3.581 \(\int \frac{\sqrt{-\frac{x}{1+x}}}{x} \, dx\)

Optimal. Leaf size=15 \[ 2 \tan ^{-1}\left (\sqrt{-\frac{x}{x+1}}\right ) \]

[Out]

2*ArcTan[Sqrt[-(x/(1 + x))]]

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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]

2*ArcTan[Sqrt[-(x/(1 + x))]]

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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]

2*atan(sqrt(-x/(x + 1)))

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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]

(2*Sqrt[-(x/(1 + x))]*Sqrt[1 + x]*ArcSinh[Sqrt[x]])/Sqrt[x]

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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]

(-x/(1+x))^(1/2)*(1+x)/(x*(1+x))^(1/2)*ln(1/2+x+(x^2+x)^(1/2))

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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]

2*arctan(sqrt(-x/(x + 1)))

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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]

2*arctan(sqrt(-x/(x + 1)))

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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]

Integral(sqrt(-x/(x + 1))/x, x)

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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]

-1/2*pi*sign(x + 1) - arcsin(2*x + 1)*sign(x + 1)