3.69 \(\int \frac{\sin ^{-1}(\frac{x}{a})^{3/2}}{\sqrt{a^2-x^2}} \, dx\)

Optimal. Leaf size=42 \[ \frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2-x^2}} \]

[Out]

(2*a*Sqrt[1 - x^2/a^2]*ArcSin[x/a]^(5/2))/(5*Sqrt[a^2 - x^2])

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Rubi [A]  time = 0.0654377, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4643, 4641} \[ \frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2-x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*Sqrt[1 - x^2/a^2]*ArcSin[x/a]^(5/2))/(5*Sqrt[a^2 - x^2])

Rule 4643

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 - c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*
d + e, 0] &&  !GtQ[d, 0]

Rule 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[c*x])^
(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rubi steps

\begin{align*} \int \frac{\sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{\sqrt{a^2-x^2}} \, dx &=\frac{\sqrt{1-\frac{x^2}{a^2}} \int \frac{\sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx}{\sqrt{a^2-x^2}}\\ &=\frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2-x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0397304, size = 42, normalized size = 1. \[ \frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2-x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*Sqrt[1 - x^2/a^2]*ArcSin[x/a]^(5/2))/(5*Sqrt[a^2 - x^2])

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Maple [A]  time = 0.047, size = 38, normalized size = 0.9 \begin{align*}{\frac{2\,a}{5} \left ( \arcsin \left ({\frac{x}{a}} \right ) \right ) ^{{\frac{5}{2}}}\sqrt{{\frac{{a}^{2}-{x}^{2}}{{a}^{2}}}}{\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(x/a)^(3/2)/(a^2-x^2)^(1/2),x)

[Out]

2/5*arcsin(x/a)^(5/2)*a/(a^2-x^2)^(1/2)*((a^2-x^2)/a^2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x/a)^(3/2)/(a^2-x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(arcsin(x/a)^(3/2)/sqrt(a^2 - x^2), x)

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Fricas [A]  time = 1.99734, size = 92, normalized size = 2.19 \begin{align*} \frac{2}{5} \, \sqrt{-\arctan \left (-\frac{x}{\sqrt{a^{2} - x^{2}}}\right )} \arctan \left (-\frac{x}{\sqrt{a^{2} - x^{2}}}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x/a)^(3/2)/(a^2-x^2)^(1/2),x, algorithm="fricas")

[Out]

2/5*sqrt(-arctan(-x/sqrt(a^2 - x^2)))*arctan(-x/sqrt(a^2 - x^2))^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(x/a)**(3/2)/(a**2-x**2)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.26984, size = 16, normalized size = 0.38 \begin{align*} \frac{2}{5} \, \arcsin \left (\frac{x}{a}\right )^{\frac{5}{2}} \mathrm{sgn}\left (a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x/a)^(3/2)/(a^2-x^2)^(1/2),x, algorithm="giac")

[Out]

2/5*arcsin(x/a)^(5/2)*sgn(a)