3.1.70 \(\int \frac {1}{\sqrt {1-x^2} \cos ^{-1}(x)^3} \, dx\) [70]

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

[Out]

1/2/arccos(x)^2

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Rubi [A]
time = 0.02, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 = {4738} \begin {gather*} \frac {1}{2 \text {ArcCos}(x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - x^2]*ArcCos[x]^3),x]

[Out]

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

Rule 4738

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

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Mathematica [A]
time = 0.01, size = 8, normalized size = 1.00 \begin {gather*} \frac {1}{2 \cos ^{-1}(x)^2} \end {gather*}

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 = 0.05, size = 7, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {1}{2 \arccos \left (x \right )^{2}}\)	\(7\)
default	\(\frac {1}{2 \arccos \left (x \right )^{2}}\)	\(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arccos(x)^3/(-x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2/arccos(x)^2

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Maxima [A]
time = 2.50, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{2 \, \arccos \left (x\right )^{2}} \end {gather*}

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 = 0.62, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{2 \, \arccos \left (x\right )^{2}} \end {gather*}

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 = 0.48, size = 7, normalized size = 0.88 \begin {gather*} \frac {1}{2 \operatorname {acos}^{2}{\left (x \right )}} \end {gather*}

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 = 0.82, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{2 \, \arccos \left (x\right )^{2}} \end {gather*}

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

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Mupad [B]
time = 0.35, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{2\,{\mathrm {acos}\left (x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(acos(x)^3*(1 - x^2)^(1/2)),x)

[Out]

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

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