3.39.50 \(\int e^{40-16 x^2+2 x^4} (32 x-8 x^3) \, dx\)

Optimal. Leaf size=20 \[ -e^{8+2 \left (4-x^2\right )^2}+\log (2) \]

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Rubi [A]  time = 0.10, antiderivative size = 16, normalized size of antiderivative = 0.80, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1593, 6706} \begin {gather*} -e^{2 x^4-16 x^2+40} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(40 - 16*x^2 + 2*x^4)*(32*x - 8*x^3),x]

[Out]

-E^(40 - 16*x^2 + 2*x^4)

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - 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} &=\int e^{40-16 x^2+2 x^4} x \left (32-8 x^2\right ) \, dx\\ &=-e^{40-16 x^2+2 x^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 16, normalized size = 0.80 \begin {gather*} -e^{40-16 x^2+2 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(40 - 16*x^2 + 2*x^4)*(32*x - 8*x^3),x]

[Out]

-E^(40 - 16*x^2 + 2*x^4)

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fricas [A]  time = 0.79, size = 15, normalized size = 0.75 \begin {gather*} -e^{\left (2 \, x^{4} - 16 \, x^{2} + 40\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^3+32*x)*exp(x^4-8*x^2+20)^2,x, algorithm="fricas")

[Out]

-e^(2*x^4 - 16*x^2 + 40)

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giac [A]  time = 0.15, size = 15, normalized size = 0.75 \begin {gather*} -e^{\left (2 \, x^{4} - 16 \, x^{2} + 40\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^3+32*x)*exp(x^4-8*x^2+20)^2,x, algorithm="giac")

[Out]

-e^(2*x^4 - 16*x^2 + 40)

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maple [A]  time = 0.03, size = 16, normalized size = 0.80

method	result	size
gosper	\(-{\mathrm e}^{2 x^{4}-16 x^{2}+40}\)	\(16\)
norman	\(-{\mathrm e}^{2 x^{4}-16 x^{2}+40}\)	\(16\)
risch	\(-{\mathrm e}^{2 x^{4}-16 x^{2}+40}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*x^3+32*x)*exp(x^4-8*x^2+20)^2,x,method=_RETURNVERBOSE)

[Out]

-exp(x^4-8*x^2+20)^2

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maxima [A]  time = 0.38, size = 15, normalized size = 0.75 \begin {gather*} -e^{\left (2 \, x^{4} - 16 \, x^{2} + 40\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^3+32*x)*exp(x^4-8*x^2+20)^2,x, algorithm="maxima")

[Out]

-e^(2*x^4 - 16*x^2 + 40)

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mupad [B]  time = 2.31, size = 16, normalized size = 0.80 \begin {gather*} -{\mathrm {e}}^{40}\,{\mathrm {e}}^{2\,x^4}\,{\mathrm {e}}^{-16\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x^4 - 16*x^2 + 40)*(32*x - 8*x^3),x)

[Out]

-exp(40)*exp(2*x^4)*exp(-16*x^2)

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sympy [A]  time = 0.10, size = 14, normalized size = 0.70 \begin {gather*} - e^{2 x^{4} - 16 x^{2} + 40} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x**3+32*x)*exp(x**4-8*x**2+20)**2,x)

[Out]

-exp(2*x**4 - 16*x**2 + 40)

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