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

Optimal. Leaf size=14 \[ \sqrt{1-x^2}+\sin ^{-1}(x) \]

[Out]

Sqrt[1 - x^2] + ArcSin[x]

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Rubi [A]  time = 0.0247177, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \sqrt{1-x^2}+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x^2]/(1 + x),x]

[Out]

Sqrt[1 - x^2] + ArcSin[x]

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Rubi in Sympy [A]  time = 3.4256, size = 10, normalized size = 0.71 \[ \sqrt{- x^{2} + 1} + \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-x**2 + 1) + asin(x)

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Mathematica [B]  time = 0.0323346, size = 46, normalized size = 3.29 \[ \sqrt{1-x^2} \left (1-\frac{2 \log \left (\sqrt{x-1}+\sqrt{x+1}\right )}{\sqrt{x-1} \sqrt{x+1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 18, normalized size = 1.3 \[ \sqrt{- \left ( 1+x \right ) ^{2}+2+2\,x}+\arcsin \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)^(1/2)/(1+x),x)

[Out]

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

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Maxima [A]  time = 0.784969, size = 16, normalized size = 1.14 \[ \sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-x^2 + 1) + arcsin(x)

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Fricas [A]  time = 0.213382, size = 65, normalized size = 4.64 \[ -\frac{x^{2} + 2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right )}{\sqrt{-x^{2} + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.37713, size = 15, normalized size = 1.07 \[ \begin{cases} \sqrt{- x^{2} + 1} + \operatorname{asin}{\left (x \right )} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211639, size = 16, normalized size = 1.14 \[ \sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-x^2 + 1) + arcsin(x)