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

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]]

_______________________________________________________________________________________

Rubi [A]  time = 0.0905389, 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 \[ \sin ^{-1}\left (\sqrt{x}\right )-\left (\sqrt{x}+2\right ) \sqrt{1-x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x]/(1 - Sqrt[x]),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.61758, size = 26, normalized size = 1.04 \[ - \sqrt{- x + 1} + \operatorname{asin}{\left (\sqrt{x} \right )} - \frac{\left (- x + 1\right )^{\frac{3}{2}}}{- \sqrt{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x + 1) + asin(sqrt(x)) - (-x + 1)**(3/2)/(-sqrt(x) + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0203013, 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]]

_______________________________________________________________________________________

Maple [B]  time = 0.004, size = 48, normalized size = 1.9 \[ -2\,\sqrt{1-x}+{\frac{1}{2}\sqrt{1-x}\sqrt{x} \left ( -2\,\sqrt{-x \left ( -1+x \right ) }+\arcsin \left ( 2\,x-1 \right ) \right ){\frac{1}{\sqrt{-x \left ( -1+x \right ) }}}} \]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.264293, size = 49, normalized size = 1.96 \[ -\sqrt{x} \sqrt{-x + 1} - 2 \, \sqrt{-x + 1} - \arctan \left (\frac{\sqrt{-x + 1}}{\sqrt{x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 8.6973, size = 87, normalized size = 3.48 \[ 2 \left (\begin{cases} - \sqrt{- x + 1} + \frac{i \operatorname{acosh}{\left (\sqrt{- x + 1} \right )}}{2} - \frac{i \left (- x + 1\right )^{\frac{3}{2}}}{2 \sqrt{- x}} + \frac{i \sqrt{- x + 1}}{2 \sqrt{- x}} & \text{for}\: \left |{x - 1}\right | > 1 \\\frac{\sqrt{x} \sqrt{- x + 1}}{2} - \sqrt{- x + 1} + \frac{\operatorname{asin}{\left (\sqrt{- x + 1} \right )}}{2} & \text{otherwise} \end{cases}\right ) \]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.265383, size = 43, normalized size = 1.72 \[ -\sqrt{x} \sqrt{-x + 1} - 2 \, \sqrt{-x + 1} - \arcsin \left (\sqrt{-x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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