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

Optimal. Leaf size=22 \[ -\sqrt{1-x^2} x-2 x-\sin ^{-1}(x) \]

[Out]

-2*x - x*Sqrt[1 - x^2] - ArcSin[x]

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Rubi [A]  time = 0.0925122, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\sqrt{1-x^2} x-2 x-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Mathematica [A]  time = 0.0226164, size = 34, normalized size = 1.55 \[ -x \left (\sqrt{1-x^2}+2\right )-2 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-2 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.002, size = 59, normalized size = 2.7 \[ -2\,x-\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}+\sqrt{1-x}\sqrt{1+x}-{\arcsin \left ( x \right ) \sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-(1-x)^(1/2)-(1+x)^(1/2))*((1-x)^(1/2)+(1+x)^(1/2)),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 = 0.768891, size = 27, normalized size = 1.23 \[ -\sqrt{-x^{2} + 1} x - 2 \, x - \arcsin \left (x\right ) \]

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 = 0.276811, size = 119, normalized size = 5.41 \[ -\frac{{\left (x^{3} + 2 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 2 \,{\left (x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 2 \, x}{x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2} \]

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 = 39.2566, size = 46, normalized size = 2.09 \[ - 2 x - 4 \left (\begin{cases} \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{4} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) - 2 \]

Verification of antiderivative is not currently implemented for this CAS.

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

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GIAC/XCAS [A]  time = 0.292131, size = 45, normalized size = 2.05 \[ -\sqrt{x + 1} x \sqrt{-x + 1} - 2 \, x - 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) - 2 \]

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