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

Optimal. Leaf size=24 \[ \sqrt{x-1} \sqrt{x}-\sinh ^{-1}\left (\sqrt{x-1}\right ) \]

[Out]

Sqrt[-1 + x]*Sqrt[x] - ArcSinh[Sqrt[-1 + x]]

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Rubi [A]  time = 0.0112379, antiderivative size = 28, normalized size of antiderivative = 1.17, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {1972, 242, 47, 63, 206} \[ \sqrt{\frac{x-1}{x}} x-\tanh ^{-1}\left (\sqrt{\frac{x-1}{x}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(-1 + x)/x],x]

[Out]

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

Rule 1972

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && BinomialQ[u, x] &&  !BinomialMatchQ[
u, x]

Rule 242

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^2, x], x, 1/x] /; FreeQ[{a, b, p},
x] && ILtQ[n, 0]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, 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 206

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

Rubi steps

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

Mathematica [A]  time = 0.0137291, size = 38, normalized size = 1.58 \[ \frac{\sqrt{x} (x-1)+\sqrt{1-x} \sin ^{-1}\left (\sqrt{1-x}\right )}{\sqrt{x-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(-1 + x)/x],x]

[Out]

((-1 + x)*Sqrt[x] + Sqrt[1 - x]*ArcSin[Sqrt[1 - x]])/Sqrt[-1 + x]

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Maple [B]  time = 0.005, size = 47, normalized size = 2. \begin{align*}{\frac{x}{2}\sqrt{{\frac{x-1}{x}}} \left ( 2\,\sqrt{{x}^{2}-x}-\ln \left ( x-{\frac{1}{2}}+\sqrt{{x}^{2}-x} \right ) \right ){\frac{1}{\sqrt{x \left ( x-1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((x-1)/x)^(1/2)*x*(2*(x^2-x)^(1/2)-ln(x-1/2+(x^2-x)^(1/2)))/(x*(x-1))^(1/2)

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Maxima [B]  time = 1.05021, size = 69, normalized size = 2.88 \begin{align*} -\frac{\sqrt{\frac{x - 1}{x}}}{\frac{x - 1}{x} - 1} - \frac{1}{2} \, \log \left (\sqrt{\frac{x - 1}{x}} + 1\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{x - 1}{x}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.43424, size = 109, normalized size = 4.54 \begin{align*} x \sqrt{\frac{x - 1}{x}} - \frac{1}{2} \, \log \left (\sqrt{\frac{x - 1}{x}} + 1\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{x - 1}{x}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{x - 1}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt((x - 1)/x), x)

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Giac [A]  time = 1.14224, size = 47, normalized size = 1.96 \begin{align*} \frac{1}{2} \, \log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} - x} + 1 \right |}\right ) \mathrm{sgn}\left (x\right ) + \sqrt{x^{2} - x} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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