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

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 = 0.050542, 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 \[ \frac{x^2}{4}+\frac{1}{2} \sqrt{1-x^2} x \cos ^{-1}(x)-\frac{1}{4} \cos ^{-1}(x)^2 \]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x \sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}}{2} - \frac{\operatorname{acos}^{2}{\left (x \right )}}{4} + \frac{\int x\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(acos(x)*(-x**2+1)**(1/2),x)

[Out]

x*sqrt(-x**2 + 1)*acos(x)/2 - acos(x)**2/4 + Integral(x, x)/2

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Mathematica [A]  time = 0.0118167, 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 = 0.065, size = 33, normalized size = 1. \[ -{\frac{\arccos \left ( x \right ) }{2} \left ( -x\sqrt{-{x}^{2}+1}+\arccos \left ( x \right ) \right ) }+{\frac{ \left ( \arccos \left ( x \right ) \right ) ^{2}}{4}}+{\frac{{x}^{2}}{4}}-{\frac{1}{4}} \]

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 = 1.49377, size = 41, normalized size = 1.21 \[ \frac{1}{4} \, x^{2} + \frac{1}{2} \,{\left (\sqrt{-x^{2} + 1} x + \arcsin \left (x\right )\right )} \arccos \left (x\right ) + \frac{1}{4} \, \arcsin \left (x\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 1)*arccos(x),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 = 0.230259, size = 35, normalized size = 1.03 \[ \frac{1}{2} \, \sqrt{-x^{2} + 1} x \arccos \left (x\right ) + \frac{1}{4} \, x^{2} - \frac{1}{4} \, \arccos \left (x\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 1)*arccos(x),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 = 31.937, size = 48, normalized size = 1.41 \[ \left (\begin{cases} \frac{x \sqrt{- x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (x \right )}}{2} & \text{for}\: x > -1 \wedge x < 1 \end{cases}\right ) \operatorname{acos}{\left (x \right )} + \begin{cases} \mathrm{NaN} & \text{for}\: x < -1 \\\frac{x^{2}}{4} + \frac{\operatorname{asin}^{2}{\left (x \right )}}{4} - \frac{\pi ^{2}}{16} - \frac{1}{4} & \text{for}\: x < 1 \\\mathrm{NaN} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(acos(x)*(-x**2+1)**(1/2),x)

[Out]

Piecewise((x*sqrt(-x**2 + 1)/2 + asin(x)/2, (x > -1) & (x < 1)))*acos(x) + Piece
wise((nan, x < -1), (x**2/4 + asin(x)**2/4 - pi**2/16 - 1/4, x < 1), (nan, True)
)

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GIAC/XCAS [A]  time = 0.211954, size = 36, normalized size = 1.06 \[ \frac{1}{2} \, \sqrt{-x^{2} + 1} x \arccos \left (x\right ) + \frac{1}{4} \, x^{2} - \frac{1}{4} \, \arccos \left (x\right )^{2} - \frac{1}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 1)*arccos(x),x, algorithm="giac")

[Out]

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