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

Optimal. Leaf size=50 \[ \frac{\log \left (2 x^2+\sqrt{2} x+1\right )}{2 \sqrt{2}}-\frac{\log \left (2 x^2-\sqrt{2} x+1\right )}{2 \sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0230111, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1164, 628} \[ \frac{\log \left (2 x^2+\sqrt{2} x+1\right )}{2 \sqrt{2}}-\frac{\log \left (2 x^2-\sqrt{2} x+1\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1164

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*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/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[b^2
- 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [A]  time = 0.0126572, size = 42, normalized size = 0.84 \[ \frac{\log \left (2 x^2+\sqrt{2} x+1\right )-\log \left (-2 x^2+\sqrt{2} x-1\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.05, size = 39, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( 1+2\,{x}^{2}-x\sqrt{2} \right ) \sqrt{2}}{4}}+{\frac{\ln \left ( 1+2\,{x}^{2}+x\sqrt{2} \right ) \sqrt{2}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [A]  time = 1.35116, size = 111, normalized size = 2.22 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{4 \, x^{4} + 6 \, x^{2} + 2 \, \sqrt{2}{\left (2 \, x^{3} + x\right )} + 1}{4 \, x^{4} + 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+2*x^2+1),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*log((4*x^4 + 6*x^2 + 2*sqrt(2)*(2*x^3 + x) + 1)/(4*x^4 + 2*x^2 + 1))

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Sympy [A]  time = 0.101103, size = 46, normalized size = 0.92 \begin{align*} - \frac{\sqrt{2} \log{\left (x^{2} - \frac{\sqrt{2} x}{2} + \frac{1}{2} \right )}}{4} + \frac{\sqrt{2} \log{\left (x^{2} + \frac{\sqrt{2} x}{2} + \frac{1}{2} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{2 \, x^{2} - 1}{4 \, x^{4} + 2 \, 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+2*x^2+1),x, algorithm="giac")

[Out]

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