3.5.7 \(\int \frac {2+e^x (-2-2 x)}{16+8 x+x^2+e^{2 x} x^2+e^x (-8 x-2 x^2)} \, dx\)

Optimal. Leaf size=14 \[ \frac {2}{-4-x+e^x x} \]

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Rubi [A]  time = 0.15, antiderivative size = 15, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {6688, 12, 6686} \begin {gather*} -\frac {2}{\left (1-e^x\right ) x+4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + E^x*(-2 - 2*x))/(16 + 8*x + x^2 + E^(2*x)*x^2 + E^x*(-8*x - 2*x^2)),x]

[Out]

-2/(4 + (1 - E^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 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 {2 \left (1-e^x-e^x x\right )}{\left (4-\left (-1+e^x\right ) x\right )^2} \, dx\\ &=2 \int \frac {1-e^x-e^x x}{\left (4-\left (-1+e^x\right ) x\right )^2} \, dx\\ &=-\frac {2}{4+\left (1-e^x\right ) x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 13, normalized size = 0.93 \begin {gather*} \frac {2}{-4+\left (-1+e^x\right ) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + E^x*(-2 - 2*x))/(16 + 8*x + x^2 + E^(2*x)*x^2 + E^x*(-8*x - 2*x^2)),x]

[Out]

2/(-4 + (-1 + E^x)*x)

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fricas [A]  time = 0.57, size = 13, normalized size = 0.93 \begin {gather*} \frac {2}{x e^{x} - x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-2)*exp(x)+2)/(exp(x)^2*x^2+(-2*x^2-8*x)*exp(x)+x^2+8*x+16),x, algorithm="fricas")

[Out]

2/(x*e^x - x - 4)

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giac [A]  time = 0.38, size = 13, normalized size = 0.93 \begin {gather*} \frac {2}{x e^{x} - x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-2)*exp(x)+2)/(exp(x)^2*x^2+(-2*x^2-8*x)*exp(x)+x^2+8*x+16),x, algorithm="giac")

[Out]

2/(x*e^x - x - 4)

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maple [A]  time = 0.06, size = 14, normalized size = 1.00

method	result	size
norman	\(\frac {2}{{\mathrm e}^{x} x -4-x}\)	\(14\)
risch	\(\frac {2}{{\mathrm e}^{x} x -4-x}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x-2)*exp(x)+2)/(exp(x)^2*x^2+(-2*x^2-8*x)*exp(x)+x^2+8*x+16),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.43, size = 13, normalized size = 0.93 \begin {gather*} \frac {2}{x e^{x} - x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-2)*exp(x)+2)/(exp(x)^2*x^2+(-2*x^2-8*x)*exp(x)+x^2+8*x+16),x, algorithm="maxima")

[Out]

2/(x*e^x - x - 4)

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mupad [B]  time = 0.10, size = 12, normalized size = 0.86 \begin {gather*} -\frac {2}{x-x\,{\mathrm {e}}^x+4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(x - x*exp(x) + 4)

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sympy [A]  time = 0.23, size = 8, normalized size = 0.57 \begin {gather*} \frac {2}{x e^{x} - x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-2)*exp(x)+2)/(exp(x)**2*x**2+(-2*x**2-8*x)*exp(x)+x**2+8*x+16),x)

[Out]

2/(x*exp(x) - x - 4)

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