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

Optimal. Leaf size=45 \[ -\sqrt{1-x^2} \tan ^{-1}(x)+\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )-\sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0868347, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\sqrt{1-x^2} \tan ^{-1}(x)+\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.01614, size = 36, normalized size = 0.8 \[ - \sqrt{- x^{2} + 1} \operatorname{atan}{\left (x \right )} - \operatorname{asin}{\left (x \right )} + \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- x^{2} + 1}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0468074, size = 45, normalized size = 1. \[ -\sqrt{1-x^2} \tan ^{-1}(x)+\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.191, size = 0, normalized size = 0. \[ \int{x\arctan \left ( x \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x*arctan(x)/(-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(x*arctan(x)/sqrt(-x^2 + 1),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.253063, size = 78, normalized size = 1.73 \[ -\sqrt{-x^{2} + 1} \arctan \left (x\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (3 \, x^{2} - 1\right )}}{4 \, \sqrt{-x^{2} + 1} x}\right ) - \arctan \left (\frac{x}{\sqrt{-x^{2} + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 1)*arctan(x) + 1/2*sqrt(2)*arctan(1/4*sqrt(2)*(3*x^2 - 1)/(sqrt(-x^
2 + 1)*x)) - arctan(x/sqrt(-x^2 + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209457, size = 146, normalized size = 3.24 \[ -\frac{1}{2} \, \pi{\rm sign}\left (x\right ) + \frac{1}{2} \, \sqrt{2}{\left (\pi{\rm sign}\left (x\right ) + 2 \, \arctan \left (-\frac{\sqrt{2} x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{4 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right )\right )} - \sqrt{-x^{2} + 1} \arctan \left (x\right ) - \arctan \left (-\frac{x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*pi*sign(x) + 1/2*sqrt(2)*(pi*sign(x) + 2*arctan(-1/4*sqrt(2)*x*((sqrt(-x^2
+ 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))) - sqrt(-x^2 + 1)*arctan(x) - arctan(
-1/2*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))