Optimal. Leaf size=25 \[ 3 x+2 e^{-x/2}-6 e^{x/2}+e^x \]
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Rubi [A] time = 0.0246283, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2248, 43} \[ 3 x+2 e^{-x/2}-6 e^{x/2}+e^x \]
Antiderivative was successfully verified.
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Rule 2248
Rule 43
Rubi steps
\begin{align*} \int e^{-x/2} \left (-1+e^{x/2}\right )^3 \, dx &=2 \operatorname{Subst}\left (\int \frac{(-1+x)^3}{x^2} \, dx,x,e^{x/2}\right )\\ &=2 \operatorname{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*}
Mathematica [A] time = 0.0130611, size = 25, normalized size = 1. \[ 3 x+2 e^{-x/2}-6 e^{x/2}+e^x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 29, normalized size = 1.2 \begin{align*} \left ({{\rm e}^{{\frac{x}{2}}}} \right ) ^{2}-6\,{{\rm e}^{x/2}}+6\,\ln \left ({{\rm e}^{x/2}} \right ) +2\, \left ({{\rm e}^{x/2}} \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955786, size = 24, normalized size = 0.96 \begin{align*} 3 \, x - 6 \, e^{\left (\frac{1}{2} \, x\right )} + 2 \, e^{\left (-\frac{1}{2} \, x\right )} + e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91503, size = 70, normalized size = 2.8 \begin{align*}{\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.098204, size = 19, normalized size = 0.76 \begin{align*} 3 x - 6 e^{\frac{x}{2}} + e^{x} + 2 e^{- \frac{x}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05295, size = 24, normalized size = 0.96 \begin{align*} 3 \, x - 6 \, e^{\left (\frac{1}{2} \, x\right )} + 2 \, e^{\left (-\frac{1}{2} \, x\right )} + e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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