3.6.29 \(\int \frac {1}{2+e^{-x}+e^x} \, dx\) [529]

Optimal. Leaf size=9 \[ -\frac {1}{1+e^x} \]

[Out]

-1/(1+exp(x))

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Rubi [A]
time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2320, 32} \begin {gather*} -\frac {1}{e^x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + E^(-x) + E^x)^(-1),x]

[Out]

-(1 + E^x)^(-1)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {align*} \int \frac {1}{2+e^{-x}+e^x} \, dx &=\text {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,e^x\right )\\ &=-\frac {1}{1+e^x}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 9, normalized size = 1.00 \begin {gather*} \frac {1}{-1-e^x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + E^(-x) + E^x)^(-1),x]

[Out]

(-1 - E^x)^(-1)

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Maple [A]
time = 0.01, size = 9, normalized size = 1.00

method	result	size
default	\(-\frac {1}{1+{\mathrm e}^{x}}\)	\(9\)
norman	\(-\frac {1}{1+{\mathrm e}^{x}}\)	\(9\)
risch	\(-\frac {1}{1+{\mathrm e}^{x}}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2+exp(-x)+exp(x)),x,method=_RETURNVERBOSE)

[Out]

-1/(1+exp(x))

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Maxima [A]
time = 0.28, size = 8, normalized size = 0.89 \begin {gather*} \frac {1}{e^{\left (-x\right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+exp(-x)+exp(x)),x, algorithm="maxima")

[Out]

1/(e^(-x) + 1)

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Fricas [A]
time = 0.38, size = 8, normalized size = 0.89 \begin {gather*} -\frac {1}{e^{x} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+exp(-x)+exp(x)),x, algorithm="fricas")

[Out]

-1/(e^x + 1)

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Sympy [A]
time = 0.02, size = 7, normalized size = 0.78 \begin {gather*} - \frac {1}{e^{x} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+exp(-x)+exp(x)),x)

[Out]

-1/(exp(x) + 1)

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Giac [A]
time = 2.88, size = 8, normalized size = 0.89 \begin {gather*} -\frac {1}{e^{x} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+exp(-x)+exp(x)),x, algorithm="giac")

[Out]

-1/(e^x + 1)

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Mupad [B]
time = 0.06, size = 8, normalized size = 0.89 \begin {gather*} -\frac {1}{{\mathrm {e}}^x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(exp(-x) + exp(x) + 2),x)

[Out]

-1/(exp(x) + 1)

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