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

Optimal. Leaf size=26 \[ -e^{-t} t^2-2 e^{-t} t-2 e^{-t} \]

[Out]

-2/E^t - (2*t)/E^t - t^2/E^t

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Rubi [A]  time = 0.0205438, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2176, 2194} \[ -e^{-t} t^2-2 e^{-t} t-2 e^{-t} \]

Antiderivative was successfully verified.

[In]

Int[t^2/E^t,t]

[Out]

-2/E^t - (2*t)/E^t - t^2/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^2 \, dt &=-e^{-t} t^2+2 \int e^{-t} t \, dt\\ &=-2 e^{-t} t-e^{-t} t^2+2 \int e^{-t} \, dt\\ &=-2 e^{-t}-2 e^{-t} t-e^{-t} t^2\\ \end{align*}

Mathematica [A]  time = 0.0048568, size = 16, normalized size = 0.62 \[ e^{-t} \left (-t^2-2 t-2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[t^2/E^t,t]

[Out]

(-2 - 2*t - t^2)/E^t

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Maple [A]  time = 0.002, size = 15, normalized size = 0.6 \begin{align*} -{\frac{{t}^{2}+2\,t+2}{{{\rm e}^{t}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(t^2/exp(t),t)

[Out]

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

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Maxima [A]  time = 0.94588, size = 19, normalized size = 0.73 \begin{align*} -{\left (t^{2} + 2 \, t + 2\right )} e^{\left (-t\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t^2/exp(t),t, algorithm="maxima")

[Out]

-(t^2 + 2*t + 2)*e^(-t)

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Fricas [A]  time = 1.07538, size = 34, normalized size = 1.31 \begin{align*} -{\left (t^{2} + 2 \, t + 2\right )} e^{\left (-t\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t^2/exp(t),t, algorithm="fricas")

[Out]

-(t^2 + 2*t + 2)*e^(-t)

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Sympy [A]  time = 0.079461, size = 12, normalized size = 0.46 \begin{align*} \left (- t^{2} - 2 t - 2\right ) e^{- t} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t**2/exp(t),t)

[Out]

(-t**2 - 2*t - 2)*exp(-t)

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Giac [A]  time = 1.09597, size = 19, normalized size = 0.73 \begin{align*} -{\left (t^{2} + 2 \, t + 2\right )} e^{\left (-t\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t^2/exp(t),t, algorithm="giac")

[Out]

-(t^2 + 2*t + 2)*e^(-t)