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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]