3.677 \(\int \sqrt {\text {csch}(x)+\sinh (x)} \, dx\)

Optimal. Leaf size=13 \[ 2 \tanh (x) \sqrt {\cosh (x) \coth (x)} \]

[Out]

2*(cosh(x)*coth(x))^(1/2)*tanh(x)

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Rubi [A]  time = 0.07, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4397, 4398, 4400, 2589} \[ 2 \tanh (x) \sqrt {\cosh (x) \coth (x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Csch[x] + Sinh[x]],x]

[Out]

2*Sqrt[Cosh[x]*Coth[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 4398

Int[(u_.)*((a_)*(v_))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = ActivateTrig[v]}, Dist[(a^IntPart[p]
*(a*vv)^FracPart[p])/vv^FracPart[p], Int[uu*vv^p, x], x]] /; FreeQ[{a, p}, x] &&  !IntegerQ[p] &&  !InertTrigF
reeQ[v]

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

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Mathematica [B]  time = 0.07, size = 35, normalized size = 2.69 \[ \frac {2 \left (\sqrt [4]{-\sinh ^2(x)}-1\right ) \tanh (x) \sqrt {\cosh (x) \coth (x)}}{\sqrt [4]{-\sinh ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Csch[x] + Sinh[x]],x]

[Out]

(2*Sqrt[Cosh[x]*Coth[x]]*(-1 + (-Sinh[x]^2)^(1/4))*Tanh[x])/(-Sinh[x]^2)^(1/4)

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fricas [B]  time = 0.41, size = 55, normalized size = 4.23 \[ \frac {2 \, \sqrt {\frac {1}{2}} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )}}{\sqrt {\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x) - \cosh \relax (x)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csch(x)+sinh(x))^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(1/2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)/sqrt(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 \[ \int \sqrt {\operatorname {csch}\relax (x) + \sinh \relax (x)}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csch(x)+sinh(x))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(csch(x) + sinh(x)), x)

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maple [B]  time = 0.58, size = 42, normalized size = 3.23 \[ \frac {\sqrt {2}\, \sqrt {\frac {\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-x}}{{\mathrm e}^{2 x}-1}}\, \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((csch(x)+sinh(x))^(1/2),x)

[Out]

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

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maxima [B]  time = 0.52, size = 54, normalized size = 4.15 \[ \frac {\sqrt {2} e^{\left (\frac {1}{2} \, x\right )}}{\sqrt {e^{\left (-x\right )} + 1} \sqrt {-e^{\left (-x\right )} + 1}} - \frac {\sqrt {2} e^{\left (-\frac {3}{2} \, x\right )}}{\sqrt {e^{\left (-x\right )} + 1} \sqrt {-e^{\left (-x\right )} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csch(x)+sinh(x))^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.54, size = 13, normalized size = 1.00 \[ 2\,\mathrm {tanh}\relax (x)\,\sqrt {\mathrm {sinh}\relax (x)+\frac {1}{\mathrm {sinh}\relax (x)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sinh(x) + 1/sinh(x))^(1/2),x)

[Out]

2*tanh(x)*(sinh(x) + 1/sinh(x))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csch(x)+sinh(x))**(1/2),x)

[Out]

Integral(sqrt(sinh(x) + csch(x)), x)

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