Integrand size = 13, antiderivative size = 117 \[ \int x^{15} \sqrt [3]{-1+x^6} \, dx=\frac {1}{324} \sqrt [3]{-1+x^6} \left (-5 x^4-3 x^{10}+18 x^{16}\right )+\frac {5 \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{162 \sqrt {3}}+\frac {5}{486} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-\frac {5}{972} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
1/324*(x^6-1)^(1/3)*(18*x^16-3*x^10-5*x^4)+5/486*arctan(3^(1/2)*x^2/(x^2+2 *(x^6-1)^(1/3)))*3^(1/2)+5/486*ln(-x^2+(x^6-1)^(1/3))-5/972*ln(x^4+x^2*(x^ 6-1)^(1/3)+(x^6-1)^(2/3))
Time = 1.11 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.96 \[ \int x^{15} \sqrt [3]{-1+x^6} \, dx=\frac {1}{972} \left (3 x^4 \sqrt [3]{-1+x^6} \left (-5-3 x^6+18 x^{12}\right )+10 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )+10 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-5 \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \]
(3*x^4*(-1 + x^6)^(1/3)*(-5 - 3*x^6 + 18*x^12) + 10*Sqrt[3]*ArcTan[(Sqrt[3 ]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))] + 10*Log[-x^2 + (-1 + x^6)^(1/3)] - 5*L og[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)])/972
Time = 0.23 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {807, 811, 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.
\(\displaystyle \int x^{15} \sqrt [3]{x^6-1} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int x^{14} \sqrt [3]{x^6-1}dx^2\) |
\(\Big \downarrow \) 811 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{9} x^{16} \sqrt [3]{x^6-1}-\frac {1}{9} \int \frac {x^{14}}{\left (x^6-1\right )^{2/3}}dx^2\right )\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (-\frac {5}{6} \int \frac {x^8}{\left (x^6-1\right )^{2/3}}dx^2-\frac {1}{6} \sqrt [3]{x^6-1} x^{10}\right )+\frac {1}{9} \sqrt [3]{x^6-1} x^{16}\right )\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (-\frac {5}{6} \left (\frac {2}{3} \int \frac {x^2}{\left (x^6-1\right )^{2/3}}dx^2+\frac {1}{3} \sqrt [3]{x^6-1} x^4\right )-\frac {1}{6} \sqrt [3]{x^6-1} x^{10}\right )+\frac {1}{9} \sqrt [3]{x^6-1} x^{16}\right )\) |
\(\Big \downarrow \) 853 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (-\frac {5}{6} \left (\frac {2}{3} \left (-\frac {\arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x^2-\sqrt [3]{x^6-1}\right )\right )+\frac {1}{3} \sqrt [3]{x^6-1} x^4\right )-\frac {1}{6} \sqrt [3]{x^6-1} x^{10}\right )+\frac {1}{9} \sqrt [3]{x^6-1} x^{16}\right )\) |
((x^16*(-1 + x^6)^(1/3))/9 + (-1/6*(x^10*(-1 + x^6)^(1/3)) - (5*((x^4*(-1 + x^6)^(1/3))/3 + (2*(-(ArcTan[(1 + (2*x^2)/(-1 + x^6)^(1/3))/Sqrt[3]]/Sqr t[3]) - Log[x^2 - (-1 + x^6)^(1/3)]/2))/3))/6)/9)/2
3.18.33.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
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 = 1.88 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28
method | result | size |
meijerg | \(\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} x^{16} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {8}{3}\right ], \left [\frac {11}{3}\right ], x^{6}\right )}{16 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}}}\) | \(33\) |
risch | \(\frac {x^{4} \left (18 x^{12}-3 x^{6}-5\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{324}-\frac {5 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {2}{3}} x^{4} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{324 \operatorname {signum}\left (x^{6}-1\right )^{\frac {2}{3}}}\) | \(58\) |
pseudoelliptic | \(\frac {-54 \left (x^{6}-1\right )^{\frac {1}{3}} x^{16}+9 \left (x^{6}-1\right )^{\frac {1}{3}} x^{10}+15 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right )+5 \ln \left (\frac {x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{4}}\right )-10 \ln \left (\frac {-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}}{x^{2}}\right )}{972 \left (x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right )^{3} \left (-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}\right )^{3}}\) | \(155\) |
trager | \(\frac {x^{4} \left (18 x^{12}-3 x^{6}-5\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{324}+\frac {5 \ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}+x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{486}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}-2 x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1\right )}{486}\) | \(199\) |
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int x^{15} \sqrt [3]{-1+x^6} \, dx=-\frac {5}{486} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) + \frac {1}{324} \, {\left (18 \, x^{16} - 3 \, x^{10} - 5 \, x^{4}\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}} + \frac {5}{486} \, \log \left (-\frac {x^{2} - {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) - \frac {5}{972} \, \log \left (\frac {x^{4} + {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \]
-5/486*sqrt(3)*arctan(1/3*(sqrt(3)*x^2 + 2*sqrt(3)*(x^6 - 1)^(1/3))/x^2) + 1/324*(18*x^16 - 3*x^10 - 5*x^4)*(x^6 - 1)^(1/3) + 5/486*log(-(x^2 - (x^6 - 1)^(1/3))/x^2) - 5/972*log((x^4 + (x^6 - 1)^(1/3)*x^2 + (x^6 - 1)^(2/3) )/x^4)
Result contains complex when optimal does not.
Time = 3.60 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.31 \[ \int x^{15} \sqrt [3]{-1+x^6} \, dx=- \frac {x^{16} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {11}{3}\right )} \]
Time = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.24 \[ \int x^{15} \sqrt [3]{-1+x^6} \, dx=-\frac {5}{486} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {\frac {10 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {13 \, {\left (x^{6} - 1\right )}^{\frac {4}{3}}}{x^{8}} - \frac {5 \, {\left (x^{6} - 1\right )}^{\frac {7}{3}}}{x^{14}}}{324 \, {\left (\frac {3 \, {\left (x^{6} - 1\right )}}{x^{6}} - \frac {3 \, {\left (x^{6} - 1\right )}^{2}}{x^{12}} + \frac {{\left (x^{6} - 1\right )}^{3}}{x^{18}} - 1\right )}} - \frac {5}{972} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) + \frac {5}{486} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \]
-5/486*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3)/x^2 + 1)) - 1/324*(10 *(x^6 - 1)^(1/3)/x^2 + 13*(x^6 - 1)^(4/3)/x^8 - 5*(x^6 - 1)^(7/3)/x^14)/(3 *(x^6 - 1)/x^6 - 3*(x^6 - 1)^2/x^12 + (x^6 - 1)^3/x^18 - 1) - 5/972*log((x ^6 - 1)^(1/3)/x^2 + (x^6 - 1)^(2/3)/x^4 + 1) + 5/486*log((x^6 - 1)^(1/3)/x ^2 - 1)
\[ \int x^{15} \sqrt [3]{-1+x^6} \, dx=\int { {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{15} \,d x } \]
Timed out. \[ \int x^{15} \sqrt [3]{-1+x^6} \, dx=\int x^{15}\,{\left (x^6-1\right )}^{1/3} \,d x \]