Integrand size = 28, antiderivative size = 91 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^2 \left (-1+x^3+x^6\right )} \, dx=\frac {\sqrt [3]{-1+x^6}}{x}+\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (x+\sqrt [3]{-1+x^6}\right )+\frac {1}{6} \log \left (x^2-x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
(x^6-1)^(1/3)/x+1/3*arctan(3^(1/2)*x/(-x+2*(x^6-1)^(1/3)))*3^(1/2)-1/3*ln( x+(x^6-1)^(1/3))+1/6*ln(x^2-x*(x^6-1)^(1/3)+(x^6-1)^(2/3))
Time = 0.95 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^2 \left (-1+x^3+x^6\right )} \, dx=\frac {\sqrt [3]{-1+x^6}}{x}+\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (x+\sqrt [3]{-1+x^6}\right )+\frac {1}{6} \log \left (x^2-x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
(-1 + x^6)^(1/3)/x + ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 + x^6)^(1/3))]/Sqrt[3] - Log[x + (-1 + x^6)^(1/3)]/3 + Log[x^2 - x*(-1 + x^6)^(1/3) + (-1 + x^6) ^(2/3)]/6
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.
| \(\displaystyle \int \frac {\sqrt [3]{x^6-1} \left (x^6+1\right )}{x^2 \left (x^6+x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x \left (-2 x^3-1\right ) \sqrt [3]{x^6-1}}{-x^6-x^3+1}-\frac {\sqrt [3]{x^6-1}}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{5} \left (5-\sqrt {5}\right ) \int \frac {x \sqrt [3]{x^6-1}}{-2 x^3+\sqrt {5}-1}dx-\frac {2 \int \frac {x \sqrt [3]{x^6-1}}{-2 x^3+\sqrt {5}-1}dx}{\sqrt {5}}+\frac {2}{5} \left (5+\sqrt {5}\right ) \int \frac {x \sqrt [3]{x^6-1}}{2 x^3+\sqrt {5}+1}dx-\frac {2 \int \frac {x \sqrt [3]{x^6-1}}{2 x^3+\sqrt {5}+1}dx}{\sqrt {5}}+\frac {\sqrt [3]{x^6-1} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{6},\frac {5}{6},x^6\right )}{x \sqrt [3]{1-x^6}}\) |
3.13.55.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 35.46 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x -2 \ln \left (\frac {x +\left (x^{6}-1\right )^{\frac {1}{3}}}{x}\right ) x +\ln \left (\frac {x^{2}-x \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x +6 \left (x^{6}-1\right )^{\frac {1}{3}}}{6 x}\) | \(87\) |
| trager | \(\frac {\left (x^{6}-1\right )^{\frac {1}{3}}}{x}-\frac {\ln \left (\frac {70909374419458055061928416 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{6}+82108206615454556761360374 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{6}+982989516466759239434865 x^{6}-558411323553232183612686276 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{3}+99846727656473353012060656 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -82085851743145991790924018 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+5062410211283585070811482 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{3}+13680975290524331965154003 x \left (x^{6}-1\right )^{\frac {2}{3}}+30322096566603224133830779 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+142980293304255889372344 x^{3}-70909374419458055061928416 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}-82108206615454556761360374 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )-982989516466759239434865}{x^{6}+x^{3}-1}\right )}{3}+2 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \ln \left (-\frac {17278099695005298031607328 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{6}-22795782064001280395934234 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{6}+155237531277710082402621 x^{6}-136065035098166721998907708 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{3}+23812614971169642412370670 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x +41780455657835417292184854 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+20729247754859831155632456 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{3}-6963409276305902882030809 x \left (x^{6}-1\right )^{\frac {2}{3}}-2994640114444295813302364 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-135524828893238960827685 x^{3}-17278099695005298031607328 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}+22795782064001280395934234 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )-155237531277710082402621}{x^{6}+x^{3}-1}\right )\) | \(401\) |
| risch | \(\frac {\left (x^{6}-1\right )^{\frac {1}{3}}}{x}+\frac {\left (-\frac {\ln \left (\frac {-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{12}-x^{12}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{9}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{9}-3 \left (x^{12}-2 x^{6}+1\right )^{\frac {1}{3}} x^{7}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}+2 x^{6}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{12}-2 x^{6}+1\right )^{\frac {2}{3}} x^{2}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-3 \left (x^{12}-2 x^{6}+1\right )^{\frac {2}{3}} x^{2}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+3 \left (x^{12}-2 x^{6}+1\right )^{\frac {1}{3}} x -3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-1}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (x^{6}+x^{3}-1\right )}\right )}{3}+\operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (-\frac {-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{12}-x^{12}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{9}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{12}-2 x^{6}+1\right )^{\frac {1}{3}} x^{7}+x^{9}-3 \left (x^{12}-2 x^{6}+1\right )^{\frac {1}{3}} x^{7}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}+2 x^{6}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{12}-2 x^{6}+1\right )^{\frac {2}{3}} x^{2}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{12}-2 x^{6}+1\right )^{\frac {1}{3}} x -x^{3}+3 \left (x^{12}-2 x^{6}+1\right )^{\frac {1}{3}} x -3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-1}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (x^{6}+x^{3}-1\right )}\right )\right ) {\left (\left (x^{6}-1\right )^{2}\right )}^{\frac {1}{3}}}{\left (x^{6}-1\right )^{\frac {2}{3}}}\) | \(543\) |
1/6*(2*3^(1/2)*arctan(1/3*3^(1/2)*(x-2*(x^6-1)^(1/3))/x)*x-2*ln((x+(x^6-1) ^(1/3))/x)*x+ln((x^2-x*(x^6-1)^(1/3)+(x^6-1)^(2/3))/x^2)*x+6*(x^6-1)^(1/3) )/x
Time = 9.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^2 \left (-1+x^3+x^6\right )} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (\frac {17707979315346691547103487078601066282657059082726673278841963389300888497059669011634 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 18779074824464902023518972945875034013564452605964125877184864112405550428883609929964 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (8791266734992875261237504664599259772605087326251698970792557525513888268399719816592 \, x^{6} + 9326814489551980499445247598236243638058784087870425269964007887066219234311690275757 \, x^{3} - 8791266734992875261237504664599259772605087326251698970792557525513888268399719816592\right )}}{3 \, {\left (9923243904393545413458713816471868889492119646716071835561526356358143878603884871272 \, x^{6} - 8320283165512251371852516195766181258618636197629327742451887394495075584367754599527 \, x^{3} - 9923243904393545413458713816471868889492119646716071835561526356358143878603884871272\right )}}\right ) - x \log \left (\frac {x^{6} + x^{3} + 3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x - 1}{x^{6} + x^{3} - 1}\right ) + 6 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{6 \, x} \]
1/6*(2*sqrt(3)*x*arctan(1/3*(177079793153466915471034870786010662826570590 82726673278841963389300888497059669011634*sqrt(3)*(x^6 - 1)^(1/3)*x^2 + 18 77907482446490202351897294587503401356445260596412587718486411240555042888 3609929964*sqrt(3)*(x^6 - 1)^(2/3)*x + sqrt(3)*(87912667349928752612375046 64599259772605087326251698970792557525513888268399719816592*x^6 + 93268144 89551980499445247598236243638058784087870425269964007887066219234311690275 757*x^3 - 8791266734992875261237504664599259772605087326251698970792557525 513888268399719816592))/(9923243904393545413458713816471868889492119646716 071835561526356358143878603884871272*x^6 - 8320283165512251371852516195766 181258618636197629327742451887394495075584367754599527*x^3 - 9923243904393 545413458713816471868889492119646716071835561526356358143878603884871272)) - x*log((x^6 + x^3 + 3*(x^6 - 1)^(1/3)*x^2 + 3*(x^6 - 1)^(2/3)*x - 1)/(x^ 6 + x^3 - 1)) + 6*(x^6 - 1)^(1/3))/x
Timed out. \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^2 \left (-1+x^3+x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^2 \left (-1+x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{{\left (x^{6} + x^{3} - 1\right )} x^{2}} \,d x } \]
\[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^2 \left (-1+x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{{\left (x^{6} + x^{3} - 1\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^2 \left (-1+x^3+x^6\right )} \, dx=\int \frac {{\left (x^6-1\right )}^{1/3}\,\left (x^6+1\right )}{x^2\,\left (x^6+x^3-1\right )} \,d x \]