3.308 \(\int \frac {1-x}{x (1+x^3)} \, dx\)

Optimal. Leaf size=42 \[ -\frac {1}{6} \log \left (x^2-x+1\right )+\log (x)-\frac {2}{3} \log (x+1)+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1834, 634, 618, 204, 628} \[ -\frac {1}{6} \log \left (x^2-x+1\right )+\log (x)-\frac {2}{3} \log (x+1)+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)/(x*(1 + x^3)),x]

[Out]

ArcTan[(1 - 2*x)/Sqrt[3]]/Sqrt[3] + Log[x] - (2*Log[1 + x])/3 - Log[1 - x + x^2]/6

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), 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 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {1-x}{x \left (1+x^3\right )} \, dx &=\int \left (\frac {1}{x}-\frac {2}{3 (1+x)}+\frac {-1-x}{3 \left (1-x+x^2\right )}\right ) \, dx\\ &=\log (x)-\frac {2}{3} \log (1+x)+\frac {1}{3} \int \frac {-1-x}{1-x+x^2} \, dx\\ &=\log (x)-\frac {2}{3} \log (1+x)-\frac {1}{6} \int \frac {-1+2 x}{1-x+x^2} \, dx-\frac {1}{2} \int \frac {1}{1-x+x^2} \, dx\\ &=\log (x)-\frac {2}{3} \log (1+x)-\frac {1}{6} \log \left (1-x+x^2\right )+\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {-1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\log (x)-\frac {2}{3} \log (1+x)-\frac {1}{6} \log \left (1-x+x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 53, normalized size = 1.26 \[ -\frac {1}{3} \log \left (x^3+1\right )+\frac {1}{6} \log \left (x^2-x+1\right )+\log (x)-\frac {1}{3} \log (x+1)-\frac {\tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)/(x*(1 + x^3)),x]

[Out]

-(ArcTan[(-1 + 2*x)/Sqrt[3]]/Sqrt[3]) + Log[x] - Log[1 + x]/3 + Log[1 - x + x^2]/6 - Log[1 + x^3]/3

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fricas [A]  time = 0.68, size = 36, normalized size = 0.86 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac {2}{3} \, \log \left (x + 1\right ) + \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)/x/(x^3+1),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 38, normalized size = 0.90 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac {2}{3} \, \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)/x/(x^3+1),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 37, normalized size = 0.88 \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}+\ln \relax (x )-\frac {2 \ln \left (x +1\right )}{3}-\frac {\ln \left (x^{2}-x +1\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)/x/(x^3+1),x)

[Out]

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

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maxima [A]  time = 2.88, size = 36, normalized size = 0.86 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac {2}{3} \, \log \left (x + 1\right ) + \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)/x/(x^3+1),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.96, size = 48, normalized size = 1.14 \[ \ln \relax (x)-\frac {2\,\ln \left (x+1\right )}{3}+\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 1)/(x*(x^3 + 1)),x)

[Out]

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

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sympy [A]  time = 0.21, size = 46, normalized size = 1.10 \[ \log {\relax (x )} - \frac {2 \log {\left (x + 1 \right )}}{3} - \frac {\log {\left (x^{2} - x + 1 \right )}}{6} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)/x/(x**3+1),x)

[Out]

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

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