Optimal. Leaf size=36 \[ \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0563378, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[(-1 + x)/(1 + x)]/x^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 2.54127, size = 29, normalized size = 0.81 \[ \operatorname{atan}{\left (\sqrt{x - 1} \sqrt{x + 1} \right )} - \frac{\sqrt{x - 1} \sqrt{x + 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((-1+x)/(1+x))**(1/2)/x**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0138223, size = 62, normalized size = 1.72 \[ -\frac{\sqrt{\frac{x-1}{x+1}} \left (\sqrt{x-1} (x+1)+x \sqrt{x+1} \tan ^{-1}\left (\frac{1}{\sqrt{x-1} \sqrt{x+1}}\right )\right )}{\sqrt{x-1} x} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[(-1 + x)/(1 + x)]/x^2,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.02, size = 59, normalized size = 1.6 \[{\frac{1+x}{x}\sqrt{{\frac{-1+x}{1+x}}} \left ( \left ({x}^{2}-1 \right ) ^{{\frac{3}{2}}}-{x}^{2}\sqrt{{x}^{2}-1}-\arctan \left ({\frac{1}{\sqrt{{x}^{2}-1}}} \right ) x \right ){\frac{1}{\sqrt{ \left ( -1+x \right ) \left ( 1+x \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((-1+x)/(1+x))^(1/2)/x^2,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.797029, size = 55, normalized size = 1.53 \[ -\frac{2 \, \sqrt{\frac{x - 1}{x + 1}}}{\frac{x - 1}{x + 1} + 1} + 2 \, \arctan \left (\sqrt{\frac{x - 1}{x + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((x - 1)/(x + 1))/x^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.284026, size = 49, normalized size = 1.36 \[ \frac{2 \, x \arctan \left (\sqrt{\frac{x - 1}{x + 1}}\right ) -{\left (x + 1\right )} \sqrt{\frac{x - 1}{x + 1}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((x - 1)/(x + 1))/x^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{x - 1}{x + 1}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((-1+x)/(1+x))**(1/2)/x**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.269447, size = 69, normalized size = 1.92 \[ -\frac{1}{2} \,{\left (\pi - 2\right )}{\rm sign}\left (x + 1\right ) + 2 \, \arctan \left (-x + \sqrt{x^{2} - 1}\right ){\rm sign}\left (x + 1\right ) - \frac{2 \,{\rm sign}\left (x + 1\right )}{{\left (x - \sqrt{x^{2} - 1}\right )}^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((x - 1)/(x + 1))/x^2,x, algorithm="giac")
[Out]