### 3.31 $$\int e^t t^3 \, dt$$

Optimal. Leaf size=27 $e^t t^3-3 e^t t^2+6 e^t t-6 e^t$

[Out]

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

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Rubi [A]  time = 0.0307144, antiderivative size = 27, normalized size of antiderivative = 1., 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 = {2176, 2194} $e^t t^3-3 e^t t^2+6 e^t t-6 e^t$

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 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^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*}

Mathematica [A]  time = 0.0060535, size = 17, normalized size = 0.63 $e^t \left (t^3-3 t^2+6 t-6\right )$

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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Maple [A]  time = 0.002, size = 17, normalized size = 0.6 \begin{align*} \left ({t}^{3}-3\,{t}^{2}+6\,t-6 \right ){{\rm e}^{t}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.93371, size = 22, normalized size = 0.81 \begin{align*}{\left (t^{3} - 3 \, t^{2} + 6 \, t - 6\right )} e^{t} \end{align*}

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 = 1.83083, size = 39, normalized size = 1.44 \begin{align*}{\left (t^{3} - 3 \, t^{2} + 6 \, t - 6\right )} e^{t} \end{align*}

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 = 0.079841, size = 15, normalized size = 0.56 \begin{align*} \left (t^{3} - 3 t^{2} + 6 t - 6\right ) e^{t} \end{align*}

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 = 1.05015, size = 22, normalized size = 0.81 \begin{align*}{\left (t^{3} - 3 \, t^{2} + 6 \, t - 6\right )} e^{t} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(t^3 - 3*t^2 + 6*t - 6)*e^t