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