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

Optimal. Leaf size=15 \[ \frac{1}{2 \left (1-e^{2 x}\right )} \]

[Out]

1/(2*(1 - E^(2*x)))

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Rubi [A]  time = 0.0139409, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2282, 261} \[ \frac{1}{2 \left (1-e^{2 x}\right )} \]

Antiderivative was successfully verified.

[In]

Int[(-E^(-x) + E^x)^(-2),x]

[Out]

1/(2*(1 - E^(2*x)))

Rule 2282

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]]]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0112566, size = 11, normalized size = 0.73 \[ \frac{1}{2-2 e^{2 x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-E^(-x) + E^x)^(-2),x]

[Out]

(2 - 2*E^(2*x))^(-1)

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Maple [A]  time = 0.018, size = 11, normalized size = 0.7 \begin{align*} -{\frac{1}{2\, \left ({{\rm e}^{x}} \right ) ^{2}-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(exp(x)^2-1)

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Maxima [A]  time = 0.961202, size = 14, normalized size = 0.93 \begin{align*} \frac{1}{2 \,{\left (e^{\left (-2 \, x\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/(e^(-2*x) - 1)

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Fricas [A]  time = 0.889163, size = 27, normalized size = 1.8 \begin{align*} -\frac{1}{2 \,{\left (e^{\left (2 \, x\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(e^(2*x) - 1)

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Sympy [A]  time = 0.074859, size = 10, normalized size = 0.67 \begin{align*} - \frac{1}{2 e^{2 x} - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*exp(2*x) - 2)

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Giac [A]  time = 1.29891, size = 14, normalized size = 0.93 \begin{align*} -\frac{1}{2 \,{\left (e^{\left (2 \, x\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(e^(2*x) - 1)