3.2524 \(\int \frac{-1+x}{(1+x) \sqrt{1+x+x^2}} \, dx\)

Optimal. Leaf size=35 \[ 2 \tanh ^{-1}\left (\frac{1-x}{2 \sqrt{x^2+x+1}}\right )+\sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0867557, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ 2 \tanh ^{-1}\left (\frac{1-x}{2 \sqrt{x^2+x+1}}\right )+\sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.4759, size = 36, normalized size = 1.03 \[ 2 \operatorname{atanh}{\left (\frac{- x + 1}{2 \sqrt{x^{2} + x + 1}} \right )} + \operatorname{atanh}{\left (\frac{2 x + 1}{2 \sqrt{x^{2} + x + 1}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0249737, size = 44, normalized size = 1.26 \[ -2 \log (x+1)+2 \log \left (-x+2 \sqrt{(x+1)^2-x}+1\right )+\sinh ^{-1}\left (\frac{2 (x+1)-1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 32, normalized size = 0.9 \[{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( x+{\frac{1}{2}} \right ) } \right ) +2\,{\it Artanh} \left ( 1/2\,{\frac{1-x}{\sqrt{ \left ( 1+x \right ) ^{2}-x}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.791916, size = 55, normalized size = 1.57 \[ \operatorname{arsinh}\left (\frac{2}{3} \, \sqrt{3} x + \frac{1}{3} \, \sqrt{3}\right ) - 2 \, \operatorname{arsinh}\left (\frac{\sqrt{3} x}{3 \,{\left | x + 1 \right |}} - \frac{\sqrt{3}}{3 \,{\left | x + 1 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.282037, size = 68, normalized size = 1.94 \[ 2 \, \log \left (-x + \sqrt{x^{2} + x + 1}\right ) - 2 \, \log \left (-x + \sqrt{x^{2} + x + 1} - 2\right ) - \log \left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(-x + sqrt(x^2 + x + 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. \[ \int \frac{x - 1}{\left (x + 1\right ) \sqrt{x^{2} + x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272655, size = 70, normalized size = 2. \[ -{\rm ln}\left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) + 2 \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + x + 1} \right |}\right ) - 2 \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + x + 1} - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(-2*x + 2*sqrt(x^2 + x + 1) - 1) + 2*ln(abs(-x + sqrt(x^2 + x + 1))) - 2*ln(a
bs(-x + sqrt(x^2 + x + 1) - 2))