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

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 = 0.0256857, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[1 - x]*(1 + x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.96166, 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-x)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0132838, size = 23, normalized size = 1. \[ -\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 19, normalized size = 0.8 \[ -{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{1-x}} \right ) \sqrt{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x)/(1-x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.787574, 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/((x + 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 = 0.222876, size = 36, normalized size = 1.57 \[ \frac{1}{2} \, \sqrt{2} \log \left (\frac{x + 2 \, \sqrt{2} \sqrt{-x + 1} - 3}{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.52683, size = 46, normalized size = 2. \[ \begin{cases} - \sqrt{2} \operatorname{acosh}{\left (\frac{\sqrt{2}}{\sqrt{x + 1}} \right )} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\\sqrt{2} i \operatorname{asin}{\left (\frac{\sqrt{2}}{\sqrt{x + 1}} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-sqrt(2)*acosh(sqrt(2)/sqrt(x + 1)), 2*Abs(1/(x + 1)) > 1), (sqrt(2)*
I*asin(sqrt(2)/sqrt(x + 1)), True))

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GIAC/XCAS [A]  time = 0.210708, size = 51, normalized size = 2.22 \[ -\frac{1}{2} \, \sqrt{2}{\rm ln}\left (\sqrt{2} + \sqrt{-x + 1}\right ) + \frac{1}{2} \, \sqrt{2}{\rm ln}\left ({\left | -\sqrt{2} + \sqrt{-x + 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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