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

Optimal. Leaf size=18 \[ x \cos ^{-1}(x)-\sqrt{1-x^2} \]

[Out]

-Sqrt[1 - x^2] + x*ArcCos[x]

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Rubi [A]  time = 0.00999595, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1. \[ x \cos ^{-1}(x)-\sqrt{1-x^2} \]

Antiderivative was successfully verified.

[In]  Int[ArcCos[x],x]

[Out]

-Sqrt[1 - x^2] + x*ArcCos[x]

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Rubi in Sympy [A]  time = 1.02105, size = 12, normalized size = 0.67 \[ x \operatorname{acos}{\left (x \right )} - \sqrt{- x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(acos(x),x)

[Out]

x*acos(x) - sqrt(-x**2 + 1)

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Mathematica [A]  time = 0.00474919, size = 18, normalized size = 1. \[ x \cos ^{-1}(x)-\sqrt{1-x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[ArcCos[x],x]

[Out]

-Sqrt[1 - x^2] + x*ArcCos[x]

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Maple [A]  time = 0.001, size = 17, normalized size = 0.9 \[ x\arccos \left ( x \right ) -\sqrt{-{x}^{2}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arccos(x),x)

[Out]

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

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Maxima [A]  time = 1.48087, size = 22, normalized size = 1.22 \[ x \arccos \left (x\right ) - \sqrt{-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*arccos(x) - sqrt(-x^2 + 1)

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Fricas [A]  time = 0.225051, size = 22, normalized size = 1.22 \[ x \arccos \left (x\right ) - \sqrt{-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*arccos(x) - sqrt(-x^2 + 1)

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Sympy [A]  time = 0.137691, size = 12, normalized size = 0.67 \[ x \operatorname{acos}{\left (x \right )} - \sqrt{- x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(acos(x),x)

[Out]

x*acos(x) - sqrt(-x**2 + 1)

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GIAC/XCAS [A]  time = 0.215181, size = 22, normalized size = 1.22 \[ x \arccos \left (x\right ) - \sqrt{-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*arccos(x) - sqrt(-x^2 + 1)