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]

-1/5*arctanh(x*2^(1/2)/(3+5^(1/2))^(1/2))*10^(1/2)/(3+5^(1/2))^(1/2)+arctanh(x*(1/2+1/2*5^(1/2)))*(1/2+1/10*5^
(1/2))

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 72, normalized size of antiderivative = 1.00, 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 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])

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]

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 = 0.04, size = 83, normalized size = 1.15 \[ \frac {1}{20} \left (-\left (\left (5+\sqrt {5}\right ) \log \left (-2 x+\sqrt {5}-1\right )\right )-\left (\sqrt {5}-5\right ) \log \left (-2 x+\sqrt {5}+1\right )+\left (5+\sqrt {5}\right ) \log \left (2 x+\sqrt {5}-1\right )+\left (\sqrt {5}-5\right ) \log \left (2 x+\sqrt {5}+1\right )\right ) \]

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

________________________________________________________________________________________

fricas [B]  time = 0.42, size = 91, normalized size = 1.26 \[ \frac {1}{20} \, \sqrt {5} \log \left (\frac {2 \, x^{2} + \sqrt {5} {\left (2 \, x + 1\right )} + 2 \, x + 3}{x^{2} + x - 1}\right ) + \frac {1}{20} \, \sqrt {5} \log \left (\frac {2 \, x^{2} + \sqrt {5} {\left (2 \, x - 1\right )} - 2 \, x + 3}{x^{2} - x - 1}\right ) - \frac {1}{4} \, \log \left (x^{2} + x - 1\right ) + \frac {1}{4} \, \log \left (x^{2} - x - 1\right ) \]

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)

________________________________________________________________________________________

giac [A]  time = 1.28, size = 81, normalized size = 1.12 \[ -\frac {1}{20} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x - \sqrt {5} + 1 \right |}}{{\left | 2 \, x + \sqrt {5} + 1 \right |}}\right ) - \frac {1}{20} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x - \sqrt {5} - 1 \right |}}{{\left | 2 \, x + \sqrt {5} - 1 \right |}}\right ) - \frac {1}{4} \, \log \left ({\left | x^{2} + x - 1 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | x^{2} - x - 1 \right |}\right ) \]

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))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 54, normalized size = 0.75 \[ \frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 x +1\right ) \sqrt {5}}{5}\right )}{10}+\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 x -1\right ) \sqrt {5}}{5}\right )}{10}+\frac {\ln \left (x^{2}-x -1\right )}{4}-\frac {\ln \left (x^{2}+x -1\right )}{4} \]

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*(2*x-1)*5^
(1/2))

________________________________________________________________________________________

maxima [A]  time = 0.96, size = 75, normalized size = 1.04 \[ -\frac {1}{20} \, \sqrt {5} \log \left (\frac {2 \, x - \sqrt {5} + 1}{2 \, x + \sqrt {5} + 1}\right ) - \frac {1}{20} \, \sqrt {5} \log \left (\frac {2 \, x - \sqrt {5} - 1}{2 \, x + \sqrt {5} - 1}\right ) - \frac {1}{4} \, \log \left (x^{2} + x - 1\right ) + \frac {1}{4} \, \log \left (x^{2} - x - 1\right ) \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 67, normalized size = 0.93 \[ \mathrm {atanh}\left (\frac {4\,x}{\sqrt {5}-3}-\frac {2\,\sqrt {5}\,x}{\sqrt {5}-3}\right )\,\left (\frac {\sqrt {5}}{10}-\frac {1}{2}\right )+\mathrm {atanh}\left (\frac {4\,x}{\sqrt {5}+3}+\frac {2\,\sqrt {5}\,x}{\sqrt {5}+3}\right )\,\left (\frac {\sqrt {5}}{10}+\frac {1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4 - 3*x^2 + 1),x)

[Out]

atanh((4*x)/(5^(1/2) - 3) - (2*5^(1/2)*x)/(5^(1/2) - 3))*(5^(1/2)/10 - 1/2) + atanh((4*x)/(5^(1/2) + 3) + (2*5
^(1/2)*x)/(5^(1/2) + 3))*(5^(1/2)/10 + 1/2)

________________________________________________________________________________________

sympy [B]  time = 0.35, size = 158, normalized size = 2.19 \[ \left (\frac {\sqrt {5}}{20} + \frac {1}{4}\right ) \log {\left (x - \frac {7}{2} - \frac {7 \sqrt {5}}{10} + 120 \left (\frac {\sqrt {5}}{20} + \frac {1}{4}\right )^{3} \right )} + \left (\frac {1}{4} - \frac {\sqrt {5}}{20}\right ) \log {\left (x - \frac {7}{2} + 120 \left (\frac {1}{4} - \frac {\sqrt {5}}{20}\right )^{3} + \frac {7 \sqrt {5}}{10} \right )} + \left (- \frac {1}{4} + \frac {\sqrt {5}}{20}\right ) \log {\left (x - \frac {7 \sqrt {5}}{10} + 120 \left (- \frac {1}{4} + \frac {\sqrt {5}}{20}\right )^{3} + \frac {7}{2} \right )} + \left (- \frac {1}{4} - \frac {\sqrt {5}}{20}\right ) \log {\left (x + 120 \left (- \frac {1}{4} - \frac {\sqrt {5}}{20}\right )^{3} + \frac {7 \sqrt {5}}{10} + \frac {7}{2} \right )} \]

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)

________________________________________________________________________________________