Integrand size = 18, antiderivative size = 111 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx=\frac {\left (-1-3 x^6\right ) \sqrt [3]{-1+x^6}}{8 x^8}-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\frac {1}{12} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
1/8*(-3*x^6-1)*(x^6-1)^(1/3)/x^8-1/6*arctan(3^(1/2)*x^2/(x^2+2*(x^6-1)^(1/ 3)))*3^(1/2)-1/6*ln(-x^2+(x^6-1)^(1/3))+1/12*ln(x^4+x^2*(x^6-1)^(1/3)+(x^6 -1)^(2/3))
Time = 0.43 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx=\frac {1}{24} \left (-\frac {3 \sqrt [3]{-1+x^6} \left (1+3 x^6\right )}{x^8}-4 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )-4 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+2 \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \]
((-3*(-1 + x^6)^(1/3)*(1 + 3*x^6))/x^8 - 4*Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x ^2 + 2*(-1 + x^6)^(1/3))] - 4*Log[-x^2 + (-1 + x^6)^(1/3)] + 2*Log[x^4 + x ^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)])/24
Time = 0.22 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.77, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {953, 807, 809, 853}
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.
| \(\displaystyle \int \frac {\sqrt [3]{x^6-1} \left (x^6+1\right )}{x^9} \, dx\) |
| \(\Big \downarrow \) 953 |
| \(\displaystyle \int \frac {\sqrt [3]{x^6-1}}{x^3}dx+\frac {\left (x^6-1\right )^{4/3}}{8 x^8}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {1}{2} \int \frac {\sqrt [3]{x^6-1}}{x^4}dx^2+\frac {\left (x^6-1\right )^{4/3}}{8 x^8}\) |
| \(\Big \downarrow \) 809 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {x^2}{\left (x^6-1\right )^{2/3}}dx^2-\frac {\sqrt [3]{x^6-1}}{x^2}\right )+\frac {\left (x^6-1\right )^{4/3}}{8 x^8}\) |
| \(\Big \downarrow \) 853 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{x^6-1}}{x^2}-\frac {1}{2} \log \left (x^2-\sqrt [3]{x^6-1}\right )\right )+\frac {\left (x^6-1\right )^{4/3}}{8 x^8}\) |
(-1 + x^6)^(4/3)/(8*x^8) + (-((-1 + x^6)^(1/3)/x^2) - ArcTan[(1 + (2*x^2)/ (-1 + x^6)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[x^2 - (-1 + x^6)^(1/3)]/2)/2
3.17.45.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Sim p[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp [Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[d/e^n Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && G tQ[m + n, -1]))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.51 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(-\frac {3 x^{12}-2 x^{6}-1}{8 x^{8} \left (x^{6}-1\right )^{\frac {2}{3}}}+\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {2}{3}} x^{4} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{4 \operatorname {signum}\left (x^{6}-1\right )^{\frac {2}{3}}}\) | \(58\) |
| meijerg | \(-\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], x^{6}\right )}{2 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{2}}-\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \left (-x^{6}+1\right )^{\frac {4}{3}}}{8 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{8}}\) | \(66\) |
| pseudoelliptic | \(\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right ) x^{8}+2 \ln \left (\frac {x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{4}}\right ) x^{8}-4 \ln \left (\frac {-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}}{x^{2}}\right ) x^{8}-9 \left (x^{6}-1\right )^{\frac {1}{3}} x^{6}-3 \left (x^{6}-1\right )^{\frac {1}{3}}}{24 x^{8}}\) | \(113\) |
| trager | \(-\frac {\left (3 x^{6}+1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{8 x^{8}}+\frac {128 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \ln \left (-102774784196608 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{6}+77455459320064 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{6}+77053995319296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}-578845773886 x^{6}+77053995319296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-577277555133 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-577277555133 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+6577586188582912 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}-100639379193600 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )+384886980542\right )}{3}+\frac {\ln \left (-102774784196608 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{6}-76652531318528 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{6}-77053995319296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}-277853604670 x^{6}-77053995319296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-276285385917 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-276285385917 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+6577586188582912 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}+49251987095296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )+92130405759\right )}{6}-\frac {128 \ln \left (-102774784196608 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{6}-76652531318528 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{6}-77053995319296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}-277853604670 x^{6}-77053995319296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-276285385917 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-276285385917 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+6577586188582912 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}+49251987095296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )+92130405759\right ) \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )}{3}\) | \(469\) |
-1/8*(3*x^12-2*x^6-1)/x^8/(x^6-1)^(2/3)+1/4/signum(x^6-1)^(2/3)*(-signum(x ^6-1))^(2/3)*x^4*hypergeom([2/3,2/3],[5/3],x^6)
Time = 0.43 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx=-\frac {4 \, \sqrt {3} x^{8} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{4} - 13720 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x^{2} + \sqrt {3} {\left (5831 \, x^{6} - 7200\right )}}{58653 \, x^{6} - 8000}\right ) + 2 \, x^{8} \log \left (-3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{4} + 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x^{2} + 1\right ) + 3 \, {\left (3 \, x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{24 \, x^{8}} \]
-1/24*(4*sqrt(3)*x^8*arctan(-(25382*sqrt(3)*(x^6 - 1)^(1/3)*x^4 - 13720*sq rt(3)*(x^6 - 1)^(2/3)*x^2 + sqrt(3)*(5831*x^6 - 7200))/(58653*x^6 - 8000)) + 2*x^8*log(-3*(x^6 - 1)^(1/3)*x^4 + 3*(x^6 - 1)^(2/3)*x^2 + 1) + 3*(3*x^ 6 + 1)*(x^6 - 1)^(1/3))/x^8
Result contains complex when optimal does not.
Time = 1.78 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx=\begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{6 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {4}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {4}{3}\right )}{6 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} + \frac {e^{\frac {i \pi }{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{2} \Gamma \left (\frac {2}{3}\right )} \]
Piecewise(((-1 + x**(-6))**(1/3)*exp(-2*I*pi/3)*gamma(-4/3)/(6*gamma(-1/3) ) - (-1 + x**(-6))**(1/3)*exp(-2*I*pi/3)*gamma(-4/3)/(6*x**6*gamma(-1/3)), 1/Abs(x**6) > 1), (-(1 - 1/x**6)**(1/3)*gamma(-4/3)/(6*gamma(-1/3)) + (1 - 1/x**6)**(1/3)*gamma(-4/3)/(6*x**6*gamma(-1/3)), True)) + exp(I*pi/3)*ga mma(-1/3)*hyper((-1/3, -1/3), (2/3,), x**6)/(6*x**2*gamma(2/3))
Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{2 \, x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {4}{3}}}{8 \, x^{8}} + \frac {1}{12} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \]
1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3)/x^2 + 1)) - 1/2*(x^6 - 1 )^(1/3)/x^2 + 1/8*(x^6 - 1)^(4/3)/x^8 + 1/12*log((x^6 - 1)^(1/3)/x^2 + (x^ 6 - 1)^(2/3)/x^4 + 1) - 1/6*log((x^6 - 1)^(1/3)/x^2 - 1)
\[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx=\int { \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{9}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx=\int \frac {{\left (x^6-1\right )}^{1/3}\,\left (x^6+1\right )}{x^9} \,d x \]