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

Optimal. Leaf size=36 \[ -e^{-t} t^3-3 e^{-t} t^2-6 e^{-t} t-6 e^{-t} \]

[Out]

-6/E^t - (6*t)/E^t - (3*t^2)/E^t - t^3/E^t

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Rubi [A]  time = 0.0338808, antiderivative size = 36, 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, Rules used = {2176, 2194} \[ -e^{-t} t^3-3 e^{-t} t^2-6 e^{-t} t-6 e^{-t} \]

Antiderivative was successfully verified.

[In]

Int[t^3/E^t,t]

[Out]

-6/E^t - (6*t)/E^t - (3*t^2)/E^t - t^3/E^t

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int e^{-t} t^3 \, dt &=-e^{-t} t^3+3 \int e^{-t} t^2 \, dt\\ &=-3 e^{-t} t^2-e^{-t} t^3+6 \int e^{-t} t \, dt\\ &=-6 e^{-t} t-3 e^{-t} t^2-e^{-t} t^3+6 \int e^{-t} \, dt\\ &=-6 e^{-t}-6 e^{-t} t-3 e^{-t} t^2-e^{-t} t^3\\ \end{align*}

Mathematica [A]  time = 0.0058054, size = 21, normalized size = 0.58 \[ e^{-t} \left (-t^3-3 t^2-6 t-6\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[t^3/E^t,t]

[Out]

(-6 - 6*t - 3*t^2 - t^3)/E^t

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Maple [A]  time = 0.001, size = 20, normalized size = 0.6 \begin{align*} -{\frac{{t}^{3}+3\,{t}^{2}+6\,t+6}{{{\rm e}^{t}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(t^3/exp(t),t)

[Out]

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

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Maxima [A]  time = 0.940438, size = 26, normalized size = 0.72 \begin{align*} -{\left (t^{3} + 3 \, t^{2} + 6 \, t + 6\right )} e^{\left (-t\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t^3/exp(t),t, algorithm="maxima")

[Out]

-(t^3 + 3*t^2 + 6*t + 6)*e^(-t)

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Fricas [A]  time = 1.19131, size = 45, normalized size = 1.25 \begin{align*} -{\left (t^{3} + 3 \, t^{2} + 6 \, t + 6\right )} e^{\left (-t\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t^3/exp(t),t, algorithm="fricas")

[Out]

-(t^3 + 3*t^2 + 6*t + 6)*e^(-t)

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Sympy [A]  time = 0.080538, size = 17, normalized size = 0.47 \begin{align*} \left (- t^{3} - 3 t^{2} - 6 t - 6\right ) e^{- t} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t**3/exp(t),t)

[Out]

(-t**3 - 3*t**2 - 6*t - 6)*exp(-t)

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Giac [A]  time = 1.08778, size = 26, normalized size = 0.72 \begin{align*} -{\left (t^{3} + 3 \, t^{2} + 6 \, t + 6\right )} e^{\left (-t\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t^3/exp(t),t, algorithm="giac")

[Out]

-(t^3 + 3*t^2 + 6*t + 6)*e^(-t)