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

Optimal. Leaf size=27 \[ -\frac{1}{2} \sqrt{1-x^2} (2-x)-\frac{1}{2} \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0681749, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{1}{2} \sqrt{1-x^2} (2-x)-\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.2659, size = 29, normalized size = 1.07 \[ - \frac{\sqrt{- x^{2} + 1}}{2} - \frac{\operatorname{asin}{\left (x \right )}}{2} - \frac{\left (- x^{2} + 1\right )^{\frac{3}{2}}}{2 \left (x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x**2 + 1)/2 - asin(x)/2 - (-x**2 + 1)**(3/2)/(2*(x + 1))

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Mathematica [A]  time = 0.0261381, size = 26, normalized size = 0.96 \[ \left (\frac{x}{2}-1\right ) \sqrt{1-x^2}-\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 34, normalized size = 1.3 \[{\frac{x}{2}\sqrt{-{x}^{2}+1}}-{\frac{\arcsin \left ( x \right ) }{2}}-\sqrt{- \left ( 1+x \right ) ^{2}+2+2\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(-x^2+1)^(1/2)/(1+x),x)

[Out]

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

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Maxima [A]  time = 0.785224, size = 38, normalized size = 1.41 \[ \frac{1}{2} \, \sqrt{-x^{2} + 1} x - \sqrt{-x^{2} + 1} - \frac{1}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.285531, size = 123, normalized size = 4.56 \[ -\frac{2 \, x^{3} - 2 \, x^{2} - 2 \,{\left (x^{2} + 2 \, \sqrt{-x^{2} + 1} - 2\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) -{\left (x^{3} - 2 \, x^{2} - 2 \, x\right )} \sqrt{-x^{2} + 1} - 2 \, x}{2 \,{\left (x^{2} + 2 \, \sqrt{-x^{2} + 1} - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.61977, size = 29, normalized size = 1.07 \[ \begin{cases} \frac{x \sqrt{- x^{2} + 1}}{2} - \sqrt{- x^{2} + 1} - \frac{\operatorname{asin}{\left (x \right )}}{2} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.283352, size = 26, normalized size = 0.96 \[ \frac{1}{2} \, \sqrt{-x^{2} + 1}{\left (x - 2\right )} - \frac{1}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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