3.722 \(\int \frac{1}{x-\sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{4} \log \left (1-2 x^2\right )-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )-\frac{1}{2} \sin ^{-1}(x) \]

[Out]

-ArcSin[x]/2 - ArcTanh[x/Sqrt[1 - x^2]]/2 + Log[1 - 2*x^2]/4

_______________________________________________________________________________________

Rubi [A]  time = 0.0916905, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{1}{4} \log \left (1-2 x^2\right )-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )-\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(x - Sqrt[1 - x^2])^(-1),x]

[Out]

-ArcSin[x]/2 - ArcTanh[x/Sqrt[1 - x^2]]/2 + Log[1 - 2*x^2]/4

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x - \sqrt{- x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x - sqrt(-x**2 + 1)), x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0210904, size = 37, normalized size = 1. \[ \frac{1}{4} \log \left (1-2 x^2\right )-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )-\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[(x - Sqrt[1 - x^2])^(-1),x]

[Out]

-ArcSin[x]/2 - ArcTanh[x/Sqrt[1 - x^2]]/2 + Log[1 - 2*x^2]/4

_______________________________________________________________________________________

Maple [B]  time = 0.047, size = 175, normalized size = 4.7 \[{\frac{\ln \left ( 2\,{x}^{2}-1 \right ) }{4}}+{\frac{\sqrt{2}}{8}\sqrt{-4\, \left ( x-1/2\,\sqrt{2} \right ) ^{2}-4\,\sqrt{2} \left ( x-1/2\,\sqrt{2} \right ) +2}}-{\frac{\arcsin \left ( x \right ) }{2}}-{\frac{1}{4}{\it Artanh} \left ({\sqrt{2} \left ( 1-\sqrt{2} \left ( x-{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{-4\, \left ( x-1/2\,\sqrt{2} \right ) ^{2}-4\,\sqrt{2} \left ( x-1/2\,\sqrt{2} \right ) +2}}}} \right ) }-{\frac{\sqrt{2}}{8}\sqrt{-4\, \left ( x+1/2\,\sqrt{2} \right ) ^{2}+4\,\sqrt{2} \left ( x+1/2\,\sqrt{2} \right ) +2}}+{\frac{1}{4}{\it Artanh} \left ({\sqrt{2} \left ( \sqrt{2} \left ( x+{\frac{\sqrt{2}}{2}} \right ) +1 \right ){\frac{1}{\sqrt{-4\, \left ( x+1/2\,\sqrt{2} \right ) ^{2}+4\,\sqrt{2} \left ( x+1/2\,\sqrt{2} \right ) +2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x-(-x^2+1)^(1/2)),x)

[Out]

1/4*ln(2*x^2-1)+1/8*2^(1/2)*(-4*(x-1/2*2^(1/2))^2-4*2^(1/2)*(x-1/2*2^(1/2))+2)^(
1/2)-1/2*arcsin(x)-1/4*arctanh((1-2^(1/2)*(x-1/2*2^(1/2)))*2^(1/2)/(-4*(x-1/2*2^
(1/2))^2-4*2^(1/2)*(x-1/2*2^(1/2))+2)^(1/2))-1/8*2^(1/2)*(-4*(x+1/2*2^(1/2))^2+4
*2^(1/2)*(x+1/2*2^(1/2))+2)^(1/2)+1/4*arctanh((2^(1/2)*(x+1/2*2^(1/2))+1)*2^(1/2
)/(-4*(x+1/2*2^(1/2))^2+4*2^(1/2)*(x+1/2*2^(1/2))+2)^(1/2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x - \sqrt{-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x - sqrt(-x^2 + 1)),x, algorithm="maxima")

[Out]

integrate(1/(x - sqrt(-x^2 + 1)), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.269286, size = 113, normalized size = 3.05 \[ \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) + \frac{1}{4} \, \log \left (2 \, x^{2} - 1\right ) + \frac{1}{4} \, \log \left (-\frac{x^{2} + \sqrt{-x^{2} + 1}{\left (x + 1\right )} - x - 1}{x^{2}}\right ) - \frac{1}{4} \, \log \left (-\frac{x^{2} - \sqrt{-x^{2} + 1}{\left (x - 1\right )} + x - 1}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x - sqrt(-x^2 + 1)),x, algorithm="fricas")

[Out]

arctan((sqrt(-x^2 + 1) - 1)/x) + 1/4*log(2*x^2 - 1) + 1/4*log(-(x^2 + sqrt(-x^2
+ 1)*(x + 1) - x - 1)/x^2) - 1/4*log(-(x^2 - sqrt(-x^2 + 1)*(x - 1) + x - 1)/x^2
)

_______________________________________________________________________________________

Sympy [A]  time = 0.37748, size = 17, normalized size = 0.46 \[ \frac{\log{\left (x - \sqrt{- x^{2} + 1} \right )}}{2} - \frac{\operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.273892, size = 189, normalized size = 5.11 \[ -\frac{1}{4} \, \pi{\rm sign}\left (x\right ) - \frac{1}{2} \, \arctan \left (-\frac{x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x + \frac{1}{2} \, \sqrt{2} \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x - \frac{1}{2} \, \sqrt{2} \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | -\frac{x}{\sqrt{-x^{2} + 1} - 1} + \frac{\sqrt{-x^{2} + 1} - 1}{x} + 2 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | -\frac{x}{\sqrt{-x^{2} + 1} - 1} + \frac{\sqrt{-x^{2} + 1} - 1}{x} - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x - sqrt(-x^2 + 1)),x, algorithm="giac")

[Out]

-1/4*pi*sign(x) - 1/2*arctan(-1/2*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2
+ 1) - 1)) + 1/4*ln(abs(x + 1/2*sqrt(2))) + 1/4*ln(abs(x - 1/2*sqrt(2))) - 1/4*l
n(abs(-x/(sqrt(-x^2 + 1) - 1) + (sqrt(-x^2 + 1) - 1)/x + 2)) + 1/4*ln(abs(-x/(sq
rt(-x^2 + 1) - 1) + (sqrt(-x^2 + 1) - 1)/x - 2))