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

Optimal. Leaf size=50 \[ \frac {\log \left (2 x^2+\sqrt {5} x+1\right )}{2 \sqrt {5}}-\frac {\log \left (2 x^2-\sqrt {5} x+1\right )}{2 \sqrt {5}} \]

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Rubi [A]  time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\log \left (2 x^2+\sqrt {5} x+1\right )}{2 \sqrt {5}}-\frac {\log \left (2 x^2-\sqrt {5} x+1\right )}{2 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x^2)/(1 - x^2 + 4*x^4),x]

[Out]

-Log[1 - Sqrt[5]*x + 2*x^2]/(2*Sqrt[5]) + Log[1 + Sqrt[5]*x + 2*x^2]/(2*Sqrt[5])

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]

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]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 42, normalized size = 0.84 \begin {gather*} \frac {\log \left (2 x^2+\sqrt {5} x+1\right )-\log \left (-2 x^2+\sqrt {5} x-1\right )}{2 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-Log[-1 + Sqrt[5]*x - 2*x^2] + Log[1 + Sqrt[5]*x + 2*x^2])/(2*Sqrt[5])

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-2 x^2}{1-x^2+4 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.99, size = 45, normalized size = 0.90 \begin {gather*} \frac {1}{10} \, \sqrt {5} \log \left (\frac {4 \, x^{4} + 9 \, x^{2} + 2 \, \sqrt {5} {\left (2 \, x^{3} + x\right )} + 1}{4 \, x^{4} - x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*sqrt(5)*log((4*x^4 + 9*x^2 + 2*sqrt(5)*(2*x^3 + x) + 1)/(4*x^4 - x^2 + 1))

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giac [A]  time = 0.24, size = 41, normalized size = 0.82 \begin {gather*} \frac {1}{10} \, \sqrt {5} \log \left (x^{2} + \frac {1}{2} \, \sqrt {10} \left (\frac {1}{4}\right )^{\frac {1}{4}} x + \frac {1}{2}\right ) - \frac {1}{10} \, \sqrt {5} \log \left (x^{2} - \frac {1}{2} \, \sqrt {10} \left (\frac {1}{4}\right )^{\frac {1}{4}} x + \frac {1}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*sqrt(5)*log(x^2 + 1/2*sqrt(10)*(1/4)^(1/4)*x + 1/2) - 1/10*sqrt(5)*log(x^2 - 1/2*sqrt(10)*(1/4)^(1/4)*x +
1/2)

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maple [A]  time = 0.01, size = 39, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {5}\, \ln \left (2 x^{2}-\sqrt {5}\, x +1\right )}{10}+\frac {\sqrt {5}\, \ln \left (2 x^{2}+\sqrt {5}\, x +1\right )}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {2 \, x^{2} - 1}{4 \, x^{4} - x^{2} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.35, size = 20, normalized size = 0.40 \begin {gather*} \frac {\sqrt {5}\,\mathrm {atanh}\left (\frac {\sqrt {5}\,x}{2\,x^2+1}\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x^2 - 1)/(4*x^4 - x^2 + 1),x)

[Out]

(5^(1/2)*atanh((5^(1/2)*x)/(2*x^2 + 1)))/5

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sympy [A]  time = 0.12, size = 46, normalized size = 0.92 \begin {gather*} - \frac {\sqrt {5} \log {\left (x^{2} - \frac {\sqrt {5} x}{2} + \frac {1}{2} \right )}}{10} + \frac {\sqrt {5} \log {\left (x^{2} + \frac {\sqrt {5} x}{2} + \frac {1}{2} \right )}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(5)*log(x**2 - sqrt(5)*x/2 + 1/2)/10 + sqrt(5)*log(x**2 + sqrt(5)*x/2 + 1/2)/10

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