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

Optimal. Leaf size=32 \[ 2 \sqrt{1-x^2}-2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )+2 \log (x) \]

[Out]

2*Sqrt[1 - x^2] - 2*ArcTanh[Sqrt[1 - x^2]] + 2*Log[x]

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Rubi [A]  time = 0.166174, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ 2 \sqrt{1-x^2}-2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )+2 \log (x) \]

Antiderivative was successfully verified.

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

[Out]

2*Sqrt[1 - x^2] - 2*ArcTanh[Sqrt[1 - x^2]] + 2*Log[x]

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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))**2/x,x)

[Out]

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

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Mathematica [B]  time = 0.039875, 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])^2/x,x]

[Out]

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

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Maple [A]  time = 0.009, 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))^2/x,x)

[Out]

2*ln(x)+2*(1-x)^(1/2)*(1+x)^(1/2)/(-x^2+1)^(1/2)*((-x^2+1)^(1/2)-arctanh(1/(-x^2
+1)^(1/2)))

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Maxima [A]  time = 0.790988, 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]

2*sqrt(-x^2 + 1) + 2*log(x) - 2*log(2*sqrt(-x^2 + 1)/abs(x) + 2/abs(x))

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Fricas [A]  time = 0.2789, 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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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]

Exception raised: NotImplementedError