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

Optimal. Leaf size=75 \[ -\frac{1}{2} \tan ^{-1}\left (\frac{3-x}{2 \sqrt{x^2-x-1}}\right )+\tanh ^{-1}\left (\frac{1-2 x}{2 \sqrt{x^2-x-1}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{3 x+1}{2 \sqrt{x^2-x-1}}\right ) \]

[Out]

-ArcTan[(3 - x)/(2*Sqrt[-1 - x + x^2])]/2 + ArcTanh[(1 - 2*x)/(2*Sqrt[-1 - x + x
^2])] + ArcTanh[(1 + 3*x)/(2*Sqrt[-1 - x + x^2])]/2

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Rubi [A]  time = 0.142708, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{1}{2} \tan ^{-1}\left (\frac{3-x}{2 \sqrt{x^2-x-1}}\right )+\tanh ^{-1}\left (\frac{1-2 x}{2 \sqrt{x^2-x-1}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{3 x+1}{2 \sqrt{x^2-x-1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-1 - x + x^2]/(1 - x^2),x]

[Out]

-ArcTan[(3 - x)/(2*Sqrt[-1 - x + x^2])]/2 + ArcTanh[(1 - 2*x)/(2*Sqrt[-1 - x + x
^2])] + ArcTanh[(1 + 3*x)/(2*Sqrt[-1 - x + x^2])]/2

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Rubi in Sympy [A]  time = 37.4938, size = 56, normalized size = 0.75 \[ - \frac{\operatorname{atan}{\left (\frac{- x + 3}{2 \sqrt{x^{2} - x - 1}} \right )}}{2} - \operatorname{atanh}{\left (\frac{2 x - 1}{2 \sqrt{x^{2} - x - 1}} \right )} + \frac{\operatorname{atanh}{\left (\frac{3 x + 1}{2 \sqrt{x^{2} - x - 1}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atan((-x + 3)/(2*sqrt(x**2 - x - 1)))/2 - atanh((2*x - 1)/(2*sqrt(x**2 - x - 1)
)) + atanh((3*x + 1)/(2*sqrt(x**2 - x - 1)))/2

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Mathematica [A]  time = 0.0214565, size = 79, normalized size = 1.05 \[ -\log \left (-2 \sqrt{x^2-x-1}-2 x+1\right )-\frac{1}{2} \log \left (-2 \sqrt{x^2-x-1}+3 x+1\right )+\frac{1}{2} \tan ^{-1}\left (\frac{x-3}{2 \sqrt{x^2-x-1}}\right )+\frac{1}{2} \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[(-3 + x)/(2*Sqrt[-1 - x + x^2])]/2 + Log[1 + x]/2 - Log[1 - 2*x - 2*Sqrt[
-1 - x + x^2]] - Log[1 + 3*x - 2*Sqrt[-1 - x + x^2]]/2

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Maple [A]  time = 0.02, size = 102, normalized size = 1.4 \[{\frac{1}{2}\sqrt{ \left ( 1+x \right ) ^{2}-2-3\,x}}-{\frac{3}{4}\ln \left ( -{\frac{1}{2}}+x+\sqrt{ \left ( 1+x \right ) ^{2}-2-3\,x} \right ) }-{\frac{1}{2}{\it Artanh} \left ({\frac{-1-3\,x}{2}{\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}-2-3\,x}}}} \right ) }-{\frac{1}{2}\sqrt{ \left ( -1+x \right ) ^{2}+x-2}}-{\frac{1}{4}\ln \left ( -{\frac{1}{2}}+x+\sqrt{ \left ( -1+x \right ) ^{2}+x-2} \right ) }+{\frac{1}{2}\arctan \left ({\frac{-3+x}{2}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}+x-2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-x-1)^(1/2)/(-x^2+1),x)

[Out]

1/2*((1+x)^2-2-3*x)^(1/2)-3/4*ln(-1/2+x+((1+x)^2-2-3*x)^(1/2))-1/2*arctanh(1/2*(
-1-3*x)/((1+x)^2-2-3*x)^(1/2))-1/2*((-1+x)^2+x-2)^(1/2)-1/4*ln(-1/2+x+((-1+x)^2+
x-2)^(1/2))+1/2*arctan(1/2*(-3+x)/((-1+x)^2+x-2)^(1/2))

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Maxima [A]  time = 0.79461, size = 112, normalized size = 1.49 \[ \frac{1}{2} \, \arcsin \left (\frac{2 \, \sqrt{5} x}{5 \,{\left | 2 \, x - 2 \right |}} - \frac{6 \, \sqrt{5}}{5 \,{\left | 2 \, x - 2 \right |}}\right ) - \log \left (x + \sqrt{x^{2} - x - 1} - \frac{1}{2}\right ) - \frac{1}{2} \, \log \left (\frac{2 \, \sqrt{x^{2} - x - 1}}{{\left | 2 \, x + 2 \right |}} + \frac{2}{{\left | 2 \, x + 2 \right |}} - \frac{3}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arcsin(2/5*sqrt(5)*x/abs(2*x - 2) - 6/5*sqrt(5)/abs(2*x - 2)) - log(x + sqrt
(x^2 - x - 1) - 1/2) - 1/2*log(2*sqrt(x^2 - x - 1)/abs(2*x + 2) + 2/abs(2*x + 2)
- 3/2)

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Fricas [A]  time = 0.299299, size = 95, normalized size = 1.27 \[ \arctan \left (-x + \sqrt{x^{2} - x - 1} + 1\right ) - \frac{1}{2} \, \log \left (-x + \sqrt{x^{2} - x - 1}\right ) + \frac{1}{2} \, \log \left (-x + \sqrt{x^{2} - x - 1} - 2\right ) + \log \left (-2 \, x + 2 \, \sqrt{x^{2} - x - 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(sqrt(x**2 - x - 1)/(x**2 - 1), x)

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GIAC/XCAS [A]  time = 0.275068, size = 99, normalized size = 1.32 \[ \arctan \left (-x + \sqrt{x^{2} - x - 1} + 1\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} - x - 1} \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} - x - 1} - 2 \right |}\right ) +{\rm ln}\left ({\left | -2 \, x + 2 \, \sqrt{x^{2} - x - 1} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(-x + sqrt(x^2 - x - 1) + 1) - 1/2*ln(abs(-x + sqrt(x^2 - x - 1))) + 1/2*l
n(abs(-x + sqrt(x^2 - x - 1) - 2)) + ln(abs(-2*x + 2*sqrt(x^2 - x - 1) + 1))