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

Optimal. Leaf size=29 \[ -\sqrt{1-x} \left (2-\sqrt{x}\right )-\sin ^{-1}\left (\sqrt{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0908416, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\sqrt{1-x} \left (2-\sqrt{x}\right )-\sin ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.26201, size = 27, normalized size = 0.93 \[ - \sqrt{- x + 1} - \operatorname{asin}{\left (\sqrt{x} \right )} - \frac{\left (- x + 1\right )^{\frac{3}{2}}}{\sqrt{x} + 1} \]

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)

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Mathematica [A]  time = 0.0172564, size = 26, normalized size = 0.9 \[ \left (\sqrt{x}-2\right ) \sqrt{1-x}-\sin ^{-1}\left (\sqrt{x}\right ) \]

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 = 0.01, size = 48, normalized size = 1.7 \[ -{\frac{1}{2}\sqrt{1-x}\sqrt{x} \left ( -2\,\sqrt{-x \left ( -1+x \right ) }+\arcsin \left ( 2\,x-1 \right ) \right ){\frac{1}{\sqrt{-x \left ( -1+x \right ) }}}}-2\,\sqrt{1-x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(1/2)/(1+x^(1/2)),x)

[Out]

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

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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)

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Fricas [A]  time = 0.269627, size = 45, normalized size = 1.55 \[ \sqrt{x} \sqrt{-x + 1} - 2 \, \sqrt{-x + 1} + \arctan \left (\frac{\sqrt{-x + 1}}{\sqrt{x}}\right ) \]

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))

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Sympy [A]  time = 6.20318, size = 32, normalized size = 1.1 \[ i \sqrt{x} \sqrt{x - 1} - 2 i \sqrt{x - 1} + i \operatorname{asinh}{\left (\sqrt{x - 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

I*sqrt(x)*sqrt(x - 1) - 2*I*sqrt(x - 1) + I*asinh(sqrt(x - 1))

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GIAC/XCAS [A]  time = 0.265805, size = 39, normalized size = 1.34 \[ \sqrt{x} \sqrt{-x + 1} - 2 \, \sqrt{-x + 1} + \arcsin \left (\sqrt{-x + 1}\right ) \]

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))