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 = 0.024774, 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, Rules used = {4642} \[ \frac{1}{2 \cos ^{-1}(x)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - x^2]*ArcCos[x]^3),x]

[Out]

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

Rule 4642

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> -Simp[(a + b*ArcCos[c*x])
^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1-x^2} \cos ^{-1}(x)^3} \, dx &=\frac{1}{2 \cos ^{-1}(x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0053038, 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 = 0.004, size = 7, normalized size = 0.9 \begin{align*}{\frac{1}{2\, \left ( \arccos \left ( x \right ) \right ) ^{2}}} \end{align*}

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 = 1.40617, size = 8, normalized size = 1. \begin{align*} \frac{1}{2 \, \arccos \left (x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccos(x)^3/(-x^2+1)^(1/2),x, algorithm="maxima")

[Out]

1/2/arccos(x)^2

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Fricas [A]  time = 1.90228, size = 23, normalized size = 2.88 \begin{align*} \frac{1}{2 \, \arccos \left (x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccos(x)^3/(-x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/2/arccos(x)^2

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Sympy [A]  time = 2.79693, size = 7, normalized size = 0.88 \begin{align*} \frac{1}{2 \operatorname{acos}^{2}{\left (x \right )}} \end{align*}

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 [A]  time = 1.10248, size = 8, normalized size = 1. \begin{align*} \frac{1}{2 \, \arccos \left (x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccos(x)^3/(-x^2+1)^(1/2),x, algorithm="giac")

[Out]

1/2/arccos(x)^2