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

Optimal. Leaf size=96 \[ \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}} x+\frac{1}{4} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{x}+1}+3}{2 \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}}}\right )-\frac{3}{4} \tanh ^{-1}\left (\frac{1-3 \sqrt{\frac{1}{x}+1}}{2 \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.19791, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}} x+\frac{1}{4} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{x}+1}+3}{2 \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}}}\right )-\frac{3}{4} \tanh ^{-1}\left (\frac{1-3 \sqrt{\frac{1}{x}+1}}{2 \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0776948, size = 0, normalized size = 0. \[ \int \sqrt{\sqrt{1+\frac{1}{x}}+\frac{1}{x}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int \sqrt{{x}^{-1}+\sqrt{1+{x}^{-1}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((1/x+(1+1/x)^(1/2))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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