### 3.560 $$\int \sqrt{\sinh (x) \tanh (x)} \, dx$$

Optimal. Leaf size=13 $2 \coth (x) \sqrt{\sinh (x) \tanh (x)}$

[Out]

2*Coth[x]*Sqrt[Sinh[x]*Tanh[x]]

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Rubi [A]  time = 0.0504159, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {4398, 4400, 2589} $2 \coth (x) \sqrt{\sinh (x) \tanh (x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[Sinh[x]*Tanh[x]],x]

[Out]

2*Coth[x]*Sqrt[Sinh[x]*Tanh[x]]

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])

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{\sinh (x) \tanh (x)} \, dx &=\frac{\sqrt{\sinh (x) \tanh (x)} \int \sqrt{-\sinh (x) \tanh (x)} \, dx}{\sqrt{-\sinh (x) \tanh (x)}}\\ &=\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 = 0.0618208, size = 13, normalized size = 1. $2 \coth (x) \sqrt{\sinh (x) \tanh (x)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[Sinh[x]*Tanh[x]],x]

[Out]

2*Coth[x]*Sqrt[Sinh[x]*Tanh[x]]

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Maple [B]  time = 0.145, size = 42, normalized size = 3.2 \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((sinh(x)*tanh(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 = 1.65304, size = 47, normalized size = 3.62 \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((sinh(x)*tanh(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 = 2.22741, size = 201, normalized size = 15.46 \begin{align*} \frac{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{\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((sinh(x)*tanh(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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sinh{\left (x \right )} \tanh{\left (x \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((sinh(x)*tanh(x))**(1/2),x)

[Out]

Integral(sqrt(sinh(x)*tanh(x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sinh \left (x\right ) \tanh \left (x\right )}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((sinh(x)*tanh(x))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(sinh(x)*tanh(x)), x)