3.37.53 \(\int \frac {e^{e^{4/x}} (-8 e^{4/x}-2 x)}{e^{8+2 e^{4/x}} x-2 e^{8+e^{4/x}} x^2+e^8 x^3} \, dx\)

Optimal. Leaf size=25 \[ 2 \left (2-\frac {x}{e^8 \left (e^{e^{4/x}}-x\right )}\right ) \]

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Rubi [A]  time = 0.51, antiderivative size = 21, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 3, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {6688, 12, 6687} \begin {gather*} -\frac {2 x}{e^8 \left (e^{e^{4/x}}-x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/(E^8*(E^E^(4/x) - 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 6687

Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y*z, u*z^(n - m), x]}, Simp[(q*y^(m +
1)*z^(m + 1))/(m + 1), x] /;  !FalseQ[q]] /; FreeQ[{m, n}, 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 {2 e^{-8+e^{4/x}} \left (-4 e^{4/x}-x\right )}{\left (e^{e^{4/x}}-x\right )^2 x} \, dx\\ &=2 \int \frac {e^{-8+e^{4/x}} \left (-4 e^{4/x}-x\right )}{\left (e^{e^{4/x}}-x\right )^2 x} \, dx\\ &=-\frac {2 x}{e^8 \left (e^{e^{4/x}}-x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 21, normalized size = 0.84 \begin {gather*} -\frac {2 x}{e^8 \left (e^{e^{4/x}}-x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/(E^8*(E^E^(4/x) - x))

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fricas [A]  time = 0.75, size = 21, normalized size = 0.84 \begin {gather*} \frac {2 \, x}{x e^{8} - e^{\left (e^{\frac {4}{x}} + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x/(x*e^8 - e^(e^(4/x) + 8))

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giac [A]  time = 0.16, size = 21, normalized size = 0.84 \begin {gather*} \frac {2 \, x}{x e^{8} - e^{\left (e^{\frac {4}{x}} + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x/(x*e^8 - e^(e^(4/x) + 8))

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maple [A]  time = 0.25, size = 19, normalized size = 0.76

method	result	size
risch	\(\frac {2 x \,{\mathrm e}^{-8}}{x -{\mathrm e}^{{\mathrm e}^{\frac {4}{x}}}}\)	\(19\)
norman	\(\frac {2 \,{\mathrm e}^{-8} {\mathrm e}^{{\mathrm e}^{\frac {4}{x}}}}{x -{\mathrm e}^{{\mathrm e}^{\frac {4}{x}}}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*exp(4/x)-2*x)*exp(exp(4/x))/(x*exp(4)^2*exp(exp(4/x))^2-2*x^2*exp(4)^2*exp(exp(4/x))+x^3*exp(4)^2),x,m
ethod=_RETURNVERBOSE)

[Out]

2*x*exp(-8)/(x-exp(exp(4/x)))

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maxima [A]  time = 0.70, size = 21, normalized size = 0.84 \begin {gather*} \frac {2 \, x}{x e^{8} - e^{\left (e^{\frac {4}{x}} + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x/(x*e^8 - e^(e^(4/x) + 8))

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mupad [B]  time = 2.26, size = 28, normalized size = 1.12 \begin {gather*} \frac {2\,x^3}{x^3\,{\mathrm {e}}^8-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{4/x}}\,{\mathrm {e}}^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x^3)/(x^3*exp(8) - x^2*exp(exp(4/x))*exp(8))

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sympy [A]  time = 0.25, size = 19, normalized size = 0.76 \begin {gather*} - \frac {2 x}{- x e^{8} + e^{8} e^{e^{\frac {4}{x}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(4/x)-2*x)*exp(exp(4/x))/(x*exp(4)**2*exp(exp(4/x))**2-2*x**2*exp(4)**2*exp(exp(4/x))+x**3*ex
p(4)**2),x)

[Out]

-2*x/(-x*exp(8) + exp(8)*exp(exp(4/x)))

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