Integrand size = 7, antiderivative size = 57 \[ \int \frac {1}{\left (-1+x^3\right )^2} \, dx=\frac {x}{3 \left (1-x^3\right )}+\frac {2 \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2}{9} \log (1-x)+\frac {1}{9} \log \left (1+x+x^2\right ) \]
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (-1+x^3\right )^2} \, dx=\frac {1}{9} \left (-\frac {3 x}{-1+x^3}+2 \sqrt {3} \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )-2 \log (1-x)+\log \left (1+x+x^2\right )\right ) \]
((-3*x)/(-1 + x^3) + 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 2*Log[1 - x] + Log[1 + x + x^2])/9
Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {749, 750, 16, 25, 1142, 1083, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (x^3-1\right )^2} \, dx\) |
\(\Big \downarrow \) 749 |
\(\displaystyle \frac {x}{3 \left (1-x^3\right )}-\frac {2}{3} \int \frac {1}{x^3-1}dx\) |
\(\Big \downarrow \) 750 |
\(\displaystyle \frac {x}{3 \left (1-x^3\right )}-\frac {2}{3} \left (\frac {1}{3} \int -\frac {x+2}{x^2+x+1}dx+\frac {1}{3} \int \frac {1}{x-1}dx\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {x}{3 \left (1-x^3\right )}-\frac {2}{3} \left (\frac {1}{3} \int -\frac {x+2}{x^2+x+1}dx+\frac {1}{3} \log (1-x)\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {x}{3 \left (1-x^3\right )}-\frac {2}{3} \left (\frac {1}{3} \log (1-x)-\frac {1}{3} \int \frac {x+2}{x^2+x+1}dx\right )\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {x}{3 \left (1-x^3\right )}-\frac {2}{3} \left (\frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{x^2+x+1}dx-\frac {1}{2} \int \frac {2 x+1}{x^2+x+1}dx\right )+\frac {1}{3} \log (1-x)\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {x}{3 \left (1-x^3\right )}-\frac {2}{3} \left (\frac {1}{3} \left (3 \int \frac {1}{-(2 x+1)^2-3}d(2 x+1)-\frac {1}{2} \int \frac {2 x+1}{x^2+x+1}dx\right )+\frac {1}{3} \log (1-x)\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {x}{3 \left (1-x^3\right )}-\frac {2}{3} \left (\frac {1}{3} \left (-\frac {1}{2} \int \frac {2 x+1}{x^2+x+1}dx-\sqrt {3} \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \log (1-x)\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {x}{3 \left (1-x^3\right )}-\frac {2}{3} \left (\frac {1}{3} \left (-\sqrt {3} \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (x^2+x+1\right )\right )+\frac {1}{3} \log (1-x)\right )\) |
x/(3*(1 - x^3)) - (2*(Log[1 - x]/3 + (-(Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]]) - Log[1 + x + x^2]/2)/3))/3
3.2.51.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(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] && (Inte gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[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]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.72
method | result | size |
risch | \(-\frac {x}{3 \left (x^{3}-1\right )}-\frac {2 \ln \left (-1+x \right )}{9}+\frac {\ln \left (x^{2}+x +1\right )}{9}+\frac {2 \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{9}\) | \(41\) |
default | \(-\frac {1}{9 \left (-1+x \right )}-\frac {2 \ln \left (-1+x \right )}{9}+\frac {-1+x}{9 x^{2}+9 x +9}+\frac {\ln \left (x^{2}+x +1\right )}{9}+\frac {2 \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{9}\) | \(53\) |
meijerg | \(-\frac {\left (-1\right )^{\frac {2}{3}} \left (\frac {3 x \left (-1\right )^{\frac {1}{3}}}{-3 x^{3}+3}-\frac {2 x \left (-1\right )^{\frac {1}{3}} \left (\ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2+\left (x^{3}\right )^{\frac {1}{3}}}\right )\right )}{3 \left (x^{3}\right )^{\frac {1}{3}}}\right )}{3}\) | \(86\) |
Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (-1+x^3\right )^2} \, dx=\frac {2 \, \sqrt {3} {\left (x^{3} - 1\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + {\left (x^{3} - 1\right )} \log \left (x^{2} + x + 1\right ) - 2 \, {\left (x^{3} - 1\right )} \log \left (x - 1\right ) - 3 \, x}{9 \, {\left (x^{3} - 1\right )}} \]
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)
Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (-1+x^3\right )^2} \, dx=- \frac {x}{3 x^{3} - 3} - \frac {2 \log {\left (x - 1 \right )}}{9} + \frac {\log {\left (x^{2} + x + 1 \right )}}{9} + \frac {2 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{9} \]
-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
Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (-1+x^3\right )^2} \, dx=\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {x}{3 \, {\left (x^{3} - 1\right )}} + \frac {1}{9} \, \log \left (x^{2} + x + 1\right ) - \frac {2}{9} \, \log \left (x - 1\right ) \]
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)
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (-1+x^3\right )^2} \, dx=\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {x}{3 \, {\left (x^{3} - 1\right )}} + \frac {1}{9} \, \log \left (x^{2} + x + 1\right ) - \frac {2}{9} \, \log \left ({\left | x - 1 \right |}\right ) \]
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))
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (-1+x^3\right )^2} \, dx=-\frac {2\,\ln \left (x-1\right )}{9}-\frac {x}{3\,\left (x^3-1\right )}-\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )+\ln \left (2\,x+1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right ) \]
log(2*x + 3^(1/2)*1i + 1)*((3^(1/2)*1i)/9 + 1/9) - x/(3*(x^3 - 1)) - log(x - (3^(1/2)*1i)/2 + 1/2)*((3^(1/2)*1i)/9 - 1/9) - (2*log(x - 1))/9
Time = 0.00 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\left (-1+x^3\right )^2} \, dx=\frac {2 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) x^{3}-2 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right )+\mathrm {log}\left (x^{2}+x +1\right ) x^{3}-\mathrm {log}\left (x^{2}+x +1\right )-2 \,\mathrm {log}\left (x -1\right ) x^{3}+2 \,\mathrm {log}\left (x -1\right )-3 x}{9 x^{3}-9} \]