \(2 x +6 x^{2}-\frac {{\mathrm e}^{{\mathrm e}^{8}}}{-1+x}+\frac {{\mathrm e}^{{\mathrm e}^{8}}}{x}-\frac {{\mathrm e}^{{\mathrm e}^{8}}}{x^{2}+x +1}\) | \(40\) |
norman | \(\frac {6 x^{6}+2 x^{5}-6 x^{3}+\left (-2 \,{\mathrm e}^{{\mathrm e}^{8}}-2\right ) x^{2}-{\mathrm e}^{{\mathrm e}^{8}}}{x \left (x^{3}-1\right )}\) | \(48\) |
gosper | \(-\frac {-6 x^{6}-2 x^{5}+2 \,{\mathrm e}^{{\mathrm e}^{8}} x^{2}+6 x^{3}+2 x^{2}+{\mathrm e}^{{\mathrm e}^{8}}}{x \left (x^{3}-1\right )}\) | \(49\) |
parallelrisch | \(-\frac {-6 x^{6}-2 x^{5}+2 \,{\mathrm e}^{{\mathrm e}^{8}} x^{2}+6 x^{3}+2 x^{2}+{\mathrm e}^{{\mathrm e}^{8}}}{x \left (x^{3}-1\right )}\) | \(49\) |
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Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=\frac {6 \, x^{6} + 2 \, x^{5} - 6 \, x^{3} - 2 \, x^{2} - {\left (2 \, x^{2} + 1\right )} e^{\left (e^{8}\right )}}{x^{4} - x} \]
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Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=6 x^{2} + 2 x + \frac {- 2 x^{2} e^{e^{8}} - e^{e^{8}}}{x^{4} - x} \]
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Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=6 \, x^{2} + 2 \, x - \frac {2 \, x^{2} e^{\left (e^{8}\right )} + e^{\left (e^{8}\right )}}{x^{4} - x} \]
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Time = 0.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=6 \, x^{2} + 2 \, x - \frac {2 \, x^{2} e^{\left (e^{8}\right )} + e^{\left (e^{8}\right )}}{x^{4} - x} \]
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Time = 8.46 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=2\,x+6\,x^2+\frac {2\,{\mathrm {e}}^{{\mathrm {e}}^8}\,x^2+{\mathrm {e}}^{{\mathrm {e}}^8}}{x-x^4} \]
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