3.493 \(\int \frac{\sinh ^{-1}(\frac{x}{a})^{3/2}}{(a^2+x^2)^{3/2}} \, dx\)

Optimal. Leaf size=83 \[ \frac{x \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{a^2 \sqrt{a^2+x^2}}-\frac{3 \sqrt{\frac{x^2}{a^2}+1} \text{Unintegrable}\left (\frac{x \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{\frac{x^2}{a^2}+1},x\right )}{2 a^3 \sqrt{a^2+x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0756793, 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 = {} \[ \int \frac{\sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{\left (a^2+x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[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*}

Mathematica [A]  time = 0.370795, size = 0, normalized size = 0. \[ \int \frac{\sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{\left (a^2+x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[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 = 0.161, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\it Arcsinh} \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(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., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\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(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., 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(arcsinh(x/a)^(3/2)/(a^2+x^2)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\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(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)