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

Optimal. Leaf size=34 \[ 2 \sqrt{x+\sqrt{x}}-2 \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}\right ) \]

[Out]

2*Sqrt[Sqrt[x] + x] - 2*ArcTanh[Sqrt[x]/Sqrt[Sqrt[x] + x]]

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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]

2*Sqrt[Sqrt[x] + x] - 2*ArcTanh[Sqrt[x]/Sqrt[Sqrt[x] + x]]

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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]

2*sqrt(sqrt(x) + x) - 2*atanh(sqrt(x)/sqrt(sqrt(x) + x))

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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]

2*Sqrt[Sqrt[x] + x] - Log[1 + 2*Sqrt[x] + 2*Sqrt[Sqrt[x] + x]]

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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]

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

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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]

2*sqrt(sqrt(x) + 1)/(x^(1/4)*((sqrt(x) + 1)/sqrt(x) - 1)) - log(sqrt(sqrt(x) + 1
)/x^(1/4) + 1) + log(sqrt(sqrt(x) + 1)/x^(1/4) - 1)

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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]

2*sqrt(x + sqrt(x)) + 1/2*log(4*sqrt(x + sqrt(x))*(2*sqrt(x) + 1) - 8*x - 8*sqrt
(x) - 1)

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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]

Integral(1/sqrt(sqrt(x) + x), x)

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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]

Timed out