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]

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

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Rubi [A]  time = 0.17, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.05, size = 50, normalized size = 1.00 \[ \frac {2 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {a^2-x^2}} \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {arcosh}\left (\frac {x}{a}\right )}}{\sqrt {a^{2} - x^{2}}}\,{d x} \]

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]

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

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maple [A]  time = 0.09, size = 44, normalized size = 0.88 \[ \frac {2 \mathrm {arccosh}\left (\frac {x}{a}\right )^{\frac {3}{2}} a \sqrt {\frac {-a +x}{a}}\, \sqrt {\frac {a +x}{a}}}{3 \sqrt {-\left (-a +x \right ) \left (a +x \right )}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {arcosh}\left (\frac {x}{a}\right )}}{\sqrt {a^{2} - x^{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {\mathrm {acosh}\left (\frac {x}{a}\right )}}{\sqrt {a^2-x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {acosh}{\left (\frac {x}{a} \right )}}}{\sqrt {- \left (- a + x\right ) \left (a + x\right )}}\, dx \]

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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