Integrand size = 30, antiderivative size = 94 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (1-x^3+x^6\right )} \, dx=\frac {\left (1+x^6\right )^{2/3}}{2 x^2}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{1+x^6}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \]
1/2*(x^6+1)^(2/3)/x^2-1/3*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 = 0.93 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (1-x^3+x^6\right )} \, dx=\frac {\left (1+x^6\right )^{2/3}}{2 x^2}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{1+x^6}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \]
(1 + x^6)^(2/3)/(2*x^2) - 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
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.87 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.63, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {7279, 2009}
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.
| \(\displaystyle \int \frac {\left (x^6-1\right ) \left (x^6+1\right )^{2/3}}{x^3 \left (x^6-x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\left (2 x^3-1\right ) \left (x^6+1\right )^{2/3}}{x^6-x^3+1}-\frac {\left (x^6+1\right )^{2/3}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (-\sqrt {3}+i\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {2 x^6}{1-i \sqrt {3}},-x^6\right )}{\sqrt {3}+i}+\frac {\left (\sqrt {3}+i\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {2 x^6}{1+i \sqrt {3}},-x^6\right )}{-\sqrt {3}+i}+\frac {x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},-x^6,-\frac {2 x^6}{1-i \sqrt {3}}\right )}{2 \left (1-i \sqrt {3}\right )}+\frac {x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},-x^6,-\frac {2 x^6}{1+i \sqrt {3}}\right )}{2 \left (1+i \sqrt {3}\right )}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{2 x^2}\) |
((I - Sqrt[3])*x*AppellF1[1/6, 1, -2/3, 7/6, (-2*x^6)/(1 - I*Sqrt[3]), -x^ 6])/(I + Sqrt[3]) + ((I + Sqrt[3])*x*AppellF1[1/6, 1, -2/3, 7/6, (-2*x^6)/ (1 + I*Sqrt[3]), -x^6])/(I - Sqrt[3]) + (x^4*AppellF1[2/3, -2/3, 1, 5/3, - x^6, (-2*x^6)/(1 - I*Sqrt[3])])/(2*(1 - I*Sqrt[3])) + (x^4*AppellF1[2/3, - 2/3, 1, 5/3, -x^6, (-2*x^6)/(1 + I*Sqrt[3])])/(2*(1 + I*Sqrt[3])) + Hyperg eometric2F1[-2/3, -1/3, 2/3, -x^6]/(2*x^2)
3.14.8.3.1 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 = 8.45 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.01
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{6}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}+2 \ln \left (\frac {-x +\left (x^{6}+1\right )^{\frac {1}{3}}}{x}\right ) x^{2}-\ln \left (\frac {x^{2}+x \left (x^{6}+1\right )^{\frac {1}{3}}+\left (x^{6}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+3 \left (x^{6}+1\right )^{\frac {2}{3}}}{6 x^{2}}\) | \(95\) |
| risch | \(\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}+18 \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^{6}+1\right )^{\frac {2}{3}} x +9 \left (x^{6}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x^{3}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}-x^{3}+1}\right )-\frac {\ln \left (\frac {6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}-18 \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^{6}+1\right )^{\frac {2}{3}} x +9 \left (x^{6}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x \left (x^{6}+1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}-x^{3}+1}\right )}{3}-\ln \left (\frac {6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}-18 \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^{6}+1\right )^{\frac {2}{3}} x +9 \left (x^{6}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x \left (x^{6}+1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}-x^{3}+1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(467\) |
| trager | \(\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {-36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{6}+27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-2 x^{6}+72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x +45 \left (x^{6}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+30 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-9 x \left (x^{6}+1\right )^{\frac {2}{3}}-9 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}-3 x^{3}-36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2}{x^{6}-x^{3}+1}\right )-\frac {\ln \left (\frac {1665 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{6}+12 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-1653 x^{6}-3330 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+4428 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x +4428 \left (x^{6}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+3861 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-2019 x \left (x^{6}+1\right )^{\frac {2}{3}}-2019 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}-551 x^{3}+1665 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+12 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1653}{x^{6}-x^{3}+1}\right )}{3}-\ln \left (\frac {1665 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{6}+12 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-1653 x^{6}-3330 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+4428 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x +4428 \left (x^{6}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+3861 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-2019 x \left (x^{6}+1\right )^{\frac {2}{3}}-2019 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}-551 x^{3}+1665 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+12 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1653}{x^{6}-x^{3}+1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(605\) |
1/6*(2*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^6+1)^(1/3)))*x^2+2*ln((-x+(x^6 +1)^(1/3))/x)*x^2-ln((x^2+x*(x^6+1)^(1/3)+(x^6+1)^(2/3))/x^2)*x^2+3*(x^6+1 )^(2/3))/x^2
Time = 5.41 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.44 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (1-x^3+x^6\right )} \, dx=-\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {1078 \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} + 196 \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (32 \, x^{6} + 605 \, x^{3} + 32\right )}}{8 \, x^{6} - 1331 \, x^{3} + 8}\right ) - x^{2} \log \left (\frac {x^{6} - x^{3} + 3 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} x + 1}{x^{6} - x^{3} + 1}\right ) - 3 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \]
-1/6*(2*sqrt(3)*x^2*arctan((1078*sqrt(3)*(x^6 + 1)^(1/3)*x^2 + 196*sqrt(3) *(x^6 + 1)^(2/3)*x + sqrt(3)*(32*x^6 + 605*x^3 + 32))/(8*x^6 - 1331*x^3 + 8)) - x^2*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)) - 3*(x^6 + 1)^(2/3))/x^2
Timed out. \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (1-x^3+x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}} {\left (x^{6} - 1\right )}}{{\left (x^{6} - x^{3} + 1\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}} {\left (x^{6} - 1\right )}}{{\left (x^{6} - x^{3} + 1\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (1-x^3+x^6\right )} \, dx=\int \frac {\left (x^6-1\right )\,{\left (x^6+1\right )}^{2/3}}{x^3\,\left (x^6-x^3+1\right )} \,d x \]