Integrand size = 23, antiderivative size = 105 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3-x^6\right )}{4 x^8}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
1/4*(x^3-1)^(2/3)*(-x^6-2*x^3+1)/x^8+1/3*arctan(3^(1/2)*x/(x+2*(x^3-1)^(1/ 3)))*3^(1/2)-1/3*ln(-x+(x^3-1)^(1/3))+1/6*ln(x^2+x*(x^3-1)^(1/3)+(x^3-1)^( 2/3))
Time = 0.19 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx=\frac {1}{12} \left (-\frac {3 \left (-1+x^3\right )^{2/3} \left (-1+2 x^3+x^6\right )}{x^8}+4 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )-4 \log \left (-x+\sqrt [3]{-1+x^3}\right )+2 \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
((-3*(-1 + x^3)^(2/3)*(-1 + 2*x^3 + x^6))/x^8 + 4*Sqrt[3]*ArcTan[(Sqrt[3]* x)/(x + 2*(-1 + x^3)^(1/3))] - 4*Log[-x + (-1 + x^3)^(1/3)] + 2*Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/12
Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1812, 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.
| \(\displaystyle \int \frac {\left (x^3-1\right )^{2/3} \left (x^6+2 x^3-2\right )}{x^9} \, dx\) |
| \(\Big \downarrow \) 1812 |
| \(\displaystyle \int \left (\frac {\left (x^3-1\right )^{2/3}}{x^3}-\frac {2 \left (x^3-1\right )^{2/3}}{x^9}+\frac {2 \left (x^3-1\right )^{2/3}}{x^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {\left (x^3-1\right )^{5/3}}{4 x^8}+\frac {\left (x^3-1\right )^{5/3}}{4 x^5}-\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\) |
-1/2*(-1 + x^3)^(2/3)/x^2 - (-1 + x^3)^(5/3)/(4*x^8) + (-1 + x^3)^(5/3)/(4 *x^5) + ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[-x + (- 1 + x^3)^(1/3)]/2
3.16.14.3.1 Defintions of rubi rules used
Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*( (d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.67 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.53
| method | result | size |
| risch | \(-\frac {x^{9}+x^{6}-3 x^{3}+1}{4 x^{8} \left (x^{3}-1\right )^{\frac {1}{3}}}+\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\) | \(56\) |
| pseudoelliptic | \(\frac {2 \ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{8}-4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{8}-4 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{8}-3 \left (x^{3}-1\right )^{\frac {2}{3}} \left (x^{6}+2 x^{3}-1\right )}{12 x^{8}}\) | \(105\) |
| meijerg | \(-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [-\frac {2}{3}, -\frac {2}{3}\right ], \left [\frac {1}{3}\right ], x^{3}\right )}{2 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2}}-\frac {2 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {5}{3}}}{5 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{5}}+\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\frac {3}{5} x^{6}-\frac {2}{5} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{8}}\) | \(110\) |
| trager | \(-\frac {\left (x^{6}+2 x^{3}-1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{4 x^{8}}-\frac {\ln \left (21954560 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{3}-2482176 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +4264704 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-1868288 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{3}+16659 x \left (x^{3}-1\right )^{\frac {2}{3}}-6963 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-9361 x^{3}-175636480 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}+452864 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )+6105\right )}{3}+\frac {\ln \left (3713319698432 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{3}+49654493184 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -386189407488 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+322029759232 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{3}-1508552373 x \left (x^{3}-1\right )^{\frac {2}{3}}+1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+250623626 x^{3}-29706557587456 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}-124866866688 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )-79743881\right )}{3}-\frac {256 \ln \left (3713319698432 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{3}+49654493184 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -386189407488 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+322029759232 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{3}-1508552373 x \left (x^{3}-1\right )^{\frac {2}{3}}+1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+250623626 x^{3}-29706557587456 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}-124866866688 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )-79743881\right ) \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )}{3}\) | \(449\) |
-1/4*(x^9+x^6-3*x^3+1)/x^8/(x^3-1)^(1/3)+1/signum(x^3-1)^(1/3)*(-signum(x^ 3-1))^(1/3)*x*hypergeom([1/3,1/3],[4/3],x^3)
Time = 0.46 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx=\frac {4 \, \sqrt {3} x^{8} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} - 7200\right )}}{58653 \, x^{3} - 8000}\right ) - 2 \, x^{8} \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 1\right ) - 3 \, {\left (x^{6} + 2 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{12 \, x^{8}} \]
1/12*(4*sqrt(3)*x^8*arctan(-(25382*sqrt(3)*(x^3 - 1)^(1/3)*x^2 - 13720*sqr t(3)*(x^3 - 1)^(2/3)*x + sqrt(3)*(5831*x^3 - 7200))/(58653*x^3 - 8000)) - 2*x^8*log(-3*(x^3 - 1)^(1/3)*x^2 + 3*(x^3 - 1)^(2/3)*x + 1) - 3*(x^6 + 2*x ^3 - 1)*(x^3 - 1)^(2/3))/x^8
Result contains complex when optimal does not.
Time = 1.97 (sec) , antiderivative size = 469, normalized size of antiderivative = 4.47 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx=2 \left (\begin {cases} \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} - \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases}\right ) - 2 \left (\begin {cases} \frac {3 x^{6} \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {x^{3} \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} + \frac {5 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{9} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{6} \Gamma \left (- \frac {2}{3}\right )} - \frac {7 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {2 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {5 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases}\right ) + \frac {e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} \]
2*Piecewise(((-1 + x**(-3))**(2/3)*exp(-I*pi/3)*gamma(-5/3)/(3*gamma(-2/3) ) - (-1 + x**(-3))**(2/3)*exp(-I*pi/3)*gamma(-5/3)/(3*x**3*gamma(-2/3)), 1 /Abs(x**3) > 1), (-(1 - 1/x**3)**(2/3)*gamma(-5/3)/(3*gamma(-2/3)) + (1 - 1/x**3)**(2/3)*gamma(-5/3)/(3*x**3*gamma(-2/3)), True)) - 2*Piecewise((3*x **6*(-1 + x**(-3))**(2/3)*exp(2*I*pi/3)*gamma(-8/3)/(9*x**6*gamma(-2/3) - 9*x**3*gamma(-2/3)) - x**3*(-1 + x**(-3))**(2/3)*exp(2*I*pi/3)*gamma(-8/3) /(9*x**6*gamma(-2/3) - 9*x**3*gamma(-2/3)) + 5*(-1 + x**(-3))**(2/3)*exp(2 *I*pi/3)*gamma(-8/3)/(9*x**9*gamma(-2/3) - 9*x**6*gamma(-2/3)) - 7*(-1 + x **(-3))**(2/3)*exp(2*I*pi/3)*gamma(-8/3)/(9*x**6*gamma(-2/3) - 9*x**3*gamm a(-2/3)), 1/Abs(x**3) > 1), ((1 - 1/x**3)**(2/3)*gamma(-8/3)/(3*gamma(-2/3 )) + 2*(1 - 1/x**3)**(2/3)*gamma(-8/3)/(9*x**3*gamma(-2/3)) - 5*(1 - 1/x** 3)**(2/3)*gamma(-8/3)/(9*x**6*gamma(-2/3)), True)) + exp(2*I*pi/3)*gamma(- 2/3)*hyper((-2/3, -2/3), (1/3,), x**3)/(3*x**2*gamma(1/3))
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} + \frac {{\left (x^{3} - 1\right )}^{\frac {8}{3}}}{4 \, x^{8}} + \frac {1}{6} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {1}{3} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3)/x + 1)) - 1/2*(x^3 - 1) ^(2/3)/x^2 + 1/4*(x^3 - 1)^(8/3)/x^8 + 1/6*log((x^3 - 1)^(1/3)/x + (x^3 - 1)^(2/3)/x^2 + 1) - 1/3*log((x^3 - 1)^(1/3)/x - 1)
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx=\int { \frac {{\left (x^{6} + 2 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{9}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+2\,x^3-2\right )}{x^9} \,d x \]