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

Optimal. Leaf size=40 \[ \frac{1}{3} x^3 \cos ^{-1}(x)+\frac{1}{9} \left (1-x^2\right )^{3/2}-\frac{\sqrt{1-x^2}}{3} \]

[Out]

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

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Rubi [A]  time = 0.0476349, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{1}{3} x^3 \cos ^{-1}(x)+\frac{1}{9} \left (1-x^2\right )^{3/2}-\frac{\sqrt{1-x^2}}{3} \]

Antiderivative was successfully verified.

[In]  Int[x^2*ArcCos[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 3.05471, size = 27, normalized size = 0.68 \[ \frac{x^{3} \operatorname{acos}{\left (x \right )}}{3} + \frac{\left (- x^{2} + 1\right )^{\frac{3}{2}}}{9} - \frac{\sqrt{- x^{2} + 1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*acos(x),x)

[Out]

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

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Mathematica [A]  time = 0.0140873, size = 30, normalized size = 0.75 \[ \frac{1}{3} x^3 \cos ^{-1}(x)-\frac{1}{9} \sqrt{1-x^2} \left (x^2+2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*ArcCos[x],x]

[Out]

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

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Maple [A]  time = 0.019, size = 34, normalized size = 0.9 \[{\frac{{x}^{3}\arccos \left ( x \right ) }{3}}-{\frac{{x}^{2}}{9}\sqrt{-{x}^{2}+1}}-{\frac{2}{9}\sqrt{-{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*arccos(x),x)

[Out]

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

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Maxima [A]  time = 1.52309, size = 45, normalized size = 1.12 \[ \frac{1}{3} \, x^{3} \arccos \left (x\right ) - \frac{1}{9} \, \sqrt{-x^{2} + 1} x^{2} - \frac{2}{9} \, \sqrt{-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.219983, size = 32, normalized size = 0.8 \[ \frac{1}{3} \, x^{3} \arccos \left (x\right ) - \frac{1}{9} \,{\left (x^{2} + 2\right )} \sqrt{-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.49457, size = 32, normalized size = 0.8 \[ \frac{x^{3} \operatorname{acos}{\left (x \right )}}{3} - \frac{x^{2} \sqrt{- x^{2} + 1}}{9} - \frac{2 \sqrt{- x^{2} + 1}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*acos(x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.21215, size = 45, normalized size = 1.12 \[ \frac{1}{3} \, x^{3} \arccos \left (x\right ) - \frac{1}{9} \, \sqrt{-x^{2} + 1} x^{2} - \frac{2}{9} \, \sqrt{-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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