### 3.683 $$\int \sqrt{-\cosh (x)+\text{sech}(x)} \, dx$$

Optimal. Leaf size=14 $2 \coth (x) \sqrt{-\sinh (x) \tanh (x)}$

[Out]

2*Coth[x]*Sqrt[-(Sinh[x]*Tanh[x])]

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Rubi [A]  time = 0.0568221, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.273, Rules used = {4397, 4400, 2589} $2 \coth (x) \sqrt{-\sinh (x) \tanh (x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[-Cosh[x] + Sech[x]],x]

[Out]

2*Coth[x]*Sqrt[-(Sinh[x]*Tanh[x])]

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rule 4400

Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = ActivateTrig[v], ww = Ac
tivateTrig[w]}, Dist[(vv^m*ww^n)^FracPart[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p])), Int[uu*vv^(m*p)*ww^(n*p)
, x], x]] /; FreeQ[{m, n, p}, x] &&  !IntegerQ[p] && ( !InertTrigFreeQ[v] ||  !InertTrigFreeQ[w])

Rule 2589

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*(a*Sin[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*m), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 1, 0]

Rubi steps

\begin{align*} \int \sqrt{-\cosh (x)+\text{sech}(x)} \, dx &=\int \sqrt{-\sinh (x) \tanh (x)} \, dx\\ &=\frac{\sqrt{-\sinh (x) \tanh (x)} \int \sqrt{i \sinh (x)} \sqrt{i \tanh (x)} \, dx}{\sqrt{i \sinh (x)} \sqrt{i \tanh (x)}}\\ &=2 \coth (x) \sqrt{-\sinh (x) \tanh (x)}\\ \end{align*}

Mathematica [A]  time = 0.0521067, size = 14, normalized size = 1. $2 \coth (x) \sqrt{-\sinh (x) \tanh (x)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[-Cosh[x] + Sech[x]],x]

[Out]

2*Coth[x]*Sqrt[-(Sinh[x]*Tanh[x])]

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Maple [B]  time = 0.185, size = 43, normalized size = 3.1 \begin{align*}{\frac{\sqrt{2} \left ({{\rm e}^{2\,x}}+1 \right ) }{{{\rm e}^{2\,x}}-1}\sqrt{-{\frac{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}{{\rm e}^{-x}}}{{{\rm e}^{2\,x}}+1}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-cosh(x)+sech(x))^(1/2),x)

[Out]

2^(1/2)*(-(exp(2*x)-1)^2*exp(-x)/(exp(2*x)+1))^(1/2)/(exp(2*x)-1)*(exp(2*x)+1)

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Maxima [B]  time = 1.78975, size = 53, normalized size = 3.79 \begin{align*} -\frac{\sqrt{2} e^{\left (\frac{1}{2} \, x\right )}}{\sqrt{-e^{\left (-2 \, x\right )} - 1}} - \frac{\sqrt{2} e^{\left (-\frac{3}{2} \, x\right )}}{\sqrt{-e^{\left (-2 \, x\right )} - 1}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cosh(x)+sech(x))^(1/2),x, algorithm="maxima")

[Out]

-sqrt(2)*e^(1/2*x)/sqrt(-e^(-2*x) - 1) - sqrt(2)*e^(-3/2*x)/sqrt(-e^(-2*x) - 1)

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Fricas [B]  time = 1.82606, size = 208, normalized size = 14.86 \begin{align*} 2 \, \sqrt{\frac{1}{2}}{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt{-\frac{1}{\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cosh(x)+sech(x))^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(1/2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(-1/(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + sinh(x)
^3 + (3*cosh(x)^2 + 1)*sinh(x) + cosh(x)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- \cosh{\left (x \right )} + \operatorname{sech}{\left (x \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cosh(x)+sech(x))**(1/2),x)

[Out]

Integral(sqrt(-cosh(x) + sech(x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-\cosh \left (x\right ) + \operatorname{sech}\left (x\right )}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cosh(x)+sech(x))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-cosh(x) + sech(x)), x)