3.6.41 \(\int \frac {e^x (-1-x)}{4+8 e^x x+4 e^{2 x} x^2} \, dx\)

Optimal. Leaf size=10 \[ \frac {1}{4+4 e^x x} \]

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Rubi [A]  time = 0.15, antiderivative size = 13, normalized size of antiderivative = 1.30, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6688, 12, 6686} \begin {gather*} \frac {1}{4 \left (e^x x+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*(-1 - x))/(4 + 8*E^x*x + 4*E^(2*x)*x^2),x]

[Out]

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

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Mathematica [A]  time = 0.02, size = 13, normalized size = 1.30 \begin {gather*} \frac {1}{4 \left (1+e^x x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-1 - x))/(4 + 8*E^x*x + 4*E^(2*x)*x^2),x]

[Out]

1/(4*(1 + E^x*x))

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fricas [A]  time = 0.66, size = 10, normalized size = 1.00 \begin {gather*} \frac {1}{4 \, {\left (x e^{x} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/(x*e^x + 1)

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giac [A]  time = 0.47, size = 10, normalized size = 1.00 \begin {gather*} \frac {1}{4 \, {\left (x e^{x} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/(x*e^x + 1)

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maple [A]  time = 0.04, size = 11, normalized size = 1.10

method	result	size
norman	\(\frac {1}{4 \,{\mathrm e}^{x} x +4}\)	\(11\)
risch	\(\frac {1}{4 \,{\mathrm e}^{x} x +4}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/(exp(x)*x+1)

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maxima [A]  time = 0.45, size = 10, normalized size = 1.00 \begin {gather*} \frac {1}{4 \, {\left (x e^{x} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/(x*e^x + 1)

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mupad [B]  time = 0.11, size = 9, normalized size = 0.90 \begin {gather*} \frac {1}{4\,x\,{\mathrm {e}}^x+4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(x + 1))/(4*x^2*exp(2*x) + 8*x*exp(x) + 4),x)

[Out]

1/(4*x*exp(x) + 4)

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sympy [A]  time = 0.09, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{4 x e^{x} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(4*x*exp(x) + 4)

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