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

Optimal. Leaf size=19 \[ -\sqrt{2} \tanh ^{-1}\left (\sqrt{2} \sqrt{x}\right ) \]

[Out]

-(Sqrt[2]*ArcTanh[Sqrt[2]*Sqrt[x]])

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Rubi [A]  time = 0.014124, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\sqrt{2} \tanh ^{-1}\left (\sqrt{2} \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[x]*(-1 + 2*x)),x]

[Out]

-(Sqrt[2]*ArcTanh[Sqrt[2]*Sqrt[x]])

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Rubi in Sympy [A]  time = 1.11609, size = 17, normalized size = 0.89 \[ - \sqrt{2} \operatorname{atanh}{\left (\sqrt{2} \sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*atanh(sqrt(2)*sqrt(x))

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Mathematica [A]  time = 0.00697179, size = 19, normalized size = 1. \[ -\sqrt{2} \tanh ^{-1}\left (\sqrt{2} \sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[2]*ArcTanh[Sqrt[2]*Sqrt[x]])

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Maple [A]  time = 0.003, size = 14, normalized size = 0.7 \[ -{\it Artanh} \left ( \sqrt{2}\sqrt{x} \right ) \sqrt{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(1/2)/(-1+2*x),x)

[Out]

-arctanh(2^(1/2)*x^(1/2))*2^(1/2)

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Maxima [A]  time = 1.50511, size = 38, normalized size = 2. \[ \frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}}{2 \, \sqrt{2} + 4 \, \sqrt{x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.208419, size = 38, normalized size = 2. \[ \frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} \sqrt{x} - 2 \, x - 1}{2 \, x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*log(-(2*sqrt(2)*sqrt(x) - 2*x - 1)/(2*x - 1))

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Sympy [A]  time = 0.75927, size = 39, normalized size = 2.05 \[ \begin{cases} - \sqrt{2} \operatorname{acoth}{\left (\sqrt{2} \sqrt{x} \right )} & \text{for}\: 2 \left |{x}\right | > 1 \\- \sqrt{2} \operatorname{atanh}{\left (\sqrt{2} \sqrt{x} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(1/2)/(2*x-1),x)

[Out]

Piecewise((-sqrt(2)*acoth(sqrt(2)*sqrt(x)), 2*Abs(x) > 1), (-sqrt(2)*atanh(sqrt(
2)*sqrt(x)), True))

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GIAC/XCAS [A]  time = 0.201376, size = 43, normalized size = 2.26 \[ -\frac{1}{2} \, \sqrt{2}{\rm ln}\left (\frac{1}{2} \, \sqrt{2} + \sqrt{x}\right ) + \frac{1}{2} \, \sqrt{2}{\rm ln}\left ({\left | -\frac{1}{2} \, \sqrt{2} + \sqrt{x} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(2)*ln(1/2*sqrt(2) + sqrt(x)) + 1/2*sqrt(2)*ln(abs(-1/2*sqrt(2) + sqrt(
x)))