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

Optimal. Leaf size=26 \[ \frac{2 \sqrt{1-x^2}}{x}+\frac{2}{x}+2 \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.411712, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{2 \sqrt{1-x^2}}{x}+\frac{2}{x}+2 \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.0407194, size = 35, normalized size = 1.35 \[ \frac{2 \left (\sqrt{1-x^2}+2 x \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )+1\right )}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.002, size = 50, normalized size = 1.9 \[ 2\,{x}^{-1}-2\,{\frac{ \left ( -\arcsin \left ( x \right ) x-\sqrt{-{x}^{2}+1} \right ) \sqrt{1-x}\sqrt{1+x}}{x\sqrt{-{x}^{2}+1}}} \]

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^2,x)

[Out]

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

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Maxima [A]  time = 0.765417, size = 32, normalized size = 1.23 \[ \frac{2 \, \sqrt{-x^{2} + 1}}{x} + \frac{2}{x} + 2 \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.276682, size = 78, normalized size = 3. \[ -\frac{2 \,{\left (2 \,{\left (\sqrt{x + 1} \sqrt{-x + 1} - 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + x\right )}}{\sqrt{x + 1} \sqrt{-x + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{2}{x^{2}}\, dx - \int \frac{2 \sqrt{- x + 1} \sqrt{x + 1}}{x^{2}}\, dx \]

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**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.316073, size = 201, normalized size = 7.73 \[ 2 \, \pi + \frac{8 \,{\left (\frac{\sqrt{2} - \sqrt{-x + 1}}{\sqrt{x + 1}} - \frac{\sqrt{x + 1}}{\sqrt{2} - \sqrt{-x + 1}}\right )}}{{\left (\frac{\sqrt{2} - \sqrt{-x + 1}}{\sqrt{x + 1}} - \frac{\sqrt{x + 1}}{\sqrt{2} - \sqrt{-x + 1}}\right )}^{2} - 4} + \frac{2}{x} + 4 \, \arctan \left (\frac{\sqrt{x + 1}{\left (\frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{2 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*pi + 8*((sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - sqrt(x + 1)/(sqrt(2) - sqrt(-x
+ 1)))/(((sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - sqrt(x + 1)/(sqrt(2) - sqrt(-x +
1)))^2 - 4) + 2/x + 4*arctan(1/2*sqrt(x + 1)*((sqrt(2) - sqrt(-x + 1))^2/(x + 1
) - 1)/(sqrt(2) - sqrt(-x + 1)))