Integrand size = 30, antiderivative size = 119 \[ \int \frac {\left (1+x^6\right ) \left (-1-x^3+x^6\right )^{2/3}}{x^3 \left (-1+x^6\right )} \, dx=\frac {\left (-1-x^3+x^6\right )^{2/3}}{2 x^2}+\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1-x^3+x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (x+\sqrt [3]{-1-x^3+x^6}\right )-\frac {1}{6} \log \left (x^2-x \sqrt [3]{-1-x^3+x^6}+\left (-1-x^3+x^6\right )^{2/3}\right ) \]
1/2*(x^6-x^3-1)^(2/3)/x^2+1/3*arctan(3^(1/2)*x/(-x+2*(x^6-x^3-1)^(1/3)))*3 ^(1/2)+1/3*ln(x+(x^6-x^3-1)^(1/3))-1/6*ln(x^2-x*(x^6-x^3-1)^(1/3)+(x^6-x^3 -1)^(2/3))
Time = 0.31 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.97 \[ \int \frac {\left (1+x^6\right ) \left (-1-x^3+x^6\right )^{2/3}}{x^3 \left (-1+x^6\right )} \, dx=\frac {1}{6} \left (\frac {3 \left (-1-x^3+x^6\right )^{2/3}}{x^2}-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{-1-x^3+x^6}}\right )+2 \log \left (x+\sqrt [3]{-1-x^3+x^6}\right )-\log \left (x^2-x \sqrt [3]{-1-x^3+x^6}+\left (-1-x^3+x^6\right )^{2/3}\right )\right ) \]
((3*(-1 - x^3 + x^6)^(2/3))/x^2 - 2*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(-1 - x^3 + x^6)^(1/3))] + 2*Log[x + (-1 - x^3 + x^6)^(1/3)] - Log[x^2 - x*(-1 - x^3 + x^6)^(1/3) + (-1 - x^3 + 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.
| \(\displaystyle \int \frac {\left (x^6+1\right ) \left (x^6-x^3-1\right )^{2/3}}{x^3 \left (x^6-1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {\left (x^6-x^3-1\right )^{2/3}}{x^3}+\frac {\left (x^6-x^3-1\right )^{2/3} (-x-2)}{3 \left (x^2+x+1\right )}+\frac {2 x \left (x^6-x^3-1\right )^{2/3}}{3 \left (x^2-1\right )}+\frac {(2-x) \left (x^6-x^3-1\right )^{2/3}}{3 \left (x^2-x+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (x^6-x^3-1\right )^{2/3}}{x-1}dx+\frac {1}{3} \int \frac {\left (x^6-x^3-1\right )^{2/3}}{x+1}dx-\frac {1}{3} \left (1+i \sqrt {3}\right ) \int \frac {\left (x^6-x^3-1\right )^{2/3}}{2 x-i \sqrt {3}-1}dx-\frac {1}{3} \left (1-i \sqrt {3}\right ) \int \frac {\left (x^6-x^3-1\right )^{2/3}}{2 x-i \sqrt {3}+1}dx-\frac {1}{3} \left (1-i \sqrt {3}\right ) \int \frac {\left (x^6-x^3-1\right )^{2/3}}{2 x+i \sqrt {3}-1}dx-\frac {1}{3} \left (1+i \sqrt {3}\right ) \int \frac {\left (x^6-x^3-1\right )^{2/3}}{2 x+i \sqrt {3}+1}dx+\frac {\left (x^6-x^3-1\right )^{2/3} \operatorname {AppellF1}\left (-\frac {2}{3},-\frac {2}{3},-\frac {2}{3},\frac {1}{3},\frac {2 x^3}{1+\sqrt {5}},\frac {2 x^3}{1-\sqrt {5}}\right )}{2 x^2 \left (1-\frac {2 x^3}{1-\sqrt {5}}\right )^{2/3} \left (1-\frac {2 x^3}{1+\sqrt {5}}\right )^{2/3}}\) |
3.18.67.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 14.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (x -2 \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x^{2}-\ln \left (\frac {x^{2}-x \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}}+\left (x^{6}-x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+2 \ln \left (\frac {x +\left (x^{6}-x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{2}+3 \left (x^{6}-x^{3}-1\right )^{\frac {2}{3}}}{6 x^{2}}\) | \(119\) |
| risch | \(\frac {\left (x^{6}-x^{3}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\frac {\ln \left (-\frac {x^{6}+3 \left (x^{6}-x^{3}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 \left (x^{6}-x^{3}-1\right )^{\frac {2}{3}} x +\left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-x^{3}-1}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{3}-\frac {\ln \left (\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \left (x^{6}-x^{3}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-\left (x^{6}-x^{3}-1\right )^{\frac {2}{3}} x -2 \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-2 x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{3}-\ln \left (\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \left (x^{6}-x^{3}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-\left (x^{6}-x^{3}-1\right )^{\frac {2}{3}} x -2 \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-2 x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (1+x \right ) \left (x^{2}-x +1\right )}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(556\) |
| trager | \(\text {Expression too large to display}\) | \(710\) |
1/6*(2*3^(1/2)*arctan(1/3*(x-2*(x^6-x^3-1)^(1/3))*3^(1/2)/x)*x^2-ln((x^2-x *(x^6-x^3-1)^(1/3)+(x^6-x^3-1)^(2/3))/x^2)*x^2+2*ln((x+(x^6-x^3-1)^(1/3))/ x)*x^2+3*(x^6-x^3-1)^(2/3))/x^2
Time = 9.48 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.25 \[ \int \frac {\left (1+x^6\right ) \left (-1-x^3+x^6\right )^{2/3}}{x^3 \left (-1+x^6\right )} \, dx=\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {37791663946489640698390389259748112672665344841760398436632573406805797258440392514 \, \sqrt {3} {\left (x^{6} - x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 42616282523552719904247910491772924807300791980535303720609605641285532900565158554 \, \sqrt {3} {\left (x^{6} - x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (18323047168343312092760155949313307647509257018220563551640555707801529868232673857 \, x^{6} + 2412309288531539602928760616012406067317723569387452641988516117239867821062383020 \, x^{3} - 18323047168343312092760155949313307647509257018220563551640555707801529868232673857\right )}}{71058247355948940593342690344230822422479089551095495524443013398313353987294270891 \, x^{6} - 120611919705063540903957449627281556219949205233443553235863268572136995238508326602 \, x^{3} - 71058247355948940593342690344230822422479089551095495524443013398313353987294270891}\right ) + x^{2} \log \left (\frac {x^{6} + 3 \, {\left (x^{6} - x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{6} - x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{x^{6} - 1}\right ) + 3 \, {\left (x^{6} - x^{3} - 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \]
1/6*(2*sqrt(3)*x^2*arctan((37791663946489640698390389259748112672665344841 760398436632573406805797258440392514*sqrt(3)*(x^6 - x^3 - 1)^(1/3)*x^2 + 4 26162825235527199042479104917729248073007919805353037206096056412855329005 65158554*sqrt(3)*(x^6 - x^3 - 1)^(2/3)*x + sqrt(3)*(1832304716834331209276 0155949313307647509257018220563551640555707801529868232673857*x^6 + 241230 92885315396029287606160124060673177235693874526419885161172398678210623830 20*x^3 - 18323047168343312092760155949313307647509257018220563551640555707 801529868232673857))/(7105824735594894059334269034423082242247908955109549 5524443013398313353987294270891*x^6 - 120611919705063540903957449627281556 219949205233443553235863268572136995238508326602*x^3 - 7105824735594894059 3342690344230822422479089551095495524443013398313353987294270891)) + x^2*l og((x^6 + 3*(x^6 - x^3 - 1)^(1/3)*x^2 + 3*(x^6 - x^3 - 1)^(2/3)*x - 1)/(x^ 6 - 1)) + 3*(x^6 - x^3 - 1)^(2/3))/x^2
\[ \int \frac {\left (1+x^6\right ) \left (-1-x^3+x^6\right )^{2/3}}{x^3 \left (-1+x^6\right )} \, dx=\int \frac {\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right ) \left (x^{6} - x^{3} - 1\right )^{\frac {2}{3}}}{x^{3} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
Integral((x**2 + 1)*(x**4 - x**2 + 1)*(x**6 - x**3 - 1)**(2/3)/(x**3*(x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1)), x)
\[ \int \frac {\left (1+x^6\right ) \left (-1-x^3+x^6\right )^{2/3}}{x^3 \left (-1+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{6} + 1\right )}}{{\left (x^{6} - 1\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (1+x^6\right ) \left (-1-x^3+x^6\right )^{2/3}}{x^3 \left (-1+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{6} + 1\right )}}{{\left (x^{6} - 1\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (1+x^6\right ) \left (-1-x^3+x^6\right )^{2/3}}{x^3 \left (-1+x^6\right )} \, dx=\int \frac {\left (x^6+1\right )\,{\left (x^6-x^3-1\right )}^{2/3}}{x^3\,\left (x^6-1\right )} \,d x \]