Integrand size = 29, antiderivative size = 13 \[ \int \frac {-2 e^{e^{\frac {1}{4 x^8}}+\frac {1}{4 x^8}}+x^9}{x^9} \, dx=e^{e^{\frac {1}{4 x^8}}}+x \]
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Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {14, 6847, 2320, 2225} \[ \int \frac {-2 e^{e^{\frac {1}{4 x^8}}+\frac {1}{4 x^8}}+x^9}{x^9} \, dx=e^{e^{\frac {1}{4 x^8}}}+x \]
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Rule 14
Rule 2225
Rule 2320
Rule 6847
Rubi steps \begin{align*} \text {integral}& = \int \left (1-\frac {2 e^{e^{\frac {1}{4 x^8}}+\frac {1}{4 x^8}}}{x^9}\right ) \, dx \\ & = x-2 \int \frac {e^{e^{\frac {1}{4 x^8}}+\frac {1}{4 x^8}}}{x^9} \, dx \\ & = x+\frac {1}{4} \text {Subst}\left (\int e^{e^{x/4}+\frac {x}{4}} \, dx,x,\frac {1}{x^8}\right ) \\ & = x+\text {Subst}\left (\int e^x \, dx,x,e^{\frac {1}{4 x^8}}\right ) \\ & = e^{e^{\frac {1}{4 x^8}}}+x \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {-2 e^{e^{\frac {1}{4 x^8}}+\frac {1}{4 x^8}}+x^9}{x^9} \, dx=e^{e^{\frac {1}{4 x^8}}}+x \]
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Time = 0.56 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77
method | result | size |
default | \({\mathrm e}^{{\mathrm e}^{\frac {1}{4 x^{8}}}}+x\) | \(10\) |
risch | \({\mathrm e}^{{\mathrm e}^{\frac {1}{4 x^{8}}}}+x\) | \(10\) |
parallelrisch | \({\mathrm e}^{{\mathrm e}^{\frac {1}{4 x^{8}}}}+x\) | \(10\) |
parts | \({\mathrm e}^{{\mathrm e}^{\frac {1}{4 x^{8}}}}+x\) | \(10\) |
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (9) = 18\).
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.69 \[ \int \frac {-2 e^{e^{\frac {1}{4 x^8}}+\frac {1}{4 x^8}}+x^9}{x^9} \, dx={\left (x e^{\left (\frac {1}{4 \, x^{8}}\right )} + e^{\left (\frac {4 \, x^{8} e^{\left (\frac {1}{4 \, x^{8}}\right )} + 1}{4 \, x^{8}}\right )}\right )} e^{\left (-\frac {1}{4 \, x^{8}}\right )} \]
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Time = 0.08 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {-2 e^{e^{\frac {1}{4 x^8}}+\frac {1}{4 x^8}}+x^9}{x^9} \, dx=x + e^{e^{\frac {1}{4 x^{8}}}} \]
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none
Time = 0.21 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {-2 e^{e^{\frac {1}{4 x^8}}+\frac {1}{4 x^8}}+x^9}{x^9} \, dx=x + e^{\left (e^{\left (\frac {1}{4 \, x^{8}}\right )}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (9) = 18\).
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.69 \[ \int \frac {-2 e^{e^{\frac {1}{4 x^8}}+\frac {1}{4 x^8}}+x^9}{x^9} \, dx={\left (x e^{\left (\frac {1}{4 \, x^{8}}\right )} + e^{\left (\frac {4 \, x^{8} e^{\left (\frac {1}{4 \, x^{8}}\right )} + 1}{4 \, x^{8}}\right )}\right )} e^{\left (-\frac {1}{4 \, x^{8}}\right )} \]
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Time = 13.39 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {-2 e^{e^{\frac {1}{4 x^8}}+\frac {1}{4 x^8}}+x^9}{x^9} \, dx=x+{\mathrm {e}}^{{\mathrm {e}}^{\frac {1}{4\,x^8}}} \]
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