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

Optimal. Leaf size=22 \[ -\tanh ^{-1}\left (\frac{1-x}{2 \sqrt{x^2+x+1}}\right ) \]

[Out]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

-log(-x + sqrt(x^2 + x + 1)) + log(-x + sqrt(x^2 + x + 1) - 2)

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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]

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

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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]

-ln(abs(-x + sqrt(x^2 + x + 1))) + ln(abs(-x + sqrt(x^2 + x + 1) - 2))