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

Optimal. Leaf size=25 \[ \sin ^{-1}\left (\sqrt{x}\right )-\left (\sqrt{x}+2\right ) \sqrt{1-x} \]

[Out]

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

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Rubi [A]  time = 0.0252974, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {1398, 785, 780, 216} \[ \sin ^{-1}\left (\sqrt{x}\right )-\left (\sqrt{x}+2\right ) \sqrt{1-x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x]/(1 - Sqrt[x]),x]

[Out]

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

Rule 1398

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rule 785

Int[(x_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^m*e^m, Int[(x*(a + c*x^2)^(m
+ p))/(a*e + c*d*x)^m, x], x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[
m, 0] && EqQ[m, -1] &&  !ILtQ[p - 1/2, 0]

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 216

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

Rubi steps

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

Mathematica [A]  time = 0.029478, size = 26, normalized size = 1.04 \[ \sqrt{1-x} \left (-\sqrt{x}-2\right )+\sin ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x]/(1 - Sqrt[x]),x]

[Out]

(-2 - Sqrt[x])*Sqrt[1 - x] + ArcSin[Sqrt[x]]

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Maple [B]  time = 0.003, size = 48, normalized size = 1.9 \begin{align*} -2\,\sqrt{1-x}+{\frac{1}{2}\sqrt{1-x}\sqrt{x} \left ( -2\,\sqrt{-x \left ( x-1 \right ) }+\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)^(1/2)/(1-x^(1/2)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate(sqrt(-x + 1)/(sqrt(x) - 1), x)

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Fricas [A]  time = 1.45369, size = 96, normalized size = 3.84 \begin{align*} -\sqrt{x} \sqrt{-x + 1} - 2 \, \sqrt{-x + 1} - \arctan \left (\frac{\sqrt{-x + 1}}{\sqrt{x}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.71859, size = 87, normalized size = 3.48 \begin{align*} 2 \left (\begin{cases} - \sqrt{1 - x} + \frac{i \operatorname{acosh}{\left (\sqrt{1 - x} \right )}}{2} - \frac{i \left (1 - x\right )^{\frac{3}{2}}}{2 \sqrt{- x}} + \frac{i \sqrt{1 - x}}{2 \sqrt{- x}} & \text{for}\: \left |{x - 1}\right | > 1 \\\frac{\sqrt{x} \sqrt{1 - x}}{2} - \sqrt{1 - x} + \frac{\operatorname{asin}{\left (\sqrt{1 - x} \right )}}{2} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*Piecewise((-sqrt(1 - x) + I*acosh(sqrt(1 - x))/2 - I*(1 - x)**(3/2)/(2*sqrt(-x)) + I*sqrt(1 - x)/(2*sqrt(-x)
), Abs(x - 1) > 1), (sqrt(x)*sqrt(1 - x)/2 - sqrt(1 - x) + asin(sqrt(1 - x))/2, True))

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Giac [A]  time = 1.17513, size = 43, normalized size = 1.72 \begin{align*} -\sqrt{x} \sqrt{-x + 1} - 2 \, \sqrt{-x + 1} - \arcsin \left (\sqrt{-x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(x)*sqrt(-x + 1) - 2*sqrt(-x + 1) - arcsin(sqrt(-x + 1))