Integrand size = 20, antiderivative size = 85 \[ \int \frac {x}{\left (-1+x^3\right ) \left (-1+2 x^3\right )^{2/3}} \, dx=\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+2 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-1+2 x^3}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+2 x^3}+\left (-1+2 x^3\right )^{2/3}\right ) \]
1/3*arctan(3^(1/2)*x/(x+2*(2*x^3-1)^(1/3)))*3^(1/2)+1/3*ln(-x+(2*x^3-1)^(1 /3))-1/6*ln(x^2+x*(2*x^3-1)^(1/3)+(2*x^3-1)^(2/3))
Time = 0.17 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (-1+x^3\right ) \left (-1+2 x^3\right )^{2/3}} \, dx=\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+2 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-1+2 x^3}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+2 x^3}+\left (-1+2 x^3\right )^{2/3}\right ) \]
ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + 2*x^3)^(1/3))]/Sqrt[3] + Log[-x + (-1 + 2* x^3)^(1/3)]/3 - Log[x^2 + x*(-1 + 2*x^3)^(1/3) + (-1 + 2*x^3)^(2/3)]/6
Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {992}
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.
\(\displaystyle \int \frac {x}{\left (x^3-1\right ) \left (2 x^3-1\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 992 |
\(\displaystyle \frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{2 x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (x^3-1\right )+\frac {1}{2} \log \left (x-\sqrt [3]{2 x^3-1}\right )\) |
ArcTan[(1 + (2*x)/(-1 + 2*x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[-1 + x^3]/6 + Log[x - (-1 + 2*x^3)^(1/3)]/2
3.12.36.3.1 Defintions of rubi rules used
Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[-ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3 ))/Sqrt[3]]/(Sqrt[3]*c*q^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c* q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Time = 3.50 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {\ln \left (\frac {-x +\left (2 x^{3}-1\right )^{\frac {1}{3}}}{x}\right )}{3}-\frac {\ln \left (\frac {x^{2}+x \left (2 x^{3}-1\right )^{\frac {1}{3}}+\left (2 x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (2 x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right )}{3}\) | \(80\) |
trager | \(\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}-1\right )^{\frac {2}{3}} x -15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-\left (2 x^{3}-1\right )^{\frac {2}{3}} x -\left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+2 x^{3}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{\left (x -1\right ) \left (x^{2}+x +1\right )}\right )-\frac {\ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-3 \left (2 x^{3}-1\right )^{\frac {2}{3}} x -3 \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-3 x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{\left (x -1\right ) \left (x^{2}+x +1\right )}\right )}{3}-\ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-3 \left (2 x^{3}-1\right )^{\frac {2}{3}} x -3 \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-3 x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{\left (x -1\right ) \left (x^{2}+x +1\right )}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(363\) |
1/3*ln((-x+(2*x^3-1)^(1/3))/x)-1/6*ln((x^2+x*(2*x^3-1)^(1/3)+(2*x^3-1)^(2/ 3))/x^2)-1/3*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(2*x^3-1)^(1/3)))
Time = 0.51 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.22 \[ \int \frac {x}{\left (-1+x^3\right ) \left (-1+2 x^3\right )^{2/3}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 2 \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (2 \, x^{3} - 1\right )}}{10 \, x^{3} - 1}\right ) + \frac {1}{6} \, \log \left (\frac {x^{3} + 3 \, {\left (2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{x^{3} - 1}\right ) \]
1/3*sqrt(3)*arctan(-(4*sqrt(3)*(2*x^3 - 1)^(1/3)*x^2 - 2*sqrt(3)*(2*x^3 - 1)^(2/3)*x + sqrt(3)*(2*x^3 - 1))/(10*x^3 - 1)) + 1/6*log((x^3 + 3*(2*x^3 - 1)^(1/3)*x^2 - 3*(2*x^3 - 1)^(2/3)*x - 1)/(x^3 - 1))
\[ \int \frac {x}{\left (-1+x^3\right ) \left (-1+2 x^3\right )^{2/3}} \, dx=\int \frac {x}{\left (x - 1\right ) \left (2 x^{3} - 1\right )^{\frac {2}{3}} \left (x^{2} + x + 1\right )}\, dx \]
\[ \int \frac {x}{\left (-1+x^3\right ) \left (-1+2 x^3\right )^{2/3}} \, dx=\int { \frac {x}{{\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}} \,d x } \]
\[ \int \frac {x}{\left (-1+x^3\right ) \left (-1+2 x^3\right )^{2/3}} \, dx=\int { \frac {x}{{\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {x}{\left (-1+x^3\right ) \left (-1+2 x^3\right )^{2/3}} \, dx=\int \frac {x}{\left (x^3-1\right )\,{\left (2\,x^3-1\right )}^{2/3}} \,d x \]