\(\int e^{-t} t^2 \, dt\) [167]

Optimal result

Integrand size = 9, antiderivative size = 26 \[ \int e^{-t} t^2 \, dt=-2 e^{-t}-2 e^{-t} t-e^{-t} t^2 \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2207, 2225} \[ \int e^{-t} t^2 \, dt=-e^{-t} t^2-2 e^{-t} t-2 e^{-t} \]

[In]

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

[Out]

-2/E^t - (2*t)/E^t - t^2/E^t

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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*} \text {integral}& = -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] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.62 \[ \int e^{-t} t^2 \, dt=e^{-t} \left (-2-2 t-t^2\right ) \]

[In]

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

[Out]

(-2 - 2*t - t^2)/E^t

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.58

method	result	size
gosper	\(-\left (t^{2}+2 t +2\right ) {\mathrm e}^{-t}\)	\(15\)
norman	\(\left (-t^{2}-2 t -2\right ) {\mathrm e}^{-t}\)	\(16\)
risch	\(\left (-t^{2}-2 t -2\right ) {\mathrm e}^{-t}\)	\(16\)
parallelrisch	\(\left (-t^{2}-2 t -2\right ) {\mathrm e}^{-t}\)	\(16\)
meijerg	\(2-\frac {\left (3 t^{2}+6 t +6\right ) {\mathrm e}^{-t}}{3}\)	\(19\)
default	\(-2 \,{\mathrm e}^{-t}-2 t \,{\mathrm e}^{-t}-t^{2} {\mathrm e}^{-t}\)	\(24\)

[In]

int(t^2/exp(t),t,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.54 \[ \int e^{-t} t^2 \, dt=-{\left (t^{2} + 2 \, t + 2\right )} e^{\left (-t\right )} \]

[In]

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

[Out]

-(t^2 + 2*t + 2)*e^(-t)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.46 \[ \int e^{-t} t^2 \, dt=\left (- t^{2} - 2 t - 2\right ) e^{- t} \]

[In]

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

[Out]

(-t**2 - 2*t - 2)*exp(-t)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.54 \[ \int e^{-t} t^2 \, dt=-{\left (t^{2} + 2 \, t + 2\right )} e^{\left (-t\right )} \]

[In]

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

[Out]

-(t^2 + 2*t + 2)*e^(-t)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.54 \[ \int e^{-t} t^2 \, dt=-{\left (t^{2} + 2 \, t + 2\right )} e^{\left (-t\right )} \]

[In]

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

[Out]

-(t^2 + 2*t + 2)*e^(-t)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.54 \[ \int e^{-t} t^2 \, dt=-{\mathrm {e}}^{-t}\,\left (t^2+2\,t+2\right ) \]

[In]

int(t^2*exp(-t),t)

[Out]

-exp(-t)*(2*t + t^2 + 2)