Integrand size = 13, antiderivative size = 119 \[ \int x^{13} \sqrt [3]{-1+x^3} \, dx=\frac {\sqrt [3]{-1+x^3} \left (-220 x^2-132 x^5-99 x^8-81 x^{11}+972 x^{14}\right )}{14580}+\frac {22 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{729 \sqrt {3}}+\frac {22 \log \left (-x+\sqrt [3]{-1+x^3}\right )}{2187}-\frac {11 \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )}{2187} \]
1/14580*(x^3-1)^(1/3)*(972*x^14-81*x^11-99*x^8-132*x^5-220*x^2)+22/2187*ar ctan(3^(1/2)*x/(x+2*(x^3-1)^(1/3)))*3^(1/2)+22/2187*ln(-x+(x^3-1)^(1/3))-1 1/2187*ln(x^2+x*(x^3-1)^(1/3)+(x^3-1)^(2/3))
Time = 0.33 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.96 \[ \int x^{13} \sqrt [3]{-1+x^3} \, dx=\frac {3 x^2 \sqrt [3]{-1+x^3} \left (-220-132 x^3-99 x^6-81 x^9+972 x^{12}\right )+440 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )+440 \log \left (-x+\sqrt [3]{-1+x^3}\right )-220 \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )}{43740} \]
(3*x^2*(-1 + x^3)^(1/3)*(-220 - 132*x^3 - 99*x^6 - 81*x^9 + 972*x^12) + 44 0*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))] + 440*Log[-x + (-1 + x^3)^(1/3)] - 220*Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/4374 0
Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.28, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {811, 843, 843, 843, 843, 853}
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.
| \(\displaystyle \int x^{13} \sqrt [3]{x^3-1} \, dx\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {1}{15} x^{14} \sqrt [3]{x^3-1}-\frac {1}{15} \int \frac {x^{13}}{\left (x^3-1\right )^{2/3}}dx\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {1}{15} \left (-\frac {11}{12} \int \frac {x^{10}}{\left (x^3-1\right )^{2/3}}dx-\frac {1}{12} \sqrt [3]{x^3-1} x^{11}\right )+\frac {1}{15} \sqrt [3]{x^3-1} x^{14}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {1}{15} \left (-\frac {11}{12} \left (\frac {8}{9} \int \frac {x^7}{\left (x^3-1\right )^{2/3}}dx+\frac {1}{9} \sqrt [3]{x^3-1} x^8\right )-\frac {1}{12} \sqrt [3]{x^3-1} x^{11}\right )+\frac {1}{15} \sqrt [3]{x^3-1} x^{14}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {1}{15} \left (-\frac {11}{12} \left (\frac {8}{9} \left (\frac {5}{6} \int \frac {x^4}{\left (x^3-1\right )^{2/3}}dx+\frac {1}{6} \sqrt [3]{x^3-1} x^5\right )+\frac {1}{9} \sqrt [3]{x^3-1} x^8\right )-\frac {1}{12} \sqrt [3]{x^3-1} x^{11}\right )+\frac {1}{15} \sqrt [3]{x^3-1} x^{14}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {1}{15} \left (-\frac {11}{12} \left (\frac {8}{9} \left (\frac {5}{6} \left (\frac {2}{3} \int \frac {x}{\left (x^3-1\right )^{2/3}}dx+\frac {1}{3} \sqrt [3]{x^3-1} x^2\right )+\frac {1}{6} \sqrt [3]{x^3-1} x^5\right )+\frac {1}{9} \sqrt [3]{x^3-1} x^8\right )-\frac {1}{12} \sqrt [3]{x^3-1} x^{11}\right )+\frac {1}{15} \sqrt [3]{x^3-1} x^{14}\) |
| \(\Big \downarrow \) 853 |
| \(\displaystyle \frac {1}{15} \left (-\frac {11}{12} \left (\frac {8}{9} \left (\frac {5}{6} \left (\frac {2}{3} \left (-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x-\sqrt [3]{x^3-1}\right )\right )+\frac {1}{3} \sqrt [3]{x^3-1} x^2\right )+\frac {1}{6} \sqrt [3]{x^3-1} x^5\right )+\frac {1}{9} \sqrt [3]{x^3-1} x^8\right )-\frac {1}{12} \sqrt [3]{x^3-1} x^{11}\right )+\frac {1}{15} \sqrt [3]{x^3-1} x^{14}\) |
(x^14*(-1 + x^3)^(1/3))/15 + (-1/12*(x^11*(-1 + x^3)^(1/3)) - (11*((x^8*(- 1 + x^3)^(1/3))/9 + (8*((x^5*(-1 + x^3)^(1/3))/6 + (5*((x^2*(-1 + x^3)^(1/ 3))/3 + (2*(-(ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3]) - Log[ x - (-1 + x^3)^(1/3)]/2))/3))/6))/9))/12)/15
3.18.62.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Sim p[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp [Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.99 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28
| method | result | size |
| meijerg | \(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} x^{14} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {14}{3}\right ], \left [\frac {17}{3}\right ], x^{3}\right )}{14 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\) | \(33\) |
| risch | \(\frac {x^{2} \left (972 x^{12}-81 x^{9}-99 x^{6}-132 x^{3}-220\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{14580}-\frac {11 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{729 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(68\) |
| pseudoelliptic | \(\frac {-220 \ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )-440 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right )+440 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )+\left (2916 x^{14}-243 x^{11}-297 x^{8}-396 x^{5}-660 x^{2}\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{43740 \left (x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )^{5} {\left (x -\left (x^{3}-1\right )^{\frac {1}{3}}\right )}^{5}}\) | \(143\) |
| trager | \(\frac {x^{2} \left (972 x^{12}-81 x^{9}-99 x^{6}-132 x^{3}-220\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{14580}+\frac {22 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{2187}-\frac {22 \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+2 x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{2187}-\frac {22 \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+2 x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{2187}\) | \(256\) |
Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.93 \[ \int x^{13} \sqrt [3]{-1+x^3} \, dx=-\frac {22}{2187} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{14580} \, {\left (972 \, x^{14} - 81 \, x^{11} - 99 \, x^{8} - 132 \, x^{5} - 220 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \frac {22}{2187} \, \log \left (-\frac {x - {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {11}{2187} \, \log \left (\frac {x^{2} + {\left (x^{3} - 1\right )}^{\frac {1}{3}} x + {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
-22/2187*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 - 1)^(1/3))/x) + 1 /14580*(972*x^14 - 81*x^11 - 99*x^8 - 132*x^5 - 220*x^2)*(x^3 - 1)^(1/3) + 22/2187*log(-(x - (x^3 - 1)^(1/3))/x) - 11/2187*log((x^2 + (x^3 - 1)^(1/3 )*x + (x^3 - 1)^(2/3))/x^2)
Result contains complex when optimal does not.
Time = 120.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.30 \[ \int x^{13} \sqrt [3]{-1+x^3} \, dx=- \frac {x^{14} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {14}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {14}{3} \\ \frac {17}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {17}{3}\right )} \]
Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.62 \[ \int x^{13} \sqrt [3]{-1+x^3} \, dx=-\frac {22}{2187} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {\frac {440 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {1555 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}}}{x^{4}} - \frac {1815 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}}}{x^{7}} + \frac {1012 \, {\left (x^{3} - 1\right )}^{\frac {10}{3}}}{x^{10}} - \frac {220 \, {\left (x^{3} - 1\right )}^{\frac {13}{3}}}{x^{13}}}{14580 \, {\left (\frac {5 \, {\left (x^{3} - 1\right )}}{x^{3}} - \frac {10 \, {\left (x^{3} - 1\right )}^{2}}{x^{6}} + \frac {10 \, {\left (x^{3} - 1\right )}^{3}}{x^{9}} - \frac {5 \, {\left (x^{3} - 1\right )}^{4}}{x^{12}} + \frac {{\left (x^{3} - 1\right )}^{5}}{x^{15}} - 1\right )}} - \frac {11}{2187} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {22}{2187} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
-22/2187*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3)/x + 1)) - 1/14580*( 440*(x^3 - 1)^(1/3)/x + 1555*(x^3 - 1)^(4/3)/x^4 - 1815*(x^3 - 1)^(7/3)/x^ 7 + 1012*(x^3 - 1)^(10/3)/x^10 - 220*(x^3 - 1)^(13/3)/x^13)/(5*(x^3 - 1)/x ^3 - 10*(x^3 - 1)^2/x^6 + 10*(x^3 - 1)^3/x^9 - 5*(x^3 - 1)^4/x^12 + (x^3 - 1)^5/x^15 - 1) - 11/2187*log((x^3 - 1)^(1/3)/x + (x^3 - 1)^(2/3)/x^2 + 1) + 22/2187*log((x^3 - 1)^(1/3)/x - 1)
\[ \int x^{13} \sqrt [3]{-1+x^3} \, dx=\int { {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{13} \,d x } \]
Timed out. \[ \int x^{13} \sqrt [3]{-1+x^3} \, dx=\int x^{13}\,{\left (x^3-1\right )}^{1/3} \,d x \]