3.2987 \(\int \frac{1}{\sqrt{1+\sqrt{\frac{1}{x}}}} \, dx\)

Optimal. Leaf size=58 \[ \sqrt{\sqrt{\frac{1}{x}}+1} x-\frac{3 \sqrt{\sqrt{\frac{1}{x}}+1}}{2 \sqrt{\frac{1}{x}}}+\frac{3}{2} \tanh ^{-1}\left (\sqrt{\sqrt{\frac{1}{x}}+1}\right ) \]

[Out]

(-3*Sqrt[1 + Sqrt[x^(-1)]])/(2*Sqrt[x^(-1)]) + Sqrt[1 + Sqrt[x^(-1)]]*x + (3*Arc
Tanh[Sqrt[1 + Sqrt[x^(-1)]]])/2

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Rubi [A]  time = 0.0437874, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \sqrt{\sqrt{\frac{1}{x}}+1} x-\frac{3 \sqrt{\sqrt{\frac{1}{x}}+1}}{2 \sqrt{\frac{1}{x}}}+\frac{3}{2} \tanh ^{-1}\left (\sqrt{\sqrt{\frac{1}{x}}+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[1 + Sqrt[x^(-1)]],x]

[Out]

(-3*Sqrt[1 + Sqrt[x^(-1)]])/(2*Sqrt[x^(-1)]) + Sqrt[1 + Sqrt[x^(-1)]]*x + (3*Arc
Tanh[Sqrt[1 + Sqrt[x^(-1)]]])/2

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Rubi in Sympy [A]  time = 4.84675, size = 51, normalized size = 0.88 \[ x \sqrt{\sqrt{\frac{1}{x}} + 1} - \frac{3 \sqrt{\sqrt{\frac{1}{x}} + 1}}{2 \sqrt{\frac{1}{x}}} + \frac{3 \operatorname{atanh}{\left (\sqrt{\sqrt{\frac{1}{x}} + 1} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(sqrt(1/x) + 1) - 3*sqrt(sqrt(1/x) + 1)/(2*sqrt(1/x)) + 3*atanh(sqrt(sqrt(
1/x) + 1))/2

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Mathematica [A]  time = 0.22486, size = 70, normalized size = 1.21 \[ \frac{1}{4} \left (2 \left (2-3 \sqrt{\frac{1}{x}}\right ) \sqrt{\sqrt{\frac{1}{x}}+1} x-3 \log \left (1-\frac{1}{\sqrt{\sqrt{\frac{1}{x}}+1}}\right )+3 \log \left (\frac{1}{\sqrt{\sqrt{\frac{1}{x}}+1}}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[1 + Sqrt[x^(-1)]],x]

[Out]

(2*(2 - 3*Sqrt[x^(-1)])*Sqrt[1 + Sqrt[x^(-1)]]*x - 3*Log[1 - 1/Sqrt[1 + Sqrt[x^(
-1)]]] + 3*Log[1 + 1/Sqrt[1 + Sqrt[x^(-1)]]])/4

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Maple [B]  time = 0.036, size = 92, normalized size = 1.6 \[ -{\frac{1}{4}\sqrt{1+\sqrt{{x}^{-1}}}\sqrt{x} \left ( 6\,\sqrt{{x}^{-1}}\sqrt{x}\sqrt{\sqrt{{x}^{-1}}x+x}-4\,\sqrt{\sqrt{{x}^{-1}}x+x}\sqrt{x}-3\,\ln \left ( 1/2\,\sqrt{{x}^{-1}}\sqrt{x}+\sqrt{x}+\sqrt{\sqrt{{x}^{-1}}x+x} \right ) \right ){\frac{1}{\sqrt{x \left ( 1+\sqrt{{x}^{-1}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+(1/x)^(1/2))^(1/2),x)

[Out]

-1/4*(1+(1/x)^(1/2))^(1/2)*x^(1/2)*(6*(1/x)^(1/2)*x^(1/2)*((1/x)^(1/2)*x+x)^(1/2
)-4*((1/x)^(1/2)*x+x)^(1/2)*x^(1/2)-3*ln(1/2*(1/x)^(1/2)*x^(1/2)+x^(1/2)+((1/x)^
(1/2)*x+x)^(1/2)))/(x*(1+(1/x)^(1/2)))^(1/2)

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Maxima [A]  time = 1.36032, size = 84, normalized size = 1.45 \[ -\frac{3 \,{\left (\frac{1}{\sqrt{x}} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{\frac{1}{\sqrt{x}} + 1}}{2 \,{\left ({\left (\frac{1}{\sqrt{x}} + 1\right )}^{2} - \frac{2}{\sqrt{x}} - 1\right )}} + \frac{3}{4} \, \log \left (\sqrt{\frac{1}{\sqrt{x}} + 1} + 1\right ) - \frac{3}{4} \, \log \left (\sqrt{\frac{1}{\sqrt{x}} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(3*(1/sqrt(x) + 1)^(3/2) - 5*sqrt(1/sqrt(x) + 1))/((1/sqrt(x) + 1)^2 - 2/sq
rt(x) - 1) + 3/4*log(sqrt(1/sqrt(x) + 1) + 1) - 3/4*log(sqrt(1/sqrt(x) + 1) - 1)

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Fricas [A]  time = 0.230017, size = 89, normalized size = 1.53 \[ \frac{2 \,{\left (2 \, x^{\frac{3}{2}} - 3 \, x\right )} \sqrt{\frac{\sqrt{x} + 1}{\sqrt{x}}} + 3 \, \sqrt{x} \log \left (\sqrt{\frac{\sqrt{x} + 1}{\sqrt{x}}} + 1\right ) - 3 \, \sqrt{x} \log \left (\sqrt{\frac{\sqrt{x} + 1}{\sqrt{x}}} - 1\right )}{4 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(2*(2*x^(3/2) - 3*x)*sqrt((sqrt(x) + 1)/sqrt(x)) + 3*sqrt(x)*log(sqrt((sqrt(
x) + 1)/sqrt(x)) + 1) - 3*sqrt(x)*log(sqrt((sqrt(x) + 1)/sqrt(x)) - 1))/sqrt(x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(sqrt(1/x) + 1), 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(1/sqrt(x) + 1),x, algorithm="giac")

[Out]

Timed out