3.77 \(\int \frac{1+x^2}{1-3 x^2+x^4} \, dx\)

Optimal. Leaf size=65 \[ \frac{1}{2} \log \left (-2 x-\sqrt{5}+1\right )+\frac{1}{2} \log \left (-2 x+\sqrt{5}+1\right )-\frac{1}{2} \log \left (2 x-\sqrt{5}+1\right )-\frac{1}{2} \log \left (2 x+\sqrt{5}+1\right ) \]

[Out]

Log[1 - Sqrt[5] - 2*x]/2 + Log[1 + Sqrt[5] - 2*x]/2 - Log[1 - Sqrt[5] + 2*x]/2 - Log[1 + Sqrt[5] + 2*x]/2

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Rubi [A]  time = 0.0319407, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1161, 616, 31} \[ \frac{1}{2} \log \left (-2 x-\sqrt{5}+1\right )+\frac{1}{2} \log \left (-2 x+\sqrt{5}+1\right )-\frac{1}{2} \log \left (2 x-\sqrt{5}+1\right )-\frac{1}{2} \log \left (2 x+\sqrt{5}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/(1 - 3*x^2 + x^4),x]

[Out]

Log[1 - Sqrt[5] - 2*x]/2 + Log[1 + Sqrt[5] - 2*x]/2 - Log[1 - Sqrt[5] + 2*x]/2 - Log[1 + Sqrt[5] + 2*x]/2

Rule 1161

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), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/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[(2*d)/e - b/c, 0] || ( !Lt
Q[(2*d)/e - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1+x^2}{1-3 x^2+x^4} \, dx &=\frac{1}{2} \int \frac{1}{1-\sqrt{5} x+x^2} \, dx+\frac{1}{2} \int \frac{1}{1+\sqrt{5} x+x^2} \, dx\\ &=\frac{1}{2} \int \frac{1}{\frac{1}{2} \left (-1-\sqrt{5}\right )+x} \, dx-\frac{1}{2} \int \frac{1}{\frac{1}{2} \left (1-\sqrt{5}\right )+x} \, dx+\frac{1}{2} \int \frac{1}{\frac{1}{2} \left (-1+\sqrt{5}\right )+x} \, dx-\frac{1}{2} \int \frac{1}{\frac{1}{2} \left (1+\sqrt{5}\right )+x} \, dx\\ &=\frac{1}{2} \log \left (1-\sqrt{5}-2 x\right )+\frac{1}{2} \log \left (1+\sqrt{5}-2 x\right )-\frac{1}{2} \log \left (1-\sqrt{5}+2 x\right )-\frac{1}{2} \log \left (1+\sqrt{5}+2 x\right )\\ \end{align*}

Mathematica [A]  time = 0.0057495, size = 29, normalized size = 0.45 \[ \frac{1}{2} \log \left (-x^2+x+1\right )-\frac{1}{2} \log \left (-x^2-x+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)/(1 - 3*x^2 + x^4),x]

[Out]

-Log[1 - x - x^2]/2 + Log[1 + x - x^2]/2

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Maple [A]  time = 0.044, size = 22, normalized size = 0.3 \begin{align*} -{\frac{\ln \left ({x}^{2}+x-1 \right ) }{2}}+{\frac{\ln \left ({x}^{2}-x-1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)/(x^4-3*x^2+1),x)

[Out]

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

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Maxima [A]  time = 0.953934, size = 28, normalized size = 0.43 \begin{align*} -\frac{1}{2} \, \log \left (x^{2} + x - 1\right ) + \frac{1}{2} \, \log \left (x^{2} - x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^4-3*x^2+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.27954, size = 62, normalized size = 0.95 \begin{align*} -\frac{1}{2} \, \log \left (x^{2} + x - 1\right ) + \frac{1}{2} \, \log \left (x^{2} - x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^4-3*x^2+1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.097323, size = 19, normalized size = 0.29 \begin{align*} \frac{\log{\left (x^{2} - x - 1 \right )}}{2} - \frac{\log{\left (x^{2} + x - 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+1)/(x**4-3*x**2+1),x)

[Out]

log(x**2 - x - 1)/2 - log(x**2 + x - 1)/2

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Giac [A]  time = 1.13753, size = 58, normalized size = 0.89 \begin{align*} -\frac{1}{4} \, \log \left ({\left | x + \frac{1}{x - \frac{1}{x}} - \frac{1}{x} + 2 \right |}\right ) + \frac{1}{4} \, \log \left ({\left | x + \frac{1}{x - \frac{1}{x}} - \frac{1}{x} - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^4-3*x^2+1),x, algorithm="giac")

[Out]

-1/4*log(abs(x + 1/(x - 1/x) - 1/x + 2)) + 1/4*log(abs(x + 1/(x - 1/x) - 1/x - 2))