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

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

[Out]

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

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Rubi [A]  time = 0.0292565, antiderivative size = 34, normalized size of antiderivative = 1., 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 = {4647, 4641, 30} \[ -\frac{x^2}{4}+\frac{1}{2} \sqrt{1-x^2} x \sin ^{-1}(x)+\frac{1}{4} \sin ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]*ArcSin[x],x]

[Out]

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

Rule 4647

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[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcSin[c*x])^n/Sqrt[1 -
c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(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]

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]

Rule 30

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

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*}

Mathematica [A]  time = 0.009158, size = 30, normalized size = 0.88 \[ \frac{1}{4} \left (-x^2+2 \sqrt{1-x^2} x \sin ^{-1}(x)+\sin ^{-1}(x)^2\right ) \]

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 = 0.038, size = 31, normalized size = 0.9 \begin{align*}{\frac{\arcsin \left ( x \right ) }{2} \left ( x\sqrt{-{x}^{2}+1}+\arcsin \left ( x \right ) \right ) }-{\frac{ \left ( \arcsin \left ( x \right ) \right ) ^{2}}{4}}-{\frac{{x}^{2}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 1.43341, size = 41, normalized size = 1.21 \begin{align*} -\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{align*}

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 = 2.40354, size = 81, normalized size = 2.38 \begin{align*} \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{align*}

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 = 18.9193, size = 48, normalized size = 1.41 \begin{align*} \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} - \frac{\pi ^{2}}{16} - \frac{1}{4} & \text{for}\: x < 1 \\\text{NaN} & \text{otherwise} \end{cases} \end{align*}

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 - pi**2/16 - 1/4, x < 1), (nan, True))

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Giac [A]  time = 1.08232, size = 36, normalized size = 1.06 \begin{align*} \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{align*}

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