### 3.291 $$\int \frac{1}{-e^{-x}+e^x} \, dx$$

Optimal. Leaf size=6 $-\tanh ^{-1}\left (e^x\right )$

[Out]

-ArcTanh[E^x]

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Rubi [A]  time = 0.0095504, antiderivative size = 6, 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, 207} $-\tanh ^{-1}\left (e^x\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(-E^(-x) + E^x)^(-1),x]

[Out]

-ArcTanh[E^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 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0027007, size = 6, normalized size = 1. $-\tanh ^{-1}\left (e^x\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-E^(-x) + E^x)^(-1),x]

[Out]

-ArcTanh[E^x]

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Maple [A]  time = 0.002, size = 6, normalized size = 1. \begin{align*} -{\it Artanh} \left ({{\rm e}^{x}} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctanh(exp(x))

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Maxima [B]  time = 0.935972, size = 26, normalized size = 4.33 \begin{align*} -\frac{1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac{1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(e^(-x) + 1) + 1/2*log(e^(-x) - 1)

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Fricas [B]  time = 1.90714, size = 51, normalized size = 8.5 \begin{align*} -\frac{1}{2} \, \log \left (e^{x} + 1\right ) + \frac{1}{2} \, \log \left (e^{x} - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(e^x + 1) + 1/2*log(e^x - 1)

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Sympy [B]  time = 0.101435, size = 19, normalized size = 3.17 \begin{align*} \frac{\log{\left (-1 + e^{- x} \right )}}{2} - \frac{\log{\left (1 + e^{- x} \right )}}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(-1 + exp(-x))/2 - log(1 + exp(-x))/2

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Giac [B]  time = 1.05135, size = 22, normalized size = 3.67 \begin{align*} -\frac{1}{2} \, \log \left (e^{x} + 1\right ) + \frac{1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(e^x + 1) + 1/2*log(abs(e^x - 1))