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

Optimal. Leaf size=51 \[ \frac{1}{10} \left (5-4 \sqrt{5}\right ) \log \left (-x-\sqrt{5}+2\right )+\frac{1}{10} \left (5+4 \sqrt{5}\right ) \log \left (-x+\sqrt{5}+2\right ) \]

[Out]

((5 - 4*Sqrt[5])*Log[2 - Sqrt[5] - x])/10 + ((5 + 4*Sqrt[5])*Log[2 + Sqrt[5] - x])/10

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Rubi [A]  time = 0.0158961, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {632, 31} \[ \frac{1}{10} \left (5-4 \sqrt{5}\right ) \log \left (-x-\sqrt{5}+2\right )+\frac{1}{10} \left (5+4 \sqrt{5}\right ) \log \left (-x+\sqrt{5}+2\right ) \]

Antiderivative was successfully verified.

[In]

Int[(2 + x)/(-1 - 4*x + x^2),x]

[Out]

((5 - 4*Sqrt[5])*Log[2 - Sqrt[5] - x])/10 + ((5 + 4*Sqrt[5])*Log[2 + Sqrt[5] - x])/10

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{2+x}{-1-4 x+x^2} \, dx &=-\left (\frac{1}{10} \left (-5+4 \sqrt{5}\right ) \int \frac{1}{-2+\sqrt{5}+x} \, dx\right )+\frac{1}{10} \left (5+4 \sqrt{5}\right ) \int \frac{1}{-2-\sqrt{5}+x} \, dx\\ &=\frac{1}{10} \left (5-4 \sqrt{5}\right ) \log \left (2-\sqrt{5}-x\right )+\frac{1}{10} \left (5+4 \sqrt{5}\right ) \log \left (2+\sqrt{5}-x\right )\\ \end{align*}

Mathematica [A]  time = 0.0256704, size = 47, normalized size = 0.92 \[ \frac{1}{10} \left (5+4 \sqrt{5}\right ) \log \left (-x+\sqrt{5}+2\right )+\frac{1}{10} \left (5-4 \sqrt{5}\right ) \log \left (x+\sqrt{5}-2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + x)/(-1 - 4*x + x^2),x]

[Out]

((5 + 4*Sqrt[5])*Log[2 + Sqrt[5] - x])/10 + ((5 - 4*Sqrt[5])*Log[-2 + Sqrt[5] + x])/10

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Maple [A]  time = 0.002, size = 29, normalized size = 0.6 \begin{align*}{\frac{\ln \left ({x}^{2}-4\,x-1 \right ) }{2}}-{\frac{4\,\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 2\,x-4 \right ) \sqrt{5}}{10}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*ln(x^2-4*x-1)-4/5*5^(1/2)*arctanh(1/10*(2*x-4)*5^(1/2))

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Maxima [A]  time = 1.48026, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{5} \, \sqrt{5} \log \left (\frac{x - \sqrt{5} - 2}{x + \sqrt{5} - 2}\right ) + \frac{1}{2} \, \log \left (x^{2} - 4 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.81323, size = 128, normalized size = 2.51 \begin{align*} \frac{2}{5} \, \sqrt{5} \log \left (\frac{x^{2} - 2 \, \sqrt{5}{\left (x - 2\right )} - 4 \, x + 9}{x^{2} - 4 \, x - 1}\right ) + \frac{1}{2} \, \log \left (x^{2} - 4 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.104519, size = 42, normalized size = 0.82 \begin{align*} \left (\frac{1}{2} - \frac{2 \sqrt{5}}{5}\right ) \log{\left (x - 2 + \sqrt{5} \right )} + \left (\frac{1}{2} + \frac{2 \sqrt{5}}{5}\right ) \log{\left (x - \sqrt{5} - 2 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.04397, size = 59, normalized size = 1.16 \begin{align*} \frac{2}{5} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{5} - 4 \right |}}{{\left | 2 \, x + 2 \, \sqrt{5} - 4 \right |}}\right ) + \frac{1}{2} \, \log \left ({\left | x^{2} - 4 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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