Optimal. Leaf size=27 \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0401559, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[(1 + x^2)/(1 - x^4)],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{x^{2} + 1}{- x^{4} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((x**2+1)/(-x**4+1))**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.00774935, size = 27, normalized size = 1. \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[(1 + x^2)/(1 - x^4)],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.005, size = 30, normalized size = 1.1 \[ \sqrt{- \left ({x}^{2}-1 \right ) ^{-1}}\sqrt{{x}^{2}-1}\ln \left ( x+\sqrt{{x}^{2}-1} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((x^2+1)/(-x^4+1))^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.792059, size = 3, normalized size = 0.11 \[ \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-(x^2 + 1)/(x^4 - 1)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.2736, size = 42, normalized size = 1.56 \[ 2 \, \arctan \left (\frac{\sqrt{-\frac{1}{x^{2} - 1}} - 1}{x \sqrt{-\frac{1}{x^{2} - 1}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-(x^2 + 1)/(x^4 - 1)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{x^{2} + 1}{- x^{4} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((x**2+1)/(-x**4+1))**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.268806, size = 14, normalized size = 0.52 \[ -\arcsin \left (x\right ){\rm sign}\left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-(x^2 + 1)/(x^4 - 1)),x, algorithm="giac")
[Out]