Optimal. Leaf size=12 \[ -2 E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0511323, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -2 E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - x]/Sqrt[-x - x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.2598, size = 14, normalized size = 1.17 \[ - 2 E\left (\operatorname{asin}{\left (\sqrt{- x} \right )}\middle | -1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-x)**(1/2)/(-x**2-x)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 0.0154677, size = 60, normalized size = 5. \[ -\frac{2 \sqrt{2} \sqrt{x} \sqrt{x+1} \left (F\left (\sin ^{-1}\left (\sqrt{1-x}\right )|\frac{1}{2}\right )-E\left (\sin ^{-1}\left (\sqrt{1-x}\right )|\frac{1}{2}\right )\right )}{\sqrt{-x (x+1)}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - x]/Sqrt[-x - x^2],x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.013, size = 38, normalized size = 3.2 \[ -2\,{\frac{{\it EllipticE} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) \sqrt{-x}\sqrt{-x \left ( 1+x \right ) }\sqrt{2}}{\sqrt{1+x}x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-x)^(1/2)/(-x^2-x)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x + 1}}{\sqrt{-x^{2} - x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x + 1)/sqrt(-x^2 - x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-x + 1}}{\sqrt{-x^{2} - x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x + 1)/sqrt(-x^2 - x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- x + 1}}{\sqrt{- x \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-x)**(1/2)/(-x**2-x)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x + 1}}{\sqrt{-x^{2} - x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x + 1)/sqrt(-x^2 - x),x, algorithm="giac")
[Out]