3.4.100 \(\int \frac {\cosh ^{-1}(\frac {x}{a})^{3/2}}{(a^2-x^2)^{3/2}} \, dx\) [400]

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

[Out]

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

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Rubi [A]
time = 0.06, 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 = {} \begin {gather*} \int \frac {\cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 0.73, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}} \, dx \end {gather*}

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 = 4.10, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {arccosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}}{\left (a^{2}-x^{2}\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(x/a)^(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

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

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]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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