3.96.7 \(\int \frac {-8-32 e^{x^2} x}{256+4 e^{2 x^2}-32 x+x^2+e^{x^2} (-64+4 x)} \, dx\)

Optimal. Leaf size=16 \[ \frac {4}{-8+e^{x^2}+\frac {x}{2}} \]

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Rubi [A]  time = 0.13, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {6688, 12, 6686} \begin {gather*} -\frac {8}{-2 e^{x^2}-x+16} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8 - 32*E^x^2*x)/(256 + 4*E^(2*x^2) - 32*x + x^2 + E^x^2*(-64 + 4*x)),x]

[Out]

-8/(16 - 2*E^x^2 - x)

Rule 12

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

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 \left (-1-4 e^{x^2} x\right )}{\left (16-2 e^{x^2}-x\right )^2} \, dx\\ &=8 \int \frac {-1-4 e^{x^2} x}{\left (16-2 e^{x^2}-x\right )^2} \, dx\\ &=-\frac {8}{16-2 e^{x^2}-x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.88 \begin {gather*} \frac {8}{-16+2 e^{x^2}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 - 32*E^x^2*x)/(256 + 4*E^(2*x^2) - 32*x + x^2 + E^x^2*(-64 + 4*x)),x]

[Out]

8/(-16 + 2*E^x^2 + x)

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fricas [A]  time = 0.92, size = 13, normalized size = 0.81 \begin {gather*} \frac {8}{x + 2 \, e^{\left (x^{2}\right )} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*exp(x^2)*x-8)/(4*exp(x^2)^2+(4*x-64)*exp(x^2)+x^2-32*x+256),x, algorithm="fricas")

[Out]

8/(x + 2*e^(x^2) - 16)

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giac [A]  time = 0.13, size = 13, normalized size = 0.81 \begin {gather*} \frac {8}{x + 2 \, e^{\left (x^{2}\right )} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*exp(x^2)*x-8)/(4*exp(x^2)^2+(4*x-64)*exp(x^2)+x^2-32*x+256),x, algorithm="giac")

[Out]

8/(x + 2*e^(x^2) - 16)

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maple [A]  time = 0.05, size = 14, normalized size = 0.88

method	result	size
norman	\(\frac {8}{-16+2 \,{\mathrm e}^{x^{2}}+x}\)	\(14\)
risch	\(\frac {8}{-16+2 \,{\mathrm e}^{x^{2}}+x}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-32*exp(x^2)*x-8)/(4*exp(x^2)^2+(4*x-64)*exp(x^2)+x^2-32*x+256),x,method=_RETURNVERBOSE)

[Out]

8/(-16+2*exp(x^2)+x)

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maxima [A]  time = 0.39, size = 13, normalized size = 0.81 \begin {gather*} \frac {8}{x + 2 \, e^{\left (x^{2}\right )} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*exp(x^2)*x-8)/(4*exp(x^2)^2+(4*x-64)*exp(x^2)+x^2-32*x+256),x, algorithm="maxima")

[Out]

8/(x + 2*e^(x^2) - 16)

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mupad [B]  time = 5.74, size = 13, normalized size = 0.81 \begin {gather*} \frac {8}{x+2\,{\mathrm {e}}^{x^2}-16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(32*x*exp(x^2) + 8)/(4*exp(2*x^2) - 32*x + exp(x^2)*(4*x - 64) + x^2 + 256),x)

[Out]

8/(x + 2*exp(x^2) - 16)

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sympy [A]  time = 0.11, size = 10, normalized size = 0.62 \begin {gather*} \frac {8}{x + 2 e^{x^{2}} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*exp(x**2)*x-8)/(4*exp(x**2)**2+(4*x-64)*exp(x**2)+x**2-32*x+256),x)

[Out]

8/(x + 2*exp(x**2) - 16)

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