3.2385 \(\int \frac{1}{\sqrt{1+\frac{1}{\sqrt{x}}}} \, dx\)

Optimal. Leaf size=50 \[ \sqrt{\frac{1}{\sqrt{x}}+1} x-\frac{3}{2} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}+\frac{3}{2} \tanh ^{-1}\left (\sqrt{\frac{1}{\sqrt{x}}+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0480614, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \sqrt{\frac{1}{\sqrt{x}}+1} x-\frac{3}{2} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}+\frac{3}{2} \tanh ^{-1}\left (\sqrt{\frac{1}{\sqrt{x}}+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[1 + 1/Sqrt[x]],x]

[Out]

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

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Rubi in Sympy [A]  time = 4.73955, size = 49, normalized size = 0.98 \[ - \frac{3 \sqrt{x} \sqrt{1 + \frac{1}{\sqrt{x}}}}{2} + x \sqrt{1 + \frac{1}{\sqrt{x}}} + \frac{3 \operatorname{atanh}{\left (\sqrt{1 + \frac{1}{\sqrt{x}}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0456849, size = 54, normalized size = 1.08 \[ \frac{1}{2} \sqrt{\frac{1}{\sqrt{x}}+1} \left (2 x-3 \sqrt{x}\right )+\frac{3}{4} \log \left (2 \sqrt{x} \left (\sqrt{\frac{1}{\sqrt{x}}+1}+1\right )+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[1 + 1/Sqrt[x]],x]

[Out]

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

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Maple [A]  time = 0.02, size = 65, normalized size = 1.3 \[{\frac{1}{4}\sqrt{{1 \left ( 1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}\sqrt{x} \left ( 4\,\sqrt{x+\sqrt{x}}\sqrt{x}-6\,\sqrt{x+\sqrt{x}}+3\,\ln \left ( 1/2+\sqrt{x}+\sqrt{x+\sqrt{x}} \right ) \right ){\frac{1}{\sqrt{ \left ( 1+\sqrt{x} \right ) \sqrt{x}}}}} \]

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

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Maxima [A]  time = 1.4299, size = 84, normalized size = 1.68 \[ -\frac{3 \,{\left (\frac{1}{\sqrt{x}} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{\frac{1}{\sqrt{x}} + 1}}{2 \,{\left ({\left (\frac{1}{\sqrt{x}} + 1\right )}^{2} - \frac{2}{\sqrt{x}} - 1\right )}} + \frac{3}{4} \, \log \left (\sqrt{\frac{1}{\sqrt{x}} + 1} + 1\right ) - \frac{3}{4} \, \log \left (\sqrt{\frac{1}{\sqrt{x}} + 1} - 1\right ) \]

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 = 0.229207, size = 89, normalized size = 1.78 \[ \frac{2 \,{\left (2 \, x^{\frac{3}{2}} - 3 \, x\right )} \sqrt{\frac{\sqrt{x} + 1}{\sqrt{x}}} + 3 \, \sqrt{x} \log \left (\sqrt{\frac{\sqrt{x} + 1}{\sqrt{x}}} + 1\right ) - 3 \, \sqrt{x} \log \left (\sqrt{\frac{\sqrt{x} + 1}{\sqrt{x}}} - 1\right )}{4 \, \sqrt{x}} \]

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 [A]  time = 10.7005, size = 60, normalized size = 1.2 \[ \frac{x^{\frac{5}{4}}}{\sqrt{\sqrt{x} + 1}} - \frac{x^{\frac{3}{4}}}{2 \sqrt{\sqrt{x} + 1}} - \frac{3 \sqrt [4]{x}}{2 \sqrt{\sqrt{x} + 1}} + \frac{3 \operatorname{asinh}{\left (\sqrt [4]{x} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(5/4)/sqrt(sqrt(x) + 1) - x**(3/4)/(2*sqrt(sqrt(x) + 1)) - 3*x**(1/4)/(2*sqrt
(sqrt(x) + 1)) + 3*asinh(x**(1/4))/2

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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