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

Optimal. Leaf size=65 \[ \frac{1}{5} \left (5-\sqrt{5}\right ) \log \left (2 \sqrt{1-x}-\sqrt{5}+1\right )+\frac{1}{5} \left (5+\sqrt{5}\right ) \log \left (2 \sqrt{1-x}+\sqrt{5}+1\right ) \]

[Out]

((5 - Sqrt[5])*Log[1 - Sqrt[5] + 2*Sqrt[1 - x]])/5 + ((5 + Sqrt[5])*Log[1 + Sqrt
[5] + 2*Sqrt[1 - x]])/5

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Rubi [A]  time = 0.073487, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{5} \left (5-\sqrt{5}\right ) \log \left (2 \sqrt{1-x}-\sqrt{5}+1\right )+\frac{1}{5} \left (5+\sqrt{5}\right ) \log \left (2 \sqrt{1-x}+\sqrt{5}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(-Sqrt[1 - x] + x)^(-1),x]

[Out]

((5 - Sqrt[5])*Log[1 - Sqrt[5] + 2*Sqrt[1 - x]])/5 + ((5 + Sqrt[5])*Log[1 + Sqrt
[5] + 2*Sqrt[1 - x]])/5

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Rubi in Sympy [A]  time = 2.5539, size = 70, normalized size = 1.08 \[ \frac{2 \sqrt{5} \left (\frac{1}{2} + \frac{\sqrt{5}}{2}\right ) \log{\left (2 \sqrt{- x + 1} + 1 + \sqrt{5} \right )}}{5} - \frac{2 \sqrt{5} \left (- \frac{\sqrt{5}}{2} + \frac{1}{2}\right ) \log{\left (2 \sqrt{- x + 1} - \sqrt{5} + 1 \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(5)*(1/2 + sqrt(5)/2)*log(2*sqrt(-x + 1) + 1 + sqrt(5))/5 - 2*sqrt(5)*(-sq
rt(5)/2 + 1/2)*log(2*sqrt(-x + 1) - sqrt(5) + 1)/5

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Mathematica [A]  time = 0.0234023, size = 42, normalized size = 0.65 \[ \log \left (x-\sqrt{1-x}\right )+\frac{2 \tanh ^{-1}\left (\frac{2 \sqrt{1-x}+1}{\sqrt{5}}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-Sqrt[1 - x] + x)^(-1),x]

[Out]

(2*ArcTanh[(1 + 2*Sqrt[1 - x])/Sqrt[5]])/Sqrt[5] + Log[-Sqrt[1 - x] + x]

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Maple [B]  time = 0.006, size = 101, normalized size = 1.6 \[{\frac{\ln \left ({x}^{2}+x-1 \right ) }{2}}+{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{5}}{5}} \right ) }-{\frac{1}{2}\ln \left ( -x-\sqrt{1-x} \right ) }+{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 2\,\sqrt{1-x}-1 \right ) } \right ) }+{\frac{1}{2}\ln \left ( -x+\sqrt{1-x} \right ) }+{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 2\,\sqrt{1-x}+1 \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x-(1-x)^(1/2)),x)

[Out]

1/2*ln(x^2+x-1)+1/5*5^(1/2)*arctanh(1/5*(1+2*x)*5^(1/2))-1/2*ln(-x-(1-x)^(1/2))+
1/5*5^(1/2)*arctanh(1/5*(2*(1-x)^(1/2)-1)*5^(1/2))+1/2*ln(-x+(1-x)^(1/2))+1/5*5^
(1/2)*arctanh(1/5*(2*(1-x)^(1/2)+1)*5^(1/2))

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Maxima [A]  time = 0.799206, size = 69, normalized size = 1.06 \[ -\frac{1}{5} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - 2 \, \sqrt{-x + 1} - 1}{\sqrt{5} + 2 \, \sqrt{-x + 1} + 1}\right ) + \log \left (-x + \sqrt{-x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*sqrt(5)*log(-(sqrt(5) - 2*sqrt(-x + 1) - 1)/(sqrt(5) + 2*sqrt(-x + 1) + 1))
+ log(-x + sqrt(-x + 1))

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Fricas [A]  time = 0.270345, size = 84, normalized size = 1.29 \[ \frac{1}{5} \, \sqrt{5}{\left (\sqrt{5} \log \left (-x + \sqrt{-x + 1}\right ) + \log \left (\frac{\sqrt{5}{\left (2 \, x - 5\right )} - 2 \, \sqrt{-x + 1}{\left (\sqrt{5} + 5\right )} - 5}{x - \sqrt{-x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*sqrt(5)*(sqrt(5)*log(-x + sqrt(-x + 1)) + log((sqrt(5)*(2*x - 5) - 2*sqrt(-x
+ 1)*(sqrt(5) + 5) - 5)/(x - sqrt(-x + 1))))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x-(1-x)**(1/2)),x)

[Out]

Integral(1/(x - sqrt(-x + 1)), x)

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GIAC/XCAS [A]  time = 0.293559, size = 73, normalized size = 1.12 \[ -\frac{1}{5} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -\sqrt{5} + 2 \, \sqrt{-x + 1} + 1 \right |}}{\sqrt{5} + 2 \, \sqrt{-x + 1} + 1}\right ) +{\rm ln}\left ({\left | -x + \sqrt{-x + 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*sqrt(5)*ln(abs(-sqrt(5) + 2*sqrt(-x + 1) + 1)/(sqrt(5) + 2*sqrt(-x + 1) + 1
)) + ln(abs(-x + sqrt(-x + 1)))