\(\int e^{-2 t} t^3 \, dt\) [318]

Optimal result

Integrand size = 9, antiderivative size = 44 \[ \int e^{-2 t} t^3 \, dt=-\frac {3}{8} e^{-2 t}-\frac {3}{4} e^{-2 t} t-\frac {3}{4} e^{-2 t} t^2-\frac {1}{2} e^{-2 t} t^3 \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2207, 2225} \[ \int e^{-2 t} t^3 \, dt=-\frac {1}{2} e^{-2 t} t^3-\frac {3}{4} e^{-2 t} t^2-\frac {3}{4} e^{-2 t} t-\frac {3 e^{-2 t}}{8} \]

[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 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}& = -\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] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.55 \[ \int e^{-2 t} t^3 \, dt=-\frac {1}{8} e^{-2 t} \left (3+6 t+6 t^2+4 t^3\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.48

method	result	size
risch	\(\left (-\frac {1}{2} t^{3}-\frac {3}{4} t^{2}-\frac {3}{4} t -\frac {3}{8}\right ) {\mathrm e}^{-2 t}\)	\(21\)
norman	\(\left (-\frac {1}{2} t^{3}-\frac {3}{4} t^{2}-\frac {3}{4} t -\frac {3}{8}\right ) {\mathrm e}^{-2 t}\)	\(23\)
gosper	\(-\frac {\left (4 t^{3}+6 t^{2}+6 t +3\right ) {\mathrm e}^{-2 t}}{8}\)	\(24\)
meijerg	\(\frac {3}{8}-\frac {\left (32 t^{3}+48 t^{2}+48 t +24\right ) {\mathrm e}^{-2 t}}{64}\)	\(24\)
parallelrisch	\(\frac {\left (-4 t^{3}-6 t^{2}-6 t -3\right ) {\mathrm e}^{-2 t}}{8}\)	\(24\)
derivativedivides	\(-\frac {3 \,{\mathrm e}^{-2 t}}{8}-\frac {3 t \,{\mathrm e}^{-2 t}}{4}-\frac {3 t^{2} {\mathrm e}^{-2 t}}{4}-\frac {t^{3} {\mathrm e}^{-2 t}}{2}\)	\(41\)
default	\(-\frac {3 \,{\mathrm e}^{-2 t}}{8}-\frac {3 t \,{\mathrm e}^{-2 t}}{4}-\frac {3 t^{2} {\mathrm e}^{-2 t}}{4}-\frac {t^{3} {\mathrm e}^{-2 t}}{2}\)	\(41\)

[In]

int(t^3/exp(2*t),t,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.48 \[ \int e^{-2 t} t^3 \, dt=-\frac {1}{8} \, {\left (4 \, t^{3} + 6 \, t^{2} + 6 \, t + 3\right )} e^{\left (-2 \, t\right )} \]

[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] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.50 \[ \int e^{-2 t} t^3 \, dt=\frac {\left (- 4 t^{3} - 6 t^{2} - 6 t - 3\right ) e^{- 2 t}}{8} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.48 \[ \int e^{-2 t} t^3 \, dt=-\frac {1}{8} \, {\left (4 \, t^{3} + 6 \, t^{2} + 6 \, t + 3\right )} e^{\left (-2 \, t\right )} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.48 \[ \int e^{-2 t} t^3 \, dt=-\frac {1}{8} \, {\left (4 \, t^{3} + 6 \, t^{2} + 6 \, t + 3\right )} e^{\left (-2 \, t\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.48 \[ \int e^{-2 t} t^3 \, dt=-\frac {{\mathrm {e}}^{-2\,t}\,\left (8\,t^3+12\,t^2+12\,t+6\right )}{16} \]

[In]

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

[Out]

-(exp(-2*t)*(12*t + 12*t^2 + 8*t^3 + 6))/16