Integrand size = 30, antiderivative size = 94 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\frac {\left (-1+x^6\right )^{2/3}}{2 x^2}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^6}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
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 = 0.94 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\frac {\left (-1+x^6\right )^{2/3}}{2 x^2}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^6}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
(-1 + x^6)^(2/3)/(2*x^2) - ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^6)^(1/3))]/Sq rt[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 = 0.98 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.33, 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.
| \(\displaystyle \int \frac {\left (x^6-1\right )^{2/3} \left (x^6+1\right )}{x^3 \left (x^6-x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle -\frac {\left (1-\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x \operatorname {AppellF1}\left (\frac {1}{6},-\frac {2}{3},1,\frac {7}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {\left (1+\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {\left (x^6-1\right )^{2/3} x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{2 \left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {\left (x^6-1\right )^{2/3} x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{2 \left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {\left (x^6-1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{2 \left (1-x^6\right )^{2/3} x^2}\) |
-(((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))) - ((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])])/(2*(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]) ])/(2*(3 + Sqrt[5])*(1 - x^6)^(2/3)) + ((-1 + x^6)^(2/3)*Hypergeometric2F1 [-2/3, -1/3, 2/3, x^6])/(2*x^2*(1 - x^6)^(2/3))
3.14.7.3.1 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 = 27.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.01
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}+2 \ln \left (\frac {-x +\left (x^{6}-1\right )^{\frac {1}{3}}}{x}\right ) x^{2}-\ln \left (\frac {x^{2}+x \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+3 \left (x^{6}-1\right )^{\frac {2}{3}}}{6 x^{2}}\) | \(95\) |
| trager | \(\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\frac {\ln \left (\frac {27960018709626208768001196672 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{6}-9793442531921313212690595108 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{6}-5614428583490269128672312324 x^{6}-220185147338306394048009423792 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-45456256378905085721353124616 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x +127282845030980663272778333724 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}-88051906839210946445401966260 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+6818882387672964795952100759 x \left (x^{6}-1\right )^{\frac {2}{3}}+3788021364908757143446093718 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-6327371895679509652948161508 x^{3}-27960018709626208768001196672 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+9793442531921313212690595108 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+5614428583490269128672312324}{x^{6}-x^{3}-1}\right )}{3}+4 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (-\frac {102663836955250635495722282496 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{6}+78258464307289633232711371152 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{6}+1723230319777309741777851531 x^{6}-808477716022598754528812974656 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-45456256378905085721353124616 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -81826588652075577551425209108 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}-9793442531921313212690595108 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}-10606903752581721939398194477 x \left (x^{6}-1\right )^{\frac {2}{3}}+3788021364908757143446093718 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+194166796594626449777786088 x^{3}-102663836955250635495722282496 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-78258464307289633232711371152 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-1723230319777309741777851531}{x^{6}-x^{3}-1}\right )\) | \(406\) |
| risch | \(\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\frac {\ln \left (\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-x^{6}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x \left (x^{6}-1\right )^{\frac {2}{3}}-3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}-x^{3}-1}\right )}{3}-\frac {\ln \left (-\frac {-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-2 x^{6}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-2 x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+2}{x^{6}-x^{3}-1}\right )}{3}-\ln \left (-\frac {-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-2 x^{6}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-2 x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+2}{x^{6}-x^{3}-1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(406\) |
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 = 9.82 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.45 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=-\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {473996388635948633452428917614298985996886224511260115036680453514888144148250 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 19325031480489228255674265966448835967818926087643600184123099965366515892788 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (771225779807741020855977802972631216428368740202755221603971931588718036144 \, x^{6} + 245889484278411189833195613987401279765924206559249102388797804808538611984375 \, x^{3} - 771225779807741020855977802972631216428368740202755221603971931588718036144\right )}}{3 \, {\left (15407513785538665202033017569552164636906896740149986002803824712402669144 \, x^{6} - 227351086091515241263579358841494627179170556108548407412281480599473216796875 \, x^{3} - 15407513785538665202033017569552164636906896740149986002803824712402669144\right )}}\right ) - x^{2} \log \left (\frac {x^{6} - x^{3} + 3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x - 1}{x^{6} - x^{3} - 1}\right ) - 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \]
-1/6*(2*sqrt(3)*x^2*arctan(1/3*(473996388635948633452428917614298985996886 224511260115036680453514888144148250*sqrt(3)*(x^6 - 1)^(1/3)*x^2 + 1932503 1480489228255674265966448835967818926087643600184123099965366515892788*sqr t(3)*(x^6 - 1)^(2/3)*x + sqrt(3)*(7712257798077410208559778029726312164283 68740202755221603971931588718036144*x^6 + 24588948427841118983319561398740 1279765924206559249102388797804808538611984375*x^3 - 771225779807741020855 977802972631216428368740202755221603971931588718036144))/(1540751378553866 5202033017569552164636906896740149986002803824712402669144*x^6 - 227351086 091515241263579358841494627179170556108548407412281480599473216796875*x^3 - 154075137855386652020330175695521646369068967401499860028038247124026691 44)) - 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. \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 1\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 1\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\int -\frac {{\left (x^6-1\right )}^{2/3}\,\left (x^6+1\right )}{x^3\,\left (-x^6+x^3+1\right )} \,d x \]