Integrand size = 13, antiderivative size = 76 \[ \int \frac {1}{x \sqrt [3]{-1+x^6}} \, dx=-\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (1+\sqrt [3]{-1+x^6}\right )+\frac {1}{12} \log \left (1-\sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
1/6*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 = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x \sqrt [3]{-1+x^6}} \, dx=\frac {1}{12} \left (-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )-2 \log \left (1+\sqrt [3]{-1+x^6}\right )+\log \left (1-\sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \]
(-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 = 55, normalized size of antiderivative = 0.72, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {798, 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.
\(\displaystyle \int \frac {1}{x \sqrt [3]{x^6-1}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{6} \int \frac {1}{x^6 \sqrt [3]{x^6-1}}dx^6\) |
\(\Big \downarrow \) 68 |
\(\displaystyle \frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{\sqrt [3]{x^6-1}+1}d\sqrt [3]{x^6-1}+\frac {3}{2} \int \frac {1}{x^{12}-\sqrt [3]{x^6-1}+1}d\sqrt [3]{x^6-1}+\frac {\log \left (x^6\right )}{2}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{6} \left (\frac {3}{2} \int \frac {1}{x^{12}-\sqrt [3]{x^6-1}+1}d\sqrt [3]{x^6-1}+\frac {\log \left (x^6\right )}{2}-\frac {3}{2} \log \left (\sqrt [3]{x^6-1}+1\right )\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{6} \left (-3 \int \frac {1}{-x^{12}-3}d\left (2 \sqrt [3]{x^6-1}-1\right )+\frac {\log \left (x^6\right )}{2}-\frac {3}{2} \log \left (\sqrt [3]{x^6-1}+1\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{6} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^6-1}-1}{\sqrt {3}}\right )+\frac {\log \left (x^6\right )}{2}-\frac {3}{2} \log \left (\sqrt [3]{x^6-1}+1\right )\right )\) |
(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
3.11.4.3.1 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[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 = 6.94 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75
method | result | size |
pseudoelliptic | \(-\frac {\ln \left (1+\left (x^{6}-1\right )^{\frac {1}{3}}\right )}{6}+\frac {\ln \left (1-\left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{6}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{6}\) | \(57\) |
meijerg | \(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{6} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+6 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{12 \pi \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\) | \(83\) |
trager | \(-\frac {\ln \left (-\frac {367799827974771138925 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}-25044213566176211456246 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+117987937437234671350173 x^{6}+68267310132857019576606 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+142296551347461340528569 \left (x^{6}-1\right )^{\frac {2}{3}}+210563861480318360105175 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}-23539188990385352891200 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-68267310132857019576606 \left (x^{6}-1\right )^{\frac {1}{3}}+165835740337846693419769 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-234103050470703712996375}{x^{6}}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {1872824403765629703971 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+115747313205494270507277 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}-22803589334435810613350 x^{6}+68267310132857019576606 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-210563861480318360105175 \left (x^{6}-1\right )^{\frac {2}{3}}-142296551347461340528569 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}-119860761841000301054144 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-68267310132857019576606 \left (x^{6}-1\right )^{\frac {1}{3}}-90703099639318059051031 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+22435789506461039474425}{x^{6}}\right )}{6}\) | \(256\) |
-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 = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x \sqrt [3]{-1+x^6}} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{12} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{6} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) \]
1/6*sqrt(3)*arctan(2/3*sqrt(3)*(x^6 - 1)^(1/3) - 1/3*sqrt(3)) + 1/12*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 = 0.54 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.42 \[ \int \frac {1}{x \sqrt [3]{-1+x^6}} \, dx=- \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{6}}} \right )}}{6 x^{2} \Gamma \left (\frac {4}{3}\right )} \]
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x \sqrt [3]{-1+x^6}} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{12} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{6} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) \]
1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3) - 1)) + 1/12*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 = 57, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x \sqrt [3]{-1+x^6}} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{12} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{6} \, \log \left ({\left | {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]
1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3) - 1)) + 1/12*log((x^6 - 1)^(2/3) - (x^6 - 1)^(1/3) + 1) - 1/6*log(abs((x^6 - 1)^(1/3) + 1))
Time = 6.41 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x \sqrt [3]{-1+x^6}} \, dx=-\frac {\ln \left (\frac {{\left (x^6-1\right )}^{1/3}}{4}+\frac {1}{4}\right )}{6}-\ln \left (9\,{\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^2+\frac {{\left (x^6-1\right )}^{1/3}}{4}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (9\,{\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^2+\frac {{\left (x^6-1\right )}^{1/3}}{4}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \]