3.1.31 \(\int e^t t^3 \, dt\) [31]

Optimal. Leaf size=27 \[ -6 e^t+6 e^t t-3 e^t t^2+e^t t^3 \]

[Out]

-6*exp(t)+6*exp(t)*t-3*exp(t)*t^2+exp(t)*t^3

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Rubi [A]
time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2207, 2225} \begin {gather*} e^t t^3-3 e^t t^2+6 e^t t-6 e^t \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^t*t^3,t]

[Out]

-6*E^t + 6*E^t*t - 3*E^t*t^2 + E^t*t^3

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*} \int e^t t^3 \, dt &=e^t t^3-3 \int e^t t^2 \, dt\\ &=-3 e^t t^2+e^t t^3+6 \int e^t t \, dt\\ &=6 e^t t-3 e^t t^2+e^t t^3-6 \int e^t \, dt\\ &=-6 e^t+6 e^t t-3 e^t t^2+e^t t^3\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 17, normalized size = 0.63 \begin {gather*} e^t \left (-6+6 t-3 t^2+t^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^t*t^3,t]

[Out]

E^t*(-6 + 6*t - 3*t^2 + t^3)

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Mathics [A]
time = 1.74, size = 17, normalized size = 0.63 \begin {gather*} \left (-6+6 t-3 t^2+t^3\right ) E^t \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[t^3*E^t,t]')

[Out]

(-6 + 6 t - 3 t ^ 2 + t ^ 3) E ^ t

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Maple [A]
time = 0.01, size = 24, normalized size = 0.89

method	result	size
gosper	\(\left (t^{3}-3 t^{2}+6 t -6\right ) {\mathrm e}^{t}\)	\(17\)
risch	\(\left (t^{3}-3 t^{2}+6 t -6\right ) {\mathrm e}^{t}\)	\(17\)
meijerg	\(6-\frac {\left (-4 t^{3}+12 t^{2}-24 t +24\right ) {\mathrm e}^{t}}{4}\)	\(22\)
default	\(-6 \,{\mathrm e}^{t}+6 \,{\mathrm e}^{t} t -3 \,{\mathrm e}^{t} t^{2}+{\mathrm e}^{t} t^{3}\)	\(24\)
norman	\(-6 \,{\mathrm e}^{t}+6 \,{\mathrm e}^{t} t -3 \,{\mathrm e}^{t} t^{2}+{\mathrm e}^{t} t^{3}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*exp(t)+6*exp(t)*t-3*exp(t)*t^2+exp(t)*t^3

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Maxima [A]
time = 0.26, size = 16, normalized size = 0.59 \begin {gather*} {\left (t^{3} - 3 \, t^{2} + 6 \, t - 6\right )} e^{t} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(t^3 - 3*t^2 + 6*t - 6)*e^t

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Fricas [A]
time = 0.34, size = 16, normalized size = 0.59 \begin {gather*} {\left (t^{3} - 3 \, t^{2} + 6 \, t - 6\right )} e^{t} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(t^3 - 3*t^2 + 6*t - 6)*e^t

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Sympy [A]
time = 0.04, size = 15, normalized size = 0.56 \begin {gather*} \left (t^{3} - 3 t^{2} + 6 t - 6\right ) e^{t} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(t**3 - 3*t**2 + 6*t - 6)*exp(t)

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Giac [A]
time = 0.00, size = 17, normalized size = 0.63 \begin {gather*} \left (t^{3}-3 t^{2}+6 t-6\right ) \mathrm {e}^{t} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t)*t^3,t)

[Out]

(t^3 - 3*t^2 + 6*t - 6)*e^t

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Mupad [B]
time = 0.02, size = 16, normalized size = 0.59 \begin {gather*} {\mathrm {e}}^t\,\left (t^3-3\,t^2+6\,t-6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(t)*(6*t - 3*t^2 + t^3 - 6)

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