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

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])

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Rubi [A]  time = 0.0210366, 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, Rules used = {1164, 628} \[ \frac{\log \left (x^2+\sqrt{3} x+1\right )}{2 \sqrt{3}}-\frac{\log \left (x^2-\sqrt{3} x+1\right )}{2 \sqrt{3}} \]

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])

Rule 1164

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e - b/c, 2]},
Dist[e/(2*c*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x
- x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] &&  !GtQ[b^2
- 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1-x^2}{1-x^2+x^4} \, dx &=-\frac{\int \frac{\sqrt{3}+2 x}{-1-\sqrt{3} x-x^2} \, dx}{2 \sqrt{3}}-\frac{\int \frac{\sqrt{3}-2 x}{-1+\sqrt{3} x-x^2} \, dx}{2 \sqrt{3}}\\ &=-\frac{\log \left (1-\sqrt{3} x+x^2\right )}{2 \sqrt{3}}+\frac{\log \left (1+\sqrt{3} x+x^2\right )}{2 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0125238, 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])

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Maple [A]  time = 0.049, size = 35, normalized size = 0.8 \begin{align*} -{\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}} \end{align*}

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)

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

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)

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Fricas [A]  time = 1.32572, size = 100, normalized size = 2.17 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (\frac{x^{4} + 5 \, x^{2} + 2 \, \sqrt{3}{\left (x^{3} + x\right )} + 1}{x^{4} - x^{2} + 1}\right ) \end{align*}

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

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Sympy [A]  time = 0.100779, size = 39, normalized size = 0.85 \begin{align*} - \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} \end{align*}

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

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Giac [A]  time = 1.12561, size = 53, normalized size = 1.15 \begin{align*} -\frac{1}{6} \, \sqrt{3} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{3} + \frac{2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt{3} + \frac{2}{x} \right |}}\right ) \end{align*}

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)*log(abs(2*x - 2*sqrt(3) + 2/x)/abs(2*x + 2*sqrt(3) + 2/x))