3.318 \(\int e^{-2 t} t^3 \, dt\)

Optimal. Leaf size=44 \[ -\frac{1}{2} e^{-2 t} t^3-\frac{3}{4} e^{-2 t} t^2-\frac{3}{4} e^{-2 t} t-\frac{3 e^{-2 t}}{8} \]

[Out]

-3/(8*E^(2*t)) - (3*t)/(4*E^(2*t)) - (3*t^2)/(4*E^(2*t)) - t^3/(2*E^(2*t))

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Rubi [A]  time = 0.0585159, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{1}{2} e^{-2 t} t^3-\frac{3}{4} e^{-2 t} t^2-\frac{3}{4} e^{-2 t} t-\frac{3 e^{-2 t}}{8} \]

Antiderivative was successfully verified.

[In]  Int[t^3/E^(2*t),t]

[Out]

-3/(8*E^(2*t)) - (3*t)/(4*E^(2*t)) - (3*t^2)/(4*E^(2*t)) - t^3/(2*E^(2*t))

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Rubi in Sympy [A]  time = 2.8235, size = 41, normalized size = 0.93 \[ - \frac{t^{3} e^{- 2 t}}{2} - \frac{3 t^{2} e^{- 2 t}}{4} - \frac{3 t e^{- 2 t}}{4} - \frac{3 e^{- 2 t}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(t**3/exp(2*t),t)

[Out]

-t**3*exp(-2*t)/2 - 3*t**2*exp(-2*t)/4 - 3*t*exp(-2*t)/4 - 3*exp(-2*t)/8

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Mathematica [A]  time = 0.00585729, size = 24, normalized size = 0.55 \[ -\frac{1}{8} e^{-2 t} \left (4 t^3+6 t^2+6 t+3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[t^3/E^(2*t),t]

[Out]

-(3 + 6*t + 6*t^2 + 4*t^3)/(8*E^(2*t))

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Maple [A]  time = 0.005, size = 24, normalized size = 0.6 \[ -{\frac{4\,{t}^{3}+6\,{t}^{2}+6\,t+3}{8\,{{\rm e}^{2\,t}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(t^3/exp(2*t),t)

[Out]

-1/8*(4*t^3+6*t^2+6*t+3)/exp(2*t)

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Maxima [A]  time = 1.33013, size = 28, normalized size = 0.64 \[ -\frac{1}{8} \,{\left (4 \, t^{3} + 6 \, t^{2} + 6 \, t + 3\right )} e^{\left (-2 \, t\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(4*t^3 + 6*t^2 + 6*t + 3)*e^(-2*t)

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Fricas [A]  time = 0.201162, size = 28, normalized size = 0.64 \[ -\frac{1}{8} \,{\left (4 \, t^{3} + 6 \, t^{2} + 6 \, t + 3\right )} e^{\left (-2 \, t\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(4*t^3 + 6*t^2 + 6*t + 3)*e^(-2*t)

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Sympy [A]  time = 0.073492, size = 22, normalized size = 0.5 \[ \frac{\left (- 4 t^{3} - 6 t^{2} - 6 t - 3\right ) e^{- 2 t}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(t**3/exp(2*t),t)

[Out]

(-4*t**3 - 6*t**2 - 6*t - 3)*exp(-2*t)/8

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GIAC/XCAS [A]  time = 0.207787, size = 28, normalized size = 0.64 \[ -\frac{1}{8} \,{\left (4 \, t^{3} + 6 \, t^{2} + 6 \, t + 3\right )} e^{\left (-2 \, t\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(4*t^3 + 6*t^2 + 6*t + 3)*e^(-2*t)