Integrand size = 43, antiderivative size = 18 \[ \int \frac {-16 e^{\frac {2}{x^8}}+20 x^9+8 x^{10}+e^{\frac {1}{x^8}} \left (-80-32 x+4 x^9\right )}{25 x^9} \, dx=\frac {1}{25} \left (-5-e^{\frac {1}{x^8}}-2 x\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(18)=36\).
Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {12, 14, 2240, 2326} \[ \int \frac {-16 e^{\frac {2}{x^8}}+20 x^9+8 x^{10}+e^{\frac {1}{x^8}} \left (-80-32 x+4 x^9\right )}{25 x^9} \, dx=\frac {2}{25} e^{\frac {1}{x^8}} (2 x+5)+\frac {e^{\frac {2}{x^8}}}{25}+\frac {1}{25} (2 x+5)^2 \]
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Rule 12
Rule 14
Rule 2240
Rule 2326
Rubi steps \begin{align*} \text {integral}& = \frac {1}{25} \int \frac {-16 e^{\frac {2}{x^8}}+20 x^9+8 x^{10}+e^{\frac {1}{x^8}} \left (-80-32 x+4 x^9\right )}{x^9} \, dx \\ & = \frac {1}{25} \int \left (-\frac {16 e^{\frac {2}{x^8}}}{x^9}+4 (5+2 x)+\frac {4 e^{\frac {1}{x^8}} \left (-20-8 x+x^9\right )}{x^9}\right ) \, dx \\ & = \frac {1}{25} (5+2 x)^2+\frac {4}{25} \int \frac {e^{\frac {1}{x^8}} \left (-20-8 x+x^9\right )}{x^9} \, dx-\frac {16}{25} \int \frac {e^{\frac {2}{x^8}}}{x^9} \, dx \\ & = \frac {e^{\frac {2}{x^8}}}{25}+\frac {2}{25} e^{\frac {1}{x^8}} (5+2 x)+\frac {1}{25} (5+2 x)^2 \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78 \[ \int \frac {-16 e^{\frac {2}{x^8}}+20 x^9+8 x^{10}+e^{\frac {1}{x^8}} \left (-80-32 x+4 x^9\right )}{25 x^9} \, dx=\frac {4}{25} \left (\frac {e^{\frac {2}{x^8}}}{4}+e^{\frac {1}{x^8}} \left (\frac {5}{2}+x\right )+x (5+x)\right ) \]
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Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.61
method | result | size |
risch | \(\frac {4 x^{2}}{25}+\frac {{\mathrm e}^{\frac {2}{x^{8}}}}{25}+\frac {4 x}{5}+\frac {\left (4 x +10\right ) {\mathrm e}^{\frac {1}{x^{8}}}}{25}\) | \(29\) |
parallelrisch | \(\frac {4 x^{2}}{25}+\frac {{\mathrm e}^{\frac {2}{x^{8}}}}{25}+\frac {4 x}{5}+\frac {4 \,{\mathrm e}^{\frac {1}{x^{8}}} x}{25}+\frac {2 \,{\mathrm e}^{\frac {1}{x^{8}}}}{5}\) | \(31\) |
parts | \(\frac {4 x}{5}+\frac {4 x^{2}}{25}+\frac {{\mathrm e}^{\frac {2}{x^{8}}}}{25}+\frac {\frac {2 \,{\mathrm e}^{\frac {1}{x^{8}}} x^{8}}{5}+\frac {4 \,{\mathrm e}^{\frac {1}{x^{8}}} x^{9}}{25}}{x^{8}}\) | \(41\) |
norman | \(\frac {\frac {4 x^{9}}{5}+\frac {4 x^{10}}{25}+\frac {2 \,{\mathrm e}^{\frac {1}{x^{8}}} x^{8}}{5}+\frac {4 \,{\mathrm e}^{\frac {1}{x^{8}}} x^{9}}{25}+\frac {{\mathrm e}^{\frac {2}{x^{8}}} x^{8}}{25}}{x^{8}}\) | \(45\) |
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Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (13) = 26\).
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.56 \[ \int \frac {-16 e^{\frac {2}{x^8}}+20 x^9+8 x^{10}+e^{\frac {1}{x^8}} \left (-80-32 x+4 x^9\right )}{25 x^9} \, dx=\frac {4}{25} \, x^{2} + \frac {2}{25} \, {\left (2 \, x + 5\right )} e^{\left (\frac {1}{x^{8}}\right )} + \frac {4}{5} \, x + \frac {1}{25} \, e^{\left (\frac {2}{x^{8}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78 \[ \int \frac {-16 e^{\frac {2}{x^8}}+20 x^9+8 x^{10}+e^{\frac {1}{x^8}} \left (-80-32 x+4 x^9\right )}{25 x^9} \, dx=\frac {4 x^{2}}{25} + \frac {4 x}{5} + \frac {\left (100 x + 250\right ) e^{\frac {1}{x^{8}}}}{625} + \frac {e^{\frac {2}{x^{8}}}}{25} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 3.28 \[ \int \frac {-16 e^{\frac {2}{x^8}}+20 x^9+8 x^{10}+e^{\frac {1}{x^8}} \left (-80-32 x+4 x^9\right )}{25 x^9} \, dx=\frac {1}{50} \, x \left (-\frac {1}{x^{8}}\right )^{\frac {1}{8}} \Gamma \left (-\frac {1}{8}, -\frac {1}{x^{8}}\right ) + \frac {4}{25} \, x^{2} + \frac {4}{5} \, x - \frac {4 \, \Gamma \left (\frac {7}{8}, -\frac {1}{x^{8}}\right )}{25 \, x^{7} \left (-\frac {1}{x^{8}}\right )^{\frac {7}{8}}} + \frac {1}{25} \, e^{\left (\frac {2}{x^{8}}\right )} + \frac {2}{5} \, e^{\left (\frac {1}{x^{8}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (13) = 26\).
Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \frac {-16 e^{\frac {2}{x^8}}+20 x^9+8 x^{10}+e^{\frac {1}{x^8}} \left (-80-32 x+4 x^9\right )}{25 x^9} \, dx=\frac {4}{25} \, x^{2} + \frac {4}{25} \, x e^{\left (\frac {1}{x^{8}}\right )} + \frac {4}{5} \, x + \frac {1}{25} \, e^{\left (\frac {2}{x^{8}}\right )} + \frac {2}{5} \, e^{\left (\frac {1}{x^{8}}\right )} \]
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Time = 9.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \frac {-16 e^{\frac {2}{x^8}}+20 x^9+8 x^{10}+e^{\frac {1}{x^8}} \left (-80-32 x+4 x^9\right )}{25 x^9} \, dx=\frac {4\,x}{5}+\frac {2\,{\mathrm {e}}^{\frac {1}{x^8}}}{5}+\frac {{\mathrm {e}}^{\frac {2}{x^8}}}{25}+\frac {4\,x\,{\mathrm {e}}^{\frac {1}{x^8}}}{25}+\frac {4\,x^2}{25} \]
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