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

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

[Out]

Sqrt[-1 + x^(-1)]*x - ArcTan[Sqrt[-1 + x^(-1)]]

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Rubi [A]  time = 0.0089003, antiderivative size = 24, 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, Rules used = {1972, 242, 47, 63, 203} \[ \sqrt{\frac{1}{x}-1} x-\tan ^{-1}\left (\sqrt{\frac{1}{x}-1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(1 - x)/x],x]

[Out]

Sqrt[-1 + x^(-1)]*x - ArcTan[Sqrt[-1 + x^(-1)]]

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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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{-1+\frac{1}{x}} x-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} x} \, dx,x,\frac{1}{x}\right )\\ &=\sqrt{-1+\frac{1}{x}} x-\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-1+\frac{1}{x}}\right )\\ &=\sqrt{-1+\frac{1}{x}} x-\tan ^{-1}\left (\sqrt{-1+\frac{1}{x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0074142, size = 24, normalized size = 1. \[ \sqrt{\frac{1}{x}-1} x-\tan ^{-1}\left (\sqrt{\frac{1}{x}-1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(1 - x)/x],x]

[Out]

Sqrt[-1 + x^(-1)]*x - ArcTan[Sqrt[-1 + x^(-1)]]

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Maple [A]  time = 0.004, size = 40, normalized size = 1.7 \begin{align*}{\frac{x}{2}\sqrt{-{\frac{x-1}{x}}} \left ( 2\,\sqrt{-{x}^{2}+x}+\arcsin \left ( 2\,x-1 \right ) \right ){\frac{1}{\sqrt{-x \left ( x-1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.63696, size = 50, normalized size = 2.08 \begin{align*} -\frac{\sqrt{-\frac{x - 1}{x}}}{\frac{x - 1}{x} - 1} - \arctan \left (\sqrt{-\frac{x - 1}{x}}\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) - arctan(sqrt(-(x - 1)/x))

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Fricas [A]  time = 1.47866, size = 63, normalized size = 2.62 \begin{align*} x \sqrt{-\frac{x - 1}{x}} - \arctan \left (\sqrt{-\frac{x - 1}{x}}\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) - arctan(sqrt(-(x - 1)/x))

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

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Giac [A]  time = 1.17534, size = 38, normalized size = 1.58 \begin{align*} \frac{1}{4} \, \pi \mathrm{sgn}\left (x\right ) + \frac{1}{2} \, \arcsin \left (2 \, x - 1\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]

1/4*pi*sgn(x) + 1/2*arcsin(2*x - 1)*sgn(x) + sqrt(-x^2 + x)*sgn(x)