3.70 \(\int \frac{1}{\sqrt{1-x^2} \cos ^{-1}(x)^3} \, dx\)

Optimal. Leaf size=8 \[ \frac{1}{2 \cos ^{-1}(x)^2} \]

[Out]

1/(2*ArcCos[x]^2)

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Rubi [A]  time = 0.0402161, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{1}{2 \cos ^{-1}(x)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[1 - x^2]*ArcCos[x]^3),x]

[Out]

1/(2*ArcCos[x]^2)

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Rubi in Sympy [A]  time = 2.95948, size = 7, normalized size = 0.88 \[ \frac{1}{2 \operatorname{acos}^{2}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/acos(x)**3/(-x**2+1)**(1/2),x)

[Out]

1/(2*acos(x)**2)

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Mathematica [A]  time = 0.003715, size = 8, normalized size = 1. \[ \frac{1}{2 \cos ^{-1}(x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[1 - x^2]*ArcCos[x]^3),x]

[Out]

1/(2*ArcCos[x]^2)

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Maple [A]  time = 0.007, size = 7, normalized size = 0.9 \[{\frac{1}{2\, \left ( \arccos \left ( x \right ) \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/arccos(x)^3/(-x^2+1)^(1/2),x)

[Out]

1/2/arccos(x)^2

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Maxima [A]  time = 1.51361, size = 8, normalized size = 1. \[ \frac{1}{2 \, \arccos \left (x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 + 1)*arccos(x)^3),x, algorithm="maxima")

[Out]

1/2/arccos(x)^2

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Fricas [A]  time = 0.219914, size = 8, normalized size = 1. \[ \frac{1}{2 \, \arccos \left (x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 + 1)*arccos(x)^3),x, algorithm="fricas")

[Out]

1/2/arccos(x)^2

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Sympy [A]  time = 4.59682, size = 7, normalized size = 0.88 \[ \frac{1}{2 \operatorname{acos}^{2}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/acos(x)**3/(-x**2+1)**(1/2),x)

[Out]

1/(2*acos(x)**2)

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GIAC/XCAS [A]  time = 0.204488, size = 8, normalized size = 1. \[ \frac{1}{2 \, \arccos \left (x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 + 1)*arccos(x)^3),x, algorithm="giac")

[Out]

1/2/arccos(x)^2