3.318 \(\int e^{-2 t} t^3 \, dt\)

Optimal. Leaf size=44 \[ -\frac{1}{2} e^{-2 t} t^3-\frac{3}{4} e^{-2 t} t^2-\frac{3}{4} e^{-2 t} t-\frac{3 e^{-2 t}}{8} \]

[Out]

-3/(8*E^(2*t)) - (3*t)/(4*E^(2*t)) - (3*t^2)/(4*E^(2*t)) - t^3/(2*E^(2*t))

________________________________________________________________________________________

Rubi [A]  time = 0.0385098, antiderivative size = 44, 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} \[ -\frac{1}{2} e^{-2 t} t^3-\frac{3}{4} e^{-2 t} t^2-\frac{3}{4} e^{-2 t} t-\frac{3 e^{-2 t}}{8} \]

Antiderivative was successfully verified.

[In]

Int[t^3/E^(2*t),t]

[Out]

-3/(8*E^(2*t)) - (3*t)/(4*E^(2*t)) - (3*t^2)/(4*E^(2*t)) - t^3/(2*E^(2*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^{-2 t} t^3 \, dt &=-\frac{1}{2} e^{-2 t} t^3+\frac{3}{2} \int e^{-2 t} t^2 \, dt\\ &=-\frac{3}{4} e^{-2 t} t^2-\frac{1}{2} e^{-2 t} t^3+\frac{3}{2} \int e^{-2 t} t \, dt\\ &=-\frac{3}{4} e^{-2 t} t-\frac{3}{4} e^{-2 t} t^2-\frac{1}{2} e^{-2 t} t^3+\frac{3}{4} \int e^{-2 t} \, dt\\ &=-\frac{3}{8} e^{-2 t}-\frac{3}{4} e^{-2 t} t-\frac{3}{4} e^{-2 t} t^2-\frac{1}{2} e^{-2 t} t^3\\ \end{align*}

Mathematica [A]  time = 0.0087274, size = 24, normalized size = 0.55 \[ -\frac{1}{8} e^{-2 t} \left (4 t^3+6 t^2+6 t+3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[t^3/E^(2*t),t]

[Out]

-(3 + 6*t + 6*t^2 + 4*t^3)/(8*E^(2*t))

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 24, normalized size = 0.6 \begin{align*} -{\frac{4\,{t}^{3}+6\,{t}^{2}+6\,t+3}{8\,{{\rm e}^{2\,t}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(t^3/exp(2*t),t)

[Out]

-1/8*(4*t^3+6*t^2+6*t+3)/exp(2*t)

________________________________________________________________________________________

Maxima [A]  time = 0.93674, size = 28, normalized size = 0.64 \begin{align*} -\frac{1}{8} \,{\left (4 \, t^{3} + 6 \, t^{2} + 6 \, t + 3\right )} e^{\left (-2 \, t\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t^3/exp(2*t),t, algorithm="maxima")

[Out]

-1/8*(4*t^3 + 6*t^2 + 6*t + 3)*e^(-2*t)

________________________________________________________________________________________

Fricas [A]  time = 2.10728, size = 55, normalized size = 1.25 \begin{align*} -\frac{1}{8} \,{\left (4 \, t^{3} + 6 \, t^{2} + 6 \, t + 3\right )} e^{\left (-2 \, t\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t^3/exp(2*t),t, algorithm="fricas")

[Out]

-1/8*(4*t^3 + 6*t^2 + 6*t + 3)*e^(-2*t)

________________________________________________________________________________________

Sympy [A]  time = 0.082717, size = 22, normalized size = 0.5 \begin{align*} \frac{\left (- 4 t^{3} - 6 t^{2} - 6 t - 3\right ) e^{- 2 t}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t**3/exp(2*t),t)

[Out]

(-4*t**3 - 6*t**2 - 6*t - 3)*exp(-2*t)/8

________________________________________________________________________________________

Giac [A]  time = 1.0501, size = 28, normalized size = 0.64 \begin{align*} -\frac{1}{8} \,{\left (4 \, t^{3} + 6 \, t^{2} + 6 \, t + 3\right )} e^{\left (-2 \, t\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t^3/exp(2*t),t, algorithm="giac")

[Out]

-1/8*(4*t^3 + 6*t^2 + 6*t + 3)*e^(-2*t)