3.604 \(\int \frac{e^x}{\cosh (x)+\sinh (x)} \, dx\)

Optimal. Leaf size=1 \[ x \]

[Out]

x

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Rubi [A]  time = 0.0149549, antiderivative size = 1, 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, 29} \[ x \]

Antiderivative was successfully verified.

[In]

Int[E^x/(Cosh[x] + Sinh[x]),x]

[Out]

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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0012358, size = 1, normalized size = 1. \[ x \]

Antiderivative was successfully verified.

[In]

Integrate[E^x/(Cosh[x] + Sinh[x]),x]

[Out]

x

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Maple [A]  time = 0.018, size = 2, normalized size = 2. \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(cosh(x)+sinh(x)),x)

[Out]

x

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Maxima [A]  time = 0.936195, size = 1, normalized size = 1. \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(cosh(x)+sinh(x)),x, algorithm="maxima")

[Out]

x

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Fricas [A]  time = 1.91013, size = 4, normalized size = 4. \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(cosh(x)+sinh(x)),x, algorithm="fricas")

[Out]

x

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Sympy [B]  time = 0.353225, size = 10, normalized size = 10. \begin{align*} \frac{x e^{x}}{\sinh{\left (x \right )} + \cosh{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(cosh(x)+sinh(x)),x)

[Out]

x*exp(x)/(sinh(x) + cosh(x))

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Giac [A]  time = 1.08285, size = 1, normalized size = 1. \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(cosh(x)+sinh(x)),x, algorithm="giac")

[Out]

x