3.7.51 \(\int \sqrt {1-x^2} \sin ^{-1}(x) \, dx\) [651]

Optimal. Leaf size=34 \[ -\frac {x^2}{4}+\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{4} \sin ^{-1}(x)^2 \]

[Out]

-1/4*x^2+1/4*arcsin(x)^2+1/2*x*arcsin(x)*(-x^2+1)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4741, 4737, 30} \begin {gather*} \frac {1}{2} \sqrt {1-x^2} x \text {ArcSin}(x)+\frac {\text {ArcSin}(x)^2}{4}-\frac {x^2}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]*ArcSin[x],x]

[Out]

-1/4*x^2 + (x*Sqrt[1 - x^2]*ArcSin[x])/2 + ArcSin[x]^2/4

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 4741

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

Rubi steps

\begin {align*} \int \sqrt {1-x^2} \sin ^{-1}(x) \, dx &=\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)-\frac {\int x \, dx}{2}+\frac {1}{2} \int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx\\ &=-\frac {x^2}{4}+\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{4} \sin ^{-1}(x)^2\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 30, normalized size = 0.88 \begin {gather*} \frac {1}{4} \left (-x^2+2 x \sqrt {1-x^2} \sin ^{-1}(x)+\sin ^{-1}(x)^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x^2]*ArcSin[x],x]

[Out]

(-x^2 + 2*x*Sqrt[1 - x^2]*ArcSin[x] + ArcSin[x]^2)/4

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Maple [A]
time = 0.09, size = 31, normalized size = 0.91

method	result	size
default	\(\frac {\arcsin \left (x \right ) \left (x \sqrt {-x^{2}+1}+\arcsin \left (x \right )\right )}{2}-\frac {\arcsin \left (x \right )^{2}}{4}-\frac {x^{2}}{4}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(x)*(-x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*arcsin(x)*(x*(-x^2+1)^(1/2)+arcsin(x))-1/4*arcsin(x)^2-1/4*x^2

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Maxima [A]
time = 2.10, size = 30, normalized size = 0.88 \begin {gather*} -\frac {1}{4} \, x^{2} + \frac {1}{2} \, {\left (\sqrt {-x^{2} + 1} x + \arcsin \left (x\right )\right )} \arcsin \left (x\right ) - \frac {1}{4} \, \arcsin \left (x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x^2 + 1/2*(sqrt(-x^2 + 1)*x + arcsin(x))*arcsin(x) - 1/4*arcsin(x)^2

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Fricas [A]
time = 0.52, size = 26, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x \arcsin \left (x\right ) - \frac {1}{4} \, x^{2} + \frac {1}{4} \, \arcsin \left (x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-x^2 + 1)*x*arcsin(x) - 1/4*x^2 + 1/4*arcsin(x)^2

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Sympy [A]
time = 0.97, size = 39, normalized size = 1.15 \begin {gather*} \left (\begin {cases} \frac {x \sqrt {1 - x^{2}}}{2} + \frac {\operatorname {asin}{\left (x \right )}}{2} & \text {for}\: x > -1 \wedge x < 1 \end {cases}\right ) \operatorname {asin}{\left (x \right )} - \begin {cases} \text {NaN} & \text {for}\: x < -1 \\\frac {x^{2}}{4} + \frac {\operatorname {asin}^{2}{\left (x \right )}}{4} & \text {for}\: x < 1 \\\text {NaN} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*sqrt(1 - x**2)/2 + asin(x)/2, (x > -1) & (x < 1)))*asin(x) - Piecewise((nan, x < -1), (x**2/4 + a
sin(x)**2/4, x < 1), (nan, True))

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Giac [A]
time = 1.15, size = 27, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x \arcsin \left (x\right ) - \frac {1}{4} \, x^{2} + \frac {1}{4} \, \arcsin \left (x\right )^{2} + \frac {1}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-x^2 + 1)*x*arcsin(x) - 1/4*x^2 + 1/4*arcsin(x)^2 + 1/8

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \mathrm {asin}\left (x\right )\,\sqrt {1-x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(x)*(1 - x^2)^(1/2),x)

[Out]

int(asin(x)*(1 - x^2)^(1/2), x)

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