Integrand size = 37, antiderivative size = 102 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=\frac {\left (1+x^6\right )^{2/3} \left (2+15 x^3+2 x^6\right )}{10 x^5}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^6}}\right )+\log \left (-x+\sqrt [3]{1+x^6}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \] Output:
1/10*(x^6+1)^(2/3)*(2*x^6+15*x^3+2)/x^5-arctan(3^(1/2)*x/(x+2*(x^6+1)^(1/3 )))*3^(1/2)+ln(-x+(x^6+1)^(1/3))-1/2*ln(x^2+x*(x^6+1)^(1/3)+(x^6+1)^(2/3))
Time = 1.79 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=\frac {\left (1+x^6\right )^{2/3} \left (2+15 x^3+2 x^6\right )}{10 x^5}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^6}}\right )+\log \left (-x+\sqrt [3]{1+x^6}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \] Input:
Integrate[((-1 + x^3)*(1 + x^3)^3*(1 + x^6)^(2/3))/(x^6*(1 - x^3 + x^6)),x ]
Output:
((1 + x^6)^(2/3)*(2 + 15*x^3 + 2*x^6))/(10*x^5) - Sqrt[3]*ArcTan[(Sqrt[3]* x)/(x + 2*(1 + x^6)^(1/3))] + Log[-x + (1 + x^6)^(1/3)] - Log[x^2 + x*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/2
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.12 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.82, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, 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.
| \(\displaystyle \int \frac {\left (x^3-1\right ) \left (x^3+1\right )^3 \left (x^6+1\right )^{2/3}}{x^6 \left (x^6-x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (-\frac {\left (x^6+1\right )^{2/3}}{x^6}+\left (x^6+1\right )^{2/3}+\frac {3 \left (x^6+1\right )^{2/3} \left (2 x^3-1\right )}{x^6-x^3+1}-\frac {3 \left (x^6+1\right )^{2/3}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \left (-\sqrt {3}+i\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {2 x^6}{1-i \sqrt {3}},-x^6\right )}{\sqrt {3}+i}+\frac {3 \left (\sqrt {3}+i\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {2 x^6}{1+i \sqrt {3}},-x^6\right )}{-\sqrt {3}+i}+\frac {3 x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},-x^6,-\frac {2 x^6}{1-i \sqrt {3}}\right )}{2 \left (1-i \sqrt {3}\right )}+\frac {3 x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},-x^6,-\frac {2 x^6}{1+i \sqrt {3}}\right )}{2 \left (1+i \sqrt {3}\right )}+x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},-x^6\right )+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},-x^6\right )}{5 x^5}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{2 x^2}\) |
Input:
Int[((-1 + x^3)*(1 + x^3)^3*(1 + x^6)^(2/3))/(x^6*(1 - x^3 + x^6)),x]
Output:
(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]) + (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]) + (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])) + (3*x^4*AppellF 1[2/3, -2/3, 1, 5/3, -x^6, (-2*x^6)/(1 + I*Sqrt[3])])/(2*(1 + I*Sqrt[3])) + Hypergeometric2F1[-5/6, -2/3, 1/6, -x^6]/(5*x^5) + (3*Hypergeometric2F1[ -2/3, -1/3, 2/3, -x^6])/(2*x^2) + x*Hypergeometric2F1[-2/3, 1/6, 7/6, -x^6 ]
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 = 5.46 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.17
| method | result | size |
| pseudoelliptic | \(\frac {2 x^{6} \left (x^{6}+1\right )^{\frac {2}{3}}+10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{6}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}-5 \ln \left (\frac {x^{2}+x \left (x^{6}+1\right )^{\frac {1}{3}}+\left (x^{6}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}+10 \ln \left (\frac {-x +\left (x^{6}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+15 \left (x^{6}+1\right )^{\frac {2}{3}} x^{3}+2 \left (x^{6}+1\right )^{\frac {2}{3}}}{10 x^{5}}\) | \(119\) |
| risch | \(\frac {2 x^{12}+15 x^{9}+4 x^{6}+15 x^{3}+2}{10 x^{5} \left (x^{6}+1\right )^{\frac {1}{3}}}+3 \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 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}} 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 )-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 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}} 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 )-\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 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}} 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 )\) | \(490\) |
| trager | \(\frac {\left (x^{6}+1\right )^{\frac {2}{3}} \left (2 x^{6}+15 x^{3}+2\right )}{10 x^{5}}-3 \ln \left (-\frac {-4959 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{6}+3861 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-370 x^{6}+9918 \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 -10485 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}} x^{2}+3849 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3495 x \left (x^{6}+1\right )^{\frac {2}{3}}-2019 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}-555 x^{3}-4959 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+3861 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-370}{x^{6}-x^{3}+1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+\ln \left (\frac {-6624 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{6}-7179 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-555 x^{6}+13248 \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 +6057 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}} x^{2}-1098 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-2019 x \left (x^{6}+1\right )^{\frac {2}{3}}+3495 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}-185 x^{3}-6624 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-7179 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-555}{x^{6}-x^{3}+1}\right )-\ln \left (-\frac {-4959 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{6}+3861 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-370 x^{6}+9918 \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 -10485 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}} x^{2}+3849 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3495 x \left (x^{6}+1\right )^{\frac {2}{3}}-2019 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}-555 x^{3}-4959 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+3861 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-370}{x^{6}-x^{3}+1}\right )\) | \(607\) |
Input:
int((x^3-1)*(x^3+1)^3*(x^6+1)^(2/3)/x^6/(x^6-x^3+1),x,method=_RETURNVERBOS E)
Output:
1/10*(2*x^6*(x^6+1)^(2/3)+10*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^6+1)^(1/ 3)))*x^5-5*ln((x^2+x*(x^6+1)^(1/3)+(x^6+1)^(2/3))/x^2)*x^5+10*ln((-x+(x^6+ 1)^(1/3))/x)*x^5+15*(x^6+1)^(2/3)*x^3+2*(x^6+1)^(2/3))/x^5
Time = 4.55 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.44 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (\frac {1078 \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} + 196 \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (32 \, x^{6} + 605 \, x^{3} + 32\right )}}{8 \, x^{6} - 1331 \, x^{3} + 8}\right ) - 5 \, x^{5} \log \left (\frac {x^{6} - x^{3} + 3 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} x + 1}{x^{6} - x^{3} + 1}\right ) - {\left (2 \, x^{6} + 15 \, x^{3} + 2\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{10 \, x^{5}} \] Input:
integrate((x^3-1)*(x^3+1)^3*(x^6+1)^(2/3)/x^6/(x^6-x^3+1),x, algorithm="fr icas")
Output:
-1/10*(10*sqrt(3)*x^5*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)) - 5*x^5*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)) - (2*x^6 + 15*x^3 + 2)*(x^6 + 1)^(2/3))/x^5
Timed out. \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=\text {Timed out} \] Input:
integrate((x**3-1)*(x**3+1)**3*(x**6+1)**(2/3)/x**6/(x**6-x**3+1),x)
Output:
Timed out
\[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{3} {\left (x^{3} - 1\right )}}{{\left (x^{6} - x^{3} + 1\right )} x^{6}} \,d x } \] Input:
integrate((x^3-1)*(x^3+1)^3*(x^6+1)^(2/3)/x^6/(x^6-x^3+1),x, algorithm="ma xima")
Output:
integrate((x^6 + 1)^(2/3)*(x^3 + 1)^3*(x^3 - 1)/((x^6 - x^3 + 1)*x^6), x)
\[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{3} {\left (x^{3} - 1\right )}}{{\left (x^{6} - x^{3} + 1\right )} x^{6}} \,d x } \] Input:
integrate((x^3-1)*(x^3+1)^3*(x^6+1)^(2/3)/x^6/(x^6-x^3+1),x, algorithm="gi ac")
Output:
integrate((x^6 + 1)^(2/3)*(x^3 + 1)^3*(x^3 - 1)/((x^6 - x^3 + 1)*x^6), x)
Timed out. \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=\int \frac {\left (x^3-1\right )\,{\left (x^3+1\right )}^3\,{\left (x^6+1\right )}^{2/3}}{x^6\,\left (x^6-x^3+1\right )} \,d x \] Input:
int(((x^3 - 1)*(x^3 + 1)^3*(x^6 + 1)^(2/3))/(x^6*(x^6 - x^3 + 1)),x)
Output:
int(((x^3 - 1)*(x^3 + 1)^3*(x^6 + 1)^(2/3))/(x^6*(x^6 - x^3 + 1)), x)
\[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=\frac {\left (x^{6}+1\right )^{\frac {2}{3}} x^{6}+\left (x^{6}+1\right )^{\frac {2}{3}}-15 \left (\int \frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{x^{9}-x^{6}+x^{3}}d x \right ) x^{5}+15 \left (\int \frac {\left (x^{6}+1\right )^{\frac {2}{3}} x^{3}}{x^{6}-x^{3}+1}d x \right ) x^{5}}{5 x^{5}} \] Input:
int((x^3-1)*(x^3+1)^3*(x^6+1)^(2/3)/x^6/(x^6-x^3+1),x)
Output:
((x**6 + 1)**(2/3)*x**6 + (x**6 + 1)**(2/3) - 15*int((x**6 + 1)**(2/3)/(x* *9 - x**6 + x**3),x)*x**5 + 15*int(((x**6 + 1)**(2/3)*x**3)/(x**6 - x**3 + 1),x)*x**5)/(5*x**5)