Integrand size = 13, antiderivative size = 104 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx=\frac {\sqrt [3]{-1+x^3} \left (-18+3 x^3+5 x^6\right )}{162 x^9}-\frac {5 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {5}{243} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {5}{486} \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
1/162*(x^3-1)^(1/3)*(5*x^6+3*x^3-18)/x^9+5/243*arctan(-1/3*3^(1/2)+2/3*(x^ 3-1)^(1/3)*3^(1/2))*3^(1/2)+5/243*ln(1+(x^3-1)^(1/3))-5/486*ln(1-(x^3-1)^( 1/3)+(x^3-1)^(2/3))
Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx=\frac {1}{486} \left (\frac {3 \sqrt [3]{-1+x^3} \left (-18+3 x^3+5 x^6\right )}{x^9}-10 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+10 \log \left (1+\sqrt [3]{-1+x^3}\right )-5 \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
((3*(-1 + x^3)^(1/3)*(-18 + 3*x^3 + 5*x^6))/x^9 - 10*Sqrt[3]*ArcTan[(1 - 2 *(-1 + x^3)^(1/3))/Sqrt[3]] + 10*Log[1 + (-1 + x^3)^(1/3)] - 5*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 = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {798, 51, 52, 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.
| \(\displaystyle \int \frac {\sqrt [3]{x^3-1}}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {1}{3} \int \frac {\sqrt [3]{x^3-1}}{x^{12}}dx^3\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{9} \int \frac {1}{x^9 \left (x^3-1\right )^{2/3}}dx^3-\frac {\sqrt [3]{x^3-1}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{9} \left (\frac {5}{6} \int \frac {1}{x^6 \left (x^3-1\right )^{2/3}}dx^3+\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\sqrt [3]{x^3-1}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{9} \left (\frac {5}{6} \left (\frac {2}{3} \int \frac {1}{x^3 \left (x^3-1\right )^{2/3}}dx^3+\frac {\sqrt [3]{x^3-1}}{x^3}\right )+\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\sqrt [3]{x^3-1}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 70 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{9} \left (\frac {5}{6} \left (\frac {2}{3} \left (\frac {3}{2} \int \frac {1}{\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}+\frac {3}{2} \int \frac {1}{x^6-\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}-\frac {1}{2} \log \left (x^3\right )\right )+\frac {\sqrt [3]{x^3-1}}{x^3}\right )+\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\sqrt [3]{x^3-1}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{9} \left (\frac {5}{6} \left (\frac {2}{3} \left (\frac {3}{2} \int \frac {1}{x^6-\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )+\frac {\sqrt [3]{x^3-1}}{x^3}\right )+\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\sqrt [3]{x^3-1}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{9} \left (\frac {5}{6} \left (\frac {2}{3} \left (-3 \int \frac {1}{-x^6-3}d\left (2 \sqrt [3]{x^3-1}-1\right )-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )+\frac {\sqrt [3]{x^3-1}}{x^3}\right )+\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\sqrt [3]{x^3-1}}{3 x^9}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{9} \left (\frac {5}{6} \left (\frac {2}{3} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^3-1}-1}{\sqrt {3}}\right )-\frac {\log \left (x^3\right )}{2}+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )+\frac {\sqrt [3]{x^3-1}}{x^3}\right )+\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\sqrt [3]{x^3-1}}{3 x^9}\right )\) |
(-1/3*(-1 + x^3)^(1/3)/x^9 + ((-1 + x^3)^(1/3)/(2*x^6) + (5*((-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
3.15.70.3.1 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_)^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_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && 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 = 1.80 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86
| method | result | size |
| meijerg | \(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], x^{3}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{27}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{9}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{6}}+\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\) | \(89\) |
| risch | \(\frac {5 x^{9}-2 x^{6}-21 x^{3}+18}{162 x^{9} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {5 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{243 \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(96\) |
| pseudoelliptic | \(\frac {\left (15 x^{6}+9 x^{3}-54\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+5 x^{9} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )+2 \ln \left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )-\ln \left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )\right )}{486 {\left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )}^{3} {\left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )}^{3}}\) | \(116\) |
| trager | \(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (5 x^{6}+3 x^{3}-18\right )}{162 x^{9}}+\frac {5 \ln \left (-\frac {55312384 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2} x^{3}-1384448 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) x^{3}+2817024 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+8628 x^{3}-442499072 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}+10111488 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+19749 \left (x^{3}-1\right )^{\frac {2}{3}}+6430208 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+14247 \left (x^{3}-1\right )^{\frac {1}{3}}-7190}{x^{3}}\right )}{243}+\frac {2560 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \ln \left (\frac {432275456 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2} x^{3}+6646272 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) x^{3}+2817024 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+10066 x^{3}-3458203648 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}-7294464 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {2}{3}}-16865792 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-19749 \left (x^{3}-1\right )^{\frac {1}{3}}-18694}{x^{3}}\right )}{243}\) | \(306\) |
1/9/GAMMA(2/3)*signum(x^3-1)^(1/3)/(-signum(x^3-1))^(1/3)*(-10/81*GAMMA(2/ 3)*x^3*hypergeom([1,1,11/3],[2,5],x^3)-5/27*(4/15+1/6*Pi*3^(1/2)-3/2*ln(3) +3*ln(x)+I*Pi)*GAMMA(2/3)-GAMMA(2/3)/x^9+1/2*GAMMA(2/3)/x^6+1/3*GAMMA(2/3) /x^3)
Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx=\frac {10 \, \sqrt {3} x^{9} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 5 \, x^{9} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 10 \, x^{9} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (5 \, x^{6} + 3 \, x^{3} - 18\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{486 \, x^{9}} \]
1/486*(10*sqrt(3)*x^9*arctan(2/3*sqrt(3)*(x^3 - 1)^(1/3) - 1/3*sqrt(3)) - 5*x^9*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 10*x^9*log((x^3 - 1)^(1 /3) + 1) + 3*(5*x^6 + 3*x^3 - 18)*(x^3 - 1)^(1/3))/x^9
Result contains complex when optimal does not.
Time = 3.72 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx=- \frac {\Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{8} \Gamma \left (\frac {11}{3}\right )} \]
Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx=\frac {5}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {5 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 13 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{162 \, {\left ({\left (x^{3} - 1\right )}^{3} + 3 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{2} - 2\right )}} - \frac {5}{486} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{243} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \]
5/243*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) - 5/486*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 5/243*log((x^3 - 1)^(1/3) + 1)
Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx=\frac {5}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {5 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 13 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{162 \, x^{9}} - \frac {5}{486} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{243} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]
5/243*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^9 - 5/486*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 5/243*log(abs((x^3 - 1)^(1/3) + 1))
Time = 6.33 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx=\frac {5\,\ln \left (\frac {25\,{\left (x^3-1\right )}^{1/3}}{6561}+\frac {25}{6561}\right )}{243}+\frac {\frac {13\,{\left (x^3-1\right )}^{4/3}}{162}-\frac {5\,{\left (x^3-1\right )}^{1/3}}{81}+\frac {5\,{\left (x^3-1\right )}^{7/3}}{162}}{3\,{\left (x^3-1\right )}^2+{\left (x^3-1\right )}^3+3\,x^3-2}-\ln \left (\frac {5}{54}-\frac {5\,{\left (x^3-1\right )}^{1/3}}{27}+\frac {\sqrt {3}\,5{}\mathrm {i}}{54}\right )\,\left (\frac {5}{486}+\frac {\sqrt {3}\,5{}\mathrm {i}}{486}\right )+\ln \left (\frac {5\,{\left (x^3-1\right )}^{1/3}}{27}-\frac {5}{54}+\frac {\sqrt {3}\,5{}\mathrm {i}}{54}\right )\,\left (-\frac {5}{486}+\frac {\sqrt {3}\,5{}\mathrm {i}}{486}\right ) \]
(5*log((25*(x^3 - 1)^(1/3))/6561 + 25/6561))/243 + ((13*(x^3 - 1)^(4/3))/1 62 - (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) - log((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)