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

Optimal. Leaf size=5 \[ -\text{CosIntegral}\left (\cos ^{-1}(x)\right ) \]

[Out]

-CosIntegral[ArcCos[x]]

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Rubi [A]  time = 0.0608316, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4724, 3302} \[ -\text{CosIntegral}\left (\cos ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[1 - x^2]*ArcCos[x]),x]

[Out]

-CosIntegral[ArcCos[x]]

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Cos[x]^m*Sin[x]^(2*p + 1), x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{1-x^2} \cos ^{-1}(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\cos ^{-1}(x)\right )\\ &=-\text{Ci}\left (\cos ^{-1}(x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0400906, size = 5, normalized size = 1. \[ -\text{CosIntegral}\left (\cos ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-CosIntegral[ArcCos[x]]

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Maple [A]  time = 0.067, size = 6, normalized size = 1.2 \begin{align*} -{\it Ci} \left ( \arccos \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/arccos(x)/(-x^2+1)^(1/2),x)

[Out]

-Ci(arccos(x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{-x^{2} + 1} \arccos \left (x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(sqrt(-x^2 + 1)*arccos(x)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{2} + 1} x}{{\left (x^{2} - 1\right )} \arccos \left (x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-x^2 + 1)*x/((x^2 - 1)*arccos(x)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )} \operatorname{acos}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/acos(x)/(-x**2+1)**(1/2),x)

[Out]

Integral(x/(sqrt(-(x - 1)*(x + 1))*acos(x)), x)

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Giac [A]  time = 1.37826, size = 7, normalized size = 1.4 \begin{align*} -\operatorname{Ci}\left (\arccos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos_integral(arccos(x))