Integrand size = 30, antiderivative size = 94 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\frac {\left (-1+x^6\right )^{2/3}}{2 x^2}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^6}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \] Output:
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 = 0.95 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\frac {\left (-1+x^6\right )^{2/3}}{2 x^2}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^6}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \] Input:
Integrate[((-1 + x^6)^(2/3)*(1 + x^6))/(x^3*(-1 - x^3 + x^6)),x]
Output:
(-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 = 0.95 (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 |
\(\displaystyle -\frac {\left (1-\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x \operatorname {AppellF1}\left (\frac {1}{6},-\frac {2}{3},1,\frac {7}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {\left (1+\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {\left (x^6-1\right )^{2/3} x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{2 \left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {\left (x^6-1\right )^{2/3} x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{2 \left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {\left (x^6-1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{2 \left (1-x^6\right )^{2/3} x^2}\) |
Input:
Int[((-1 + x^6)^(2/3)*(1 + x^6))/(x^3*(-1 - x^3 + x^6)),x]
Output:
-(((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))
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 = 17.69 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.01
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (x +2 \left (x^{6}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x^{2}-\ln \left (\frac {x^{2}+x \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+2 \ln \left (\frac {-x +\left (x^{6}-1\right )^{\frac {1}{3}}}{x}\right ) x^{2}+3 \left (x^{6}-1\right )^{\frac {2}{3}}}{6 x^{2}}\) | \(95\) |
risch | \(\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{2 x^{2}}-\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 \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 \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 \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 \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 )+\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 \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}+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 )\) | \(473\) |
trager | \(\text {Expression too large to display}\) | \(597\) |
Input:
int((x^6-1)^(2/3)*(x^6+1)/x^3/(x^6-x^3-1),x,method=_RETURNVERBOSE)
Output:
1/6*(2*3^(1/2)*arctan(1/3*(x+2*(x^6-1)^(1/3))*3^(1/2)/x)*x^2-ln((x^2+x*(x^ 6-1)^(1/3)+(x^6-1)^(2/3))/x^2)*x^2+2*ln((-x+(x^6-1)^(1/3))/x)*x^2+3*(x^6-1 )^(2/3))/x^2
Time = 9.31 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.45 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=-\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {473996388635948633452428917614298985996886224511260115036680453514888144148250 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 19325031480489228255674265966448835967818926087643600184123099965366515892788 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (771225779807741020855977802972631216428368740202755221603971931588718036144 \, x^{6} + 245889484278411189833195613987401279765924206559249102388797804808538611984375 \, x^{3} - 771225779807741020855977802972631216428368740202755221603971931588718036144\right )}}{3 \, {\left (15407513785538665202033017569552164636906896740149986002803824712402669144 \, x^{6} - 227351086091515241263579358841494627179170556108548407412281480599473216796875 \, x^{3} - 15407513785538665202033017569552164636906896740149986002803824712402669144\right )}}\right ) - x^{2} \log \left (\frac {x^{6} - x^{3} + 3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x - 1}{x^{6} - x^{3} - 1}\right ) - 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \] Input:
integrate((x^6-1)^(2/3)*(x^6+1)/x^3/(x^6-x^3-1),x, algorithm="fricas")
Output:
-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. \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\text {Timed out} \] Input:
integrate((x**6-1)**(2/3)*(x**6+1)/x**3/(x**6-x**3-1),x)
Output:
Timed out
\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 1\right )} x^{3}} \,d x } \] Input:
integrate((x^6-1)^(2/3)*(x^6+1)/x^3/(x^6-x^3-1),x, algorithm="maxima")
Output:
integrate((x^6 + 1)*(x^6 - 1)^(2/3)/((x^6 - x^3 - 1)*x^3), x)
\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 1\right )} x^{3}} \,d x } \] Input:
integrate((x^6-1)^(2/3)*(x^6+1)/x^3/(x^6-x^3-1),x, algorithm="giac")
Output:
integrate((x^6 + 1)*(x^6 - 1)^(2/3)/((x^6 - x^3 - 1)*x^3), x)
Timed out. \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\int -\frac {{\left (x^6-1\right )}^{2/3}\,\left (x^6+1\right )}{x^3\,\left (-x^6+x^3+1\right )} \,d x \] Input:
int(-((x^6 - 1)^(2/3)*(x^6 + 1))/(x^3*(x^3 - x^6 + 1)),x)
Output:
int(-((x^6 - 1)^(2/3)*(x^6 + 1))/(x^3*(x^3 - x^6 + 1)), x)
\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\frac {\left (x^{6}-1\right )^{\frac {2}{3}}+2 \left (\int \frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{12}-x^{9}-2 x^{6}+x^{3}+1}d x \right ) x^{2}+2 \left (\int \frac {\left (x^{6}-1\right )^{\frac {2}{3}} x^{6}}{x^{12}-x^{9}-2 x^{6}+x^{3}+1}d x \right ) x^{2}}{2 x^{2}} \] Input:
int((x^6-1)^(2/3)*(x^6+1)/x^3/(x^6-x^3-1),x)
Output:
((x**6 - 1)**(2/3) + 2*int((x**6 - 1)**(2/3)/(x**12 - x**9 - 2*x**6 + x**3 + 1),x)*x**2 + 2*int(((x**6 - 1)**(2/3)*x**6)/(x**12 - x**9 - 2*x**6 + x* *3 + 1),x)*x**2)/(2*x**2)