3.62.98 \(\int \frac {1}{9} e^{-6+\frac {4 x^2-4 x^3+x^4}{9 e^6}} (-8 x+12 x^2-4 x^3) \, dx\)

Optimal. Leaf size=21 \[ -e^{\frac {(2-x)^2 x^2}{9 e^6}} \]

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Rubi [A]  time = 0.17, antiderivative size = 25, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 3, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {12, 1594, 6706} \begin {gather*} -e^{\frac {x^4-4 x^3+4 x^2}{9 e^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-6 + (4*x^2 - 4*x^3 + x^4)/(9*E^6))*(-8*x + 12*x^2 - 4*x^3))/9,x]

[Out]

-E^((4*x^2 - 4*x^3 + x^4)/(9*E^6))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int e^{-6+\frac {4 x^2-4 x^3+x^4}{9 e^6}} \left (-8 x+12 x^2-4 x^3\right ) \, dx\\ &=\frac {1}{9} \int e^{-6+\frac {4 x^2-4 x^3+x^4}{9 e^6}} x \left (-8+12 x-4 x^2\right ) \, dx\\ &=-e^{\frac {4 x^2-4 x^3+x^4}{9 e^6}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 19, normalized size = 0.90 \begin {gather*} -e^{\frac {(-2+x)^2 x^2}{9 e^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-6 + (4*x^2 - 4*x^3 + x^4)/(9*E^6))*(-8*x + 12*x^2 - 4*x^3))/9,x]

[Out]

-E^(((-2 + x)^2*x^2)/(9*E^6))

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fricas [A]  time = 0.85, size = 27, normalized size = 1.29 \begin {gather*} -e^{\left (\frac {1}{9} \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2} - 54 \, e^{6}\right )} e^{\left (-6\right )} + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-4*x^3+12*x^2-8*x)*exp(1/9*(x^4-4*x^3+4*x^2)/exp(3)^2)/exp(3)^2,x, algorithm="fricas")

[Out]

-e^(1/9*(x^4 - 4*x^3 + 4*x^2 - 54*e^6)*e^(-6) + 6)

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giac [A]  time = 0.15, size = 25, normalized size = 1.19 \begin {gather*} -e^{\left (\frac {1}{9} \, x^{4} e^{\left (-6\right )} - \frac {4}{9} \, x^{3} e^{\left (-6\right )} + \frac {4}{9} \, x^{2} e^{\left (-6\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-4*x^3+12*x^2-8*x)*exp(1/9*(x^4-4*x^3+4*x^2)/exp(3)^2)/exp(3)^2,x, algorithm="giac")

[Out]

-e^(1/9*x^4*e^(-6) - 4/9*x^3*e^(-6) + 4/9*x^2*e^(-6))

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maple [A]  time = 0.13, size = 16, normalized size = 0.76

method	result	size
risch	\(-{\mathrm e}^{\frac {x^{2} \left (x -2\right )^{2} {\mathrm e}^{-6}}{9}}\)	\(16\)
gosper	\(-{\mathrm e}^{\frac {x^{2} \left (x^{2}-4 x +4\right ) {\mathrm e}^{-6}}{9}}\)	\(21\)
norman	\(-{\mathrm e}^{\frac {\left (x^{4}-4 x^{3}+4 x^{2}\right ) {\mathrm e}^{-6}}{9}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*(-4*x^3+12*x^2-8*x)*exp(1/9*(x^4-4*x^3+4*x^2)/exp(3)^2)/exp(3)^2,x,method=_RETURNVERBOSE)

[Out]

-exp(1/9*x^2*(x-2)^2*exp(-6))

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maxima [A]  time = 0.49, size = 25, normalized size = 1.19 \begin {gather*} -e^{\left (\frac {1}{9} \, x^{4} e^{\left (-6\right )} - \frac {4}{9} \, x^{3} e^{\left (-6\right )} + \frac {4}{9} \, x^{2} e^{\left (-6\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-4*x^3+12*x^2-8*x)*exp(1/9*(x^4-4*x^3+4*x^2)/exp(3)^2)/exp(3)^2,x, algorithm="maxima")

[Out]

-e^(1/9*x^4*e^(-6) - 4/9*x^3*e^(-6) + 4/9*x^2*e^(-6))

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mupad [B]  time = 0.12, size = 26, normalized size = 1.24 \begin {gather*} -{\mathrm {e}}^{\frac {x^4\,{\mathrm {e}}^{-6}}{9}}\,{\mathrm {e}}^{\frac {4\,x^2\,{\mathrm {e}}^{-6}}{9}}\,{\mathrm {e}}^{-\frac {4\,x^3\,{\mathrm {e}}^{-6}}{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(-6)*((4*x^2)/9 - (4*x^3)/9 + x^4/9))*exp(-6)*(8*x - 12*x^2 + 4*x^3))/9,x)

[Out]

-exp((x^4*exp(-6))/9)*exp((4*x^2*exp(-6))/9)*exp(-(4*x^3*exp(-6))/9)

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sympy [A]  time = 0.15, size = 24, normalized size = 1.14 \begin {gather*} - e^{\frac {\frac {x^{4}}{9} - \frac {4 x^{3}}{9} + \frac {4 x^{2}}{9}}{e^{6}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-4*x**3+12*x**2-8*x)*exp(1/9*(x**4-4*x**3+4*x**2)/exp(3)**2)/exp(3)**2,x)

[Out]

-exp((x**4/9 - 4*x**3/9 + 4*x**2/9)*exp(-6))

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