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

Optimal. Leaf size=97 \[ \frac{3 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \text{Unintegrable}\left (\frac{x \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}}{1-\frac{x^2}{a^2}},x\right )}{2 a^3 \sqrt{a^2-x^2}}+\frac{x \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{a^2 \sqrt{a^2-x^2}} \]

[Out]

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

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Rubi [A]  time = 0.24817, 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 = {} \[ \int \frac{\cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.96622, size = 0, normalized size = 0. \[ \int \frac{\cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.236, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\rm arccosh} \left ({\frac{x}{a}}\right ) \right ) ^{{\frac{3}{2}}} \left ({a}^{2}-{x}^{2} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcosh}\left (\frac{x}{a}\right )^{\frac{3}{2}}}{{\left (a^{2} - x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError