3.394 \(\int \frac{\sqrt{\cosh ^{-1}(\frac{x}{a})}}{\sqrt{a^2-x^2}} \, dx\)

Optimal. Leaf size=50 \[ \frac{2 a \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{a^2-x^2}} \]

[Out]

(2*a*Sqrt[-1 + x/a]*Sqrt[1 + x/a]*ArcCosh[x/a]^(3/2))/(3*Sqrt[a^2 - x^2])

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Rubi [A]  time = 0.174926, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5713, 5676} \[ \frac{2 a \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{a^2-x^2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[ArcCosh[x/a]]/Sqrt[a^2 - x^2],x]

[Out]

(2*a*Sqrt[-1 + x/a]*Sqrt[1 + x/a]*ArcCosh[x/a]^(3/2))/(3*Sqrt[a^2 - x^2])

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((-d)^IntPart[p]*(
d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{a^2-x^2}} \, dx &=\frac{\left (\sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}\right ) \int \frac{\sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx}{\sqrt{a^2-x^2}}\\ &=\frac{2 a \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{a^2-x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0466195, size = 50, normalized size = 1. \[ \frac{2 a \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{a^2-x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[ArcCosh[x/a]]/Sqrt[a^2 - x^2],x]

[Out]

(2*a*Sqrt[-1 + x/a]*Sqrt[1 + x/a]*ArcCosh[x/a]^(3/2))/(3*Sqrt[a^2 - x^2])

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Maple [A]  time = 0.055, size = 44, normalized size = 0.9 \begin{align*}{\frac{2\,a}{3} \left ({\rm arccosh} \left ({\frac{x}{a}}\right ) \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{-a+x}{a}}}\sqrt{{\frac{a+x}{a}}}{\frac{1}{\sqrt{- \left ( -a+x \right ) \left ( a+x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(x/a)^(1/2)/(a^2-x^2)^(1/2),x)

[Out]

2/3*arccosh(x/a)^(3/2)*a/(-(-a+x)*(a+x))^(1/2)*((-a+x)/a)^(1/2)*((a+x)/a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{arcosh}\left (\frac{x}{a}\right )}}{\sqrt{a^{2} - x^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(x/a)^(1/2)/(a^2-x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(arccosh(x/a))/sqrt(a^2 - x^2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(x/a)^(1/2)/(a^2-x^2)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(x/a)**(1/2)/(a**2-x**2)**(1/2),x)

[Out]

Integral(sqrt(acosh(x/a))/sqrt(-(-a + x)*(a + x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(x/a)^(1/2)/(a^2-x^2)^(1/2),x, algorithm="giac")

[Out]

sage0*x