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

Optimal. Leaf size=23 \[ -\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

[Out]

-(Sqrt[2]*ArcTanh[Sqrt[1 - x]/Sqrt[2]])

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Rubi [A]  time = 0.0420026, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[2]*ArcTanh[Sqrt[1 - x]/Sqrt[2]])

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Rubi in Sympy [A]  time = 5.07769, size = 20, normalized size = 0.87 \[ - \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{- x + 1}}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0206994, size = 45, normalized size = 1.96 \[ \frac{\sqrt{2} \sqrt{x-1} \sqrt{x+1} \tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right )}{\sqrt{1-x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.012, size = 40, normalized size = 1.7 \[ -{\sqrt{2}\sqrt{-{x}^{2}+1}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{1-x}} \right ){\frac{1}{\sqrt{1+x}}}{\frac{1}{\sqrt{1-x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.806912, size = 49, normalized size = 2.13 \[ \frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{2 \, \sqrt{2} + 2 \, \sqrt{-x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.212675, size = 61, normalized size = 2.65 \[ \frac{1}{2} \, \sqrt{2} \log \left (-\frac{x^{2} + 2 \, \sqrt{2} \sqrt{-x^{2} + 1} \sqrt{x + 1} - 2 \, x - 3}{x^{2} + 2 \, x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*log(-(x^2 + 2*sqrt(2)*sqrt(-x^2 + 1)*sqrt(x + 1) - 2*x - 3)/(x^2 + 2
*x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.214985, size = 50, normalized size = 2.17 \[ -\frac{1}{2} \, \sqrt{2}{\rm ln}\left (\sqrt{2} + \sqrt{-x + 1}\right ) + \frac{1}{2} \, \sqrt{2}{\rm ln}\left (\sqrt{2} - \sqrt{-x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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