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

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 = 0.051085, 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 \[ \frac{\cos ^{-1}(x)}{\sqrt{1-x^2}}+\tanh ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

ArcCos[x]/Sqrt[1 - x^2] + ArcTanh[x]

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Rubi in Sympy [A]  time = 2.95559, size = 14, normalized size = 0.82 \[ \operatorname{atanh}{\left (x \right )} + \frac{\operatorname{acos}{\left (x \right )}}{\sqrt{- x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*acos(x)/(-x**2+1)**(3/2),x)

[Out]

atanh(x) + acos(x)/sqrt(-x**2 + 1)

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Mathematica [A]  time = 0.0503308, 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 = 0.075, size = 47, normalized size = 2.8 \[ -{\frac{\arccos \left ( x \right ) }{{x}^{2}-1}\sqrt{-{x}^{2}+1}}-\ln \left ({\frac{1}{\sqrt{-{x}^{2}+1}}}-{x{\frac{1}{\sqrt{-{x}^{2}+1}}}} \right ) \]

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 = 1.49289, size = 34, normalized size = 2. \[ \frac{\arccos \left (x\right )}{\sqrt{-x^{2} + 1}} + \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \]

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 [A]  time = 0.226343, size = 59, normalized size = 3.47 \[ \frac{{\left (x^{2} - 1\right )} \log \left (x + 1\right ) -{\left (x^{2} - 1\right )} \log \left (x - 1\right ) - 2 \, \sqrt{-x^{2} + 1} \arccos \left (x\right )}{2 \,{\left (x^{2} - 1\right )}} \]

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 = 7.73623, size = 20, normalized size = 1.18 \[ \begin{cases} \operatorname{acoth}{\left (x \right )} & \text{for}\: x^{2} > 1 \\\operatorname{atanh}{\left (x \right )} & \text{for}\: x^{2} < 1 \end{cases} + \frac{\operatorname{acos}{\left (x \right )}}{\sqrt{- x^{2} + 1}} \]

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(-x**2 + 1)

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GIAC/XCAS [A]  time = 0.214696, size = 36, normalized size = 2.12 \[ \frac{\arccos \left (x\right )}{\sqrt{-x^{2} + 1}} + \frac{1}{2} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

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*ln(abs(x + 1)) - 1/2*ln(abs(x - 1))