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

Optimal. Leaf size=33 \[ \sqrt{\frac{1}{x^2-1}} \sqrt{x^2-1} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0290945, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \sqrt{\frac{1}{x^2-1}} \sqrt{x^2-1} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(-1 + x^2)^(-1)],x]

[Out]

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

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Rubi in Sympy [A]  time = 0.717235, size = 29, normalized size = 0.88 \[ \sqrt{x^{2} - 1} \sqrt{\frac{1}{x^{2} - 1}} \operatorname{atanh}{\left (\frac{x}{\sqrt{x^{2} - 1}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0341611, size = 56, normalized size = 1.7 \[ \frac{1}{2} \sqrt{\frac{1}{x^2-1}} \sqrt{x^2-1} \left (\log \left (\frac{x}{\sqrt{x^2-1}}+1\right )-\log \left (1-\frac{x}{\sqrt{x^2-1}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 28, normalized size = 0.9 \[ \sqrt{ \left ({x}^{2}-1 \right ) ^{-1}}\sqrt{{x}^{2}-1}\ln \left ( x+\sqrt{{x}^{2}-1} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1/(x^2-1))^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.677104, size = 19, normalized size = 0.58 \[ \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(2*x + 2*sqrt(x^2 - 1))

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Fricas [A]  time = 0.266237, size = 19, normalized size = 0.58 \[ -\log \left (-x + \sqrt{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(-x + sqrt(x^2 - 1))

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Sympy [A]  time = 3.47764, size = 15, normalized size = 0.45 \[ \begin{cases} \log{\left (x + \sqrt{x^{2} - 1} \right )} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((log(x + sqrt(x**2 - 1)), (x > -1) & (x < 1)))

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GIAC/XCAS [A]  time = 0.265338, size = 20, normalized size = 0.61 \[ -{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(abs(-x + sqrt(x^2 - 1)))