3.29 \(\int \frac{\tan ^{-1}(x)}{x^2 \sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=57 \[ -\frac{\sqrt{1-x^2} \tan ^{-1}(x)}{x}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )+\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-x^2}}{\sqrt{2}}\right ) \]

[Out]

-((Sqrt[1 - x^2]*ArcTan[x])/x) - ArcTanh[Sqrt[1 - x^2]] + Sqrt[2]*ArcTanh[Sqrt[1
- x^2]/Sqrt[2]]

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Rubi [A]  time = 0.179349, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{\sqrt{1-x^2} \tan ^{-1}(x)}{x}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )+\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-x^2}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[1 - x^2]*ArcTan[x])/x) - ArcTanh[Sqrt[1 - x^2]] + Sqrt[2]*ArcTanh[Sqrt[1
- x^2]/Sqrt[2]]

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Rubi in Sympy [A]  time = 11.1068, size = 44, normalized size = 0.77 \[ \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{- x^{2} + 1}}{2} \right )} - \operatorname{atanh}{\left (\sqrt{- x^{2} + 1} \right )} - \frac{\sqrt{- x^{2} + 1} \operatorname{atan}{\left (x \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*atanh(sqrt(2)*sqrt(-x**2 + 1)/2) - atanh(sqrt(-x**2 + 1)) - sqrt(-x**2 +
1)*atan(x)/x

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Mathematica [A]  time = 0.0978162, size = 77, normalized size = 1.35 \[ -\frac{\log \left (x^2+1\right )}{\sqrt{2}}+\frac{\log \left (-x^2+2 \sqrt{2-2 x^2}+3\right )}{\sqrt{2}}-\log \left (\sqrt{1-x^2}+1\right )-\frac{\sqrt{1-x^2} \tan ^{-1}(x)}{x}+\log (x) \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[1 - x^2]*ArcTan[x])/x) + Log[x] - Log[1 + x^2]/Sqrt[2] + Log[3 - x^2 + 2
*Sqrt[2 - 2*x^2]]/Sqrt[2] - Log[1 + Sqrt[1 - x^2]]

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Maple [F]  time = 0.38, size = 0, normalized size = 0. \[ \int{\frac{\arctan \left ( x \right ) }{{x}^{2}}{\frac{1}{\sqrt{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arctan(x)/x^2/(-x^2+1)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.255414, size = 109, normalized size = 1.91 \[ \frac{\sqrt{2} x \log \left (\frac{x^{2} - 2 \, \sqrt{2} \sqrt{-x^{2} + 1} - 3}{x^{2} + 1}\right ) - x \log \left (\sqrt{-x^{2} + 1} + 1\right ) + x \log \left (\sqrt{-x^{2} + 1} - 1\right ) - 2 \, \sqrt{-x^{2} + 1} \arctan \left (x\right )}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(sqrt(2)*x*log((x^2 - 2*sqrt(2)*sqrt(-x^2 + 1) - 3)/(x^2 + 1)) - x*log(sqrt(
-x^2 + 1) + 1) + x*log(sqrt(-x^2 + 1) - 1) - 2*sqrt(-x^2 + 1)*arctan(x))/x

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.210697, size = 140, normalized size = 2.46 \[ \frac{1}{2} \,{\left (\frac{x}{\sqrt{-x^{2} + 1} - 1} - \frac{\sqrt{-x^{2} + 1} - 1}{x}\right )} \arctan \left (x\right ) - \frac{1}{2} \, \sqrt{2}{\rm ln}\left (\frac{\sqrt{2} - \sqrt{-x^{2} + 1}}{\sqrt{2} + \sqrt{-x^{2} + 1}}\right ) - \frac{1}{2} \,{\rm ln}\left (\sqrt{-x^{2} + 1} + 1\right ) + \frac{1}{2} \,{\rm ln}\left (-\sqrt{-x^{2} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(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)*arctan(x) - 1/2*sqrt(2)*ln
((sqrt(2) - sqrt(-x^2 + 1))/(sqrt(2) + sqrt(-x^2 + 1))) - 1/2*ln(sqrt(-x^2 + 1)
+ 1) + 1/2*ln(-sqrt(-x^2 + 1) + 1)