\(\int e^{-t} t \, dt\) [166]

Optimal result

Integrand size = 7, antiderivative size = 16 \[ \int e^{-t} t \, dt=-e^{-t}-e^{-t} t \]

[Out]

-1/exp(t)-t/exp(t)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2207, 2225} \[ \int e^{-t} t \, dt=-e^{-t} t-e^{-t} \]

[In]

Int[t/E^t,t]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69 \[ \int e^{-t} t \, dt=e^{-t} (-1-t) \]

[In]

Integrate[t/E^t,t]

[Out]

(-1 - t)/E^t

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62

method	result	size
gosper	\(-\left (1+t \right ) {\mathrm e}^{-t}\)	\(10\)
norman	\(\left (-1-t \right ) {\mathrm e}^{-t}\)	\(11\)
risch	\(\left (-1-t \right ) {\mathrm e}^{-t}\)	\(11\)
parallelrisch	\(\left (-1-t \right ) {\mathrm e}^{-t}\)	\(11\)
meijerg	\(1-\frac {\left (2+2 t \right ) {\mathrm e}^{-t}}{2}\)	\(14\)
default	\(-{\mathrm e}^{-t}-t \,{\mathrm e}^{-t}\)	\(15\)

[In]

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

[Out]

-(1+t)/exp(t)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int e^{-t} t \, dt=-{\left (t + 1\right )} e^{\left (-t\right )} \]

[In]

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

[Out]

-(t + 1)*e^(-t)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.44 \[ \int e^{-t} t \, dt=\left (- t - 1\right ) e^{- t} \]

[In]

integrate(t/exp(t),t)

[Out]

(-t - 1)*exp(-t)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int e^{-t} t \, dt=-{\left (t + 1\right )} e^{\left (-t\right )} \]

[In]

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

[Out]

-(t + 1)*e^(-t)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int e^{-t} t \, dt=-{\left (t + 1\right )} e^{\left (-t\right )} \]

[In]

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

[Out]

-(t + 1)*e^(-t)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int e^{-t} t \, dt=-{\mathrm {e}}^{-t}\,\left (t+1\right ) \]

[In]

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

[Out]

-exp(-t)*(t + 1)