### 3.36 $$\int e^{-t} t \, dt$$

Optimal. Leaf size=16 $-e^{-t} t-e^{-t}$

[Out]

-E^(-t) - t/E^t

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Rubi [A]  time = 0.0086793, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {2176, 2194} $-e^{-t} t-e^{-t}$

Antiderivative was successfully veriﬁed.

[In]

Int[t/E^t,t]

[Out]

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

Mathematica [A]  time = 0.0043066, size = 11, normalized size = 0.69 $e^{-t} (-t-1)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[t/E^t,t]

[Out]

(-1 - t)/E^t

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Maple [A]  time = 0.001, size = 10, normalized size = 0.6 \begin{align*} -{\frac{1+t}{{{\rm e}^{t}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(t/exp(t),t)

[Out]

-(1+t)/exp(t)

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Maxima [A]  time = 0.924738, size = 12, normalized size = 0.75 \begin{align*} -{\left (t + 1\right )} e^{\left (-t\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(t + 1)*e^(-t)

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Fricas [A]  time = 1.9907, size = 23, normalized size = 1.44 \begin{align*} -{\left (t + 1\right )} e^{\left (-t\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(t + 1)*e^(-t)

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Sympy [A]  time = 0.073225, size = 7, normalized size = 0.44 \begin{align*} \left (- t - 1\right ) e^{- t} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(t/exp(t),t)

[Out]

(-t - 1)*exp(-t)

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Giac [A]  time = 1.04802, size = 12, normalized size = 0.75 \begin{align*} -{\left (t + 1\right )} e^{\left (-t\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(t + 1)*e^(-t)