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

Optimal. Leaf size=49 \[ \frac{1}{10} \left (5-\sqrt{5}\right ) \log \left (2 x-\sqrt{5}+1\right )+\frac{1}{10} \left (5+\sqrt{5}\right ) \log \left (2 x+\sqrt{5}+1\right ) \]

[Out]

((5 - Sqrt[5])*Log[1 - Sqrt[5] + 2*x])/10 + ((5 + Sqrt[5])*Log[1 + Sqrt[5] + 2*x])/10

________________________________________________________________________________________

Rubi [A]  time = 0.0117783, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {632, 31} \[ \frac{1}{10} \left (5-\sqrt{5}\right ) \log \left (2 x-\sqrt{5}+1\right )+\frac{1}{10} \left (5+\sqrt{5}\right ) \log \left (2 x+\sqrt{5}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(-1 + x + x^2),x]

[Out]

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

Mathematica [A]  time = 0.0169402, size = 44, normalized size = 0.9 \[ \frac{1}{10} \left (\left (5+\sqrt{5}\right ) \log \left (2 x+\sqrt{5}+1\right )-\left (\sqrt{5}-5\right ) \log \left (-2 x+\sqrt{5}-1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(-1 + x + x^2),x]

[Out]

(-((-5 + Sqrt[5])*Log[-1 + Sqrt[5] - 2*x]) + (5 + Sqrt[5])*Log[1 + Sqrt[5] + 2*x])/10

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 27, normalized size = 0.6 \begin{align*}{\frac{\ln \left ({x}^{2}+x-1 \right ) }{2}}+{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{5}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.40364, size = 50, normalized size = 1.02 \begin{align*} -\frac{1}{10} \, \sqrt{5} \log \left (\frac{2 \, x - \sqrt{5} + 1}{2 \, x + \sqrt{5} + 1}\right ) + \frac{1}{2} \, \log \left (x^{2} + x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.89853, size = 127, normalized size = 2.59 \begin{align*} \frac{1}{10} \, \sqrt{5} \log \left (\frac{2 \, x^{2} + \sqrt{5}{\left (2 \, x + 1\right )} + 2 \, x + 3}{x^{2} + x - 1}\right ) + \frac{1}{2} \, \log \left (x^{2} + x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.106135, size = 46, normalized size = 0.94 \begin{align*} \left (\frac{\sqrt{5}}{10} + \frac{1}{2}\right ) \log{\left (x + \frac{1}{2} + \frac{\sqrt{5}}{2} \right )} + \left (\frac{1}{2} - \frac{\sqrt{5}}{10}\right ) \log{\left (x - \frac{\sqrt{5}}{2} + \frac{1}{2} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.04988, size = 54, normalized size = 1.1 \begin{align*} -\frac{1}{10} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x - \sqrt{5} + 1 \right |}}{{\left | 2 \, x + \sqrt{5} + 1 \right |}}\right ) + \frac{1}{2} \, \log \left ({\left | x^{2} + x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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