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

Optimal. Leaf size=4 \[ \tan ^{-1}\left (e^x\right ) \]

[Out]

ArcTan[E^x]

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Rubi [A]  time = 0.0106405, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2282, 203} \[ \tan ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Int[(E^(-x) + E^x)^(-1),x]

[Out]

ArcTan[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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[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 )\\ &=\tan ^{-1}\left (e^x\right )\\ \end{align*}

Mathematica [A]  time = 0.0026222, size = 4, normalized size = 1. \[ \tan ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(-x) + E^x)^(-1),x]

[Out]

ArcTan[E^x]

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Maple [A]  time = 0.02, size = 4, normalized size = 1. \begin{align*} \arctan \left ({{\rm e}^{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(exp(x))

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Maxima [B]  time = 1.45054, size = 9, normalized size = 2.25 \begin{align*} -\arctan \left (e^{\left (-x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(e^(-x))

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Fricas [A]  time = 0.760628, size = 18, normalized size = 4.5 \begin{align*} \arctan \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(e^x)

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Sympy [B]  time = 0.10334, size = 15, normalized size = 3.75 \begin{align*} \operatorname{RootSum}{\left (4 z^{2} + 1, \left ( i \mapsto i \log{\left (2 i + e^{x} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(4*_z**2 + 1, Lambda(_i, _i*log(2*_i + exp(x))))

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Giac [A]  time = 1.222, size = 4, normalized size = 1. \begin{align*} \arctan \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(e^x)