3.282 \(\int \sqrt{1-x} \left (\sqrt{1-x}+\sqrt{1+x}\right ) \, dx\)

Optimal. Leaf size=31 \[ -\frac{x^2}{2}+\frac{1}{2} \sqrt{1-x^2} x+x+\frac{1}{2} \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0824615, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{x^2}{2}+\frac{1}{2} \sqrt{1-x^2} x+x+\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0220164, size = 31, normalized size = 1. \[ -\frac{x^2}{2}+\frac{1}{2} \sqrt{1-x^2} x+x+\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.002, size = 63, normalized size = 2. \[{\frac{1}{2}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{1}{2}\sqrt{1-x}\sqrt{1+x}}+{\frac{\arcsin \left ( x \right ) }{2}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}}-{\frac{{x}^{2}}{2}}+x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(1-x)^(1/2)*(1+x)^(3/2)-1/2*(1-x)^(1/2)*(1+x)^(1/2)+1/2*((1+x)*(1-x))^(1/2)/
(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)-1/2*x^2+x

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Maxima [A]  time = 0.766558, size = 31, normalized size = 1. \[ -\frac{1}{2} \, x^{2} + \frac{1}{2} \, \sqrt{-x^{2} + 1} x + x + \frac{1}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.298677, size = 138, normalized size = 4.45 \[ -\frac{x^{4} - 2 \, x^{2} -{\left (x^{3} - 2 \, x^{2} + 2 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 2 \,{\left (x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 2 \, x}{2 \,{\left (x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.65975, size = 48, normalized size = 1.55 \[ - \frac{\left (- x + 1\right )^{2}}{2} - 2 \left (\begin{cases} - \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{4} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{- x + 1}}{2} \right )}}{2} & \text{for}\: x \leq 1 \wedge x > -1 \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-x + 1)**2/2 - 2*Piecewise((-x*sqrt(-x + 1)*sqrt(x + 1)/4 + asin(sqrt(2)*sqrt(
-x + 1)/2)/2, (x <= 1) & (x > -1)))

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GIAC/XCAS [A]  time = 0.302178, size = 51, normalized size = 1.65 \[ -\frac{1}{2} \,{\left (x - 1\right )}^{2} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} - \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{-x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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