3.602 \(\int \frac{e^x}{\cosh (x)-\sinh (x)} \, dx\)

Optimal. Leaf size=9 \[ \frac{e^{2 x}}{2} \]

[Out]

E^(2*x)/2

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Rubi [A]  time = 0.016691, antiderivative size = 9, 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, 30} \[ \frac{e^{2 x}}{2} \]

Antiderivative was successfully verified.

[In]

Int[E^x/(Cosh[x] - Sinh[x]),x]

[Out]

E^(2*x)/2

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0030923, size = 9, normalized size = 1. \[ \frac{e^{2 x}}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x/(Cosh[x] - Sinh[x]),x]

[Out]

E^(2*x)/2

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Maple [B]  time = 0.002, size = 14, normalized size = 1.6 \begin{align*} -{\frac{{{\rm e}^{x}}}{-2\,\cosh \left ( x \right ) +2\,\sinh \left ( x \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*exp(x)/(-cosh(x)+sinh(x))

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Maxima [A]  time = 0.943977, size = 8, normalized size = 0.89 \begin{align*} \frac{1}{2} \, e^{\left (2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*e^(2*x)

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Fricas [B]  time = 2.02849, size = 61, normalized size = 6.78 \begin{align*} \frac{\cosh \left (x\right ) + \sinh \left (x\right )}{2 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(cosh(x) + sinh(x))/(cosh(x) - sinh(x))

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Sympy [B]  time = 0.380086, size = 12, normalized size = 1.33 \begin{align*} \frac{e^{x}}{- 2 \sinh{\left (x \right )} + 2 \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]

exp(x)/(-2*sinh(x) + 2*cosh(x))

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Giac [A]  time = 1.09636, size = 8, normalized size = 0.89 \begin{align*} \frac{1}{2} \, e^{\left (2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*e^(2*x)