Integrand size = 35, antiderivative size = 78 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (1-x^3+x^6\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-2+7 x^3\right )}{10 x^5}-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]
Time = 0.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (1-x^3+x^6\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-2+7 x^3\right )}{10 x^5}-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]
((-1 + x^3)^(2/3)*(-2 + 7*x^3))/(10*x^5) - RootSum[1 - #1^3 + #1^6 & , (-L og[x] + Log[(-1 + x^3)^(1/3) - x*#1])/(-#1 + 2*#1^4) & ]/3
Result contains complex when optimal does not.
Time = 0.77 (sec) , antiderivative size = 703, normalized size of antiderivative = 9.01, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {1380, 1844, 25, 809, 809, 769, 1758, 25, 933, 25, 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.
\(\displaystyle \int \frac {\left (x^3-1\right )^{2/3} \left (x^6-2 x^3+1\right )}{x^6 \left (x^6-x^3+1\right )} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \int \frac {\left (x^3-1\right )^{8/3}}{x^6 \left (x^6-x^3+1\right )}dx\) |
\(\Big \downarrow \) 1844 |
\(\displaystyle -\int \frac {\left (x^3-1\right )^{5/3}}{x^6}dx-\int -\frac {\left (x^3-1\right )^{5/3}}{x^6-x^3+1}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\left (x^3-1\right )^{5/3}}{x^6-x^3+1}dx-\int \frac {\left (x^3-1\right )^{5/3}}{x^6}dx\) |
\(\Big \downarrow \) 809 |
\(\displaystyle -\int \frac {\left (x^3-1\right )^{2/3}}{x^3}dx+\int \frac {\left (x^3-1\right )^{5/3}}{x^6-x^3+1}dx+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}\) |
\(\Big \downarrow \) 809 |
\(\displaystyle -\int \frac {1}{\sqrt [3]{x^3-1}}dx+\int \frac {\left (x^3-1\right )^{5/3}}{x^6-x^3+1}dx+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\) |
\(\Big \downarrow \) 769 |
\(\displaystyle \int \frac {\left (x^3-1\right )^{5/3}}{x^6-x^3+1}dx-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\) |
\(\Big \downarrow \) 1758 |
\(\displaystyle \frac {2 i \int -\frac {\left (x^3-1\right )^{5/3}}{-2 x^3-i \sqrt {3}+1}dx}{\sqrt {3}}-\frac {2 i \int -\frac {\left (x^3-1\right )^{5/3}}{-2 x^3+i \sqrt {3}+1}dx}{\sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2 i \int \frac {\left (x^3-1\right )^{5/3}}{-2 x^3-i \sqrt {3}+1}dx}{\sqrt {3}}+\frac {2 i \int \frac {\left (x^3-1\right )^{5/3}}{-2 x^3+i \sqrt {3}+1}dx}{\sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\) |
\(\Big \downarrow \) 933 |
\(\displaystyle \frac {2 i \left (-\frac {1}{6} \left (x^3-1\right )^{2/3} x-\frac {1}{6} \int -\frac {-\left (\left (7-3 i \sqrt {3}\right ) x^3\right )-i \sqrt {3}+5}{\left (-2 x^3+i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )}{\sqrt {3}}-\frac {2 i \left (-\frac {1}{6} \left (x^3-1\right )^{2/3} x-\frac {1}{6} \int -\frac {-\left (\left (7+3 i \sqrt {3}\right ) x^3\right )+i \sqrt {3}+5}{\left (-2 x^3-i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \int \frac {-\left (\left (7-3 i \sqrt {3}\right ) x^3\right )-i \sqrt {3}+5}{\left (-2 x^3+i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )}{\sqrt {3}}-\frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \int \frac {-\left (\left (7+3 i \sqrt {3}\right ) x^3\right )+i \sqrt {3}+5}{\left (-2 x^3-i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\) |
\(\Big \downarrow \) 1026 |
\(\displaystyle -\frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \left (\frac {1}{2} \left (7+3 i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{x^3-1}}dx-3 \left (1-i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^3-i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )\right )}{\sqrt {3}}+\frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \left (\frac {1}{2} \left (7-3 i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{x^3-1}}dx-3 \left (1+i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^3+i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\) |
\(\Big \downarrow \) 769 |
\(\displaystyle -\frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \left (\frac {1}{2} \left (7+3 i \sqrt {3}\right ) \left (\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )\right )-3 \left (1-i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^3-i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )\right )}{\sqrt {3}}+\frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \left (\frac {1}{2} \left (7-3 i \sqrt {3}\right ) \left (\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )\right )-3 \left (1+i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^3+i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\) |
\(\Big \downarrow \) 901 |
\(\displaystyle -\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \left (\frac {1}{2} \left (7-3 i \sqrt {3}\right ) \left (\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )\right )-3 \left (1+i \sqrt {3}\right ) \left (-\frac {i \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}-i\right )^{2/3} \sqrt [3]{\sqrt {3}+i}}-\frac {i \log \left (-2 x^3+i \sqrt {3}+1\right )}{6 \left (\sqrt {3}-i\right )^{2/3} \sqrt [3]{\sqrt {3}+i}}+\frac {i \log \left (-\sqrt [3]{x^3-1}+\frac {x}{\sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}}}\right )}{2 \left (\sqrt {3}-i\right )^{2/3} \sqrt [3]{\sqrt {3}+i}}\right )\right )\right )}{\sqrt {3}}-\frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \left (\frac {1}{2} \left (7+3 i \sqrt {3}\right ) \left (\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )\right )-3 \left (1-i \sqrt {3}\right ) \left (\frac {i \arctan \left (\frac {1+\frac {2 \sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{\sqrt {3}-i} \left (\sqrt {3}+i\right )^{2/3}}+\frac {i \log \left (-2 x^3-i \sqrt {3}+1\right )}{6 \sqrt [3]{\sqrt {3}-i} \left (\sqrt {3}+i\right )^{2/3}}-\frac {i \log \left (-\sqrt [3]{x^3-1}+\sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x\right )}{2 \sqrt [3]{\sqrt {3}-i} \left (\sqrt {3}+i\right )^{2/3}}\right )\right )\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\) |
(-1 + x^3)^(2/3)/(2*x^2) + (-1 + x^3)^(5/3)/(5*x^5) - ArcTan[(1 + (2*x)/(- 1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] + ((2*I)*(-1/6*(x*(-1 + x^3)^(2/3)) + (-3 *(1 + I*Sqrt[3])*(((-I)*ArcTan[(1 + (2*x)/((-((I - Sqrt[3])/(I + Sqrt[3])) )^(1/3)*(-1 + x^3)^(1/3)))/Sqrt[3]])/(Sqrt[3]*(-I + Sqrt[3])^(2/3)*(I + Sq rt[3])^(1/3)) - ((I/6)*Log[1 + I*Sqrt[3] - 2*x^3])/((-I + Sqrt[3])^(2/3)*( I + Sqrt[3])^(1/3)) + ((I/2)*Log[x/(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3) - (-1 + x^3)^(1/3)])/((-I + Sqrt[3])^(2/3)*(I + Sqrt[3])^(1/3))) + ((7 - ( 3*I)*Sqrt[3])*(ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[ -x + (-1 + x^3)^(1/3)]/2))/2)/6))/Sqrt[3] - ((2*I)*(-1/6*(x*(-1 + x^3)^(2/ 3)) + (-3*(1 - I*Sqrt[3])*((I*ArcTan[(1 + (2*(-((I - Sqrt[3])/(I + Sqrt[3] )))^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*(-I + Sqrt[3])^(1/3)*(I + Sqrt[3])^(2/3)) + ((I/6)*Log[1 - I*Sqrt[3] - 2*x^3])/((-I + Sqrt[3])^(1/ 3)*(I + Sqrt[3])^(2/3)) - ((I/2)*Log[(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3 )*x - (-1 + x^3)^(1/3)])/((-I + Sqrt[3])^(1/3)*(I + Sqrt[3])^(2/3))) + ((7 + (3*I)*Sqrt[3])*(ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[-x + (-1 + x^3)^(1/3)]/2))/2)/6))/Sqrt[3] + Log[-x + (-1 + x^3)^(1/3)] /2
3.11.34.3.1 Defintions of rubi rules used
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, 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_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Simp[1/(b*(n*(p + q) + 1)) Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Sim p[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d , 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[ a, b, c, d, n, p, q, x]
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[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ )), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/r) Int[(d + e*x ^n)^q/(b - r + 2*c*x^n), x], x] - Simp[2*(c/r) Int[(d + e*x^n)^q/(b + r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && Ne Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[q]
Int[(((f_.)*(x_))^(m_)*((d_.) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^( n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[d/a Int[(f*x)^m*(d + e*x^n)^( q - 1), x], x] - Simp[1/(a*f^n) Int[(f*x)^(m + n)*(d + e*x^n)^(q - 1)*(Si mp[b*d - a*e + c*d*x^n, x]/(a + b*x^n + c*x^(2*n))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && ! IntegerQ[q] && GtQ[q, 0] && LtQ[m, 0]
Time = 22.45 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.97
method | result | size |
pseudoelliptic | \(\frac {-10 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )}{2 \textit {\_R}^{4}-\textit {\_R}}\right ) x^{5}+21 x^{3} \left (x^{3}-1\right )^{\frac {2}{3}}-6 \left (x^{3}-1\right )^{\frac {2}{3}}}{30 x^{5}}\) | \(76\) |
risch | \(\text {Expression too large to display}\) | \(1631\) |
trager | \(\text {Expression too large to display}\) | \(4238\) |
1/30*(-10*sum(ln((-_R*x+(x^3-1)^(1/3))/x)/(2*_R^4-_R),_R=RootOf(_Z^6-_Z^3+ 1))*x^5+21*x^3*(x^3-1)^(2/3)-6*(x^3-1)^(2/3))/x^5
Exception generated. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (1-x^3+x^6\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (trace 0)
Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (1-x^3+x^6\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 0.32 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.45 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 2 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} + 1\right )} x^{6}} \,d x } \]
Not integrable
Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.45 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 2 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} + 1\right )} x^{6}} \,d x } \]
Not integrable
Time = 6.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.45 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (1-x^3+x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6-2\,x^3+1\right )}{x^6\,\left (x^6-x^3+1\right )} \,d x \]