3.5.93 \(\int \frac {\sinh ^{-1}(\frac {x}{a})^{3/2}}{(a^2+x^2)^{3/2}} \, dx\) [493]

Optimal. Leaf size=84 \[ \frac {x \sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{a^2 \sqrt {a^2+x^2}}-\frac {3 \sqrt {1+\frac {x^2}{a^2}} \text {Int}\left (\frac {x \sqrt {\sinh ^{-1}\left (\frac {x}{a}\right )}}{1+\frac {x^2}{a^2}},x\right )}{2 a^3 \sqrt {a^2+x^2}} \]

[Out]

x*arcsinh(x/a)^(3/2)/a^2/(a^2+x^2)^(1/2)-3/2*(1+x^2/a^2)^(1/2)*Unintegrable(x*arcsinh(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 {\sinh ^{-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[ArcSinh[x/a]^(3/2)/(a^2 + x^2)^(3/2),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2+x^2\right )^{3/2}} \, dx &=\frac {x \sinh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{a^2 \sqrt {a^2+x^2}}-\frac {\left (3 \sqrt {1+\frac {x^2}{a^2}}\right ) \int \frac {x \sqrt {\sinh ^{-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 = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^{-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[ArcSinh[x/a]^(3/2)/(a^2 + x^2)^(3/2),x]

[Out]

Integrate[ArcSinh[x/a]^(3/2)/(a^2 + x^2)^(3/2), x]

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Maple [A]
time = 4.87, size = 0, normalized size = 0.00 \[\int \frac {\arcsinh \left (\frac {x}{a}\right )^{\frac {3}{2}}}{\left (a^{2}+x^{2}\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(x/a)^(3/2)/(a^2+x^2)^(3/2),x)

[Out]

int(arcsinh(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(arcsinh(x/a)^(3/2)/(a^2+x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(arcsinh(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(arcsinh(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 {asinh}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}}{\left (a^{2} + x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(x/a)**(3/2)/(a**2+x**2)**(3/2),x)

[Out]

Integral(asinh(x/a)**(3/2)/(a**2 + x**2)**(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(arcsinh(x/a)^(3/2)/(a^2+x^2)^(3/2),x, algorithm="giac")

[Out]

integrate(arcsinh(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 {asinh}\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(asinh(x/a)^(3/2)/(a^2 + x^2)^(3/2),x)

[Out]

int(asinh(x/a)^(3/2)/(a^2 + x^2)^(3/2), x)

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