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