\(\int \frac {e^x}{\cosh (x)+\sinh (x)} \, dx\) [604]

Optimal result

Integrand size = 11, antiderivative size = 1 \[ \int \frac {e^x}{\cosh (x)+\sinh (x)} \, dx=x \]

[Out]

x

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2320, 29} \[ \int \frac {e^x}{\cosh (x)+\sinh (x)} \, dx=x \]

[In]

Int[E^x/(Cosh[x] + Sinh[x]),x]

[Out]

x

Rule 29

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

Rule 2320

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]]]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right ) \\ & = x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {e^x}{\cosh (x)+\sinh (x)} \, dx=x \]

[In]

Integrate[E^x/(Cosh[x] + Sinh[x]),x]

[Out]

x

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 2, normalized size of antiderivative = 2.00

method	result	size
default	\(x\)	\(2\)

[In]

int(exp(x)/(cosh(x)+sinh(x)),x,method=_RETURNVERBOSE)

[Out]

x

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {e^x}{\cosh (x)+\sinh (x)} \, dx=x \]

[In]

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

[Out]

x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10 vs. \(2 (0) = 0\).

Time = 0.17 (sec) , antiderivative size = 10, normalized size of antiderivative = 10.00 \[ \int \frac {e^x}{\cosh (x)+\sinh (x)} \, dx=\frac {x e^{x}}{\sinh {\left (x \right )} + \cosh {\left (x \right )}} \]

[In]

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

[Out]

x*exp(x)/(sinh(x) + cosh(x))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {e^x}{\cosh (x)+\sinh (x)} \, dx=x \]

[In]

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

[Out]

x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {e^x}{\cosh (x)+\sinh (x)} \, dx=x \]

[In]

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

[Out]

x

Mupad [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {e^x}{\cosh (x)+\sinh (x)} \, dx=x \]

[In]

int(exp(x)/(cosh(x) + sinh(x)),x)

[Out]

x