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

Optimal. Leaf size=30 \[ \frac{x^3}{9}-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)-\frac{x}{3} \]

[Out]

-x/3 + x^3/9 - ((1 - x^2)^(3/2)*ArcCos[x])/3

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Rubi [A]  time = 0.0505711, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{x^3}{9}-\frac{1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)-\frac{x}{3} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[1 - x^2]*ArcCos[x],x]

[Out]

-x/3 + x^3/9 - ((1 - x^2)^(3/2)*ArcCos[x])/3

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Rubi in Sympy [A]  time = 3.02564, size = 20, normalized size = 0.67 \[ \frac{x^{3}}{9} - \frac{x}{3} - \frac{\left (- x^{2} + 1\right )^{\frac{3}{2}} \operatorname{acos}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3/9 - x/3 - (-x**2 + 1)**(3/2)*acos(x)/3

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Mathematica [A]  time = 0.0248096, size = 26, normalized size = 0.87 \[ \frac{1}{9} \left (x^3-3 \left (1-x^2\right )^{3/2} \cos ^{-1}(x)-3 x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[1 - x^2]*ArcCos[x],x]

[Out]

(-3*x + x^3 - 3*(1 - x^2)^(3/2)*ArcCos[x])/9

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Maple [C]  time = 0.208, size = 134, normalized size = 4.5 \[ -{\frac{i+3\,\arccos \left ( x \right ) }{72} \left ( 4\,i{x}^{3}-4\,{x}^{2}\sqrt{-{x}^{2}+1}-3\,ix+\sqrt{-{x}^{2}+1} \right ) }+{\frac{\arccos \left ( x \right ) +i}{8} \left ( ix-\sqrt{-{x}^{2}+1} \right ) }-{\frac{\arccos \left ( x \right ) -i}{8} \left ( ix+\sqrt{-{x}^{2}+1} \right ) }+{\frac{-i+3\,\arccos \left ( x \right ) }{72} \left ( 4\,i{x}^{3}+4\,{x}^{2}\sqrt{-{x}^{2}+1}-3\,ix-\sqrt{-{x}^{2}+1} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/72*(I+3*arccos(x))*(4*I*x^3-4*x^2*(-x^2+1)^(1/2)-3*I*x+(-x^2+1)^(1/2))+1/8*(a
rccos(x)+I)*(I*x-(-x^2+1)^(1/2))-1/8*(arccos(x)-I)*(I*x+(-x^2+1)^(1/2))+1/72*(-I
+3*arccos(x))*(4*I*x^3+4*x^2*(-x^2+1)^(1/2)-3*I*x-(-x^2+1)^(1/2))

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Maxima [A]  time = 1.49901, size = 30, normalized size = 1. \[ \frac{1}{9} \, x^{3} - \frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \arccos \left (x\right ) - \frac{1}{3} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*x^3 - 1/3*(-x^2 + 1)^(3/2)*arccos(x) - 1/3*x

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Fricas [A]  time = 0.234238, size = 36, normalized size = 1.2 \[ \frac{1}{9} \, x^{3} + \frac{1}{3} \,{\left (x^{2} - 1\right )} \sqrt{-x^{2} + 1} \arccos \left (x\right ) - \frac{1}{3} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*x^3 + 1/3*(x^2 - 1)*sqrt(-x^2 + 1)*arccos(x) - 1/3*x

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Sympy [A]  time = 1.42644, size = 37, normalized size = 1.23 \[ \frac{x^{3}}{9} + \frac{x^{2} \sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}}{3} - \frac{x}{3} - \frac{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3/9 + x**2*sqrt(-x**2 + 1)*acos(x)/3 - x/3 - sqrt(-x**2 + 1)*acos(x)/3

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GIAC/XCAS [A]  time = 0.211673, size = 30, normalized size = 1. \[ \frac{1}{9} \, x^{3} - \frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \arccos \left (x\right ) - \frac{1}{3} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*x^3 - 1/3*(-x^2 + 1)^(3/2)*arccos(x) - 1/3*x