3.66 \(\int \frac{1-2 x^2}{1-5 x^2+4 x^4} \, dx\)

Optimal. Leaf size=39 \[ -\frac{1}{6} \log (1-2 x)-\frac{1}{6} \log (1-x)+\frac{1}{6} \log (x+1)+\frac{1}{6} \log (2 x+1) \]

[Out]

-Log[1 - 2*x]/6 - Log[1 - x]/6 + Log[1 + x]/6 + Log[1 + 2*x]/6

________________________________________________________________________________________

Rubi [A]  time = 0.0171199, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1161, 616, 31} \[ -\frac{1}{6} \log (1-2 x)-\frac{1}{6} \log (1-x)+\frac{1}{6} \log (x+1)+\frac{1}{6} \log (2 x+1) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x^2)/(1 - 5*x^2 + 4*x^4),x]

[Out]

-Log[1 - 2*x]/6 - Log[1 - x]/6 + Log[1 + x]/6 + Log[1 + 2*x]/6

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 616

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0056717, size = 31, normalized size = 0.79 \[ \frac{1}{6} \log \left (2 x^2+3 x+1\right )-\frac{1}{6} \log \left (2 x^2-3 x+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x^2)/(1 - 5*x^2 + 4*x^4),x]

[Out]

-Log[1 - 3*x + 2*x^2]/6 + Log[1 + 3*x + 2*x^2]/6

________________________________________________________________________________________

Maple [A]  time = 0.052, size = 30, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 1+x \right ) }{6}}-{\frac{\ln \left ( 2\,x-1 \right ) }{6}}-{\frac{\ln \left ( -1+x \right ) }{6}}+{\frac{\ln \left ( 1+2\,x \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^2+1)/(4*x^4-5*x^2+1),x)

[Out]

1/6*ln(1+x)-1/6*ln(2*x-1)-1/6*ln(-1+x)+1/6*ln(1+2*x)

________________________________________________________________________________________

Maxima [A]  time = 0.960764, size = 39, normalized size = 1. \begin{align*} \frac{1}{6} \, \log \left (2 \, x + 1\right ) - \frac{1}{6} \, \log \left (2 \, x - 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) - \frac{1}{6} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+1)/(4*x^4-5*x^2+1),x, algorithm="maxima")

[Out]

1/6*log(2*x + 1) - 1/6*log(2*x - 1) + 1/6*log(x + 1) - 1/6*log(x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.37242, size = 72, normalized size = 1.85 \begin{align*} \frac{1}{6} \, \log \left (2 \, x^{2} + 3 \, x + 1\right ) - \frac{1}{6} \, \log \left (2 \, x^{2} - 3 \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+1)/(4*x^4-5*x^2+1),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.102137, size = 29, normalized size = 0.74 \begin{align*} - \frac{\log{\left (x^{2} - \frac{3 x}{2} + \frac{1}{2} \right )}}{6} + \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{1}{2} \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**2+1)/(4*x**4-5*x**2+1),x)

[Out]

-log(x**2 - 3*x/2 + 1/2)/6 + log(x**2 + 3*x/2 + 1/2)/6

________________________________________________________________________________________

Giac [A]  time = 1.13605, size = 45, normalized size = 1.15 \begin{align*} \frac{1}{6} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) - \frac{1}{6} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) + \frac{1}{6} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{6} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+1)/(4*x^4-5*x^2+1),x, algorithm="giac")

[Out]

1/6*log(abs(2*x + 1)) - 1/6*log(abs(2*x - 1)) + 1/6*log(abs(x + 1)) - 1/6*log(abs(x - 1))