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