3.95 \(\int \frac{1-x^2}{1-x^2+x^4} \, dx\)

Optimal. Leaf size=46 \[ \frac{\log \left (x^2+\sqrt{3} x+1\right )}{2 \sqrt{3}}-\frac{\log \left (x^2-\sqrt{3} x+1\right )}{2 \sqrt{3}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0400427, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\log \left (x^2+\sqrt{3} x+1\right )}{2 \sqrt{3}}-\frac{\log \left (x^2-\sqrt{3} x+1\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^2)/(1 - x^2 + x^4),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.8177, size = 39, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{6} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+1)/(x**4-x**2+1),x)

[Out]

-sqrt(3)*log(x**2 - sqrt(3)*x + 1)/6 + sqrt(3)*log(x**2 + sqrt(3)*x + 1)/6

_______________________________________________________________________________________

Mathematica [A]  time = 0.02027, size = 40, normalized size = 0.87 \[ \frac{\log \left (x^2+\sqrt{3} x+1\right )-\log \left (-x^2+\sqrt{3} x-1\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^2)/(1 - x^2 + x^4),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 35, normalized size = 0.8 \[ -{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{6}}+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)/(x^4-x^2+1),x)

[Out]

-1/6*ln(1+x^2-x*3^(1/2))*3^(1/2)+1/6*ln(1+x^2+x*3^(1/2))*3^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{2} - 1}{x^{4} - x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 1)/(x^4 - x^2 + 1),x, algorithm="maxima")

[Out]

-integrate((x^2 - 1)/(x^4 - x^2 + 1), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.290388, size = 57, normalized size = 1.24 \[ \frac{1}{6} \, \sqrt{3} \log \left (\frac{6 \, x^{3} + \sqrt{3}{\left (x^{4} + 5 \, x^{2} + 1\right )} + 6 \, x}{x^{4} - x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 1)/(x^4 - x^2 + 1),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*log((6*x^3 + sqrt(3)*(x^4 + 5*x^2 + 1) + 6*x)/(x^4 - x^2 + 1))

_______________________________________________________________________________________

Sympy [A]  time = 0.197104, size = 39, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{6} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+1)/(x**4-x**2+1),x)

[Out]

-sqrt(3)*log(x**2 - sqrt(3)*x + 1)/6 + sqrt(3)*log(x**2 + sqrt(3)*x + 1)/6

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.27022, size = 53, normalized size = 1.15 \[ -\frac{1}{6} \, \sqrt{3}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{3} + \frac{2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt{3} + \frac{2}{x} \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 1)/(x^4 - x^2 + 1),x, algorithm="giac")

[Out]

-1/6*sqrt(3)*ln(abs(2*x - 2*sqrt(3) + 2/x)/abs(2*x + 2*sqrt(3) + 2/x))