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