3.525 \(\int x \sqrt{\frac{1-x^2}{1+x^2}} \, dx\)

Optimal. Leaf size=51 \[ \frac{1}{2} \sqrt{\frac{1-x^2}{x^2+1}} \left (x^2+1\right )-\tan ^{-1}\left (\sqrt{\frac{1-x^2}{x^2+1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0451144, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{2} \sqrt{\frac{1-x^2}{x^2+1}} \left (x^2+1\right )-\tan ^{-1}\left (\sqrt{\frac{1-x^2}{x^2+1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.30847, size = 39, normalized size = 0.76 \[ \frac{\sqrt{\frac{- x^{2} + 1}{x^{2} + 1}}}{\frac{- x^{2} + 1}{x^{2} + 1} + 1} - \operatorname{atan}{\left (\sqrt{\frac{- x^{2} + 1}{x^{2} + 1}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0465447, size = 79, normalized size = 1.55 \[ \frac{\sqrt{\frac{1-x^2}{x^2+1}} \left (\sqrt{1-x^2} \left (x^2+1\right )+2 \sqrt{x^2+1} \sin ^{-1}\left (\frac{\sqrt{x^2+1}}{\sqrt{2}}\right )\right )}{2 \sqrt{1-x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.024, size = 52, normalized size = 1. \[{\frac{{x}^{2}+1}{2}\sqrt{-{\frac{{x}^{2}-1}{{x}^{2}+1}}} \left ( \sqrt{-{x}^{4}+1}+\arcsin \left ({x}^{2} \right ) \right ){\frac{1}{\sqrt{- \left ({x}^{2}-1 \right ) \left ({x}^{2}+1 \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(-(x^2-1)/(x^2+1))^(1/2)*(x^2+1)*((-x^4+1)^(1/2)+arcsin(x^2))/(-(x^2-1)*(x^2
+1))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{-\frac{x^{2} - 1}{x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x*sqrt(-(x^2 - 1)/(x^2 + 1)), x)

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Fricas [A]  time = 0.272159, size = 117, normalized size = 2.29 \[ -\frac{x^{4} + 2 \,{\left ({\left (x^{2} + 1\right )} \sqrt{-\frac{x^{2} - 1}{x^{2} + 1}} - 1\right )} \arctan \left (\frac{{\left (x^{2} + 1\right )} \sqrt{-\frac{x^{2} - 1}{x^{2} + 1}} - 1}{x^{2}}\right )}{2 \,{\left ({\left (x^{2} + 1\right )} \sqrt{-\frac{x^{2} - 1}{x^{2} + 1}} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 178.468, size = 39, normalized size = 0.76 \[ \begin{cases} \frac{\sqrt{- x^{2} + 1} \sqrt{x^{2} + 1}}{2} - \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{- x^{2} + 1}}{2} \right )} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((sqrt(-x**2 + 1)*sqrt(x**2 + 1)/2 - asin(sqrt(2)*sqrt(-x**2 + 1)/2), (
x > -1) & (x < 1)))

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GIAC/XCAS [A]  time = 0.266698, size = 24, normalized size = 0.47 \[ \frac{1}{2} \, \sqrt{-x^{4} + 1} + \frac{1}{2} \, \arcsin \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(-x^4 + 1) + 1/2*arcsin(x^2)