Integrand size = 27, antiderivative size = 174 \[ \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx=-\frac {\left (-1+x^3\right )^{2/3}}{x^2}-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^3}}\right )+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \log \left (x+\sqrt [3]{-1+x^3}\right )+\frac {1}{4} \log \left (x^2-x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )+\frac {1}{12} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
-(x^3-1)^(2/3)/x^2-1/2*arctan(3^(1/2)*x/(-x+2*(x^3-1)^(1/3)))*3^(1/2)+1/6* arctan(3^(1/2)*x/(x+2*(x^3-1)^(1/3)))*3^(1/2)-1/6*ln(-x+(x^3-1)^(1/3))-1/2 *ln(x+(x^3-1)^(1/3))+1/4*ln(x^2-x*(x^3-1)^(1/3)+(x^3-1)^(2/3))+1/12*ln(x^2 +x*(x^3-1)^(1/3)+(x^3-1)^(2/3))
Time = 0.26 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.93 \[ \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx=\frac {1}{12} \left (-\frac {12 \left (-1+x^3\right )^{2/3}}{x^2}+6 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{-1+x^3}}\right )+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )-2 \log \left (-x+\sqrt [3]{-1+x^3}\right )-6 \log \left (x+\sqrt [3]{-1+x^3}\right )+3 \log \left (x^2-x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )+\log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
((-12*(-1 + x^3)^(2/3))/x^2 + 6*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(-1 + x^ 3)^(1/3))] + 2*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))] - 2*Lo g[-x + (-1 + x^3)^(1/3)] - 6*Log[x + (-1 + x^3)^(1/3)] + 3*Log[x^2 - x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)] + Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x ^3)^(2/3)])/12
Time = 0.26 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.75, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1050, 27, 1026, 769, 901}
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.
| \(\displaystyle \int \frac {\left (x^3-2\right ) \left (x^3-1\right )^{2/3}}{x^3 \left (2 x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 1050 |
| \(\displaystyle -\frac {1}{2} \int \frac {2 \left (x^3+1\right )}{\left (1-2 x^3\right ) \sqrt [3]{x^3-1}}dx-\frac {\left (x^3-1\right )^{2/3}}{x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\int \frac {x^3+1}{\left (1-2 x^3\right ) \sqrt [3]{x^3-1}}dx-\frac {\left (x^3-1\right )^{2/3}}{x^2}\) |
| \(\Big \downarrow \) 1026 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt [3]{x^3-1}}dx-\frac {3}{2} \int \frac {1}{\left (1-2 x^3\right ) \sqrt [3]{x^3-1}}dx-\frac {\left (x^3-1\right )^{2/3}}{x^2}\) |
| \(\Big \downarrow \) 769 |
| \(\displaystyle -\frac {3}{2} \int \frac {1}{\left (1-2 x^3\right ) \sqrt [3]{x^3-1}}dx+\frac {1}{2} \left (\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )\right )-\frac {\left (x^3-1\right )^{2/3}}{x^2}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle -\frac {3}{2} \left (-\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (1-2 x^3\right )+\frac {1}{2} \log \left (-\sqrt [3]{x^3-1}-x\right )\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )\right )-\frac {\left (x^3-1\right )^{2/3}}{x^2}\) |
-((-1 + x^3)^(2/3)/x^2) - (3*(-(ArcTan[(1 - (2*x)/(-1 + x^3)^(1/3))/Sqrt[3 ]]/Sqrt[3]) - Log[1 - 2*x^3]/6 + Log[-x - (-1 + x^3)^(1/3)]/2))/2 + (ArcTa n[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[-x + (-1 + x^3)^(1/3 )]/2)/2
3.23.85.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* (x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S qrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)* (x_)^(n_)), x_Symbol] :> Simp[f/d Int[(a + b*x^n)^p, x], x] + Simp[(d*e - c*f)/d Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, p, n}, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^n*(m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1 ))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n, 0] && G tQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])
Time = 16.86 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.99
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}-6 \sqrt {3}\, \arctan \left (\frac {\left (x -2 \left (x^{3}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x^{2}+\ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}-2 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{2}+3 \ln \left (\frac {x^{2}-x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}-6 \ln \left (\frac {x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{2}-12 \left (x^{3}-1\right )^{\frac {2}{3}}}{12 x^{2}}\) | \(172\) |
| risch | \(\text {Expression too large to display}\) | \(1191\) |
| trager | \(\text {Expression too large to display}\) | \(1241\) |
1/12*(-2*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^3-1)^(1/3)))*x^2-6*3^(1/2)*a rctan(1/3*(x-2*(x^3-1)^(1/3))*3^(1/2)/x)*x^2+ln((x^2+x*(x^3-1)^(1/3)+(x^3- 1)^(2/3))/x^2)*x^2-2*ln((-x+(x^3-1)^(1/3))/x)*x^2+3*ln((x^2-x*(x^3-1)^(1/3 )+(x^3-1)^(2/3))/x^2)*x^2-6*ln((x+(x^3-1)^(1/3))/x)*x^2-12*(x^3-1)^(2/3))/ x^2
Time = 4.19 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.23 \[ \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx=\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {383838 \, \sqrt {3} {\left (x^{10} - 3 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 13468 \, \sqrt {3} {\left (x^{11} - 3 \, x^{8} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (198653 \, x^{12} + 393594 \, x^{9} + 5568 \, x^{6} - 400090 \, x^{3} - 198189\right )}}{3 \, {\left (185185 \, x^{12} + 370434 \, x^{9} - 96 \, x^{6} - 370322 \, x^{3} - 185193\right )}}\right ) - x^{2} \log \left (\frac {8 \, x^{9} - 12 \, x^{6} + 6 \, x^{3} - 3 \, {\left (x^{10} - 3 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 3 \, {\left (x^{11} - 3 \, x^{8} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1}{8 \, x^{9} - 12 \, x^{6} + 6 \, x^{3} - 1}\right ) - 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{12 \, x^{2}} \]
1/12*(2*sqrt(3)*x^2*arctan(1/3*(383838*sqrt(3)*(x^10 - 3*x^4 - 2*x)*(x^3 - 1)^(2/3) + 13468*sqrt(3)*(x^11 - 3*x^8 + 4*x^2)*(x^3 - 1)^(1/3) + sqrt(3) *(198653*x^12 + 393594*x^9 + 5568*x^6 - 400090*x^3 - 198189))/(185185*x^12 + 370434*x^9 - 96*x^6 - 370322*x^3 - 185193)) - x^2*log((8*x^9 - 12*x^6 + 6*x^3 - 3*(x^10 - 3*x^4 - 2*x)*(x^3 - 1)^(2/3) + 3*(x^11 - 3*x^8 + 4*x^2) *(x^3 - 1)^(1/3) - 1)/(8*x^9 - 12*x^6 + 6*x^3 - 1)) - 12*(x^3 - 1)^(2/3))/ x^2
\[ \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} - 2\right )}{x^{3} \cdot \left (2 x^{3} - 1\right )}\, dx \]
\[ \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (2 \, x^{3} - 1\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (2 \, x^{3} - 1\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^3-2\right )}{x^3\,\left (2\,x^3-1\right )} \,d x \]