3.2384 \(\int \sqrt{3-\frac{1}{\sqrt{x}}} \, dx\)

Optimal. Leaf size=67 \[ \sqrt{3-\frac{1}{\sqrt{x}}} x-\frac{1}{6} \sqrt{3-\frac{1}{\sqrt{x}}} \sqrt{x}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3-\frac{1}{\sqrt{x}}}}{\sqrt{3}}\right )}{6 \sqrt{3}} \]

[Out]

-(Sqrt[3 - 1/Sqrt[x]]*Sqrt[x])/6 + Sqrt[3 - 1/Sqrt[x]]*x - ArcTanh[Sqrt[3 - 1/Sq
rt[x]]/Sqrt[3]]/(6*Sqrt[3])

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Rubi [A]  time = 0.0752693, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \sqrt{3-\frac{1}{\sqrt{x}}} x-\frac{1}{6} \sqrt{3-\frac{1}{\sqrt{x}}} \sqrt{x}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3-\frac{1}{\sqrt{x}}}}{\sqrt{3}}\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[3 - 1/Sqrt[x]],x]

[Out]

-(Sqrt[3 - 1/Sqrt[x]]*Sqrt[x])/6 + Sqrt[3 - 1/Sqrt[x]]*x - ArcTanh[Sqrt[3 - 1/Sq
rt[x]]/Sqrt[3]]/(6*Sqrt[3])

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Rubi in Sympy [A]  time = 6.19566, size = 58, normalized size = 0.87 \[ - \frac{\sqrt{x} \sqrt{3 - \frac{1}{\sqrt{x}}}}{6} + x \sqrt{3 - \frac{1}{\sqrt{x}}} - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{3 - \frac{1}{\sqrt{x}}}}{3} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3-1/x**(1/2))**(1/2),x)

[Out]

-sqrt(x)*sqrt(3 - 1/sqrt(x))/6 + x*sqrt(3 - 1/sqrt(x)) - sqrt(3)*atanh(sqrt(3)*s
qrt(3 - 1/sqrt(x))/3)/18

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Mathematica [A]  time = 0.0970739, size = 63, normalized size = 0.94 \[ \frac{1}{36} \left (6 \sqrt{3-\frac{1}{\sqrt{x}}} \left (6 x-\sqrt{x}\right )-\sqrt{3} \log \left (1-2 \left (\sqrt{9-\frac{3}{\sqrt{x}}}+3\right ) \sqrt{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[3 - 1/Sqrt[x]],x]

[Out]

(6*Sqrt[3 - 1/Sqrt[x]]*(-Sqrt[x] + 6*x) - Sqrt[3]*Log[1 - 2*(3 + Sqrt[9 - 3/Sqrt
[x]])*Sqrt[x]])/36

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Maple [A]  time = 0.02, size = 91, normalized size = 1.4 \[ -{\frac{1}{36}\sqrt{{1 \left ( 3\,\sqrt{x}-1 \right ){\frac{1}{\sqrt{x}}}}}\sqrt{x} \left ( \ln \left ( -{\frac{\sqrt{3}}{6}}+\sqrt{3}\sqrt{x}+\sqrt{3\,x-\sqrt{x}} \right ) \sqrt{3}-36\,\sqrt{3\,x-\sqrt{x}}\sqrt{x}+6\,\sqrt{3\,x-\sqrt{x}} \right ){\frac{1}{\sqrt{ \left ( 3\,\sqrt{x}-1 \right ) \sqrt{x}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3-1/x^(1/2))^(1/2),x)

[Out]

-1/36*((3*x^(1/2)-1)/x^(1/2))^(1/2)*x^(1/2)*(ln(-1/6*3^(1/2)+3^(1/2)*x^(1/2)+(3*
x-x^(1/2))^(1/2))*3^(1/2)-36*(3*x-x^(1/2))^(1/2)*x^(1/2)+6*(3*x-x^(1/2))^(1/2))/
((3*x^(1/2)-1)*x^(1/2))^(1/2)

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Maxima [A]  time = 1.61529, size = 105, normalized size = 1.57 \[ \frac{1}{36} \, \sqrt{3} \log \left (-\frac{\sqrt{3} - \sqrt{-\frac{1}{\sqrt{x}} + 3}}{\sqrt{3} + \sqrt{-\frac{1}{\sqrt{x}} + 3}}\right ) + \frac{{\left (-\frac{1}{\sqrt{x}} + 3\right )}^{\frac{3}{2}} + 3 \, \sqrt{-\frac{1}{\sqrt{x}} + 3}}{6 \,{\left ({\left (\frac{1}{\sqrt{x}} - 3\right )}^{2} + \frac{6}{\sqrt{x}} - 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-1/sqrt(x) + 3),x, algorithm="maxima")

[Out]

1/36*sqrt(3)*log(-(sqrt(3) - sqrt(-1/sqrt(x) + 3))/(sqrt(3) + sqrt(-1/sqrt(x) +
3))) + 1/6*((-1/sqrt(x) + 3)^(3/2) + 3*sqrt(-1/sqrt(x) + 3))/((1/sqrt(x) - 3)^2
+ 6/sqrt(x) - 9)

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Fricas [A]  time = 0.233806, size = 100, normalized size = 1.49 \[ \frac{\sqrt{3}{\left (2 \,{\left (6 \, \sqrt{3} x^{\frac{3}{2}} - \sqrt{3} x\right )} \sqrt{\frac{3 \, \sqrt{x} - 1}{\sqrt{x}}} + \sqrt{x} \log \left (-6 \, \sqrt{3} \sqrt{x} + 6 \, \sqrt{x} \sqrt{\frac{3 \, \sqrt{x} - 1}{\sqrt{x}}} + \sqrt{3}\right )\right )}}{36 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-1/sqrt(x) + 3),x, algorithm="fricas")

[Out]

1/36*sqrt(3)*(2*(6*sqrt(3)*x^(3/2) - sqrt(3)*x)*sqrt((3*sqrt(x) - 1)/sqrt(x)) +
sqrt(x)*log(-6*sqrt(3)*sqrt(x) + 6*sqrt(x)*sqrt((3*sqrt(x) - 1)/sqrt(x)) + sqrt(
3)))/sqrt(x)

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Sympy [A]  time = 10.0706, size = 165, normalized size = 2.46 \[ \begin{cases} \frac{3 x^{\frac{5}{4}}}{\sqrt{3 \sqrt{x} - 1}} - \frac{3 x^{\frac{3}{4}}}{2 \sqrt{3 \sqrt{x} - 1}} + \frac{\sqrt [4]{x}}{6 \sqrt{3 \sqrt{x} - 1}} - \frac{\sqrt{3} \operatorname{acosh}{\left (\sqrt{3} \sqrt [4]{x} \right )}}{18} & \text{for}\: 3 \left |{\sqrt{x}}\right | > 1 \\- \frac{3 i x^{\frac{5}{4}}}{\sqrt{- 3 \sqrt{x} + 1}} + \frac{3 i x^{\frac{3}{4}}}{2 \sqrt{- 3 \sqrt{x} + 1}} - \frac{i \sqrt [4]{x}}{6 \sqrt{- 3 \sqrt{x} + 1}} + \frac{\sqrt{3} i \operatorname{asin}{\left (\sqrt{3} \sqrt [4]{x} \right )}}{18} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3-1/x**(1/2))**(1/2),x)

[Out]

Piecewise((3*x**(5/4)/sqrt(3*sqrt(x) - 1) - 3*x**(3/4)/(2*sqrt(3*sqrt(x) - 1)) +
x**(1/4)/(6*sqrt(3*sqrt(x) - 1)) - sqrt(3)*acosh(sqrt(3)*x**(1/4))/18, 3*Abs(sq
rt(x)) > 1), (-3*I*x**(5/4)/sqrt(-3*sqrt(x) + 1) + 3*I*x**(3/4)/(2*sqrt(-3*sqrt(
x) + 1)) - I*x**(1/4)/(6*sqrt(-3*sqrt(x) + 1)) + sqrt(3)*I*asin(sqrt(3)*x**(1/4)
)/18, True))

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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(sqrt(-1/sqrt(x) + 3),x, algorithm="giac")

[Out]

Timed out