Optimal. Leaf size=15 \[ 2 \tan ^{-1}\left (\sqrt{-\frac{x}{x+1}}\right ) \]
[Out]
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Rubi [A] time = 0.0222439, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ 2 \tan ^{-1}\left (\sqrt{-\frac{x}{x+1}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[-(x/(1 + x))]/x,x]
[Out]
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Rubi in Sympy [A] time = 1.32897, size = 12, normalized size = 0.8 \[ 2 \operatorname{atan}{\left (\sqrt{- \frac{x}{x + 1}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x/(1+x))**(1/2)/x,x)
[Out]
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Mathematica [B] time = 0.0233332, size = 32, normalized size = 2.13 \[ \frac{2 \sqrt{-\frac{x}{x+1}} \sqrt{x+1} \sinh ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[-(x/(1 + x))]/x,x]
[Out]
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Maple [B] time = 0.006, size = 33, normalized size = 2.2 \[{(1+x)\sqrt{-{\frac{x}{1+x}}}\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ){\frac{1}{\sqrt{x \left ( 1+x \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x/(1+x))^(1/2)/x,x)
[Out]
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Maxima [A] time = 0.797968, size = 18, normalized size = 1.2 \[ 2 \, \arctan \left (\sqrt{-\frac{x}{x + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x/(x + 1))/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.277425, size = 18, normalized size = 1.2 \[ 2 \, \arctan \left (\sqrt{-\frac{x}{x + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x/(x + 1))/x,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \frac{x}{x + 1}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x/(1+x))**(1/2)/x,x)
[Out]
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GIAC/XCAS [A] time = 0.269677, size = 27, normalized size = 1.8 \[ -\frac{1}{2} \, \pi{\rm sign}\left (x + 1\right ) - \arcsin \left (2 \, x + 1\right ){\rm sign}\left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x/(x + 1))/x,x, algorithm="giac")
[Out]