Integrand size = 38, antiderivative size = 105 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=\frac {\left (-1+x^6\right )^{2/3} \left (-1-5 x^3+x^6\right )}{5 x^5}-\frac {2 \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}-\frac {2}{3} \log \left (x+\sqrt [3]{-1+x^6}\right )+\frac {1}{3} \log \left (x^2-x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \] Output:
1/5*(x^6-1)^(2/3)*(x^6-5*x^3-1)/x^5-2/3*arctan(3^(1/2)*x/(-x+2*(x^6-1)^(1/ 3)))*3^(1/2)-2/3*ln(x+(x^6-1)^(1/3))+1/3*ln(x^2-x*(x^6-1)^(1/3)+(x^6-1)^(2 /3))
Time = 1.85 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=\frac {\left (-1+x^6\right )^{2/3} \left (-1-5 x^3+x^6\right )}{5 x^5}-\frac {2 \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}-\frac {2}{3} \log \left (x+\sqrt [3]{-1+x^6}\right )+\frac {1}{3} \log \left (x^2-x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \] Input:
Integrate[((-1 + x^6)^(2/3)*(1 + x^6)*(-1 - x^3 + x^6))/(x^6*(-1 + x^3 + x ^6)),x]
Output:
((-1 + x^6)^(2/3)*(-1 - 5*x^3 + x^6))/(5*x^5) - (2*ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 + x^6)^(1/3))])/Sqrt[3] - (2*Log[x + (-1 + x^6)^(1/3)])/3 + Log[x^ 2 - x*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/3
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.15 (sec) , antiderivative size = 380, normalized size of antiderivative = 3.62, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, 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.
| \(\displaystyle \int \frac {\left (x^6-1\right )^{2/3} \left (x^6+1\right ) \left (x^6-x^3-1\right )}{x^6 \left (x^6+x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\left (x^6-1\right )^{2/3}}{x^6}+\left (x^6-1\right )^{2/3}-\frac {2 \left (x^6-1\right )^{2/3} \left (2 x^3+1\right )}{x^6+x^3-1}+\frac {2 \left (x^6-1\right )^{2/3}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (1-\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x \operatorname {AppellF1}\left (\frac {1}{6},-\frac {2}{3},1,\frac {7}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {2 \left (1+\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {\left (x^6-1\right )^{2/3} x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {\left (x^6-1\right )^{2/3} x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {\left (x^6-1\right )^{2/3} x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}}-\frac {\left (x^6-1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 \left (1-x^6\right )^{2/3} x^5}-\frac {\left (x^6-1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{\left (1-x^6\right )^{2/3} x^2}\) |
Input:
Int[((-1 + x^6)^(2/3)*(1 + x^6)*(-1 - x^3 + x^6))/(x^6*(-1 + x^3 + x^6)),x ]
Output:
(-2*(1 - Sqrt[5])*x*(-1 + x^6)^(2/3)*AppellF1[1/6, -2/3, 1, 7/6, x^6, (2*x ^6)/(3 - Sqrt[5])])/((3 - Sqrt[5])*(1 - x^6)^(2/3)) - (2*(1 + Sqrt[5])*x*( -1 + x^6)^(2/3)*AppellF1[1/6, 1, -2/3, 7/6, (2*x^6)/(3 + Sqrt[5]), x^6])/( (3 + Sqrt[5])*(1 - x^6)^(2/3)) + (x^4*(-1 + x^6)^(2/3)*AppellF1[2/3, -2/3, 1, 5/3, x^6, (2*x^6)/(3 - Sqrt[5])])/((3 - Sqrt[5])*(1 - x^6)^(2/3)) + (x ^4*(-1 + x^6)^(2/3)*AppellF1[2/3, -2/3, 1, 5/3, x^6, (2*x^6)/(3 + Sqrt[5]) ])/((3 + Sqrt[5])*(1 - x^6)^(2/3)) - ((-1 + x^6)^(2/3)*Hypergeometric2F1[- 5/6, -2/3, 1/6, x^6])/(5*x^5*(1 - x^6)^(2/3)) - ((-1 + x^6)^(2/3)*Hypergeo metric2F1[-2/3, -1/3, 2/3, x^6])/(x^2*(1 - x^6)^(2/3)) + (x*(-1 + x^6)^(2/ 3)*Hypergeometric2F1[-2/3, 1/6, 7/6, x^6])/(1 - x^6)^(2/3)
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 = 17.99 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.97
| method | result | size |
| pseudoelliptic | \(\frac {-10 \ln \left (\frac {x +\left (x^{6}-1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+3 \left (x^{6}-5 x^{3}-1\right ) \left (x^{6}-1\right )^{\frac {2}{3}}+5 x^{5} \left (-2 \sqrt {3}\, \arctan \left (\frac {\left (x -2 \left (x^{6}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right )+\ln \left (\frac {x^{2}-x \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{15 x^{5}}\) | \(102\) |
| risch | \(\frac {x^{12}-5 x^{9}-2 x^{6}+5 x^{3}+1}{5 x^{5} \left (x^{6}-1\right )^{\frac {1}{3}}}-\frac {2 \ln \left (-\frac {-3 \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 )^{2} x^{3}-9 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}-6 \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 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-2}{x^{6}+x^{3}-1}\right )}{3}+\frac {2 \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}-9 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) 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 x \left (x^{6}-1\right )^{\frac {2}{3}}-x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-1}{x^{6}+x^{3}-1}\right )}{3}-2 \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}-9 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) 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 x \left (x^{6}-1\right )^{\frac {2}{3}}-x^{3}-3 \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 )\) | \(434\) |
| trager | \(\text {Expression too large to display}\) | \(600\) |
Input:
int((x^6-1)^(2/3)*(x^6+1)*(x^6-x^3-1)/x^6/(x^6+x^3-1),x,method=_RETURNVERB OSE)
Output:
1/15*(-10*ln((x+(x^6-1)^(1/3))/x)*x^5+3*(x^6-5*x^3-1)*(x^6-1)^(2/3)+5*x^5* (-2*3^(1/2)*arctan(1/3*(x-2*(x^6-1)^(1/3))*3^(1/2)/x)+ln((x^2-x*(x^6-1)^(1 /3)+(x^6-1)^(2/3))/x^2)))/x^5
Time = 12.43 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.35 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (\frac {17707979315346691547103487078601066282657059082726673278841963389300888497059669011634 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 18779074824464902023518972945875034013564452605964125877184864112405550428883609929964 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (8791266734992875261237504664599259772605087326251698970792557525513888268399719816592 \, x^{6} + 9326814489551980499445247598236243638058784087870425269964007887066219234311690275757 \, x^{3} - 8791266734992875261237504664599259772605087326251698970792557525513888268399719816592\right )}}{3 \, {\left (9923243904393545413458713816471868889492119646716071835561526356358143878603884871272 \, x^{6} - 8320283165512251371852516195766181258618636197629327742451887394495075584367754599527 \, x^{3} - 9923243904393545413458713816471868889492119646716071835561526356358143878603884871272\right )}}\right ) + 5 \, x^{5} \log \left (\frac {x^{6} + x^{3} + 3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x - 1}{x^{6} + x^{3} - 1}\right ) - 3 \, {\left (x^{6} - 5 \, x^{3} - 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{15 \, x^{5}} \] Input:
integrate((x^6-1)^(2/3)*(x^6+1)*(x^6-x^3-1)/x^6/(x^6+x^3-1),x, algorithm=" fricas")
Output:
-1/15*(10*sqrt(3)*x^5*arctan(1/3*(1770797931534669154710348707860106628265 7059082726673278841963389300888497059669011634*sqrt(3)*(x^6 - 1)^(1/3)*x^2 + 18779074824464902023518972945875034013564452605964125877184864112405550 428883609929964*sqrt(3)*(x^6 - 1)^(2/3)*x + sqrt(3)*(879126673499287526123 7504664599259772605087326251698970792557525513888268399719816592*x^6 + 932 68144895519804994452475982362436380587840878704252699640078870662192343116 90275757*x^3 - 87912667349928752612375046645992597726050873262516989707925 57525513888268399719816592))/(99232439043935454134587138164718688894921196 46716071835561526356358143878603884871272*x^6 - 83202831655122513718525161 95766181258618636197629327742451887394495075584367754599527*x^3 - 99232439 04393545413458713816471868889492119646716071835561526356358143878603884871 272)) + 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)) - 3*(x^6 - 5*x^3 - 1)*(x^6 - 1)^(2/3))/x^5
Timed out. \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=\text {Timed out} \] Input:
integrate((x**6-1)**(2/3)*(x**6+1)*(x**6-x**3-1)/x**6/(x**6+x**3-1),x)
Output:
Timed out
\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - x^{3} - 1\right )} {\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + x^{3} - 1\right )} x^{6}} \,d x } \] Input:
integrate((x^6-1)^(2/3)*(x^6+1)*(x^6-x^3-1)/x^6/(x^6+x^3-1),x, algorithm=" maxima")
Output:
integrate((x^6 - x^3 - 1)*(x^6 + 1)*(x^6 - 1)^(2/3)/((x^6 + x^3 - 1)*x^6), x)
\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - x^{3} - 1\right )} {\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + x^{3} - 1\right )} x^{6}} \,d x } \] Input:
integrate((x^6-1)^(2/3)*(x^6+1)*(x^6-x^3-1)/x^6/(x^6+x^3-1),x, algorithm=" giac")
Output:
integrate((x^6 - x^3 - 1)*(x^6 + 1)*(x^6 - 1)^(2/3)/((x^6 + x^3 - 1)*x^6), x)
Timed out. \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=\int -\frac {{\left (x^6-1\right )}^{2/3}\,\left (x^6+1\right )\,\left (-x^6+x^3+1\right )}{x^6\,\left (x^6+x^3-1\right )} \,d x \] Input:
int(-((x^6 - 1)^(2/3)*(x^6 + 1)*(x^3 - x^6 + 1))/(x^6*(x^3 + x^6 - 1)),x)
Output:
int(-((x^6 - 1)^(2/3)*(x^6 + 1)*(x^3 - x^6 + 1))/(x^6*(x^3 + x^6 - 1)), x)
\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=\frac {\left (x^{6}-1\right )^{\frac {2}{3}} x^{6}-\left (x^{6}-1\right )^{\frac {2}{3}}-10 \left (\int \frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{9}+x^{6}-x^{3}}d x \right ) x^{5}-10 \left (\int \frac {\left (x^{6}-1\right )^{\frac {2}{3}} x^{3}}{x^{6}+x^{3}-1}d x \right ) x^{5}}{5 x^{5}} \] Input:
int((x^6-1)^(2/3)*(x^6+1)*(x^6-x^3-1)/x^6/(x^6+x^3-1),x)
Output:
((x**6 - 1)**(2/3)*x**6 - (x**6 - 1)**(2/3) - 10*int((x**6 - 1)**(2/3)/(x* *9 + x**6 - x**3),x)*x**5 - 10*int(((x**6 - 1)**(2/3)*x**3)/(x**6 + x**3 - 1),x)*x**5)/(5*x**5)