3.7.99 \(\int \frac {\sin ^{-1}(\sqrt {1-x^2})}{\sqrt {1-x^2}} \, dx\) [699]

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

[Out]

-1/2*arcsin((-x^2+1)^(1/2))^2*(x^2)^(1/2)/x

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Rubi [A]
time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 = {4918, 4737} \begin {gather*} -\frac {\sqrt {x^2} \text {ArcSin}\left (\sqrt {1-x^2}\right )^2}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4737

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

Rule 4918

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]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {x^2} \sin ^{-1}\left (\sqrt {1-x^2}\right )^2}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\arcsin \left (\sqrt {-x^{2}+1}\right )}{\sqrt {-x^{2}+1}}\, dx\]

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 = 0.00, 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(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 = 0.61, size = 14, normalized size = 0.50 \begin {gather*} -\frac {1}{2} \, \arcsin \left (\sqrt {-x^{2} + 1}\right )^{2} \end {gather*}

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 = 0.41, size = 22, normalized size = 0.79 \begin {gather*} - \frac {\sqrt {x^{2}} \operatorname {asin}^{2}{\left (\sqrt {1 - x^{2}} \right )}}{2 x} \end {gather*}

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]

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

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Giac [F]
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(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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\mathrm {asin}\left (\sqrt {1-x^2}\right )}{\sqrt {1-x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin((1 - x^2)^(1/2))/(1 - x^2)^(1/2),x)

[Out]

int(asin((1 - x^2)^(1/2))/(1 - x^2)^(1/2), x)

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