3.2358 \(\int \frac{\sqrt{-1-x+x^2}}{1+x} \, dx\)

Optimal. Leaf size=61 \[ \sqrt{x^2-x-1}+\frac{3}{2} \tanh ^{-1}\left (\frac{1-2 x}{2 \sqrt{x^2-x-1}}\right )+\tanh ^{-1}\left (\frac{3 x+1}{2 \sqrt{x^2-x-1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.123732, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \sqrt{x^2-x-1}+\frac{3}{2} \tanh ^{-1}\left (\frac{1-2 x}{2 \sqrt{x^2-x-1}}\right )+\tanh ^{-1}\left (\frac{3 x+1}{2 \sqrt{x^2-x-1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.0223, size = 51, normalized size = 0.84 \[ \sqrt{x^{2} - x - 1} - \operatorname{atanh}{\left (\frac{- 3 x - 1}{2 \sqrt{x^{2} - x - 1}} \right )} - \frac{3 \operatorname{atanh}{\left (\frac{2 x - 1}{2 \sqrt{x^{2} - x - 1}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0391467, size = 63, normalized size = 1.03 \[ \sqrt{x^2-x-1}-\frac{3}{2} \log \left (-2 \sqrt{x^2-x-1}-2 x+1\right )-\log \left (2 \sqrt{x^2-x-1}-3 x-1\right )+\log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 54, normalized size = 0.9 \[ \sqrt{ \left ( 1+x \right ) ^{2}-2-3\,x}-{\frac{3}{2}\ln \left ( -{\frac{1}{2}}+x+\sqrt{ \left ( 1+x \right ) ^{2}-2-3\,x} \right ) }-{\it Artanh} \left ({\frac{-3\,x-1}{2}{\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}-2-3\,x}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-x-1)^(1/2)/(1+x),x)

[Out]

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

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Maxima [A]  time = 0.6778, size = 84, normalized size = 1.38 \[ \sqrt{x^{2} - x - 1} - \frac{3}{2} \, \log \left (2 \, x + 2 \, \sqrt{x^{2} - x - 1} - 1\right ) - \log \left (\frac{2 \, \sqrt{x^{2} - x - 1}}{{\left | x + 1 \right |}} + \frac{2}{{\left | x + 1 \right |}} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.23652, size = 208, normalized size = 3.41 \[ -\frac{8 \, x^{2} + 4 \,{\left (2 \, x - 2 \, \sqrt{x^{2} - x - 1} - 1\right )} \log \left (-x + \sqrt{x^{2} - x - 1}\right ) - 4 \,{\left (2 \, x - 2 \, \sqrt{x^{2} - x - 1} - 1\right )} \log \left (-x + \sqrt{x^{2} - x - 1} - 2\right ) - 6 \,{\left (2 \, x - 2 \, \sqrt{x^{2} - x - 1} - 1\right )} \log \left (-2 \, x + 2 \, \sqrt{x^{2} - x - 1} + 1\right ) - 2 \, \sqrt{x^{2} - x - 1}{\left (4 \, x - 1\right )} - 6 \, x - 9}{4 \,{\left (2 \, x - 2 \, \sqrt{x^{2} - x - 1} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(8*x^2 + 4*(2*x - 2*sqrt(x^2 - x - 1) - 1)*log(-x + sqrt(x^2 - x - 1)) - 4*
(2*x - 2*sqrt(x^2 - x - 1) - 1)*log(-x + sqrt(x^2 - x - 1) - 2) - 6*(2*x - 2*sqr
t(x^2 - x - 1) - 1)*log(-2*x + 2*sqrt(x^2 - x - 1) + 1) - 2*sqrt(x^2 - x - 1)*(4
*x - 1) - 6*x - 9)/(2*x - 2*sqrt(x^2 - x - 1) - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.214519, size = 90, normalized size = 1.48 \[ \sqrt{x^{2} - x - 1} -{\rm ln}\left ({\left | -x + \sqrt{x^{2} - x - 1} \right |}\right ) +{\rm ln}\left ({\left | -x + \sqrt{x^{2} - x - 1} - 2 \right |}\right ) + \frac{3}{2} \,{\rm ln}\left ({\left | -2 \, x + 2 \, \sqrt{x^{2} - x - 1} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x^2 - x - 1) - ln(abs(-x + sqrt(x^2 - x - 1))) + ln(abs(-x + sqrt(x^2 - x -
1) - 2)) + 3/2*ln(abs(-2*x + 2*sqrt(x^2 - x - 1) + 1))