3.261 \(\int \left (\sqrt{1-x}+\sqrt{1+x}\right )^2 \, dx\)

Optimal. Leaf size=19 \[ \sqrt{1-x^2} x+2 x+\sin ^{-1}(x) \]

[Out]

2*x + x*Sqrt[1 - x^2] + ArcSin[x]

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Rubi [A]  time = 0.0436127, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \sqrt{1-x^2} x+2 x+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

2*x + x*Sqrt[1 - x^2] + ArcSin[x]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0224705, size = 33, normalized size = 1.74 \[ x \left (\sqrt{1-x^2}+2\right )+2 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )+2 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.007, size = 58, normalized size = 3.1 \[ 2\,x+\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}-\sqrt{1-x}\sqrt{1+x}+{\arcsin \left ( x \right ) \sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.783593, size = 23, normalized size = 1.21 \[ \sqrt{-x^{2} + 1} x + 2 \, x + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-x^2 + 1)*x + 2*x + arcsin(x)

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Fricas [A]  time = 0.274992, size = 117, normalized size = 6.16 \[ \frac{{\left (x^{3} + 2 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 2 \,{\left (x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 2 \, x}{x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((x^3 + 2*x)*sqrt(x + 1)*sqrt(-x + 1) - 2*(x^2 + 2*sqrt(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 = 30.0789, size = 42, normalized size = 2.21 \[ 2 x + 4 \left (\begin{cases} \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{4} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x + 4*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 = 0.282978, size = 43, normalized size = 2.26 \[ \sqrt{x + 1} x \sqrt{-x + 1} + 2 \, x + 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) + 2 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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