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

Optimal. Leaf size=28 \[ \sqrt{1-x^2}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )+\log (x) \]

[Out]

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

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Rubi [A]  time = 0.586311, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.18 \[ \sqrt{1-x^2}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )+\log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [B]  time = 0.0432623, size = 82, normalized size = 2.93 \[ \sqrt{1-x^2}+\log (-x)+\log \left (1-\sqrt{x+1}\right )-\log \left (\sqrt{1-x}-\sqrt{x+1}+2\right )-\log \left (\sqrt{x+1}+1\right )+\log \left (\sqrt{1-x}+\sqrt{x+1}+2\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

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 = 0.004, size = 48, normalized size = 1.7 \[ \ln \left ( x \right ) +{1\sqrt{1-x}\sqrt{1+x} \left ( \sqrt{-{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

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)

[Out]

ln(x)+(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 [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x + 1} + \sqrt{-x + 1}}{\sqrt{x + 1} - \sqrt{-x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((sqrt(x + 1) + sqrt(-x + 1))/(sqrt(x + 1) - sqrt(-x + 1)), x)

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Fricas [A]  time = 0.266512, size = 105, normalized size = 3.75 \[ -\frac{x^{2} - \sqrt{x + 1} \sqrt{-x + 1} \log \left (x\right ) -{\left (\sqrt{x + 1} \sqrt{-x + 1} - 1\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + \log \left (x\right )}{\sqrt{x + 1} \sqrt{-x + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(x^2 - sqrt(x + 1)*sqrt(-x + 1)*log(x) - (sqrt(x + 1)*sqrt(-x + 1) - 1)*log((sq
rt(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. \[ - \int \frac{\sqrt{- x + 1}}{\sqrt{- x + 1} - \sqrt{x + 1}}\, dx - \int \frac{\sqrt{x + 1}}{\sqrt{- x + 1} - \sqrt{x + 1}}\, dx \]

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)

[Out]

-Integral(sqrt(-x + 1)/(sqrt(-x + 1) - sqrt(x + 1)), x) - Integral(sqrt(x + 1)/(
sqrt(-x + 1) - sqrt(x + 1)), 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))/(sqrt(x + 1) - sqrt(-x + 1)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError