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 = 0.0052372, 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, Rules used = {63, 207} \[ -\sqrt{2} \tanh ^{-1}\left (\sqrt{2} \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(-1 + 2*x)),x]

[Out]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{x} (-1+2 x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{-1+2 x^2} \, dx,x,\sqrt{x}\right )\\ &=-\sqrt{2} \tanh ^{-1}\left (\sqrt{2} \sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0028453, 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 \begin{align*} -{\it Artanh} \left ( \sqrt{2}\sqrt{x} \right ) \sqrt{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(1/2)/(2*x-1),x)

[Out]

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

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Maxima [B]  time = 1.41823, size = 38, normalized size = 2. \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - 2 \, \sqrt{x}}{\sqrt{2} + 2 \, \sqrt{x}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(-1+2*x),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.61785, size = 80, normalized size = 4.21 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} \sqrt{x} - 2 \, x - 1}{2 \, x - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(-1+2*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 [B]  time = 0.323808, size = 39, normalized size = 2.05 \begin{align*} \frac{\sqrt{2} \log{\left (\sqrt{x} - \frac{\sqrt{2}}{2} \right )}}{2} - \frac{\sqrt{2} \log{\left (\sqrt{x} + \frac{\sqrt{2}}{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(1/2)/(-1+2*x),x)

[Out]

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

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Giac [B]  time = 1.08088, size = 43, normalized size = 2.26 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left (\frac{1}{2} \, \sqrt{2} + \sqrt{x}\right ) + \frac{1}{2} \, \sqrt{2} \log \left ({\left | -\frac{1}{2} \, \sqrt{2} + \sqrt{x} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(-1+2*x),x, algorithm="giac")

[Out]

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