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

Optimal. Leaf size=12 \[ \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]

[Out]

ArcTanh[x/Sqrt[-1 + x^2]]

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Rubi [A]  time = 0.00633214, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-1 + x^2],x]

[Out]

ArcTanh[x/Sqrt[-1 + x^2]]

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Rubi in Sympy [A]  time = 0.149264, size = 10, normalized size = 0.83 \[ \operatorname{atanh}{\left (\frac{x}{\sqrt{x^{2} - 1}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh(x/sqrt(x**2 - 1))

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Mathematica [B]  time = 0.00413546, size = 38, normalized size = 3.17 \[ \frac{1}{2} \log \left (\frac{x}{\sqrt{x^2-1}}+1\right )-\frac{1}{2} \log \left (1-\frac{x}{\sqrt{x^2-1}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[-1 + x^2],x]

[Out]

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

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Maple [A]  time = 0.002, size = 11, normalized size = 0.9 \[ \ln \left ( x+\sqrt{{x}^{2}-1} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2-1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.41897, size = 19, normalized size = 1.58 \[ \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \]

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 = 0.203619, size = 19, normalized size = 1.58 \[ -\log \left (-x + \sqrt{x^{2} - 1}\right ) \]

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 = 0.148382, size = 2, normalized size = 0.17 \[ \operatorname{acosh}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**2-1)**(1/2),x)

[Out]

acosh(x)

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GIAC/XCAS [A]  time = 0.20183, size = 20, normalized size = 1.67 \[ -{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 1} \right |}\right ) \]

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)))