Integrand size = 20, antiderivative size = 104 \[ \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{10}} \, dx=\frac {\sqrt [3]{-1+x^3} \left (18-57 x^3+13 x^6\right )}{162 x^9}-\frac {13 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {13}{243} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {13}{486} \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
1/162*(x^3-1)^(1/3)*(13*x^6-57*x^3+18)/x^9+13/243*arctan(-1/3*3^(1/2)+2/3* (x^3-1)^(1/3)*3^(1/2))*3^(1/2)+13/243*ln(1+(x^3-1)^(1/3))-13/486*ln(1-(x^3 -1)^(1/3)+(x^3-1)^(2/3))
Time = 0.16 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{10}} \, dx=\frac {1}{486} \left (\frac {3 \sqrt [3]{-1+x^3} \left (18-57 x^3+13 x^6\right )}{x^9}-26 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+26 \log \left (1+\sqrt [3]{-1+x^3}\right )-13 \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
((3*(-1 + x^3)^(1/3)*(18 - 57*x^3 + 13*x^6))/x^9 - 26*Sqrt[3]*ArcTan[(1 - 2*(-1 + x^3)^(1/3))/Sqrt[3]] + 26*Log[1 + (-1 + x^3)^(1/3)] - 13*Log[1 - ( -1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/486
Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {948, 25, 87, 51, 52, 70, 16, 1083, 217}
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.
| \(\displaystyle \int \frac {\sqrt [3]{x^3-1} \left (2 x^3-1\right )}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int -\frac {\left (1-2 x^3\right ) \sqrt [3]{x^3-1}}{x^{12}}dx^3\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{3} \int \frac {\left (1-2 x^3\right ) \sqrt [3]{x^3-1}}{x^{12}}dx^3\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{3} \left (\frac {13}{9} \int \frac {\sqrt [3]{x^3-1}}{x^9}dx^3-\frac {\left (x^3-1\right )^{4/3}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{3} \left (\frac {13}{9} \left (\frac {1}{6} \int \frac {1}{x^6 \left (x^3-1\right )^{2/3}}dx^3-\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\left (x^3-1\right )^{4/3}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{3} \left (\frac {13}{9} \left (\frac {1}{6} \left (\frac {2}{3} \int \frac {1}{x^3 \left (x^3-1\right )^{2/3}}dx^3+\frac {\sqrt [3]{x^3-1}}{x^3}\right )-\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\left (x^3-1\right )^{4/3}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 70 |
| \(\displaystyle \frac {1}{3} \left (\frac {13}{9} \left (\frac {1}{6} \left (\frac {2}{3} \left (\frac {3}{2} \int \frac {1}{\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}+\frac {3}{2} \int \frac {1}{x^6-\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}-\frac {1}{2} \log \left (x^3\right )\right )+\frac {\sqrt [3]{x^3-1}}{x^3}\right )-\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\left (x^3-1\right )^{4/3}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} \left (\frac {13}{9} \left (\frac {1}{6} \left (\frac {2}{3} \left (\frac {3}{2} \int \frac {1}{x^6-\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )+\frac {\sqrt [3]{x^3-1}}{x^3}\right )-\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\left (x^3-1\right )^{4/3}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{3} \left (\frac {13}{9} \left (\frac {1}{6} \left (\frac {2}{3} \left (-3 \int \frac {1}{-x^6-3}d\left (2 \sqrt [3]{x^3-1}-1\right )-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )+\frac {\sqrt [3]{x^3-1}}{x^3}\right )-\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\left (x^3-1\right )^{4/3}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (\frac {13}{9} \left (\frac {1}{6} \left (\frac {2}{3} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^3-1}-1}{\sqrt {3}}\right )-\frac {\log \left (x^3\right )}{2}+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )+\frac {\sqrt [3]{x^3-1}}{x^3}\right )-\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\left (x^3-1\right )^{4/3}}{3 x^9}\right )\) |
(-1/3*(-1 + x^3)^(4/3)/x^9 + (13*(-1/2*(-1 + x^3)^(1/3)/x^6 + ((-1 + x^3)^ (1/3)/x^3 + (2*(Sqrt[3]*ArcTan[(-1 + 2*(-1 + x^3)^(1/3))/Sqrt[3]] - Log[x^ 3]/2 + (3*Log[1 + (-1 + x^3)^(1/3)])/2))/3)/6))/9)/3
3.15.73.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[-(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2) , x] + (Simp[3/(2*b*q) Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1 /3)], x] + Simp[3/(2*b*q^2) Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.61 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \(\frac {13 x^{9}-70 x^{6}+75 x^{3}-18}{162 x^{9} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {13 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{243 \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(96\) |
| pseudoelliptic | \(\frac {\left (39 x^{6}-171 x^{3}+54\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+13 x^{9} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )+2 \ln \left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )-\ln \left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )\right )}{486 {\left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )}^{3} {\left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )}^{3}}\) | \(116\) |
| meijerg | \(-\frac {2 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], x^{3}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{6}}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], x^{3}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{27}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{9}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{6}}+\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\) | \(170\) |
| trager | \(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (13 x^{6}-57 x^{3}+18\right )}{162 x^{9}}+\frac {6656 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \ln \left (\frac {-1310720 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2} x^{3}+8704 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) x^{3}+19968 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+78 x^{3}-105 \left (x^{3}-1\right )^{\frac {2}{3}}-19968 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+10485760 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}+105 \left (x^{3}-1\right )^{\frac {1}{3}}+40448 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-65}{x^{3}}\right )}{243}-\frac {13 \ln \left (\frac {-376963072 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2} x^{3}-5045760 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) x^{3}+2817024 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+1477 x^{3}-14247 \left (x^{3}-1\right )^{\frac {2}{3}}-2817024 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+3015704576 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}+14247 \left (x^{3}-1\right )^{\frac {1}{3}}+8707072 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-2743}{x^{3}}\right )}{243}-\frac {6656 \ln \left (\frac {-376963072 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2} x^{3}-5045760 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) x^{3}+2817024 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+1477 x^{3}-14247 \left (x^{3}-1\right )^{\frac {2}{3}}-2817024 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+3015704576 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}+14247 \left (x^{3}-1\right )^{\frac {1}{3}}+8707072 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-2743}{x^{3}}\right ) \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )}{243}\) | \(450\) |
1/162*(13*x^9-70*x^6+75*x^3-18)/x^9/(x^3-1)^(2/3)+13/243/GAMMA(2/3)/signum (x^3-1)^(2/3)*(-signum(x^3-1))^(2/3)*(2/3*GAMMA(2/3)*x^3*hypergeom([1,1,5/ 3],[2,2],x^3)+(1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x)+I*Pi)*GAMMA(2/3))
Time = 0.43 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{10}} \, dx=\frac {26 \, \sqrt {3} x^{9} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 13 \, x^{9} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 26 \, x^{9} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (13 \, x^{6} - 57 \, x^{3} + 18\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{486 \, x^{9}} \]
1/486*(26*sqrt(3)*x^9*arctan(2/3*sqrt(3)*(x^3 - 1)^(1/3) - 1/3*sqrt(3)) - 13*x^9*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 26*x^9*log((x^3 - 1)^( 1/3) + 1) + 3*(13*x^6 - 57*x^3 + 18)*(x^3 - 1)^(1/3))/x^9
Result contains complex when optimal does not.
Time = 145.35 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{10}} \, dx=- \frac {2 \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{5} \Gamma \left (\frac {8}{3}\right )} + \frac {\Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{8} \Gamma \left (\frac {11}{3}\right )} \]
-2*gamma(5/3)*hyper((-1/3, 5/3), (8/3,), exp_polar(2*I*pi)/x**3)/(3*x**5*g amma(8/3)) + gamma(8/3)*hyper((-1/3, 8/3), (11/3,), exp_polar(2*I*pi)/x**3 )/(3*x**8*gamma(11/3))
Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{10}} \, dx=\frac {13}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {5 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 13 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{162 \, {\left ({\left (x^{3} - 1\right )}^{3} + 3 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{2} - 2\right )}} + \frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}} - 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{9 \, {\left (2 \, x^{3} + {\left (x^{3} - 1\right )}^{2} - 1\right )}} - \frac {13}{486} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {13}{243} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \]
13/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) - 1/162*(5*(x^3 - 1)^(7/3) + 13*(x^3 - 1)^(4/3) - 10*(x^3 - 1)^(1/3))/((x^3 - 1)^3 + 3*x^ 3 + 3*(x^3 - 1)^2 - 2) + 1/9*((x^3 - 1)^(4/3) - 2*(x^3 - 1)^(1/3))/(2*x^3 + (x^3 - 1)^2 - 1) - 13/486*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 1 3/243*log((x^3 - 1)^(1/3) + 1)
Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{10}} \, dx=\frac {13}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {13 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} - 31 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 26 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{162 \, x^{9}} - \frac {13}{486} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {13}{243} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]
13/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) + 1/162*(13*(x^ 3 - 1)^(7/3) - 31*(x^3 - 1)^(4/3) - 26*(x^3 - 1)^(1/3))/x^9 - 13/486*log(( x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 13/243*log(abs((x^3 - 1)^(1/3) + 1 ))
Time = 6.70 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.22 \[ \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{10}} \, dx=\frac {2\,\ln \left (\frac {4\,{\left (x^3-1\right )}^{1/3}}{81}+\frac {4}{81}\right )}{27}-\frac {5\,\ln \left (\frac {25\,{\left (x^3-1\right )}^{1/3}}{6561}+\frac {25}{6561}\right )}{243}-\frac {\frac {13\,{\left (x^3-1\right )}^{4/3}}{162}-\frac {5\,{\left (x^3-1\right )}^{1/3}}{81}+\frac {5\,{\left (x^3-1\right )}^{7/3}}{162}}{3\,{\left (x^3-1\right )}^2+{\left (x^3-1\right )}^3+3\,x^3-2}-\frac {\frac {2\,{\left (x^3-1\right )}^{1/3}}{9}-\frac {{\left (x^3-1\right )}^{4/3}}{9}}{{\left (x^3-1\right )}^2+2\,x^3-1}-\ln \left (\frac {1}{3}-\frac {2\,{\left (x^3-1\right )}^{1/3}}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )\,\left (\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )+\ln \left (\frac {2\,{\left (x^3-1\right )}^{1/3}}{3}-\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )\,\left (-\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )+\ln \left (\frac {5}{54}-\frac {5\,{\left (x^3-1\right )}^{1/3}}{27}+\frac {\sqrt {3}\,5{}\mathrm {i}}{54}\right )\,\left (\frac {5}{486}+\frac {\sqrt {3}\,5{}\mathrm {i}}{486}\right )-\ln \left (\frac {5\,{\left (x^3-1\right )}^{1/3}}{27}-\frac {5}{54}+\frac {\sqrt {3}\,5{}\mathrm {i}}{54}\right )\,\left (-\frac {5}{486}+\frac {\sqrt {3}\,5{}\mathrm {i}}{486}\right ) \]
(2*log((4*(x^3 - 1)^(1/3))/81 + 4/81))/27 - (5*log((25*(x^3 - 1)^(1/3))/65 61 + 25/6561))/243 - ((13*(x^3 - 1)^(4/3))/162 - (5*(x^3 - 1)^(1/3))/81 + (5*(x^3 - 1)^(7/3))/162)/(3*(x^3 - 1)^2 + (x^3 - 1)^3 + 3*x^3 - 2) - ((2*( x^3 - 1)^(1/3))/9 - (x^3 - 1)^(4/3)/9)/((x^3 - 1)^2 + 2*x^3 - 1) - log((3^ (1/2)*1i)/3 - (2*(x^3 - 1)^(1/3))/3 + 1/3)*((3^(1/2)*1i)/27 + 1/27) + log( (3^(1/2)*1i)/3 + (2*(x^3 - 1)^(1/3))/3 - 1/3)*((3^(1/2)*1i)/27 - 1/27) + l og((3^(1/2)*5i)/54 - (5*(x^3 - 1)^(1/3))/27 + 5/54)*((3^(1/2)*5i)/486 + 5/ 486) - log((3^(1/2)*5i)/54 + (5*(x^3 - 1)^(1/3))/27 - 5/54)*((3^(1/2)*5i)/ 486 - 5/486)