Integrand size = 11, antiderivative size = 104 \[ \int x \left (-1+x^6\right )^{2/3} \, dx=\frac {1}{6} x^2 \left (-1+x^6\right )^{2/3}-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{3 \sqrt {3}}+\frac {1}{9} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-\frac {1}{18} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
1/6*x^2*(x^6-1)^(2/3)-1/9*arctan(3^(1/2)*x^2/(x^2+2*(x^6-1)^(1/3)))*3^(1/2 )+1/9*ln(-x^2+(x^6-1)^(1/3))-1/18*ln(x^4+x^2*(x^6-1)^(1/3)+(x^6-1)^(2/3))
Time = 0.47 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.96 \[ \int x \left (-1+x^6\right )^{2/3} \, dx=\frac {1}{18} \left (3 x^2 \left (-1+x^6\right )^{2/3}-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )+2 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-\log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \]
(3*x^2*(-1 + x^6)^(2/3) - 2*Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^ 6)^(1/3))] + 2*Log[-x^2 + (-1 + x^6)^(1/3)] - Log[x^4 + x^2*(-1 + x^6)^(1/ 3) + (-1 + x^6)^(2/3)])/18
Time = 0.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.72, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {807, 748, 769}
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.
\(\displaystyle \int x \left (x^6-1\right )^{2/3} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int \left (x^6-1\right )^{2/3}dx^2\) |
\(\Big \downarrow \) 748 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^2 \left (x^6-1\right )^{2/3}-\frac {2}{3} \int \frac {1}{\sqrt [3]{x^6-1}}dx^2\right )\) |
\(\Big \downarrow \) 769 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^2 \left (x^6-1\right )^{2/3}-\frac {2}{3} \left (\frac {\arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^6-1}-x^2\right )\right )\right )\) |
((x^2*(-1 + x^6)^(2/3))/3 - (2*(ArcTan[(1 + (2*x^2)/(-1 + x^6)^(1/3))/Sqrt [3]]/Sqrt[3] - Log[-x^2 + (-1 + x^6)^(1/3)]/2))/3)/2
3.15.87.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Simp[a*n*(p/(n*p + 1)) Int[(a + b*x^n)^(p - 1), x], x] /; Fre eQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || LtQ[Denominat or[p + 1/n], Denominator[p]])
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[(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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.32
method | result | size |
meijerg | \(\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {2}{3}} x^{2} \operatorname {hypergeom}\left (\left [-\frac {2}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right )}{2 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {2}{3}}}\) | \(33\) |
risch | \(\frac {x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}}{6}-\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right )}{3 \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\) | \(46\) |
pseudoelliptic | \(\frac {-3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right )+\ln \left (\frac {x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{4}}\right )-2 \ln \left (\frac {-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}}{x^{2}}\right )}{18 \left (x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right ) \left (-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}\right )}\) | \(129\) |
trager | \(\frac {x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}}{6}+\frac {\ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}-2 x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1\right )}{9}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}-2 x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2\right )}{9}\) | \(200\) |
Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int x \left (-1+x^6\right )^{2/3} \, dx=\frac {1}{6} \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x^{2} + \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) + \frac {1}{9} \, \log \left (-\frac {x^{2} - {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) - \frac {1}{18} \, \log \left (\frac {x^{4} + {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \]
1/6*(x^6 - 1)^(2/3)*x^2 + 1/9*sqrt(3)*arctan(1/3*(sqrt(3)*x^2 + 2*sqrt(3)* (x^6 - 1)^(1/3))/x^2) + 1/9*log(-(x^2 - (x^6 - 1)^(1/3))/x^2) - 1/18*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 = 0.62 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.33 \[ \int x \left (-1+x^6\right )^{2/3} \, dx=- \frac {x^{2} e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {4}{3}\right )} \]
Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int x \left (-1+x^6\right )^{2/3} \, dx=\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{6 \, x^{4} {\left (\frac {x^{6} - 1}{x^{6}} - 1\right )}} - \frac {1}{18} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) + \frac {1}{9} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \]
1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3)/x^2 + 1)) - 1/6*(x^6 - 1 )^(2/3)/(x^4*((x^6 - 1)/x^6 - 1)) - 1/18*log((x^6 - 1)^(1/3)/x^2 + (x^6 - 1)^(2/3)/x^4 + 1) + 1/9*log((x^6 - 1)^(1/3)/x^2 - 1)
\[ \int x \left (-1+x^6\right )^{2/3} \, dx=\int { {\left (x^{6} - 1\right )}^{\frac {2}{3}} x \,d x } \]
Timed out. \[ \int x \left (-1+x^6\right )^{2/3} \, dx=\int x\,{\left (x^6-1\right )}^{2/3} \,d x \]