3.699 \(\int \frac{\sin ^{-1}(\sqrt{1-x^2})}{\sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=28 \[ -\frac{\sqrt{x^2} \sin ^{-1}\left (\sqrt{1-x^2}\right )^2}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.0332714, antiderivative size = 28, 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 = {4834, 4641} \[ -\frac{\sqrt{x^2} \sin ^{-1}\left (\sqrt{1-x^2}\right )^2}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[Sqrt[1 - x^2]]/Sqrt[1 - x^2],x]

[Out]

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

Rule 4834

Int[ArcSin[Sqrt[1 + (b_.)*(x_)^2]]^(n_.)/Sqrt[1 + (b_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[-(b*x^2)]/(b*x), Subst
[Int[ArcSin[x]^n/Sqrt[1 - x^2], x], x, Sqrt[1 + b*x^2]], x] /; FreeQ[{b, n}, x]

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 (\sqrt{1-x^2}\right )}{\sqrt{1-x^2}} \, dx &=-\frac{\sqrt{x^2} \operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,\sqrt{1-x^2}\right )}{x}\\ &=-\frac{\sqrt{x^2} \sin ^{-1}\left (\sqrt{1-x^2}\right )^2}{2 x}\\ \end{align*}

Mathematica [A]  time = 0.0166199, size = 28, normalized size = 1. \[ -\frac{\sqrt{x^2} \sin ^{-1}\left (\sqrt{1-x^2}\right )^2}{2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[Sqrt[1 - x^2]]/Sqrt[1 - x^2],x]

[Out]

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

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\arcsin \left ( \sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin((-x^2+1)^(1/2))/(-x^2+1)^(1/2),x)

[Out]

int(arcsin((-x^2+1)^(1/2))/(-x^2+1)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (\sqrt{-x^{2} + 1}\right )}{\sqrt{-x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsin(sqrt(-x^2 + 1))/sqrt(-x^2 + 1), x)

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Fricas [A]  time = 2.55128, size = 42, normalized size = 1.5 \begin{align*} -\frac{1}{2} \, \arcsin \left (\sqrt{-x^{2} + 1}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arcsin(sqrt(-x^2 + 1))^2

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Sympy [A]  time = 2.92055, size = 27, normalized size = 0.96 \begin{align*} \frac{x \operatorname{asin}^{2}{\left (x \right )}}{2 \sqrt{x^{2}}} + \operatorname{asin}{\left (x \right )} \operatorname{asin}{\left (\sqrt{1 - x^{2}} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin((-x**2+1)**(1/2))/(-x**2+1)**(1/2),x)

[Out]

x*asin(x)**2/(2*sqrt(x**2)) + asin(x)*asin(sqrt(1 - x**2))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (\sqrt{-x^{2} + 1}\right )}{\sqrt{-x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsin(sqrt(-x^2 + 1))/sqrt(-x^2 + 1), x)