3.151 \(\int \frac{1}{(-1+x^3)^2} \, dx\)

Optimal. Leaf size=57 \[ \frac{x}{3 \left (1-x^3\right )}+\frac{1}{9} \log \left (x^2+x+1\right )-\frac{2}{9} \log (1-x)+\frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

x/(3*(1 - x^3)) + (2*ArcTan[(1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) - (2*Log[1 - x])/9 + Log[1 + x + x^2]/9

________________________________________________________________________________________

Rubi [A]  time = 0.0244686, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {199, 200, 31, 634, 618, 204, 628} \[ \frac{x}{3 \left (1-x^3\right )}+\frac{1}{9} \log \left (x^2+x+1\right )-\frac{2}{9} \log (1-x)+\frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x^3)^(-2),x]

[Out]

x/(3*(1 - x^3)) + (2*ArcTan[(1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) - (2*Log[1 - x])/9 + Log[1 + x + x^2]/9

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

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

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 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 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 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]

Rubi steps

\begin{align*} \int \frac{1}{\left (-1+x^3\right )^2} \, dx &=\frac{x}{3 \left (1-x^3\right )}-\frac{2}{3} \int \frac{1}{-1+x^3} \, dx\\ &=\frac{x}{3 \left (1-x^3\right )}-\frac{2}{9} \int \frac{1}{-1+x} \, dx-\frac{2}{9} \int \frac{-2-x}{1+x+x^2} \, dx\\ &=\frac{x}{3 \left (1-x^3\right )}-\frac{2}{9} \log (1-x)+\frac{1}{9} \int \frac{1+2 x}{1+x+x^2} \, dx+\frac{1}{3} \int \frac{1}{1+x+x^2} \, dx\\ &=\frac{x}{3 \left (1-x^3\right )}-\frac{2}{9} \log (1-x)+\frac{1}{9} \log \left (1+x+x^2\right )-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\frac{x}{3 \left (1-x^3\right )}+\frac{2 \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{2}{9} \log (1-x)+\frac{1}{9} \log \left (1+x+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0231122, size = 49, normalized size = 0.86 \[ \frac{1}{9} \left (-\frac{3 x}{x^3-1}+\log \left (x^2+x+1\right )-2 \log (1-x)+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^3)^(-2),x]

[Out]

((-3*x)/(-1 + x^3) + 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 2*Log[1 - x] + Log[1 + x + x^2])/9

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 53, normalized size = 0.9 \begin{align*} -{\frac{1}{-9+9\,x}}-{\frac{2\,\ln \left ( -1+x \right ) }{9}}+{\frac{-1+x}{9\,{x}^{2}+9\,x+9}}+{\frac{\ln \left ({x}^{2}+x+1 \right ) }{9}}+{\frac{2\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3-1)^2,x)

[Out]

-1/9/(-1+x)-2/9*ln(-1+x)+1/9*(-1+x)/(x^2+x+1)+1/9*ln(x^2+x+1)+2/9*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.47755, size = 57, normalized size = 1. \begin{align*} \frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{x}{3 \,{\left (x^{3} - 1\right )}} + \frac{1}{9} \, \log \left (x^{2} + x + 1\right ) - \frac{2}{9} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^3-1)^2,x, algorithm="maxima")

[Out]

2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/3*x/(x^3 - 1) + 1/9*log(x^2 + x + 1) - 2/9*log(x - 1)

________________________________________________________________________________________

Fricas [A]  time = 2.09021, size = 171, normalized size = 3. \begin{align*} \frac{2 \, \sqrt{3}{\left (x^{3} - 1\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) +{\left (x^{3} - 1\right )} \log \left (x^{2} + x + 1\right ) - 2 \,{\left (x^{3} - 1\right )} \log \left (x - 1\right ) - 3 \, x}{9 \,{\left (x^{3} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^3-1)^2,x, algorithm="fricas")

[Out]

1/9*(2*sqrt(3)*(x^3 - 1)*arctan(1/3*sqrt(3)*(2*x + 1)) + (x^3 - 1)*log(x^2 + x + 1) - 2*(x^3 - 1)*log(x - 1) -
3*x)/(x^3 - 1)

________________________________________________________________________________________

Sympy [A]  time = 0.141168, size = 53, normalized size = 0.93 \begin{align*} - \frac{x}{3 x^{3} - 3} - \frac{2 \log{\left (x - 1 \right )}}{9} + \frac{\log{\left (x^{2} + x + 1 \right )}}{9} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**3-1)**2,x)

[Out]

-x/(3*x**3 - 3) - 2*log(x - 1)/9 + log(x**2 + x + 1)/9 + 2*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/9

________________________________________________________________________________________

Giac [A]  time = 1.04441, size = 58, normalized size = 1.02 \begin{align*} \frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{x}{3 \,{\left (x^{3} - 1\right )}} + \frac{1}{9} \, \log \left (x^{2} + x + 1\right ) - \frac{2}{9} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^3-1)^2,x, algorithm="giac")

[Out]

2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/3*x/(x^3 - 1) + 1/9*log(x^2 + x + 1) - 2/9*log(abs(x - 1))