Optimal. Leaf size=32 \[ \sqrt{-\frac{x}{x+1}} (x+1)-\tan ^{-1}\left (\sqrt{-\frac{x}{x+1}}\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0308614, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \sqrt{-\frac{x}{x+1}} (x+1)-\tan ^{-1}\left (\sqrt{-\frac{x}{x+1}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[-(x/(1 + x))],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 2.0556, size = 27, normalized size = 0.84 \[ \frac{\sqrt{- \frac{x}{x + 1}}}{- \frac{x}{x + 1} + 1} - \operatorname{atan}{\left (\sqrt{- \frac{x}{x + 1}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x/(1+x))**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0283316, size = 43, normalized size = 1.34 \[ \frac{\sqrt{-\frac{x}{x+1}} \left (\sqrt{x} (x+1)-\sqrt{x+1} \sinh ^{-1}\left (\sqrt{x}\right )\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[-(x/(1 + x))],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.005, size = 46, normalized size = 1.4 \[{\frac{1+x}{2}\sqrt{-{\frac{x}{1+x}}} \left ( 2\,\sqrt{{x}^{2}+x}-\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ) \right ){\frac{1}{\sqrt{x \left ( 1+x \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x/(1+x))^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.798798, size = 50, normalized size = 1.56 \[ -\frac{\sqrt{-\frac{x}{x + 1}}}{\frac{x}{x + 1} - 1} - \arctan \left (\sqrt{-\frac{x}{x + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x/(x + 1)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.274071, size = 38, normalized size = 1.19 \[{\left (x + 1\right )} \sqrt{-\frac{x}{x + 1}} - \arctan \left (\sqrt{-\frac{x}{x + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x/(x + 1)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- \frac{x}{x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x/(1+x))**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.274055, size = 49, normalized size = 1.53 \[ \frac{1}{4} \, \pi{\rm sign}\left (x + 1\right ) + \frac{1}{2} \, \arcsin \left (2 \, x + 1\right ){\rm sign}\left (x + 1\right ) + \sqrt{-x^{2} - x}{\rm sign}\left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x/(x + 1)),x, algorithm="giac")
[Out]