Optimal. Leaf size=22 \[ -\tanh ^{-1}\left (\frac{1-x}{2 \sqrt{x^2+x+1}}\right ) \]
[Out]
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Rubi [A] time = 0.0247833, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\tanh ^{-1}\left (\frac{1-x}{2 \sqrt{x^2+x+1}}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/((1 + x)*Sqrt[1 + x + x^2]),x]
[Out]
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Rubi in Sympy [A] time = 2.35862, size = 17, normalized size = 0.77 \[ - \operatorname{atanh}{\left (\frac{- x + 1}{2 \sqrt{x^{2} + x + 1}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1+x)/(x**2+x+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.014181, size = 28, normalized size = 1.27 \[ \log (x+1)-\log \left (-x+2 \sqrt{(x+1)^2-x}+1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 + x)*Sqrt[1 + x + x^2]),x]
[Out]
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Maple [A] time = 0.007, size = 22, normalized size = 1. \[ -{\it Artanh} \left ({\frac{1-x}{2}{\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}-x}}}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1+x)/(x^2+x+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.59763, size = 34, normalized size = 1.55 \[ \operatorname{arsinh}\left (\frac{\sqrt{3} x}{3 \,{\left | x + 1 \right |}} - \frac{\sqrt{3}}{3 \,{\left | x + 1 \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^2 + x + 1)*(x + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209753, size = 41, normalized size = 1.86 \[ -\log \left (-x + \sqrt{x^{2} + x + 1}\right ) + \log \left (-x + \sqrt{x^{2} + x + 1} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(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. \[ \int \frac{1}{\left (x + 1\right ) \sqrt{x^{2} + x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1+x)/(x**2+x+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.20579, size = 43, normalized size = 1.95 \[ -{\rm ln}\left ({\left | -x + \sqrt{x^{2} + x + 1} \right |}\right ) +{\rm ln}\left ({\left | -x + \sqrt{x^{2} + x + 1} - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^2 + x + 1)*(x + 1)),x, algorithm="giac")
[Out]