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

Optimal. Leaf size=17 \[ \frac{\cos ^{-1}(x)}{\sqrt{1-x^2}}+\tanh ^{-1}(x) \]

[Out]

ArcCos[x]/Sqrt[1 - x^2] + ArcTanh[x]

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Rubi [A]  time = 0.0348905, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4678, 206} \[ \frac{\cos ^{-1}(x)}{\sqrt{1-x^2}}+\tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(x*ArcCos[x])/(1 - x^2)^(3/2),x]

[Out]

ArcCos[x]/Sqrt[1 - x^2] + ArcTanh[x]

Rule 4678

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0539185, size = 32, normalized size = 1.88 \[ \frac{1}{2} \left (\frac{2 \cos ^{-1}(x)}{\sqrt{1-x^2}}-\log (1-x)+\log (x+1)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*ArcCos[x])/Sqrt[1 - x^2] - Log[1 - x] + Log[1 + x])/2

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Maple [B]  time = 0.038, size = 47, normalized size = 2.8 \begin{align*} -{\frac{\arccos \left ( x \right ) }{{x}^{2}-1}\sqrt{-{x}^{2}+1}}-\ln \left ({\frac{1}{\sqrt{-{x}^{2}+1}}}-{x{\frac{1}{\sqrt{-{x}^{2}+1}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arccos(x)/(-x^2+1)^(3/2),x)

[Out]

-(-x^2+1)^(1/2)/(x^2-1)*arccos(x)-ln(1/(-x^2+1)^(1/2)-x/(-x^2+1)^(1/2))

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Maxima [A]  time = 1.40954, size = 34, normalized size = 2. \begin{align*} \frac{\arccos \left (x\right )}{\sqrt{-x^{2} + 1}} + \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.40786, size = 122, normalized size = 7.18 \begin{align*} \frac{{\left (x^{2} - 1\right )} \log \left (x + 1\right ) -{\left (x^{2} - 1\right )} \log \left (x - 1\right ) - 2 \, \sqrt{-x^{2} + 1} \arccos \left (x\right )}{2 \,{\left (x^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 6.04356, size = 20, normalized size = 1.18 \begin{align*} \begin{cases} \operatorname{acoth}{\left (x \right )} & \text{for}\: x^{2} > 1 \\\operatorname{atanh}{\left (x \right )} & \text{for}\: x^{2} < 1 \end{cases} + \frac{\operatorname{acos}{\left (x \right )}}{\sqrt{1 - x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((acoth(x), x**2 > 1), (atanh(x), x**2 < 1)) + acos(x)/sqrt(1 - x**2)

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Giac [A]  time = 1.08046, size = 36, normalized size = 2.12 \begin{align*} \frac{\arccos \left (x\right )}{\sqrt{-x^{2} + 1}} + \frac{1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arccos(x)/sqrt(-x^2 + 1) + 1/2*log(abs(x + 1)) - 1/2*log(abs(x - 1))