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

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

________________________________________________________________________________________

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 = {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 = 0.01, size = 8, normalized size = 1.00 \[ \frac {1}{2 \cos ^{-1}(x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {1-x^2} \cos ^{-1}(x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

________________________________________________________________________________________

fricas [A]  time = 0.90, size = 6, normalized size = 0.75 \[ \frac {1}{2 \, \arccos \relax (x)^{2}} \]

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

________________________________________________________________________________________

giac [A]  time = 0.93, size = 6, normalized size = 0.75 \[ \frac {1}{2 \, \arccos \relax (x)^{2}} \]

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

________________________________________________________________________________________

maple [A]  time = 0.26, size = 7, normalized size = 0.88

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

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

________________________________________________________________________________________

maxima [A]  time = 0.99, size = 6, normalized size = 0.75 \[ \frac {1}{2 \, \arccos \relax (x)^{2}} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.35, size = 6, normalized size = 0.75 \[ \frac {1}{2\,{\mathrm {acos}\relax (x)}^2} \]

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)

________________________________________________________________________________________

sympy [A]  time = 2.68, size = 7, normalized size = 0.88 \[ \frac {1}{2 \operatorname {acos}^{2}{\relax (x )}} \]

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)

________________________________________________________________________________________