3.607 \(\int \frac{e^x (1-\sinh (x))}{1-\cosh (x)} \, dx\)

Optimal. Leaf size=15 \[ e^x-\frac{2}{1-e^x} \]

[Out]

E^x - 2/(1 - E^x)

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Rubi [A]  time = 0.0332623, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2282, 683} \[ e^x-\frac{2}{1-e^x} \]

Antiderivative was successfully verified.

[In]

Int[(E^x*(1 - Sinh[x]))/(1 - Cosh[x]),x]

[Out]

E^x - 2/(1 - 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 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

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

Mathematica [A]  time = 0.0288623, size = 20, normalized size = 1.33 \[ \frac{-e^x+e^{2 x}+2}{e^x-1} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(1 - Sinh[x]))/(1 - Cosh[x]),x]

[Out]

(2 - E^x + E^(2*x))/(-1 + E^x)

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Maple [A]  time = 0.017, size = 18, normalized size = 1.2 \begin{align*} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}-2\, \left ( -1+\tanh \left ( x/2 \right ) \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*(1-sinh(x))/(1-cosh(x)),x)

[Out]

1/tanh(1/2*x)-2/(-1+tanh(1/2*x))

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Maxima [A]  time = 0.973092, size = 15, normalized size = 1. \begin{align*} \frac{2}{e^{x} - 1} + e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(1-sinh(x))/(1-cosh(x)),x, algorithm="maxima")

[Out]

2/(e^x - 1) + e^x

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Fricas [A]  time = 1.98821, size = 70, normalized size = 4.67 \begin{align*} -\frac{3 \, \cosh \left (x\right ) - \sinh \left (x\right ) - 1}{\cosh \left (x\right ) - \sinh \left (x\right ) - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(1-sinh(x))/(1-cosh(x)),x, algorithm="fricas")

[Out]

-(3*cosh(x) - sinh(x) - 1)/(cosh(x) - sinh(x) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\sinh{\left (x \right )} - 1\right ) e^{x}}{\cosh{\left (x \right )} - 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(1-sinh(x))/(1-cosh(x)),x)

[Out]

Integral((sinh(x) - 1)*exp(x)/(cosh(x) - 1), x)

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Giac [A]  time = 1.09202, size = 15, normalized size = 1. \begin{align*} \frac{2}{e^{x} - 1} + e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(1-sinh(x))/(1-cosh(x)),x, algorithm="giac")

[Out]

2/(e^x - 1) + e^x