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

Optimal. Leaf size=14 \[ \frac{\tanh ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

[Out]

ArcTanh[Sqrt[2]*x]/Sqrt[2]

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Rubi [A]  time = 0.005621, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {28, 21, 206} \[ \frac{\tanh ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Sqrt[2]*x]/Sqrt[2]

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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

Mathematica [B]  time = 0.0074046, size = 32, normalized size = 2.29 \[ \frac{\log \left (2 x+\sqrt{2}\right )-\log \left (\sqrt{2}-2 x\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 12, normalized size = 0.9 \begin{align*}{\frac{{\it Artanh} \left ( x\sqrt{2} \right ) \sqrt{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctanh(x*2^(1/2))*2^(1/2)

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Maxima [B]  time = 1.49258, size = 34, normalized size = 2.43 \begin{align*} -\frac{1}{4} \, \sqrt{2} \log \left (\frac{2 \, x - \sqrt{2}}{2 \, x + \sqrt{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.2824, size = 76, normalized size = 5.43 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{2 \, x^{2} + 2 \, \sqrt{2} x + 1}{2 \, x^{2} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.091748, size = 32, normalized size = 2.29 \begin{align*} - \frac{\sqrt{2} \log{\left (x - \frac{\sqrt{2}}{2} \right )}}{4} + \frac{\sqrt{2} \log{\left (x + \frac{\sqrt{2}}{2} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*log(x - sqrt(2)/2)/4 + sqrt(2)*log(x + sqrt(2)/2)/4

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Giac [B]  time = 1.13631, size = 39, normalized size = 2.79 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left ({\left | x + \frac{1}{2} \, \sqrt{2} \right |}\right ) - \frac{1}{4} \, \sqrt{2} \log \left ({\left | x - \frac{1}{2} \, \sqrt{2} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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