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

Optimal. Leaf size=72 \[ \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )-\sqrt{\frac{2}{5 \left (3+\sqrt{5}\right )}} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right ) \]

[Out]

-(Sqrt[2/(5*(3 + Sqrt[5]))]*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*x]) + Sqrt[(3 + Sqrt[5])/10]*ArcTanh[Sqrt[(3 + Sqrt[
5])/2]*x]

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Rubi [A]  time = 0.0653318, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1093, 207} \[ \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )-\sqrt{\frac{2}{5 \left (3+\sqrt{5}\right )}} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 3*x^2 + x^4)^(-1),x]

[Out]

-(Sqrt[2/(5*(3 + Sqrt[5]))]*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*x]) + Sqrt[(3 + Sqrt[5])/10]*ArcTanh[Sqrt[(3 + Sqrt[
5])/2]*x]

Rule 1093

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

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{1-3 x^2+x^4} \, dx &=\frac{\int \frac{1}{-\frac{3}{2}-\frac{\sqrt{5}}{2}+x^2} \, dx}{\sqrt{5}}-\frac{\int \frac{1}{-\frac{3}{2}+\frac{\sqrt{5}}{2}+x^2} \, dx}{\sqrt{5}}\\ &=-\sqrt{\frac{2}{5 \left (3+\sqrt{5}\right )}} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right )+\sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )\\ \end{align*}

Mathematica [A]  time = 0.0351519, size = 83, normalized size = 1.15 \[ \frac{1}{20} \left (-\left (5+\sqrt{5}\right ) \log \left (-2 x+\sqrt{5}-1\right )-\left (\sqrt{5}-5\right ) \log \left (-2 x+\sqrt{5}+1\right )+\left (5+\sqrt{5}\right ) \log \left (2 x+\sqrt{5}-1\right )+\left (\sqrt{5}-5\right ) \log \left (2 x+\sqrt{5}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 3*x^2 + x^4)^(-1),x]

[Out]

(-((5 + Sqrt[5])*Log[-1 + Sqrt[5] - 2*x]) - (-5 + Sqrt[5])*Log[1 + Sqrt[5] - 2*x] + (5 + Sqrt[5])*Log[-1 + Sqr
t[5] + 2*x] + (-5 + Sqrt[5])*Log[1 + Sqrt[5] + 2*x])/20

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Maple [A]  time = 0.006, size = 54, normalized size = 0.8 \begin{align*}{\frac{\ln \left ({x}^{2}-x-1 \right ) }{4}}+{\frac{\sqrt{5}}{10}{\it Artanh} \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{5}}{5}} \right ) }-{\frac{\ln \left ({x}^{2}+x-1 \right ) }{4}}+{\frac{\sqrt{5}}{10}{\it Artanh} \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{5}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*ln(x^2-x-1)+1/10*5^(1/2)*arctanh(1/5*(2*x-1)*5^(1/2))-1/4*ln(x^2+x-1)+1/10*5^(1/2)*arctanh(1/5*(1+2*x)*5^(
1/2))

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Maxima [A]  time = 1.44582, size = 101, normalized size = 1.4 \begin{align*} -\frac{1}{20} \, \sqrt{5} \log \left (\frac{2 \, x - \sqrt{5} + 1}{2 \, x + \sqrt{5} + 1}\right ) - \frac{1}{20} \, \sqrt{5} \log \left (\frac{2 \, x - \sqrt{5} - 1}{2 \, x + \sqrt{5} - 1}\right ) - \frac{1}{4} \, \log \left (x^{2} + x - 1\right ) + \frac{1}{4} \, \log \left (x^{2} - x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*sqrt(5)*log((2*x - sqrt(5) + 1)/(2*x + sqrt(5) + 1)) - 1/20*sqrt(5)*log((2*x - sqrt(5) - 1)/(2*x + sqrt(
5) - 1)) - 1/4*log(x^2 + x - 1) + 1/4*log(x^2 - x - 1)

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Fricas [B]  time = 1.89264, size = 255, normalized size = 3.54 \begin{align*} \frac{1}{20} \, \sqrt{5} \log \left (\frac{2 \, x^{2} + \sqrt{5}{\left (2 \, x + 1\right )} + 2 \, x + 3}{x^{2} + x - 1}\right ) + \frac{1}{20} \, \sqrt{5} \log \left (\frac{2 \, x^{2} + \sqrt{5}{\left (2 \, x - 1\right )} - 2 \, x + 3}{x^{2} - x - 1}\right ) - \frac{1}{4} \, \log \left (x^{2} + x - 1\right ) + \frac{1}{4} \, \log \left (x^{2} - x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*sqrt(5)*log((2*x^2 + sqrt(5)*(2*x + 1) + 2*x + 3)/(x^2 + x - 1)) + 1/20*sqrt(5)*log((2*x^2 + sqrt(5)*(2*x
- 1) - 2*x + 3)/(x^2 - x - 1)) - 1/4*log(x^2 + x - 1) + 1/4*log(x^2 - x - 1)

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Sympy [B]  time = 0.413806, size = 158, normalized size = 2.19 \begin{align*} \left (\frac{\sqrt{5}}{20} + \frac{1}{4}\right ) \log{\left (x - \frac{7}{2} - \frac{7 \sqrt{5}}{10} + 120 \left (\frac{\sqrt{5}}{20} + \frac{1}{4}\right )^{3} \right )} + \left (\frac{1}{4} - \frac{\sqrt{5}}{20}\right ) \log{\left (x - \frac{7}{2} + 120 \left (\frac{1}{4} - \frac{\sqrt{5}}{20}\right )^{3} + \frac{7 \sqrt{5}}{10} \right )} + \left (- \frac{1}{4} + \frac{\sqrt{5}}{20}\right ) \log{\left (x - \frac{7 \sqrt{5}}{10} + 120 \left (- \frac{1}{4} + \frac{\sqrt{5}}{20}\right )^{3} + \frac{7}{2} \right )} + \left (- \frac{1}{4} - \frac{\sqrt{5}}{20}\right ) \log{\left (x + 120 \left (- \frac{1}{4} - \frac{\sqrt{5}}{20}\right )^{3} + \frac{7 \sqrt{5}}{10} + \frac{7}{2} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(5)/20 + 1/4)*log(x - 7/2 - 7*sqrt(5)/10 + 120*(sqrt(5)/20 + 1/4)**3) + (1/4 - sqrt(5)/20)*log(x - 7/2 +
120*(1/4 - sqrt(5)/20)**3 + 7*sqrt(5)/10) + (-1/4 + sqrt(5)/20)*log(x - 7*sqrt(5)/10 + 120*(-1/4 + sqrt(5)/20)
**3 + 7/2) + (-1/4 - sqrt(5)/20)*log(x + 120*(-1/4 - sqrt(5)/20)**3 + 7*sqrt(5)/10 + 7/2)

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Giac [A]  time = 1.08602, size = 109, normalized size = 1.51 \begin{align*} -\frac{1}{20} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x - \sqrt{5} + 1 \right |}}{{\left | 2 \, x + \sqrt{5} + 1 \right |}}\right ) - \frac{1}{20} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x - \sqrt{5} - 1 \right |}}{{\left | 2 \, x + \sqrt{5} - 1 \right |}}\right ) - \frac{1}{4} \, \log \left ({\left | x^{2} + x - 1 \right |}\right ) + \frac{1}{4} \, \log \left ({\left | x^{2} - x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*sqrt(5)*log(abs(2*x - sqrt(5) + 1)/abs(2*x + sqrt(5) + 1)) - 1/20*sqrt(5)*log(abs(2*x - sqrt(5) - 1)/abs
(2*x + sqrt(5) - 1)) - 1/4*log(abs(x^2 + x - 1)) + 1/4*log(abs(x^2 - x - 1))