Integrand size = 16, antiderivative size = 115 \[ \int \frac {-1+x}{x^{10} \sqrt [3]{1+x^3}} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (360-405 x-420 x^3+486 x^4+560 x^6-729 x^7\right )}{3240 x^9}+\frac {14 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {14}{243} \log \left (-1+\sqrt [3]{1+x^3}\right )-\frac {7}{243} \log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
1/3240*(x^3+1)^(2/3)*(-729*x^7+560*x^6+486*x^4-420*x^3-405*x+360)/x^9+14/2 43*arctan(1/3*3^(1/2)+2/3*(x^3+1)^(1/3)*3^(1/2))*3^(1/2)+14/243*ln(-1+(x^3 +1)^(1/3))-7/243*ln(1+(x^3+1)^(1/3)+(x^3+1)^(2/3))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.40 \[ \int \frac {-1+x}{x^{10} \sqrt [3]{1+x^3}} \, dx=-\frac {\left (1+x^3\right )^{2/3} \left (5-6 x^3+9 x^6+20 x^8 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},4,\frac {5}{3},1+x^3\right )\right )}{40 x^8} \]
-1/40*((1 + x^3)^(2/3)*(5 - 6*x^3 + 9*x^6 + 20*x^8*Hypergeometric2F1[2/3, 4, 5/3, 1 + x^3]))/x^8
Time = 0.33 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.30, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2383, 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.
| \(\displaystyle \int \frac {x-1}{x^{10} \sqrt [3]{x^3+1}} \, dx\) |
| \(\Big \downarrow \) 2383 |
| \(\displaystyle \int \left (\frac {1}{x^9 \sqrt [3]{x^3+1}}-\frac {1}{x^{10} \sqrt [3]{x^3+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {14 \arctan \left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {14 \left (x^3+1\right )^{2/3}}{81 x^3}+\frac {7}{81} \log \left (1-\sqrt [3]{x^3+1}\right )+\frac {\left (x^3+1\right )^{2/3}}{9 x^9}-\frac {\left (x^3+1\right )^{2/3}}{8 x^8}-\frac {7 \left (x^3+1\right )^{2/3}}{54 x^6}+\frac {3 \left (x^3+1\right )^{2/3}}{20 x^5}-\frac {9 \left (x^3+1\right )^{2/3}}{40 x^2}-\frac {7 \log (x)}{81}\) |
(1 + x^3)^(2/3)/(9*x^9) - (1 + x^3)^(2/3)/(8*x^8) - (7*(1 + x^3)^(2/3))/(5 4*x^6) + (3*(1 + x^3)^(2/3))/(20*x^5) + (14*(1 + x^3)^(2/3))/(81*x^3) - (9 *(1 + x^3)^(2/3))/(40*x^2) + (14*ArcTan[(1 + 2*(1 + x^3)^(1/3))/Sqrt[3]])/ (81*Sqrt[3]) - (7*Log[x])/81 + (7*Log[1 - (1 + x^3)^(1/3)])/81
3.18.7.3.1 Defintions of rubi rules used
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> I nt[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n , p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !IGtQ[m, 0]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.77 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(-\frac {729 x^{10}-560 x^{9}+243 x^{7}-140 x^{6}-81 x^{4}+60 x^{3}+405 x -360}{3240 x^{9} \left (x^{3}+1\right )^{\frac {1}{3}}}+\frac {7 \sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{243 \pi }\) | \(111\) |
| meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {70 \pi \sqrt {3}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {13}{3}\right ], \left [2, 5\right ], -x^{3}\right )}{729 \Gamma \left (\frac {2}{3}\right )}-\frac {28 \left (\frac {197}{84}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )\right ) \pi \sqrt {3}}{243 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{9}}+\frac {\pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{6}}-\frac {4 \pi \sqrt {3}}{27 \Gamma \left (\frac {2}{3}\right ) x^{3}}\right )}{6 \pi }-\frac {\left (\frac {9}{5} x^{6}-\frac {6}{5} x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{8 x^{8}}\) | \(128\) |
| trager | \(-\frac {\left (729 x^{7}-560 x^{6}-486 x^{4}+420 x^{3}+405 x -360\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{3240 x^{9}}+\frac {28 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (-\frac {16 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}-18 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}-30 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}-16 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-30 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+9 \left (x^{3}+1\right )^{\frac {2}{3}}-38 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {1}{3}}+5}{x^{3}}\right )}{243}-\frac {14 \ln \left (-\frac {16 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+34 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+30 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}-16 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+30 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+24 \left (x^{3}+1\right )^{\frac {2}{3}}+22 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {1}{3}}+20}{x^{3}}\right )}{243}-\frac {28 \ln \left (-\frac {16 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+34 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+30 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}-16 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+30 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+24 \left (x^{3}+1\right )^{\frac {2}{3}}+22 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {1}{3}}+20}{x^{3}}\right ) \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{243}\) | \(466\) |
-1/3240*(729*x^10-560*x^9+243*x^7-140*x^6-81*x^4+60*x^3+405*x-360)/x^9/(x^ 3+1)^(1/3)+7/243/Pi*3^(1/2)*GAMMA(2/3)*(-2/9*Pi*3^(1/2)/GAMMA(2/3)*x^3*hyp ergeom([1,1,4/3],[2,2],-x^3)+2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x))*Pi*3^ (1/2)/GAMMA(2/3))
Time = 0.50 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.08 \[ \int \frac {-1+x}{x^{10} \sqrt [3]{1+x^3}} \, dx=-\frac {560 \, \sqrt {3} x^{9} \arctan \left (-\frac {\sqrt {3} {\left (x^{3} + 1\right )} - 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3} + 9}\right ) - 280 \, x^{9} \log \left (\frac {x^{3} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3}}\right ) + 3 \, {\left (729 \, x^{7} - 560 \, x^{6} - 486 \, x^{4} + 420 \, x^{3} + 405 \, x - 360\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{9720 \, x^{9}} \]
-1/9720*(560*sqrt(3)*x^9*arctan(-(sqrt(3)*(x^3 + 1) - 2*sqrt(3)*(x^3 + 1)^ (2/3) + 4*sqrt(3)*(x^3 + 1)^(1/3))/(x^3 + 9)) - 280*x^9*log((x^3 - 3*(x^3 + 1)^(2/3) + 3*(x^3 + 1)^(1/3))/x^3) + 3*(729*x^7 - 560*x^6 - 486*x^4 + 42 0*x^3 + 405*x - 360)*(x^3 + 1)^(2/3))/x^9
Result contains complex when optimal does not.
Time = 6.76 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {-1+x}{x^{10} \sqrt [3]{1+x^3}} \, dx=\frac {2 \left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{3 \Gamma \left (\frac {1}{3}\right )} - \frac {4 \left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{3} \Gamma \left (\frac {1}{3}\right )} + \frac {10 \left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{27 x^{6} \Gamma \left (\frac {1}{3}\right )} + \frac {\Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{10} \Gamma \left (\frac {13}{3}\right )} \]
2*(1 + x**(-3))**(2/3)*gamma(-8/3)/(3*gamma(1/3)) - 4*(1 + x**(-3))**(2/3) *gamma(-8/3)/(9*x**3*gamma(1/3)) + 10*(1 + x**(-3))**(2/3)*gamma(-8/3)/(27 *x**6*gamma(1/3)) + gamma(10/3)*hyper((1/3, 10/3), (13/3,), exp_polar(I*pi )/x**3)/(3*x**10*gamma(13/3))
Time = 0.33 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.26 \[ \int \frac {-1+x}{x^{10} \sqrt [3]{1+x^3}} \, dx=\frac {14}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {28 \, {\left (x^{3} + 1\right )}^{\frac {8}{3}} - 77 \, {\left (x^{3} + 1\right )}^{\frac {5}{3}} + 67 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{162 \, {\left ({\left (x^{3} + 1\right )}^{3} + 3 \, x^{3} - 3 \, {\left (x^{3} + 1\right )}^{2} + 2\right )}} - \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} + \frac {2 \, {\left (x^{3} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} - \frac {{\left (x^{3} + 1\right )}^{\frac {8}{3}}}{8 \, x^{8}} - \frac {7}{243} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {14}{243} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
14/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3) + 1)) + 1/162*(28*(x^ 3 + 1)^(8/3) - 77*(x^3 + 1)^(5/3) + 67*(x^3 + 1)^(2/3))/((x^3 + 1)^3 + 3*x ^3 - 3*(x^3 + 1)^2 + 2) - 1/2*(x^3 + 1)^(2/3)/x^2 + 2/5*(x^3 + 1)^(5/3)/x^ 5 - 1/8*(x^3 + 1)^(8/3)/x^8 - 7/243*log((x^3 + 1)^(2/3) + (x^3 + 1)^(1/3) + 1) + 14/243*log((x^3 + 1)^(1/3) - 1)
\[ \int \frac {-1+x}{x^{10} \sqrt [3]{1+x^3}} \, dx=\int { \frac {x - 1}{{\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{10}} \,d x } \]
Time = 6.00 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.37 \[ \int \frac {-1+x}{x^{10} \sqrt [3]{1+x^3}} \, dx=\frac {14\,\ln \left (\frac {196\,{\left (x^3+1\right )}^{1/3}}{6561}-\frac {196}{6561}\right )}{243}+\ln \left (\frac {196\,{\left (x^3+1\right )}^{1/3}}{6561}-9\,{\left (-\frac {7}{243}+\frac {\sqrt {3}\,7{}\mathrm {i}}{243}\right )}^2\right )\,\left (-\frac {7}{243}+\frac {\sqrt {3}\,7{}\mathrm {i}}{243}\right )-\ln \left (\frac {196\,{\left (x^3+1\right )}^{1/3}}{6561}-9\,{\left (\frac {7}{243}+\frac {\sqrt {3}\,7{}\mathrm {i}}{243}\right )}^2\right )\,\left (\frac {7}{243}+\frac {\sqrt {3}\,7{}\mathrm {i}}{243}\right )+\frac {\frac {67\,{\left (x^3+1\right )}^{2/3}}{162}-\frac {77\,{\left (x^3+1\right )}^{5/3}}{162}+\frac {14\,{\left (x^3+1\right )}^{8/3}}{81}}{{\left (x^3+1\right )}^3-3\,{\left (x^3+1\right )}^2+3\,x^3+2}-\frac {{\left (x^3+1\right )}^{2/3}\,\left (9\,x^6-6\,x^3+5\right )}{40\,x^8} \]
(14*log((196*(x^3 + 1)^(1/3))/6561 - 196/6561))/243 + log((196*(x^3 + 1)^( 1/3))/6561 - 9*((3^(1/2)*7i)/243 - 7/243)^2)*((3^(1/2)*7i)/243 - 7/243) - log((196*(x^3 + 1)^(1/3))/6561 - 9*((3^(1/2)*7i)/243 + 7/243)^2)*((3^(1/2) *7i)/243 + 7/243) + ((67*(x^3 + 1)^(2/3))/162 - (77*(x^3 + 1)^(5/3))/162 + (14*(x^3 + 1)^(8/3))/81)/((x^3 + 1)^3 - 3*(x^3 + 1)^2 + 3*x^3 + 2) - ((x^ 3 + 1)^(2/3)*(9*x^6 - 6*x^3 + 5))/(40*x^8)