3.395 \(\int \frac {\sqrt {\cosh ^{-1}(\frac {x}{a})}}{(a^2-x^2)^{3/2}} \, dx\)

Optimal. Leaf size=98 \[ \frac {\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \text {Int}\left (\frac {x}{\left (1-\frac {x^2}{a^2}\right ) \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}},x\right )}{2 a^3 \sqrt {a^2-x^2}}+\frac {x \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{a^2 \sqrt {a^2-x^2}} \]

[Out]

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

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Rubi [A]  time = 0.26, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[ArcCosh[x/a]]/(a^2 - x^2)^(3/2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.96, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[ArcCosh[x/a]]/(a^2 - x^2)^(3/2),x]

[Out]

Integrate[Sqrt[ArcCosh[x/a]]/(a^2 - x^2)^(3/2), x]

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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)^(3/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {arcosh}\left (\frac {x}{a}\right )}}{{\left (a^{2} - x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(x/a)^(1/2)/(a^2-x^2)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(arccosh(x/a))/(a^2 - x^2)^(3/2), x)

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maple [A]  time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\mathrm {arccosh}\left (\frac {x}{a}\right )}}{\left (a^{2}-x^{2}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(x/a)^(1/2)/(a^2-x^2)^(3/2),x)

[Out]

int(arccosh(x/a)^(1/2)/(a^2-x^2)^(3/2),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {arcosh}\left (\frac {x}{a}\right )}}{{\left (a^{2} - x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(x/a)^(1/2)/(a^2-x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(arccosh(x/a))/(a^2 - x^2)^(3/2), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\mathrm {acosh}\left (\frac {x}{a}\right )}}{{\left (a^2-x^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(x/a)^(1/2)/(a^2 - x^2)^(3/2),x)

[Out]

int(acosh(x/a)^(1/2)/(a^2 - x^2)^(3/2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {acosh}{\left (\frac {x}{a} \right )}}}{\left (- \left (- a + x\right ) \left (a + x\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(x/a)**(1/2)/(a**2-x**2)**(3/2),x)

[Out]

Integral(sqrt(acosh(x/a))/(-(-a + x)*(a + x))**(3/2), x)

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