3.3.16 \(\int \frac {x}{-1+x+x^2} \, dx\) [216]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {646, 31} \begin {gather*} \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 ) \end {gather*}

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 31

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

Rule 646

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]

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*}

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Mathematica [A]
time = 0.01, size = 44, normalized size = 0.90 \begin {gather*} \frac {1}{10} \left (-\left (\left (-5+\sqrt {5}\right ) \log \left (-1+\sqrt {5}-2 x\right )\right )+\left (5+\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right )\right ) \end {gather*}

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

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Maple [A]
time = 0.09, size = 27, normalized size = 0.55

method	result	size
default	\(\frac {\ln \left (x^{2}+x -1\right )}{2}+\frac {\sqrt {5}\, \arctanh \left (\frac {\left (1+2 x \right ) \sqrt {5}}{5}\right )}{5}\)	\(27\)
risch	\(\frac {\ln \left (2 x +\sqrt {5}+1\right )}{2}+\frac {\ln \left (2 x +\sqrt {5}+1\right ) \sqrt {5}}{10}+\frac {\ln \left (2 x +1-\sqrt {5}\right )}{2}-\frac {\ln \left (2 x +1-\sqrt {5}\right ) \sqrt {5}}{10}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2+x-1),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.55, size = 37, normalized size = 0.76 \begin {gather*} -\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 {gather*}

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)

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Fricas [A]
time = 0.72, size = 44, normalized size = 0.90 \begin {gather*} \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 {gather*}

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)

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Sympy [A]
time = 0.04, size = 46, normalized size = 0.94 \begin {gather*} \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 {gather*}

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)

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Giac [A]
time = 0.45, size = 40, normalized size = 0.82 \begin {gather*} -\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 {gather*}

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))

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Mupad [B]
time = 0.18, size = 36, normalized size = 0.73 \begin {gather*} \ln \left (x+\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{10}+\frac {1}{2}\right )-\ln \left (x-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{10}-\frac {1}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x + x^2 - 1),x)

[Out]

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

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