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

Optimal. Leaf size=22 \[ \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{x}+1}}{\sqrt{2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0994283, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{x}+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.16368, size = 20, normalized size = 0.91 \[ \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{1 + \frac{1}{x}}}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0327234, size = 38, normalized size = 1.73 \[ \frac{\log \left (\left (2 \sqrt{2} \sqrt{\frac{1}{x}+1}+3\right ) x+1\right )-\log (1-x)}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-Log[1 - x] + Log[1 + (3 + 2*Sqrt[2]*Sqrt[1 + x^(-1)])*x])/Sqrt[2]

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Maple [B]  time = 0.019, size = 41, normalized size = 1.9 \[{\frac{\sqrt{2}x}{2}\sqrt{{\frac{1+x}{x}}}{\it Artanh} \left ({\frac{ \left ( 3\,x+1 \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{{x}^{2}+x}}}} \right ){\frac{1}{\sqrt{x \left ( 1+x \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+1/x)^(1/2)/(-x^2+1),x)

[Out]

1/2*((1+x)/x)^(1/2)*x/(x*(1+x))^(1/2)*2^(1/2)*arctanh(1/4*(3*x+1)*2^(1/2)/(x^2+x
)^(1/2))

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Maxima [A]  time = 0.777649, size = 54, normalized size = 2.45 \[ -\frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \sqrt{\frac{x + 1}{x}}\right )}}{2 \, \sqrt{2} + 2 \, \sqrt{\frac{x + 1}{x}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.27427, size = 45, normalized size = 2.05 \[ \frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} x \sqrt{\frac{x + 1}{x}} + 3 \, x + 1}{x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.290333, size = 99, normalized size = 4.5 \[ \frac{1}{2} \, \sqrt{2}{\rm ln}\left (\frac{\sqrt{2} - 1}{\sqrt{2} + 1}\right ){\rm sign}\left (x\right ) - \frac{1}{2} \, \sqrt{2}{\rm ln}\left (\frac{{\left | -2 \, x - 2 \, \sqrt{2} + 2 \, \sqrt{x^{2} + x} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt{2} + 2 \, \sqrt{x^{2} + x} + 2 \right |}}\right ){\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*ln((sqrt(2) - 1)/(sqrt(2) + 1))*sign(x) - 1/2*sqrt(2)*ln(abs(-2*x -
2*sqrt(2) + 2*sqrt(x^2 + x) + 2)/abs(-2*x + 2*sqrt(2) + 2*sqrt(x^2 + x) + 2))*si
gn(x)