Integrand size = 19, antiderivative size = 25 \[ \int e^{-x/2} \left (-1+e^{x/2}\right )^3 \, dx=2 e^{-x/2}-6 e^{x/2}+e^x+3 x \]
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Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2280, 45} \[ \int e^{-x/2} \left (-1+e^{x/2}\right )^3 \, dx=3 x+2 e^{-x/2}-6 e^{x/2}+e^x \]
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Rule 45
Rule 2280
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {(-1+x)^3}{x^2} \, dx,x,e^{x/2}\right ) \\ & = 2 \text {Subst}\left (\int \left (-3-\frac {1}{x^2}+\frac {3}{x}+x\right ) \, dx,x,e^{x/2}\right ) \\ & = 2 e^{-x/2}-6 e^{x/2}+e^x+3 x \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int e^{-x/2} \left (-1+e^{x/2}\right )^3 \, dx=e^{-x/2} \left (2-6 e^x+e^{3 x/2}\right )+6 \log \left (e^{x/2}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76
method | result | size |
risch | \({\mathrm e}^{x}+3 x -6 \,{\mathrm e}^{\frac {x}{2}}+2 \,{\mathrm e}^{-\frac {x}{2}}\) | \(19\) |
parts | \({\mathrm e}^{x}+3 x -6 \,{\mathrm e}^{\frac {x}{2}}+2 \,{\mathrm e}^{-\frac {x}{2}}\) | \(25\) |
derivativedivides | \({\mathrm e}^{x}-6 \,{\mathrm e}^{\frac {x}{2}}+6 \ln \left ({\mathrm e}^{\frac {x}{2}}\right )+2 \,{\mathrm e}^{-\frac {x}{2}}\) | \(29\) |
default | \({\mathrm e}^{x}-6 \,{\mathrm e}^{\frac {x}{2}}+6 \ln \left ({\mathrm e}^{\frac {x}{2}}\right )+2 \,{\mathrm e}^{-\frac {x}{2}}\) | \(29\) |
norman | \(\left (2+{\mathrm e}^{\frac {3 x}{2}}-6 \,{\mathrm e}^{x}+3 x \,{\mathrm e}^{\frac {x}{2}}\right ) {\mathrm e}^{-\frac {x}{2}}\) | \(31\) |
parallelrisch | \(-\left (-2-{\mathrm e}^{\frac {3 x}{2}}-6 \ln \left ({\mathrm e}^{\frac {x}{2}}\right ) {\mathrm e}^{\frac {x}{2}}+6 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {x}{2}}\) | \(38\) |
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Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int e^{-x/2} \left (-1+e^{x/2}\right )^3 \, dx={\left (3 \, x e^{\left (\frac {1}{2} \, x\right )} + e^{\left (\frac {3}{2} \, x\right )} - 6 \, e^{x} + 2\right )} e^{\left (-\frac {1}{2} \, x\right )} \]
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Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int e^{-x/2} \left (-1+e^{x/2}\right )^3 \, dx=3 x - 6 e^{\frac {x}{2}} + e^{x} + 2 e^{- \frac {x}{2}} \]
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Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int e^{-x/2} \left (-1+e^{x/2}\right )^3 \, dx=3 \, x - 6 \, e^{\left (\frac {1}{2} \, x\right )} + 2 \, e^{\left (-\frac {1}{2} \, x\right )} + e^{x} \]
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Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int e^{-x/2} \left (-1+e^{x/2}\right )^3 \, dx=3 \, x - 6 \, e^{\left (\frac {1}{2} \, x\right )} + 2 \, e^{\left (-\frac {1}{2} \, x\right )} + e^{x} \]
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Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int e^{-x/2} \left (-1+e^{x/2}\right )^3 \, dx=3\,x+2\,{\mathrm {e}}^{-\frac {x}{2}}-6\,{\mathrm {e}}^{x/2}+{\mathrm {e}}^x \]
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