Integrand size = 29, antiderivative size = 101 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-2+7 x^3\right )}{10 x^5}+\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (x+\sqrt [3]{-1+x^3}\right )-\frac {1}{6} \log \left (x^2-x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
1/10*(x^3-1)^(2/3)*(7*x^3-2)/x^5+1/3*arctan(3^(1/2)*x/(-x+2*(x^3-1)^(1/3)) )*3^(1/2)+1/3*ln(x+(x^3-1)^(1/3))-1/6*ln(x^2-x*(x^3-1)^(1/3)+(x^3-1)^(2/3) )
Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\frac {1}{30} \left (\frac {3 \left (-1+x^3\right )^{2/3} \left (-2+7 x^3\right )}{x^5}-10 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{-1+x^3}}\right )+10 \log \left (x+\sqrt [3]{-1+x^3}\right )-5 \log \left (x^2-x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
((3*(-1 + x^3)^(2/3)*(-2 + 7*x^3))/x^5 - 10*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(-1 + x^3)^(1/3))] + 10*Log[x + (-1 + x^3)^(1/3)] - 5*Log[x^2 - x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/30
Time = 0.25 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1050, 25, 1053, 27, 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.
| \(\displaystyle \int \frac {\left (x^3-1\right )^{2/3} \left (3 x^3-1\right )}{x^6 \left (2 x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 1050 |
| \(\displaystyle -\frac {1}{5} \int -\frac {7-9 x^3}{x^3 \left (1-2 x^3\right ) \sqrt [3]{x^3-1}}dx-\frac {\left (x^3-1\right )^{2/3}}{5 x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \int \frac {7-9 x^3}{x^3 \left (1-2 x^3\right ) \sqrt [3]{x^3-1}}dx-\frac {\left (x^3-1\right )^{2/3}}{5 x^5}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{2} \int \frac {10}{\left (1-2 x^3\right ) \sqrt [3]{x^3-1}}dx+\frac {7 \left (x^3-1\right )^{2/3}}{2 x^2}\right )-\frac {\left (x^3-1\right )^{2/3}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (5 \int \frac {1}{\left (1-2 x^3\right ) \sqrt [3]{x^3-1}}dx+\frac {7 \left (x^3-1\right )^{2/3}}{2 x^2}\right )-\frac {\left (x^3-1\right )^{2/3}}{5 x^5}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle \frac {1}{5} \left (5 \left (-\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (1-2 x^3\right )+\frac {1}{2} \log \left (-\sqrt [3]{x^3-1}-x\right )\right )+\frac {7 \left (x^3-1\right )^{2/3}}{2 x^2}\right )-\frac {\left (x^3-1\right )^{2/3}}{5 x^5}\) |
-1/5*(-1 + x^3)^(2/3)/x^5 + ((7*(-1 + x^3)^(2/3))/(2*x^2) + 5*(-(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))/5
3.15.16.3.1 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[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[((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])
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 + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
Time = 2.61 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99
| method | result | size |
| pseudoelliptic | \(\frac {10 \ln \left (\frac {x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (21 x^{3}-6\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+5 x^{5} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (x -2 \left (x^{3}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right )-\ln \left (\frac {x^{2}-x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{30 x^{5}}\) | \(100\) |
| risch | \(\frac {7 x^{6}-9 x^{3}+2}{10 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}}}-\frac {\ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-2 x \left (x^{3}-1\right )^{\frac {2}{3}}+2 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{2 x^{3}-1}\right )}{3}-\ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-2 x \left (x^{3}-1\right )^{\frac {2}{3}}+2 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{2 x^{3}-1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-x \left (x^{3}-1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{2 x^{3}-1}\right )\) | \(419\) |
| trager | \(\frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (7 x^{3}-2\right )}{10 x^{5}}+32 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \ln \left (-\frac {-374952960 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+10252800 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -10252800 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}-34273056 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}-421089 x \left (x^{3}-1\right )^{\frac {2}{3}}+421089 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-252248 x^{3}+2999623680 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}+51556128 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )+220717}{2 x^{3}-1}\right )-\frac {\ln \left (\frac {374952960 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+10252800 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -10252800 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}-26461536 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}+527889 x \left (x^{3}-1\right )^{\frac {2}{3}}-527889 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-64078 x^{3}-2999623680 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-10936032 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )-9154}{2 x^{3}-1}\right )}{3}-32 \ln \left (\frac {374952960 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+10252800 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -10252800 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}-26461536 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}+527889 x \left (x^{3}-1\right )^{\frac {2}{3}}-527889 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-64078 x^{3}-2999623680 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-10936032 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )-9154}{2 x^{3}-1}\right ) \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )\) | \(488\) |
1/30*(10*ln((x+(x^3-1)^(1/3))/x)*x^5+(21*x^3-6)*(x^3-1)^(2/3)+5*x^5*(2*3^( 1/2)*arctan(1/3*(x-2*(x^3-1)^(1/3))*3^(1/2)/x)-ln((x^2-x*(x^3-1)^(1/3)+(x^ 3-1)^(2/3))/x^2)))/x^5
Time = 0.53 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.23 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} - 1\right )}}{7 \, x^{3} + 1}\right ) - 5 \, x^{5} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{2 \, x^{3} - 1}\right ) - 3 \, {\left (7 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \]
-1/30*(10*sqrt(3)*x^5*arctan((4*sqrt(3)*(x^3 - 1)^(1/3)*x^2 + 2*sqrt(3)*(x ^3 - 1)^(2/3)*x + sqrt(3)*(x^3 - 1))/(7*x^3 + 1)) - 5*x^5*log((2*x^3 + 3*( x^3 - 1)^(1/3)*x^2 + 3*(x^3 - 1)^(2/3)*x - 1)/(2*x^3 - 1)) - 3*(7*x^3 - 2) *(x^3 - 1)^(2/3))/x^5
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \cdot \left (3 x^{3} - 1\right )}{x^{6} \cdot \left (2 x^{3} - 1\right )}\, dx \]
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\int { \frac {{\left (3 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )} x^{6}} \,d x } \]
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\int { \frac {{\left (3 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (3\,x^3-1\right )}{x^6\,\left (2\,x^3-1\right )} \,d x \]