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

Optimal. Leaf size=8 \[ 2 \sinh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

2*ArcSinh[Sqrt[x]]

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Rubi [A]  time = 0.0217844, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ 2 \sinh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x/(1 + x)]/x,x]

[Out]

2*ArcSinh[Sqrt[x]]

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Rubi in Sympy [A]  time = 1.65361, size = 7, normalized size = 0.88 \[ 2 \operatorname{asinh}{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x/(1+x))**(1/2)/x,x)

[Out]

2*asinh(sqrt(x))

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Mathematica [A]  time = 0.0097038, size = 8, normalized size = 1. \[ 2 \sinh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x/(1 + x)]/x,x]

[Out]

2*ArcSinh[Sqrt[x]]

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Maple [B]  time = 0.015, size = 32, normalized size = 4. \[{(1+x)\sqrt{{\frac{x}{1+x}}}\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ){\frac{1}{\sqrt{x \left ( 1+x \right ) }}}} \]

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

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Maxima [A]  time = 0.722257, size = 36, normalized size = 4.5 \[ \log \left (\sqrt{\frac{x}{x + 1}} + 1\right ) - \log \left (\sqrt{\frac{x}{x + 1}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x/(x + 1))/x,x, algorithm="maxima")

[Out]

log(sqrt(x/(x + 1)) + 1) - log(sqrt(x/(x + 1)) - 1)

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Fricas [A]  time = 0.273894, size = 36, normalized size = 4.5 \[ \log \left (\sqrt{\frac{x}{x + 1}} + 1\right ) - \log \left (\sqrt{\frac{x}{x + 1}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x/(x + 1))/x,x, algorithm="fricas")

[Out]

log(sqrt(x/(x + 1)) + 1) - log(sqrt(x/(x + 1)) - 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 = 0.26987, size = 30, normalized size = 3.75 \[ -{\rm ln}\left ({\left | -2 \, x + 2 \, \sqrt{x^{2} + x} - 1 \right |}\right ){\rm sign}\left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x/(x + 1))/x,x, algorithm="giac")

[Out]

-ln(abs(-2*x + 2*sqrt(x^2 + x) - 1))*sign(x + 1)