Integrand size = 34, antiderivative size = 118 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (6-33 x^3+62 x^6\right )}{30 x^5 \left (-1+2 x^3\right )}+\frac {7 \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^3}}\right )}{3 \sqrt {3}}+\frac {7}{9} \log \left (x+\sqrt [3]{-1+x^3}\right )-\frac {7}{18} \log \left (x^2-x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
1/30*(x^3-1)^(2/3)*(62*x^6-33*x^3+6)/x^5/(2*x^3-1)+7/9*arctan(3^(1/2)*x/(- x+2*(x^3-1)^(1/3)))*3^(1/2)+7/9*ln(x+(x^3-1)^(1/3))-7/18*ln(x^2-x*(x^3-1)^ (1/3)+(x^3-1)^(2/3))
Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx=\frac {1}{90} \left (\frac {3 \left (-1+x^3\right )^{2/3} \left (6-33 x^3+62 x^6\right )}{x^5 \left (-1+2 x^3\right )}-70 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{-1+x^3}}\right )+70 \log \left (x+\sqrt [3]{-1+x^3}\right )-35 \log \left (x^2-x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
((3*(-1 + x^3)^(2/3)*(6 - 33*x^3 + 62*x^6))/(x^5*(-1 + 2*x^3)) - 70*Sqrt[3 ]*ArcTan[(Sqrt[3]*x)/(x - 2*(-1 + x^3)^(1/3))] + 70*Log[x + (-1 + x^3)^(1/ 3)] - 35*Log[x^2 - x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/90
Time = 0.33 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1387, 1047, 27, 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.
| \(\displaystyle \int \frac {\left (x^3-1\right )^{2/3} \left (4 x^6-5 x^3+1\right )}{x^6 \left (2 x^3-1\right )^2} \, dx\) |
| \(\Big \downarrow \) 1387 |
| \(\displaystyle \int \frac {\left (x^3-1\right )^{5/3} \left (4 x^3-1\right )}{x^6 \left (2 x^3-1\right )^2}dx\) |
| \(\Big \downarrow \) 1047 |
| \(\displaystyle \frac {1}{6} \int -\frac {2 \left (x^3-1\right )^{2/3} \left (3 x^3+2\right )}{x^6 \left (1-2 x^3\right )}dx+\frac {\left (x^3-1\right )^{5/3}}{3 x^5 \left (1-2 x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (x^3-1\right )^{5/3}}{3 x^5 \left (1-2 x^3\right )}-\frac {1}{3} \int \frac {\left (x^3-1\right )^{2/3} \left (3 x^3+2\right )}{x^6 \left (1-2 x^3\right )}dx\) |
| \(\Big \downarrow \) 1050 |
| \(\displaystyle \frac {1}{3} \left (\frac {2 \left (x^3-1\right )^{2/3}}{5 x^5}-\frac {1}{5} \int -\frac {31-27 x^3}{x^3 \left (1-2 x^3\right ) \sqrt [3]{x^3-1}}dx\right )+\frac {\left (x^3-1\right )^{5/3}}{3 x^5 \left (1-2 x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \int \frac {31-27 x^3}{x^3 \left (1-2 x^3\right ) \sqrt [3]{x^3-1}}dx+\frac {2 \left (x^3-1\right )^{2/3}}{5 x^5}\right )+\frac {\left (x^3-1\right )^{5/3}}{3 x^5 \left (1-2 x^3\right )}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \int \frac {70}{\left (1-2 x^3\right ) \sqrt [3]{x^3-1}}dx+\frac {31 \left (x^3-1\right )^{2/3}}{2 x^2}\right )+\frac {2 \left (x^3-1\right )^{2/3}}{5 x^5}\right )+\frac {\left (x^3-1\right )^{5/3}}{3 x^5 \left (1-2 x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (35 \int \frac {1}{\left (1-2 x^3\right ) \sqrt [3]{x^3-1}}dx+\frac {31 \left (x^3-1\right )^{2/3}}{2 x^2}\right )+\frac {2 \left (x^3-1\right )^{2/3}}{5 x^5}\right )+\frac {\left (x^3-1\right )^{5/3}}{3 x^5 \left (1-2 x^3\right )}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{5} \left (35 \left (-\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (1-2 x^3\right )+\frac {1}{2} \log \left (-\sqrt [3]{x^3-1}-x\right )\right )+\frac {31 \left (x^3-1\right )^{2/3}}{2 x^2}\right )+\frac {2 \left (x^3-1\right )^{2/3}}{5 x^5}\right )+\frac {\left (x^3-1\right )^{5/3}}{3 x^5 \left (1-2 x^3\right )}\) |
(-1 + x^3)^(5/3)/(3*x^5*(1 - 2*x^3)) + ((2*(-1 + x^3)^(2/3))/(5*x^5) + ((3 1*(-1 + x^3)^(2/3))/(2*x^2) + 35*(-(ArcTan[(1 - (2*x)/(-1 + x^3)^(1/3))/Sq rt[3]]/Sqrt[3]) - Log[1 - 2*x^3]/6 + Log[-x - (-1 + x^3)^(1/3)]/2))/5)/3
3.18.56.3.1 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[(-(b*e - a*f))*(g*x)^( m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*b*g*n*(p + 1))), x] + Simp[1/( a*b*n*(p + 1)) Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c *(b*e*n*(p + 1) + (b*e - a*f)*(m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && IGtQ[n , 0] && LtQ[p, -1] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
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]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)* (x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^ p, x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 2.55 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.30
| method | result | size |
| pseudoelliptic | \(\frac {\left (140 x^{8}-70 x^{5}\right ) \ln \left (\frac {x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )+\left (186 x^{6}-99 x^{3}+18\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+35 x^{5} \left (2 x^{3}-1\right ) \left (2 \sqrt {3}\, \arctan \left (\frac {\left (x -2 \left (x^{3}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right )-\ln \left (\frac {x^{2}-x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{90 \left (x +\left (x^{3}-1\right )^{\frac {1}{3}}\right ) \left (\left (x^{3}-1\right )^{\frac {2}{3}}+x \left (x -\left (x^{3}-1\right )^{\frac {1}{3}}\right )\right ) x^{5}}\) | \(153\) |
| risch | \(\frac {62 x^{9}-95 x^{6}+39 x^{3}-6}{30 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}} \left (2 x^{3}-1\right )}+\frac {7 \ln \left (\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-x \left (x^{3}-1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+1}{2 x^{3}-1}\right )}{9}+\frac {7 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 x \left (x^{3}-1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{2 x^{3}-1}\right )}{3}\) | \(274\) |
| trager | \(\frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (62 x^{6}-33 x^{3}+6\right )}{30 x^{5} \left (2 x^{3}-1\right )}+\frac {7 \ln \left (-\frac {65382681600 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )^{2} x^{3}+153792000 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +760160160 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) x^{2}+350055360 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) x^{3}-421089 x \left (x^{3}-1\right )^{\frac {2}{3}}+106800 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-73232 x^{3}-523061452800 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )^{2}-212378400 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )+64078}{2 x^{3}-1}\right )}{9}+1120 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) \ln \left (\frac {-18981734400 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )^{2} x^{3}+153792000 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -606368160 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) x^{2}+137676960 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) x^{3}+527889 x \left (x^{3}-1\right )^{\frac {2}{3}}+106800 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-220717 x^{3}+151853875200 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )^{2}-350055360 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )-31531}{2 x^{3}-1}\right )\) | \(343\) |
1/90*((140*x^8-70*x^5)*ln((x+(x^3-1)^(1/3))/x)+(186*x^6-99*x^3+18)*(x^3-1) ^(2/3)+35*x^5*(2*x^3-1)*(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+(x^3-1)^(1/3))/((x^3-1 )^(2/3)+x*(x-(x^3-1)^(1/3)))/x^5
Time = 0.60 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.31 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx=-\frac {70 \, \sqrt {3} {\left (2 \, x^{8} - x^{5}\right )} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} - 1\right )}}{7 \, x^{3} + 1}\right ) - 35 \, {\left (2 \, x^{8} - x^{5}\right )} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{2 \, x^{3} - 1}\right ) - 3 \, {\left (62 \, x^{6} - 33 \, x^{3} + 6\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, {\left (2 \, x^{8} - x^{5}\right )}} \]
-1/90*(70*sqrt(3)*(2*x^8 - 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)) - 35*(2*x^8 - 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*(62*x^6 - 33*x^3 + 6)*(x^3 - 1)^(2/3))/(2*x^8 - x^5)
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x - 1\right ) \left (4 x^{3} - 1\right ) \left (x^{2} + x + 1\right )}{x^{6} \left (2 x^{3} - 1\right )^{2}}\, dx \]
Integral(((x - 1)*(x**2 + x + 1))**(2/3)*(x - 1)*(4*x**3 - 1)*(x**2 + x + 1)/(x**6*(2*x**3 - 1)**2), x)
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx=\int { \frac {{\left (4 \, x^{6} - 5 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )}^{2} x^{6}} \,d x } \]
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx=\int { \frac {{\left (4 \, x^{6} - 5 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )}^{2} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (4\,x^6-5\,x^3+1\right )}{x^6\,{\left (2\,x^3-1\right )}^2} \,d x \]