3.1067 \(\int \sqrt{1-x} \sqrt{1+x} \, dx\)

Optimal. Leaf size=28 \[ \frac{1}{2} \sqrt{1-x} \sqrt{x+1} x+\frac{1}{2} \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.021347, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{1}{2} \sqrt{1-x} \sqrt{x+1} x+\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x]*Sqrt[1 + x],x]

[Out]

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

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Rubi in Sympy [A]  time = 4.25747, size = 20, normalized size = 0.71 \[ \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{2} + \frac{\operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.010211, size = 20, normalized size = 0.71 \[ \frac{1}{2} \left (\sqrt{1-x^2} x+\sin ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[1 - x^2] + ArcSin[x])/2

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Maple [B]  time = 0.006, size = 57, normalized size = 2. \[ -{\frac{1}{2} \left ( 1-x \right ) ^{{\frac{3}{2}}}\sqrt{1+x}}+{\frac{1}{2}\sqrt{1-x}\sqrt{1+x}}+{\frac{\arcsin \left ( x \right ) }{2}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(1/2)*(1+x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.49146, size = 23, normalized size = 0.82 \[ \frac{1}{2} \, \sqrt{-x^{2} + 1} x + \frac{1}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(-x^2 + 1)*x + 1/2*arcsin(x)

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Fricas [A]  time = 0.208777, size = 127, normalized size = 4.54 \[ -\frac{2 \, x^{3} -{\left (x^{3} - 2 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 2 \,{\left (x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 2 \, x}{2 \,{\left (x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*x^3 - (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 = 8.64077, size = 133, normalized size = 4.75 \[ \begin{cases} - i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{i \left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{x - 1}} - \frac{3 i \left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{x - 1}} + \frac{i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{\left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{- x + 1}} + \frac{3 \left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{- x + 1}} - \frac{\sqrt{x + 1}}{\sqrt{- x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-I*acosh(sqrt(2)*sqrt(x + 1)/2) + I*(x + 1)**(5/2)/(2*sqrt(x - 1)) -
3*I*(x + 1)**(3/2)/(2*sqrt(x - 1)) + I*sqrt(x + 1)/sqrt(x - 1), Abs(x + 1)/2 > 1
), (asin(sqrt(2)*sqrt(x + 1)/2) - (x + 1)**(5/2)/(2*sqrt(-x + 1)) + 3*(x + 1)**(
3/2)/(2*sqrt(-x + 1)) - sqrt(x + 1)/sqrt(-x + 1), True))

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GIAC/XCAS [A]  time = 0.209828, size = 36, normalized size = 1.29 \[ \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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