3.3062 \(\int \sqrt{2+\sqrt{\frac{1}{x}}+\frac{1}{x}} \, dx\)

Optimal. Leaf size=75 \[ \frac{1}{4} \left (\sqrt{\frac{1}{x}}+4\right ) \sqrt{\sqrt{\frac{1}{x}}+\frac{1}{x}+2} x+\frac{7 \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}+4}{2 \sqrt{2} \sqrt{\sqrt{\frac{1}{x}}+\frac{1}{x}+2}}\right )}{8 \sqrt{2}} \]

[Out]

((4 + Sqrt[x^(-1)])*Sqrt[2 + Sqrt[x^(-1)] + x^(-1)]*x)/4 + (7*ArcTanh[(4 + Sqrt[
x^(-1)])/(2*Sqrt[2]*Sqrt[2 + Sqrt[x^(-1)] + x^(-1)])])/(8*Sqrt[2])

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Rubi [A]  time = 0.0890058, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ \frac{1}{4} \left (\sqrt{\frac{1}{x}}+4\right ) \sqrt{\sqrt{\frac{1}{x}}+\frac{1}{x}+2} x+\frac{7 \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}+4}{2 \sqrt{2} \sqrt{\sqrt{\frac{1}{x}}+\frac{1}{x}+2}}\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 + Sqrt[x^(-1)] + x^(-1)],x]

[Out]

((4 + Sqrt[x^(-1)])*Sqrt[2 + Sqrt[x^(-1)] + x^(-1)]*x)/4 + (7*ArcTanh[(4 + Sqrt[
x^(-1)])/(2*Sqrt[2]*Sqrt[2 + Sqrt[x^(-1)] + x^(-1)])])/(8*Sqrt[2])

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Rubi in Sympy [A]  time = 9.35362, size = 66, normalized size = 0.88 \[ \frac{x \left (\sqrt{\frac{1}{x}} + 4\right ) \sqrt{\sqrt{\frac{1}{x}} + 2 + \frac{1}{x}}}{4} + \frac{7 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (\sqrt{\frac{1}{x}} + 4\right )}{4 \sqrt{\sqrt{\frac{1}{x}} + 2 + \frac{1}{x}}} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(sqrt(1/x) + 4)*sqrt(sqrt(1/x) + 2 + 1/x)/4 + 7*sqrt(2)*atanh(sqrt(2)*(sqrt(1/
x) + 4)/(4*sqrt(sqrt(1/x) + 2 + 1/x)))/16

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Mathematica [A]  time = 1.63907, size = 0, normalized size = 0. \[ \int \sqrt{2+\sqrt{\frac{1}{x}}+\frac{1}{x}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[2 + Sqrt[x^(-1)] + x^(-1)],x]

[Out]

Integrate[Sqrt[2 + Sqrt[x^(-1)] + x^(-1)], x]

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Maple [B]  time = 0.034, size = 123, normalized size = 1.6 \[{\frac{1}{16}\sqrt{{\frac{1}{x} \left ( \sqrt{{x}^{-1}}x+2\,x+1 \right ) }}\sqrt{x} \left ( 4\,\sqrt{\sqrt{{x}^{-1}}x+2\,x+1}\sqrt{{x}^{-1}}\sqrt{x}+16\,\sqrt{\sqrt{{x}^{-1}}x+2\,x+1}\sqrt{x}+7\,\ln \left ( 1/4\,\sqrt{2}\sqrt{{x}^{-1}}\sqrt{x}+\sqrt{x}\sqrt{2}+\sqrt{\sqrt{{x}^{-1}}x+2\,x+1} \right ) \sqrt{2} \right ){\frac{1}{\sqrt{\sqrt{{x}^{-1}}x+2\,x+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+1/x+(1/x)^(1/2))^(1/2),x)

[Out]

1/16*(((1/x)^(1/2)*x+2*x+1)/x)^(1/2)*x^(1/2)*(4*((1/x)^(1/2)*x+2*x+1)^(1/2)*(1/x
)^(1/2)*x^(1/2)+16*((1/x)^(1/2)*x+2*x+1)^(1/2)*x^(1/2)+7*ln(1/4*2^(1/2)*(1/x)^(1
/2)*x^(1/2)+x^(1/2)*2^(1/2)+((1/x)^(1/2)*x+2*x+1)^(1/2))*2^(1/2))/((1/x)^(1/2)*x
+2*x+1)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{1}{\sqrt{x}} + \frac{1}{x} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(1/sqrt(x) + 1/x + 2),x, algorithm="maxima")

[Out]

integrate(sqrt(1/sqrt(x) + 1/x + 2), x)

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Fricas [A]  time = 1.51762, size = 171, normalized size = 2.28 \[ \frac{\sqrt{2}{\left (8 \,{\left (4 \, \sqrt{2} x^{\frac{3}{2}} + \sqrt{2} x\right )} \sqrt{\frac{{\left (2 \, x + 1\right )} \sqrt{x} + x}{x^{\frac{3}{2}}}} + 7 \, \sqrt{x} \log \left (-\frac{\sqrt{2}{\left (2048 \, x^{2} + 1664 \, x + 113\right )} \sqrt{x} + 64 \, \sqrt{2}{\left (32 \, x^{2} + 9 \, x\right )} + 16 \,{\left (96 \, x^{2} + 4 \,{\left (32 \, x^{2} + 13 \, x\right )} \sqrt{x} + 9 \, x\right )} \sqrt{\frac{{\left (2 \, x + 1\right )} \sqrt{x} + x}{x^{\frac{3}{2}}}}}{\sqrt{x}}\right )\right )}}{64 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(1/sqrt(x) + 1/x + 2),x, algorithm="fricas")

[Out]

1/64*sqrt(2)*(8*(4*sqrt(2)*x^(3/2) + sqrt(2)*x)*sqrt(((2*x + 1)*sqrt(x) + x)/x^(
3/2)) + 7*sqrt(x)*log(-(sqrt(2)*(2048*x^2 + 1664*x + 113)*sqrt(x) + 64*sqrt(2)*(
32*x^2 + 9*x) + 16*(96*x^2 + 4*(32*x^2 + 13*x)*sqrt(x) + 9*x)*sqrt(((2*x + 1)*sq
rt(x) + x)/x^(3/2)))/sqrt(x)))/sqrt(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\sqrt{\frac{1}{x}} + 2 + \frac{1}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(sqrt(1/x) + 2 + 1/x), x)

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GIAC/XCAS [A]  time = 0.222735, size = 100, normalized size = 1.33 \[ -\frac{1}{16} \, \sqrt{2}{\left (2 \, \sqrt{2} - 7 \,{\rm ln}\left (2 \, \sqrt{2} - 1\right )\right )} + \frac{1}{4} \, \sqrt{2 \, x + \sqrt{x} + 1}{\left (4 \, \sqrt{x} + 1\right )} - \frac{7}{16} \, \sqrt{2}{\rm ln}\left (-2 \, \sqrt{2}{\left (\sqrt{2} \sqrt{x} - \sqrt{2 \, x + \sqrt{x} + 1}\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(1/sqrt(x) + 1/x + 2),x, algorithm="giac")

[Out]

-1/16*sqrt(2)*(2*sqrt(2) - 7*ln(2*sqrt(2) - 1)) + 1/4*sqrt(2*x + sqrt(x) + 1)*(4
*sqrt(x) + 1) - 7/16*sqrt(2)*ln(-2*sqrt(2)*(sqrt(2)*sqrt(x) - sqrt(2*x + sqrt(x)
+ 1)) - 1)