3.46 \(\int \sqrt{1-x^2} \cos ^{-1}(x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0297441, 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 = {4648, 4642, 30} \[ \frac{x^2}{4}+\frac{1}{2} \sqrt{1-x^2} x \cos ^{-1}(x)-\frac{1}{4} \cos ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]*ArcCos[x],x]

[Out]

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

Rule 4648

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

Rule 4642

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> -Simp[(a + b*ArcCos[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} \cos ^{-1}(x) \, dx &=\frac{1}{2} x \sqrt{1-x^2} \cos ^{-1}(x)+\frac{\int x \, dx}{2}+\frac{1}{2} \int \frac{\cos ^{-1}(x)}{\sqrt{1-x^2}} \, dx\\ &=\frac{x^2}{4}+\frac{1}{2} x \sqrt{1-x^2} \cos ^{-1}(x)-\frac{1}{4} \cos ^{-1}(x)^2\\ \end{align*}

Mathematica [A]  time = 0.0157414, size = 30, normalized size = 0.88 \[ \frac{1}{4} \left (x^2+2 \sqrt{1-x^2} x \cos ^{-1}(x)-\cos ^{-1}(x)^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x^2]*ArcCos[x],x]

[Out]

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

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Maple [A]  time = 0.083, size = 33, normalized size = 1. \begin{align*} -{\frac{\arccos \left ( x \right ) }{2} \left ( -x\sqrt{-{x}^{2}+1}+\arccos \left ( x \right ) \right ) }+{\frac{ \left ( \arccos \left ( x \right ) \right ) ^{2}}{4}}+{\frac{{x}^{2}}{4}}-{\frac{1}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.49029, 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 )} \arccos \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(arccos(x)*(-x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.38116, size = 81, normalized size = 2.38 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 1} x \arccos \left (x\right ) + \frac{1}{4} \, x^{2} - \frac{1}{4} \, \arccos \left (x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 21.5284, 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{acos}{\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(acos(x)*(-x**2+1)**(1/2),x)

[Out]

Piecewise((x*sqrt(1 - x**2)/2 + asin(x)/2, (x > -1) & (x < 1)))*acos(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 = 1.32261, size = 36, normalized size = 1.06 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 1} x \arccos \left (x\right ) + \frac{1}{4} \, x^{2} - \frac{1}{4} \, \arccos \left (x\right )^{2} - \frac{1}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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