3.43 \(\int e^{-x/2} \left (-1+e^{x/2}\right )^3 \, dx\)

Optimal. Leaf size=25 \[ 3 x+2 e^{-x/2}-6 e^{x/2}+e^x \]

[Out]

2/E^(x/2) - 6*E^(x/2) + E^x + 3*x

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Rubi [A]  time = 0.0464107, 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 \[ 3 x+2 e^{-x/2}-6 e^{x/2}+e^x \]

Antiderivative was successfully verified.

[In]  Int[(-1 + E^(x/2))^3/E^(x/2),x]

[Out]

2/E^(x/2) - 6*E^(x/2) + E^x + 3*x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 6 e^{\frac{x}{2}} + 6 \log{\left (e^{\frac{x}{2}} \right )} + 2 \int ^{e^{\frac{x}{2}}} x\, dx + 2 e^{- \frac{x}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-1+exp(1/2*x))**3/exp(1/2*x),x)

[Out]

-6*exp(x/2) + 6*log(exp(x/2)) + 2*Integral(x, (x, exp(x/2))) + 2*exp(-x/2)

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Mathematica [A]  time = 0.015117, size = 25, normalized size = 1. \[ 3 x+2 e^{-x/2}-6 e^{x/2}+e^x \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + E^(x/2))^3/E^(x/2),x]

[Out]

2/E^(x/2) - 6*E^(x/2) + E^x + 3*x

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Maple [A]  time = 0.008, size = 29, normalized size = 1.2 \[ \left ({{\rm e}^{{\frac{x}{2}}}} \right ) ^{2}-6\,{{\rm e}^{x/2}}+2\, \left ({{\rm e}^{x/2}} \right ) ^{-1}+6\,\ln \left ({{\rm e}^{x/2}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-1+exp(1/2*x))^3/exp(1/2*x),x)

[Out]

exp(1/2*x)^2-6*exp(1/2*x)+2/exp(1/2*x)+6*ln(exp(1/2*x))

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Maxima [A]  time = 1.33878, size = 24, normalized size = 0.96 \[ 3 \, x - 6 \, e^{\left (\frac{1}{2} \, x\right )} + 2 \, e^{\left (-\frac{1}{2} \, x\right )} + e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e^(1/2*x) - 1)^3*e^(-1/2*x),x, algorithm="maxima")

[Out]

3*x - 6*e^(1/2*x) + 2*e^(-1/2*x) + e^x

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Fricas [A]  time = 0.224021, size = 30, normalized size = 1.2 \[{\left (3 \, x e^{\left (\frac{1}{2} \, x\right )} + e^{\left (\frac{3}{2} \, x\right )} - 6 \, e^{x} + 2\right )} e^{\left (-\frac{1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e^(1/2*x) - 1)^3*e^(-1/2*x),x, algorithm="fricas")

[Out]

(3*x*e^(1/2*x) + e^(3/2*x) - 6*e^x + 2)*e^(-1/2*x)

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Sympy [A]  time = 0.101972, size = 19, normalized size = 0.76 \[ 3 x - 6 e^{\frac{x}{2}} + e^{x} + 2 e^{- \frac{x}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-1+exp(1/2*x))**3/exp(1/2*x),x)

[Out]

3*x - 6*exp(x/2) + exp(x) + 2*exp(-x/2)

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GIAC/XCAS [A]  time = 0.20845, size = 24, normalized size = 0.96 \[ 3 \, x - 6 \, e^{\left (\frac{1}{2} \, x\right )} + 2 \, e^{\left (-\frac{1}{2} \, x\right )} + e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e^(1/2*x) - 1)^3*e^(-1/2*x),x, algorithm="giac")

[Out]

3*x - 6*e^(1/2*x) + 2*e^(-1/2*x) + e^x