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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]