3.396 \(\int \frac{\sqrt{\cosh ^{-1}(\frac{x}{a})}}{(a^2-x^2)^{5/2}} \, dx\)

Optimal. Leaf size=198 \[ \frac{\sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \text{Unintegrable}\left (\frac{x}{\left (1-\frac{x^2}{a^2}\right ) \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}},x\right )}{3 a^5 \sqrt{a^2-x^2}}+\frac{\sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \text{Unintegrable}\left (\frac{x}{\left (\frac{x^2}{a^2}-1\right )^2 \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}},x\right )}{6 a^5 \sqrt{a^2-x^2}}+\frac{2 x \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{3 a^4 \sqrt{a^2-x^2}}+\frac{x \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{3 a^2 \left (a^2-x^2\right )^{3/2}} \]

[Out]

(x*Sqrt[ArcCosh[x/a]])/(3*a^2*(a^2 - x^2)^(3/2)) + (2*x*Sqrt[ArcCosh[x/a]])/(3*a^4*Sqrt[a^2 - x^2]) + (Sqrt[-1
+ x/a]*Sqrt[1 + x/a]*Unintegrable[x/((1 - x^2/a^2)*Sqrt[ArcCosh[x/a]]), x])/(3*a^5*Sqrt[a^2 - x^2]) + (Sqrt[-
1 + x/a]*Sqrt[1 + x/a]*Unintegrable[x/((-1 + x^2/a^2)^2*Sqrt[ArcCosh[x/a]]), x])/(6*a^5*Sqrt[a^2 - x^2])

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Rubi [A]  time = 0.47886, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{\left (a^2-x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[ArcCosh[x/a]]/(a^2 - x^2)^(5/2),x]

[Out]

(2*x*Sqrt[ArcCosh[x/a]])/(3*a^4*Sqrt[a^2 - x^2]) + (x*Sqrt[ArcCosh[x/a]])/(3*a^2*(a - x)*(a + x)*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])/(3*a^5*Sqrt[a^2 - x^2
]) + (Sqrt[-1 + x/a]*Sqrt[1 + x/a]*Defer[Int][x/((-1 + x^2/a^2)^2*Sqrt[ArcCosh[x/a]]), x])/(6*a^5*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 )^{5/2}} \, 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 )^{5/2} \left (1+\frac{x}{a}\right )^{5/2}} \, dx}{a^4 \sqrt{a^2-x^2}}\\ &=\frac{x \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{3 a^2 (a-x) (a+x) \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 )^2 \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}} \, dx}{6 a^5 \sqrt{a^2-x^2}}-\frac{\left (2 \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}{3 a^4 \sqrt{a^2-x^2}}\\ &=\frac{2 x \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{3 a^4 \sqrt{a^2-x^2}}+\frac{x \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{3 a^2 (a-x) (a+x) \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 )^2 \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}} \, dx}{6 a^5 \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}{3 a^5 \sqrt{a^2-x^2}}\\ \end{align*}

Mathematica [A]  time = 2.0983, size = 0, normalized size = 0. \[ \int \frac{\sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{\left (a^2-x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[ArcCosh[x/a]]/(a^2 - x^2)^(5/2),x]

[Out]

Integrate[Sqrt[ArcCosh[x/a]]/(a^2 - x^2)^(5/2), x]

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Maple [A]  time = 0.321, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{{\rm arccosh} \left ({\frac{x}{a}}\right )} \left ({a}^{2}-{x}^{2} \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(x/a)^(1/2)/(a^2-x^2)^(5/2),x)

[Out]

int(arccosh(x/a)^(1/2)/(a^2-x^2)^(5/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{arcosh}\left (\frac{x}{a}\right )}}{{\left (a^{2} - x^{2}\right )}^{\frac{5}{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)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(arccosh(x/a))/(a^2 - x^2)^(5/2), 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)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(x/a)**(1/2)/(a**2-x**2)**(5/2),x)

[Out]

Timed out

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Giac [A]  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)^(5/2),x, algorithm="giac")

[Out]

sage0*x