Integrand size = 30, antiderivative size = 16 \[ \int \frac {e^{-2/x} \left (x+x^2+e^{\frac {1}{x}} (6+6 x)\right )}{18 x} \, dx=\left (1+\frac {1}{6} e^{-1/x} x\right )^2 \]
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Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {12, 6873, 6874, 2237, 2241, 2245, 2326} \[ \int \frac {e^{-2/x} \left (x+x^2+e^{\frac {1}{x}} (6+6 x)\right )}{18 x} \, dx=\frac {1}{36} e^{-2/x} x^2+\frac {1}{3} e^{-1/x} x \]
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Rule 12
Rule 2237
Rule 2241
Rule 2245
Rule 2326
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{18} \int \frac {e^{-2/x} \left (x+x^2+e^{\frac {1}{x}} (6+6 x)\right )}{x} \, dx \\ & = \frac {1}{18} \int \frac {e^{-2/x} (1+x) \left (6 e^{\frac {1}{x}}+x\right )}{x} \, dx \\ & = \frac {1}{18} \int \left (e^{-2/x}+e^{-2/x} x+\frac {6 e^{-1/x} (1+x)}{x}\right ) \, dx \\ & = \frac {1}{18} \int e^{-2/x} \, dx+\frac {1}{18} \int e^{-2/x} x \, dx+\frac {1}{3} \int \frac {e^{-1/x} (1+x)}{x} \, dx \\ & = \frac {1}{18} e^{-2/x} x+\frac {1}{3} e^{-1/x} x+\frac {1}{36} e^{-2/x} x^2-\frac {1}{18} \int e^{-2/x} \, dx-\frac {1}{9} \int \frac {e^{-2/x}}{x} \, dx \\ & = \frac {1}{3} e^{-1/x} x+\frac {1}{36} e^{-2/x} x^2+\frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{x}\right )}{9}+\frac {1}{9} \int \frac {e^{-2/x}}{x} \, dx \\ & = \frac {1}{3} e^{-1/x} x+\frac {1}{36} e^{-2/x} x^2 \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int \frac {e^{-2/x} \left (x+x^2+e^{\frac {1}{x}} (6+6 x)\right )}{18 x} \, dx=\frac {1}{36} e^{-2/x} x \left (12 e^{\frac {1}{x}}+x\right ) \]
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Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31
method | result | size |
norman | \(\left (\frac {x^{2}}{36}+\frac {x \,{\mathrm e}^{\frac {1}{x}}}{3}\right ) {\mathrm e}^{-\frac {2}{x}}\) | \(21\) |
derivativedivides | \(\frac {x^{2} {\mathrm e}^{-\frac {2}{x}}}{36}+\frac {x \,{\mathrm e}^{-\frac {1}{x}}}{3}\) | \(22\) |
default | \(\frac {x^{2} {\mathrm e}^{-\frac {2}{x}}}{36}+\frac {x \,{\mathrm e}^{-\frac {1}{x}}}{3}\) | \(22\) |
risch | \(\frac {x^{2} {\mathrm e}^{-\frac {2}{x}}}{36}+\frac {x \,{\mathrm e}^{-\frac {1}{x}}}{3}\) | \(22\) |
parts | \(\frac {x^{2} {\mathrm e}^{-\frac {2}{x}}}{36}+\frac {x \,{\mathrm e}^{-\frac {1}{x}}}{3}\) | \(22\) |
parallelrisch | \(\frac {\left (x^{3}+12 x^{2} {\mathrm e}^{\frac {1}{x}}\right ) {\mathrm e}^{-\frac {2}{x}}}{36 x}\) | \(25\) |
meijerg | \(\frac {5 x}{18}+\frac {x^{2}}{36}-\frac {5}{18}+\frac {x^{2} \left (3-\frac {6}{x}\right ) {\mathrm e}^{-\frac {2}{x}}}{108}-\frac {x \left (2-\frac {2}{x}\right )}{6}+\frac {x \,{\mathrm e}^{-\frac {2}{x}}}{18}+\frac {x \,{\mathrm e}^{-\frac {1}{x}}}{3}-\frac {x \left (2-\frac {4}{x}\right )}{36}-\frac {x^{2} \left (\frac {36}{x^{2}}-\frac {24}{x}+6\right )}{216}\) | \(84\) |
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-2/x} \left (x+x^2+e^{\frac {1}{x}} (6+6 x)\right )}{18 x} \, dx=\frac {1}{36} \, {\left (x^{2} + 12 \, x e^{\frac {1}{x}}\right )} e^{\left (-\frac {2}{x}\right )} \]
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Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-2/x} \left (x+x^2+e^{\frac {1}{x}} (6+6 x)\right )}{18 x} \, dx=\frac {x^{2} e^{- \frac {2}{x}}}{36} + \frac {x e^{- \frac {1}{x}}}{3} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.12 \[ \int \frac {e^{-2/x} \left (x+x^2+e^{\frac {1}{x}} (6+6 x)\right )}{18 x} \, dx=-\frac {1}{3} \, {\rm Ei}\left (-\frac {1}{x}\right ) + \frac {1}{9} \, \Gamma \left (-1, \frac {2}{x}\right ) + \frac {1}{3} \, \Gamma \left (-1, \frac {1}{x}\right ) + \frac {2}{9} \, \Gamma \left (-2, \frac {2}{x}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-2/x} \left (x+x^2+e^{\frac {1}{x}} (6+6 x)\right )}{18 x} \, dx=\frac {1}{36} \, x^{2} {\left (\frac {12 \, e^{\left (-\frac {1}{x}\right )}}{x} + e^{\left (-\frac {2}{x}\right )}\right )} \]
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Time = 13.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int \frac {e^{-2/x} \left (x+x^2+e^{\frac {1}{x}} (6+6 x)\right )}{18 x} \, dx=\frac {x\,{\mathrm {e}}^{-\frac {1}{x}}}{3}+\frac {x^2\,{\mathrm {e}}^{-\frac {2}{x}}}{36} \]
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