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

Optimal. Leaf size=22 \[ \sqrt{\frac{1}{x}+1} x+\tanh ^{-1}\left (\sqrt{\frac{1}{x}+1}\right ) \]

[Out]

Sqrt[1 + x^(-1)]*x + ArcTanh[Sqrt[1 + x^(-1)]]

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Rubi [A]  time = 0.0292551, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ \sqrt{\frac{1}{x}+1} x+\tanh ^{-1}\left (\sqrt{\frac{1}{x}+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(1 + x)/x],x]

[Out]

Sqrt[1 + x^(-1)]*x + ArcTanh[Sqrt[1 + x^(-1)]]

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Rubi in Sympy [A]  time = 1.59993, size = 19, normalized size = 0.86 \[ x \sqrt{1 + \frac{1}{x}} + \operatorname{atanh}{\left (\sqrt{1 + \frac{1}{x}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(1 + 1/x) + atanh(sqrt(1 + 1/x))

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Mathematica [A]  time = 0.0226183, size = 34, normalized size = 1.55 \[ \sqrt{\frac{1}{x}+1} x+\frac{1}{2} \log \left (\left (2 \sqrt{\frac{1}{x}+1}+2\right ) x+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[(1 + x)/x],x]

[Out]

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

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Maple [B]  time = 0.005, size = 41, normalized size = 1.9 \[{\frac{x}{2}\sqrt{{\frac{1+x}{x}}} \left ( 2\,\sqrt{{x}^{2}+x}+\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ) \right ){\frac{1}{\sqrt{x \left ( 1+x \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((1+x)/x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.698558, size = 68, normalized size = 3.09 \[ \frac{\sqrt{\frac{x + 1}{x}}}{\frac{x + 1}{x} - 1} + \frac{1}{2} \, \log \left (\sqrt{\frac{x + 1}{x}} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{\frac{x + 1}{x}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt((x + 1)/x)/((x + 1)/x - 1) + 1/2*log(sqrt((x + 1)/x) + 1) - 1/2*log(sqrt((x
+ 1)/x) - 1)

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Fricas [A]  time = 0.26469, size = 54, normalized size = 2.45 \[ x \sqrt{\frac{x + 1}{x}} + \frac{1}{2} \, \log \left (\sqrt{\frac{x + 1}{x}} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{\frac{x + 1}{x}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt((x + 1)/x) + 1/2*log(sqrt((x + 1)/x) + 1) - 1/2*log(sqrt((x + 1)/x) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{x + 1}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((1+x)/x)**(1/2),x)

[Out]

Integral(sqrt((x + 1)/x), x)

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GIAC/XCAS [A]  time = 0.267479, size = 42, normalized size = 1.91 \[ -\frac{1}{2} \,{\rm ln}\left ({\left | -2 \, x + 2 \, \sqrt{x^{2} + x} - 1 \right |}\right ){\rm sign}\left (x\right ) + \sqrt{x^{2} + x}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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