3.27 \(\int \frac{\log (x)}{x^2 \sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{\sqrt{1-x^2}}{x}-\frac{\sqrt{1-x^2} \log (x)}{x}-\sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0884065, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{\sqrt{1-x^2}}{x}-\frac{\sqrt{1-x^2} \log (x)}{x}-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[Log[x]/(x^2*Sqrt[1 - x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.54087, size = 27, normalized size = 0.69 \[ - \operatorname{asin}{\left (x \right )} - \frac{\sqrt{- x^{2} + 1} \log{\left (x \right )}}{x} - \frac{\sqrt{- x^{2} + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-asin(x) - sqrt(-x**2 + 1)*log(x)/x - sqrt(-x**2 + 1)/x

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Mathematica [A]  time = 0.0339758, size = 25, normalized size = 0.64 \[ -\frac{\sqrt{1-x^2} (\log (x)+1)}{x}-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.063, size = 35, normalized size = 0.9 \[ -\arcsin \left ( x \right ) +{\frac{1}{x} \left ( -\ln \left ( x \right ) \sqrt{-{x}^{2}+1}-\sqrt{-{x}^{2}+1} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(x)/x^2/(-x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.65262, size = 47, normalized size = 1.21 \[ -\frac{\sqrt{-x^{2} + 1} \log \left (x\right )}{x} - \frac{\sqrt{-x^{2} + 1}}{x} - \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(x)/(sqrt(-x^2 + 1)*x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.233965, size = 105, normalized size = 2.69 \[ \frac{x^{2} + 2 \,{\left (\sqrt{-x^{2} + 1} x - x\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) +{\left (x^{2} - 1\right )} \log \left (x\right ) + \sqrt{-x^{2} + 1}{\left (\log \left (x\right ) + 1\right )} - 1}{\sqrt{-x^{2} + 1} x - x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(x)/(sqrt(-x^2 + 1)*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\log{\left (x \right )}}{x^{2} \sqrt{- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(x)/x**2/(-x**2+1)**(1/2),x)

[Out]

Integral(log(x)/(x**2*sqrt(-(x - 1)*(x + 1))), x)

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GIAC/XCAS [A]  time = 0.209299, size = 99, normalized size = 2.54 \[ \frac{1}{2} \,{\left (\frac{x}{\sqrt{-x^{2} + 1} - 1} - \frac{\sqrt{-x^{2} + 1} - 1}{x}\right )}{\rm ln}\left (x\right ) + \frac{x}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}} - \frac{\sqrt{-x^{2} + 1} - 1}{2 \, x} - \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(x)/(sqrt(-x^2 + 1)*x^2),x, algorithm="giac")

[Out]

1/2*(x/(sqrt(-x^2 + 1) - 1) - (sqrt(-x^2 + 1) - 1)/x)*ln(x) + 1/2*x/(sqrt(-x^2 +
1) - 1) - 1/2*(sqrt(-x^2 + 1) - 1)/x - arcsin(x)