Optimal. Leaf size=32 \[ -2 \sqrt{1-x^2}+2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )-2 \log (x) \]
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Rubi [A] time = 0.381187, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -2 \sqrt{1-x^2}+2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )-2 \log (x) \]
Antiderivative was successfully verified.
[In] Int[((-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]))/x,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - 2 \int ^{\sqrt{x + 1}} \frac{x \sqrt{- x^{2} + 2} + 1}{x - 1}\, dx - 2 \int ^{\sqrt{x + 1}} \frac{x \sqrt{- x^{2} + 2} + 1}{x + 1}\, dx \]
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,x)
[Out]
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Mathematica [B] time = 0.04144, size = 84, normalized size = 2.62 \[ -2 \left (\sqrt{1-x^2}+\log (-x)+\log \left (1-\sqrt{x+1}\right )-\log \left (\sqrt{1-x}-\sqrt{x+1}+2\right )-\log \left (\sqrt{x+1}+1\right )+\log \left (\sqrt{1-x}+\sqrt{x+1}+2\right )\right ) \]
Warning: Unable to verify antiderivative.
[In] Integrate[((-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]))/x,x]
[Out]
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Maple [A] time = 0.003, size = 51, normalized size = 1.6 \[ -2\,\ln \left ( x \right ) -2\,{\frac{\sqrt{1-x}\sqrt{1+x} \left ( \sqrt{-{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) \right ) }{\sqrt{-{x}^{2}+1}}} \]
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,x)
[Out]
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Maxima [A] time = 0.768683, size = 55, normalized size = 1.72 \[ -2 \, \sqrt{-x^{2} + 1} - 2 \, \log \left (x\right ) + 2 \, \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(sqrt(x + 1) + sqrt(-x + 1))^2/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.265748, size = 105, normalized size = 3.28 \[ \frac{2 \,{\left (x^{2} - \sqrt{x + 1} \sqrt{-x + 1} \log \left (x\right ) -{\left (\sqrt{x + 1} \sqrt{-x + 1} - 1\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + \log \left (x\right )\right )}}{\sqrt{x + 1} \sqrt{-x + 1} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(sqrt(x + 1) + sqrt(-x + 1))^2/x,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{2}{x}\, dx - \int \frac{2 \sqrt{- x + 1} \sqrt{x + 1}}{x}\, dx \]
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,x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(sqrt(x + 1) + sqrt(-x + 1))^2/x,x, algorithm="giac")
[Out]