3.67 \(\int \frac{1-2 x^2}{1-6 x^2+4 x^4} \, dx\)

Optimal. Leaf size=48 \[ \frac{\tanh ^{-1}\left (\frac{2 \sqrt{2} x+1}{\sqrt{5}}\right )}{\sqrt{10}}-\frac{\tanh ^{-1}\left (\frac{1-2 \sqrt{2} x}{\sqrt{5}}\right )}{\sqrt{10}} \]

[Out]

-(ArcTanh[(1 - 2*Sqrt[2]*x)/Sqrt[5]]/Sqrt[10]) + ArcTanh[(1 + 2*Sqrt[2]*x)/Sqrt[5]]/Sqrt[10]

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Rubi [A]  time = 0.0386751, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1161, 618, 206} \[ \frac{\tanh ^{-1}\left (\frac{2 \sqrt{2} x+1}{\sqrt{5}}\right )}{\sqrt{10}}-\frac{\tanh ^{-1}\left (\frac{1-2 \sqrt{2} x}{\sqrt{5}}\right )}{\sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x^2)/(1 - 6*x^2 + 4*x^4),x]

[Out]

-(ArcTanh[(1 - 2*Sqrt[2]*x)/Sqrt[5]]/Sqrt[10]) + ArcTanh[(1 + 2*Sqrt[2]*x)/Sqrt[5]]/Sqrt[10]

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 618

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

Rule 206

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

Rubi steps

\begin{align*} \int \frac{1-2 x^2}{1-6 x^2+4 x^4} \, dx &=-\left (\frac{1}{4} \int \frac{1}{-\frac{1}{2}-\frac{x}{\sqrt{2}}+x^2} \, dx\right )-\frac{1}{4} \int \frac{1}{-\frac{1}{2}+\frac{x}{\sqrt{2}}+x^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{5}{2}-x^2} \, dx,x,-\frac{1}{\sqrt{2}}+2 x\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{5}{2}-x^2} \, dx,x,\frac{1}{\sqrt{2}}+2 x\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{1-2 \sqrt{2} x}{\sqrt{5}}\right )}{\sqrt{10}}+\frac{\tanh ^{-1}\left (\frac{1+2 \sqrt{2} x}{\sqrt{5}}\right )}{\sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0190844, size = 42, normalized size = 0.88 \[ \frac{\log \left (2 x^2+\sqrt{10} x+1\right )-\log \left (-2 x^2+\sqrt{10} x-1\right )}{2 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x^2)/(1 - 6*x^2 + 4*x^4),x]

[Out]

(-Log[-1 + Sqrt[10]*x - 2*x^2] + Log[1 + Sqrt[10]*x + 2*x^2])/(2*Sqrt[10])

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Maple [B]  time = 0.055, size = 82, normalized size = 1.7 \begin{align*}{\frac{ \left ( -2+2\,\sqrt{5} \right ) \sqrt{5}}{10\,\sqrt{10}-10\,\sqrt{2}}{\it Artanh} \left ( 8\,{\frac{x}{2\,\sqrt{10}-2\,\sqrt{2}}} \right ) }+{\frac{ \left ( 2+2\,\sqrt{5} \right ) \sqrt{5}}{10\,\sqrt{10}+10\,\sqrt{2}}{\it Artanh} \left ( 8\,{\frac{x}{2\,\sqrt{10}+2\,\sqrt{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^2+1)/(4*x^4-6*x^2+1),x)

[Out]

2/5*(5^(1/2)-1)*5^(1/2)/(2*10^(1/2)-2*2^(1/2))*arctanh(8*x/(2*10^(1/2)-2*2^(1/2)))+2/5*(5^(1/2)+1)*5^(1/2)/(2*
10^(1/2)+2*2^(1/2))*arctanh(8*x/(2*10^(1/2)+2*2^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{2 \, x^{2} - 1}{4 \, x^{4} - 6 \, x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+1)/(4*x^4-6*x^2+1),x, algorithm="maxima")

[Out]

-integrate((2*x^2 - 1)/(4*x^4 - 6*x^2 + 1), x)

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Fricas [A]  time = 1.27381, size = 116, normalized size = 2.42 \begin{align*} \frac{1}{20} \, \sqrt{10} \log \left (\frac{4 \, x^{4} + 14 \, x^{2} + 2 \, \sqrt{10}{\left (2 \, x^{3} + x\right )} + 1}{4 \, x^{4} - 6 \, x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+1)/(4*x^4-6*x^2+1),x, algorithm="fricas")

[Out]

1/20*sqrt(10)*log((4*x^4 + 14*x^2 + 2*sqrt(10)*(2*x^3 + x) + 1)/(4*x^4 - 6*x^2 + 1))

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Sympy [A]  time = 0.10442, size = 46, normalized size = 0.96 \begin{align*} - \frac{\sqrt{10} \log{\left (x^{2} - \frac{\sqrt{10} x}{2} + \frac{1}{2} \right )}}{20} + \frac{\sqrt{10} \log{\left (x^{2} + \frac{\sqrt{10} x}{2} + \frac{1}{2} \right )}}{20} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**2+1)/(4*x**4-6*x**2+1),x)

[Out]

-sqrt(10)*log(x**2 - sqrt(10)*x/2 + 1/2)/20 + sqrt(10)*log(x**2 + sqrt(10)*x/2 + 1/2)/20

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Giac [A]  time = 1.18294, size = 104, normalized size = 2.17 \begin{align*} \frac{1}{20} \, \sqrt{10} \log \left ({\left | x + \frac{1}{4} \, \sqrt{10} + \frac{1}{4} \, \sqrt{2} \right |}\right ) + \frac{1}{20} \, \sqrt{10} \log \left ({\left | x + \frac{1}{4} \, \sqrt{10} - \frac{1}{4} \, \sqrt{2} \right |}\right ) - \frac{1}{20} \, \sqrt{10} \log \left ({\left | x - \frac{1}{4} \, \sqrt{10} + \frac{1}{4} \, \sqrt{2} \right |}\right ) - \frac{1}{20} \, \sqrt{10} \log \left ({\left | x - \frac{1}{4} \, \sqrt{10} - \frac{1}{4} \, \sqrt{2} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+1)/(4*x^4-6*x^2+1),x, algorithm="giac")

[Out]

1/20*sqrt(10)*log(abs(x + 1/4*sqrt(10) + 1/4*sqrt(2))) + 1/20*sqrt(10)*log(abs(x + 1/4*sqrt(10) - 1/4*sqrt(2))
) - 1/20*sqrt(10)*log(abs(x - 1/4*sqrt(10) + 1/4*sqrt(2))) - 1/20*sqrt(10)*log(abs(x - 1/4*sqrt(10) - 1/4*sqrt
(2)))