3.903 \(\int \frac{-1+2 x}{1+8 x+4 x^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{1}{4} \left (1-\sqrt{3}\right ) \log \left (2 x-\sqrt{3}+2\right )+\frac{1}{4} \left (1+\sqrt{3}\right ) \log \left (2 x+\sqrt{3}+2\right ) \]

[Out]

((1 - Sqrt[3])*Log[2 - Sqrt[3] + 2*x])/4 + ((1 + Sqrt[3])*Log[2 + Sqrt[3] + 2*x])/4

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Rubi [A]  time = 0.0257659, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {632, 31} \[ \frac{1}{4} \left (1-\sqrt{3}\right ) \log \left (2 x-\sqrt{3}+2\right )+\frac{1}{4} \left (1+\sqrt{3}\right ) \log \left (2 x+\sqrt{3}+2\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + 2*x)/(1 + 8*x + 4*x^2),x]

[Out]

((1 - Sqrt[3])*Log[2 - Sqrt[3] + 2*x])/4 + ((1 + Sqrt[3])*Log[2 + Sqrt[3] + 2*x])/4

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[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}{1+8 x+4 x^2} \, dx &=-\left (\left (-1+\sqrt{3}\right ) \int \frac{1}{4-2 \sqrt{3}+4 x} \, dx\right )+\left (1+\sqrt{3}\right ) \int \frac{1}{4+2 \sqrt{3}+4 x} \, dx\\ &=\frac{1}{4} \left (1-\sqrt{3}\right ) \log \left (2-\sqrt{3}+2 x\right )+\frac{1}{4} \left (1+\sqrt{3}\right ) \log \left (2+\sqrt{3}+2 x\right )\\ \end{align*}

Mathematica [A]  time = 0.0207772, size = 44, normalized size = 0.9 \[ \frac{1}{4} \left (\left (1+\sqrt{3}\right ) \log \left (2 x+\sqrt{3}+2\right )-\left (\sqrt{3}-1\right ) \log \left (-2 x+\sqrt{3}-2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 2*x)/(1 + 8*x + 4*x^2),x]

[Out]

(-((-1 + Sqrt[3])*Log[-2 + Sqrt[3] - 2*x]) + (1 + Sqrt[3])*Log[2 + Sqrt[3] + 2*x])/4

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Maple [A]  time = 0.004, size = 31, normalized size = 0.6 \begin{align*}{\frac{\ln \left ( 4\,{x}^{2}+8\,x+1 \right ) }{4}}+{\frac{\sqrt{3}}{2}{\it Artanh} \left ({\frac{ \left ( 8\,x+8 \right ) \sqrt{3}}{12}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+2*x)/(4*x^2+8*x+1),x)

[Out]

1/4*ln(4*x^2+8*x+1)+1/2*3^(1/2)*arctanh(1/12*(8*x+8)*3^(1/2))

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Maxima [A]  time = 1.62243, size = 55, normalized size = 1.12 \begin{align*} -\frac{1}{4} \, \sqrt{3} \log \left (\frac{2 \, x - \sqrt{3} + 2}{2 \, x + \sqrt{3} + 2}\right ) + \frac{1}{4} \, \log \left (4 \, x^{2} + 8 \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)/(4*x^2+8*x+1),x, algorithm="maxima")

[Out]

-1/4*sqrt(3)*log((2*x - sqrt(3) + 2)/(2*x + sqrt(3) + 2)) + 1/4*log(4*x^2 + 8*x + 1)

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Fricas [A]  time = 1.5568, size = 136, normalized size = 2.78 \begin{align*} \frac{1}{4} \, \sqrt{3} \log \left (\frac{4 \, x^{2} + 4 \, \sqrt{3}{\left (x + 1\right )} + 8 \, x + 7}{4 \, x^{2} + 8 \, x + 1}\right ) + \frac{1}{4} \, \log \left (4 \, x^{2} + 8 \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)/(4*x^2+8*x+1),x, algorithm="fricas")

[Out]

1/4*sqrt(3)*log((4*x^2 + 4*sqrt(3)*(x + 1) + 8*x + 7)/(4*x^2 + 8*x + 1)) + 1/4*log(4*x^2 + 8*x + 1)

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Sympy [A]  time = 0.253882, size = 42, normalized size = 0.86 \begin{align*} \left (\frac{1}{4} - \frac{\sqrt{3}}{4}\right ) \log{\left (x - \frac{\sqrt{3}}{2} + 1 \right )} + \left (\frac{1}{4} + \frac{\sqrt{3}}{4}\right ) \log{\left (x + \frac{\sqrt{3}}{2} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)/(4*x**2+8*x+1),x)

[Out]

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

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Giac [A]  time = 1.30801, size = 62, normalized size = 1.27 \begin{align*} -\frac{1}{4} \, \sqrt{3} \log \left (\frac{{\left | 8 \, x - 4 \, \sqrt{3} + 8 \right |}}{{\left | 8 \, x + 4 \, \sqrt{3} + 8 \right |}}\right ) + \frac{1}{4} \, \log \left ({\left | 4 \, x^{2} + 8 \, x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)/(4*x^2+8*x+1),x, algorithm="giac")

[Out]

-1/4*sqrt(3)*log(abs(8*x - 4*sqrt(3) + 8)/abs(8*x + 4*sqrt(3) + 8)) + 1/4*log(abs(4*x^2 + 8*x + 1))