Optimal. Leaf size=34 \[ 2 \sqrt{x+\sqrt{x}}-2 \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}\right ) \]
[Out]
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Rubi [A] time = 0.053362, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ 2 \sqrt{x+\sqrt{x}}-2 \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[Sqrt[x] + x],x]
[Out]
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Rubi in Sympy [A] time = 3.04009, size = 29, normalized size = 0.85 \[ 2 \sqrt{\sqrt{x} + x} - 2 \operatorname{atanh}{\left (\frac{\sqrt{x}}{\sqrt{\sqrt{x} + x}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x+x**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0245814, size = 39, normalized size = 1.15 \[ 2 \sqrt{x+\sqrt{x}}-\log \left (2 \sqrt{x}+2 \sqrt{x+\sqrt{x}}+1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[Sqrt[x] + x],x]
[Out]
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Maple [A] time = 0.015, size = 44, normalized size = 1.3 \[ -{1\sqrt{x+\sqrt{x}} \left ( -2\,\sqrt{x+\sqrt{x}}+\ln \left ({\frac{1}{2}}+\sqrt{x}+\sqrt{x+\sqrt{x}} \right ) \right ){\frac{1}{\sqrt{\sqrt{x} \left ( 1+\sqrt{x} \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x+x^(1/2))^(1/2),x)
[Out]
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Maxima [A] time = 0.768714, size = 76, normalized size = 2.24 \[ \frac{2 \, \sqrt{\sqrt{x} + 1}}{x^{\frac{1}{4}}{\left (\frac{\sqrt{x} + 1}{\sqrt{x}} - 1\right )}} - \log \left (\frac{\sqrt{\sqrt{x} + 1}}{x^{\frac{1}{4}}} + 1\right ) + \log \left (\frac{\sqrt{\sqrt{x} + 1}}{x^{\frac{1}{4}}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(x + sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.503679, size = 53, normalized size = 1.56 \[ 2 \, \sqrt{x + \sqrt{x}} + \frac{1}{2} \, \log \left (4 \, \sqrt{x + \sqrt{x}}{\left (2 \, \sqrt{x} + 1\right )} - 8 \, x - 8 \, \sqrt{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(x + sqrt(x)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\sqrt{x} + x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x+x**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(x + sqrt(x)),x, algorithm="giac")
[Out]