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)*(-sinh(x)*tanh(x))^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [A]  time = 0.06, size = 14, normalized size = 1.00 \[ 2 \coth (x) \sqrt {-\sinh (x) \tanh (x)} \]

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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fricas [B]  time = 0.42, size = 57, normalized size = 4.07 \[ 2 \, \sqrt {\frac {1}{2}} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \sqrt {-\frac {1}{\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x) + \cosh \relax (x)}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-\cosh \relax (x) + \operatorname {sech}\relax (x)}\,{d x} \]

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)

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maple [B]  time = 0.62, size = 43, normalized size = 3.07 \[ \frac {\sqrt {2}\, \sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-x}}{1+{\mathrm e}^{2 x}}}\, \left (1+{\mathrm e}^{2 x}\right )}{{\mathrm e}^{2 x}-1} \]

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

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maxima [B]  time = 0.88, size = 39, normalized size = 2.79 \[ -\frac {\sqrt {2} e^{\left (\frac {1}{2} \, x\right )}}{\sqrt {-e^{\left (-2 \, x\right )} - 1}} - \frac {\sqrt {2} e^{\left (-\frac {3}{2} \, x\right )}}{\sqrt {-e^{\left (-2 \, x\right )} - 1}} \]

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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mupad [B]  time = 1.60, size = 15, normalized size = 1.07 \[ 2\,\mathrm {coth}\relax (x)\,\sqrt {\frac {1}{\mathrm {cosh}\relax (x)}-\mathrm {cosh}\relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/cosh(x) - cosh(x))^(1/2),x)

[Out]

2*coth(x)*(1/cosh(x) - cosh(x))^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- \cosh {\relax (x )} + \operatorname {sech}{\relax (x )}}\, dx \]

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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