Integrand size = 13, antiderivative size = 89 \[ \int \frac {\left (-1+x^6\right )^{2/3}}{x} \, dx=\frac {1}{4} \left (-1+x^6\right )^{2/3}+\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{6} \log \left (1+\sqrt [3]{-1+x^6}\right )-\frac {1}{12} \log \left (1-\sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
1/4*(x^6-1)^(2/3)-1/6*arctan(-1/3*3^(1/2)+2/3*(x^6-1)^(1/3)*3^(1/2))*3^(1/ 2)+1/6*ln(1+(x^6-1)^(1/3))-1/12*ln(1-(x^6-1)^(1/3)+(x^6-1)^(2/3))
Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-1+x^6\right )^{2/3}}{x} \, dx=\frac {1}{12} \left (3 \left (-1+x^6\right )^{2/3}+2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )+2 \log \left (1+\sqrt [3]{-1+x^6}\right )-\log \left (1-\sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \]
(3*(-1 + x^6)^(2/3) + 2*Sqrt[3]*ArcTan[(1 - 2*(-1 + x^6)^(1/3))/Sqrt[3]] + 2*Log[1 + (-1 + x^6)^(1/3)] - Log[1 - (-1 + x^6)^(1/3) + (-1 + x^6)^(2/3) ])/12
Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {798, 60, 68, 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.
| \(\displaystyle \int \frac {\left (x^6-1\right )^{2/3}}{x} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {1}{6} \int \frac {\left (x^6-1\right )^{2/3}}{x^6}dx^6\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{2} \left (x^6-1\right )^{2/3}-\int \frac {1}{x^6 \sqrt [3]{x^6-1}}dx^6\right )\) |
| \(\Big \downarrow \) 68 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{2} \int \frac {1}{\sqrt [3]{x^6-1}+1}d\sqrt [3]{x^6-1}-\frac {3}{2} \int \frac {1}{x^{12}-\sqrt [3]{x^6-1}+1}d\sqrt [3]{x^6-1}+\frac {3}{2} \left (x^6-1\right )^{2/3}-\frac {1}{2} \log \left (x^6\right )\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{x^{12}-\sqrt [3]{x^6-1}+1}d\sqrt [3]{x^6-1}+\frac {3}{2} \left (x^6-1\right )^{2/3}-\frac {1}{2} \log \left (x^6\right )+\frac {3}{2} \log \left (\sqrt [3]{x^6-1}+1\right )\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{6} \left (3 \int \frac {1}{-x^{12}-3}d\left (2 \sqrt [3]{x^6-1}-1\right )+\frac {3}{2} \left (x^6-1\right )^{2/3}-\frac {1}{2} \log \left (x^6\right )+\frac {3}{2} \log \left (\sqrt [3]{x^6-1}+1\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^6-1}-1}{\sqrt {3}}\right )+\frac {3}{2} \left (x^6-1\right )^{2/3}-\frac {\log \left (x^6\right )}{2}+\frac {3}{2} \log \left (\sqrt [3]{x^6-1}+1\right )\right )\) |
((3*(-1 + x^6)^(2/3))/2 - Sqrt[3]*ArcTan[(-1 + 2*(-1 + x^6)^(1/3))/Sqrt[3] ] - Log[x^6]/2 + (3*Log[1 + (-1 + x^6)^(1/3)])/2)/6
3.13.24.3.1 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 + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[-(b*c - a*d)/b, 3]}, Simp[Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) 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]
Time = 7.36 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.74
| method | result | size |
| pseudoelliptic | \(\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{4}+\frac {\ln \left (1+\left (x^{6}-1\right )^{\frac {1}{3}}\right )}{6}-\frac {\ln \left (1-\left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{6}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{6}\) | \(66\) |
| meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{6}-1\right )^{\frac {2}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{6} \operatorname {hypergeom}\left (\left [\frac {1}{3}, 1, 1\right ], \left [2, 2\right ], x^{6}\right )}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+6 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right )}\right )}{18 \pi {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {2}{3}}}\) | \(84\) |
| trager | \(\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {1505024575790858565046 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+94448748446849318458973 x^{6} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-23171389162410581752275 x^{6}-68267310132857019576606 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-96321572850614948162944 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+68267310132857019576606 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+142296551347461340528569 \left (x^{6}-1\right )^{\frac {2}{3}}-164588882983471967739550 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-142296551347461340528569 \left (x^{6}-1\right )^{\frac {1}{3}}+45974978496846392365625}{x^{6}}\right )}{6}-\frac {\ln \left (-\frac {1505024575790858565046 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-91438699295267601328881 x^{6} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-116115113033469041646202 x^{6}+68267310132857019576606 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-96321572850614948162944 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-68267310132857019576606 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+210563861480318360105175 \left (x^{6}-1\right )^{\frac {2}{3}}-28054262717757928586338 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-210563861480318360105175 \left (x^{6}-1\right )^{\frac {1}{3}}+114242288629703411942231}{x^{6}}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{6}-\frac {\ln \left (-\frac {1505024575790858565046 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-91438699295267601328881 x^{6} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-116115113033469041646202 x^{6}+68267310132857019576606 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-96321572850614948162944 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-68267310132857019576606 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+210563861480318360105175 \left (x^{6}-1\right )^{\frac {2}{3}}-28054262717757928586338 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-210563861480318360105175 \left (x^{6}-1\right )^{\frac {1}{3}}+114242288629703411942231}{x^{6}}\right )}{6}\) | \(358\) |
1/4*(x^6-1)^(2/3)+1/6*ln(1+(x^6-1)^(1/3))-1/12*ln(1-(x^6-1)^(1/3)+(x^6-1)^ (2/3))-1/6*3^(1/2)*arctan(1/3*(2*(x^6-1)^(1/3)-1)*3^(1/2))
Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^6\right )^{2/3}}{x} \, dx=-\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{4} \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} - \frac {1}{12} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) \]
-1/6*sqrt(3)*arctan(2/3*sqrt(3)*(x^6 - 1)^(1/3) - 1/3*sqrt(3)) + 1/4*(x^6 - 1)^(2/3) - 1/12*log((x^6 - 1)^(2/3) - (x^6 - 1)^(1/3) + 1) + 1/6*log((x^ 6 - 1)^(1/3) + 1)
Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.42 \[ \int \frac {\left (-1+x^6\right )^{2/3}}{x} \, dx=- \frac {x^{4} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{6}}} \right )}}{6 \Gamma \left (\frac {1}{3}\right )} \]
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.73 \[ \int \frac {\left (-1+x^6\right )^{2/3}}{x} \, dx=-\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{4} \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} - \frac {1}{12} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) \]
-1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3) - 1)) + 1/4*(x^6 - 1)^( 2/3) - 1/12*log((x^6 - 1)^(2/3) - (x^6 - 1)^(1/3) + 1) + 1/6*log((x^6 - 1) ^(1/3) + 1)
Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-1+x^6\right )^{2/3}}{x} \, dx=-\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{4} \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} - \frac {1}{12} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left | {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]
-1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3) - 1)) + 1/4*(x^6 - 1)^( 2/3) - 1/12*log((x^6 - 1)^(2/3) - (x^6 - 1)^(1/3) + 1) + 1/6*log(abs((x^6 - 1)^(1/3) + 1))
Time = 5.91 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^6\right )^{2/3}}{x} \, dx=\frac {\ln \left (\frac {{\left (x^6-1\right )}^{1/3}}{4}+\frac {1}{4}\right )}{6}+\ln \left (9\,{\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^2+\frac {{\left (x^6-1\right )}^{1/3}}{4}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (9\,{\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^2+\frac {{\left (x^6-1\right )}^{1/3}}{4}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\frac {{\left (x^6-1\right )}^{2/3}}{4} \]