Integrand size = 13, antiderivative size = 104 \[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=\frac {\left (-1+x^6\right )^{2/3} \left (18+21 x^6+28 x^{12}\right )}{324 x^{18}}-\frac {7 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {7}{243} \log \left (1+\sqrt [3]{-1+x^6}\right )+\frac {7}{486} \log \left (1-\sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \] Output:
1/324*(x^6-1)^(2/3)*(28*x^12+21*x^6+18)/x^18+7/243*arctan(-1/3*3^(1/2)+2/3 *(x^6-1)^(1/3)*3^(1/2))*3^(1/2)-7/243*ln(1+(x^6-1)^(1/3))+7/486*ln(1-(x^6- 1)^(1/3)+(x^6-1)^(2/3))
Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=\frac {1}{972} \left (\frac {3 \left (-1+x^6\right )^{2/3} \left (18+21 x^6+28 x^{12}\right )}{x^{18}}-28 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )-28 \log \left (1+\sqrt [3]{-1+x^6}\right )+14 \log \left (1-\sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \] Input:
Integrate[1/(x^19*(-1 + x^6)^(1/3)),x]
Output:
((3*(-1 + x^6)^(2/3)*(18 + 21*x^6 + 28*x^12))/x^18 - 28*Sqrt[3]*ArcTan[(1 - 2*(-1 + x^6)^(1/3))/Sqrt[3]] - 28*Log[1 + (-1 + x^6)^(1/3)] + 14*Log[1 - (-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)])/972
Time = 0.21 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {798, 52, 52, 52, 68, 16, 1083, 217}
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.
| \(\displaystyle \int \frac {1}{x^{19} \sqrt [3]{x^6-1}} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {1}{6} \int \frac {1}{x^{24} \sqrt [3]{x^6-1}}dx^6\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{6} \left (\frac {7}{9} \int \frac {1}{x^{18} \sqrt [3]{x^6-1}}dx^6+\frac {\left (x^6-1\right )^{2/3}}{3 x^{18}}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{6} \left (\frac {7}{9} \left (\frac {2}{3} \int \frac {1}{x^{12} \sqrt [3]{x^6-1}}dx^6+\frac {\left (x^6-1\right )^{2/3}}{2 x^{12}}\right )+\frac {\left (x^6-1\right )^{2/3}}{3 x^{18}}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{6} \left (\frac {7}{9} \left (\frac {2}{3} \left (\frac {1}{3} \int \frac {1}{x^6 \sqrt [3]{x^6-1}}dx^6+\frac {\left (x^6-1\right )^{2/3}}{x^6}\right )+\frac {\left (x^6-1\right )^{2/3}}{2 x^{12}}\right )+\frac {\left (x^6-1\right )^{2/3}}{3 x^{18}}\right )\) |
| \(\Big \downarrow \) 68 |
| \(\displaystyle \frac {1}{6} \left (\frac {7}{9} \left (\frac {2}{3} \left (\frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{\sqrt [3]{x^6-1}+1}d\sqrt [3]{x^6-1}+\frac {3}{2} \int \frac {1}{x^{12}-\sqrt [3]{x^6-1}+1}d\sqrt [3]{x^6-1}+\frac {\log \left (x^6\right )}{2}\right )+\frac {\left (x^6-1\right )^{2/3}}{x^6}\right )+\frac {\left (x^6-1\right )^{2/3}}{2 x^{12}}\right )+\frac {\left (x^6-1\right )^{2/3}}{3 x^{18}}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{6} \left (\frac {7}{9} \left (\frac {2}{3} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^{12}-\sqrt [3]{x^6-1}+1}d\sqrt [3]{x^6-1}+\frac {\log \left (x^6\right )}{2}-\frac {3}{2} \log \left (\sqrt [3]{x^6-1}+1\right )\right )+\frac {\left (x^6-1\right )^{2/3}}{x^6}\right )+\frac {\left (x^6-1\right )^{2/3}}{2 x^{12}}\right )+\frac {\left (x^6-1\right )^{2/3}}{3 x^{18}}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{6} \left (\frac {7}{9} \left (\frac {2}{3} \left (\frac {1}{3} \left (-3 \int \frac {1}{-x^{12}-3}d\left (2 \sqrt [3]{x^6-1}-1\right )+\frac {\log \left (x^6\right )}{2}-\frac {3}{2} \log \left (\sqrt [3]{x^6-1}+1\right )\right )+\frac {\left (x^6-1\right )^{2/3}}{x^6}\right )+\frac {\left (x^6-1\right )^{2/3}}{2 x^{12}}\right )+\frac {\left (x^6-1\right )^{2/3}}{3 x^{18}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{6} \left (\frac {7}{9} \left (\frac {2}{3} \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^6-1}-1}{\sqrt {3}}\right )+\frac {\log \left (x^6\right )}{2}-\frac {3}{2} \log \left (\sqrt [3]{x^6-1}+1\right )\right )+\frac {\left (x^6-1\right )^{2/3}}{x^6}\right )+\frac {\left (x^6-1\right )^{2/3}}{2 x^{12}}\right )+\frac {\left (x^6-1\right )^{2/3}}{3 x^{18}}\right )\) |
Input:
Int[1/(x^19*(-1 + x^6)^(1/3)),x]
Output:
((-1 + x^6)^(2/3)/(3*x^18) + (7*((-1 + x^6)^(2/3)/(2*x^12) + (2*((-1 + x^6 )^(2/3)/x^6 + (Sqrt[3]*ArcTan[(-1 + 2*(-1 + x^6)^(1/3))/Sqrt[3]] + Log[x^6 ]/2 - (3*Log[1 + (-1 + x^6)^(1/3)])/2)/3))/3))/9)/6
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[-(b*c - a*d)/b, 3]}, Simp[Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 5.43 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.09
| method | result | size |
| risch | \(\frac {28 x^{18}-7 x^{12}-3 x^{6}-18}{324 x^{18} \left (x^{6}-1\right )^{\frac {1}{3}}}+\frac {7 \sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{6} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+6 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{486 \pi \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\) | \(113\) |
| pseudoelliptic | \(\frac {-14 \ln \left (1-\left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right ) x^{18}-28 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{6}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right ) x^{18}+28 \ln \left (1+\left (x^{6}-1\right )^{\frac {1}{3}}\right ) x^{18}+\left (-84 x^{12}-63 x^{6}-54\right ) \left (x^{6}-1\right )^{\frac {2}{3}}}{972 {\left (-1+\left (x^{6}-1\right )^{\frac {1}{3}}-\left (x^{6}-1\right )^{\frac {2}{3}}\right )}^{3} {\left (1+\left (x^{6}-1\right )^{\frac {1}{3}}\right )}^{3}}\) | \(119\) |
| meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} \left (-\frac {70 \pi \sqrt {3}\, x^{6} \operatorname {hypergeom}\left (\left [1, 1, \frac {13}{3}\right ], \left [2, 5\right ], x^{6}\right )}{729 \Gamma \left (\frac {2}{3}\right )}-\frac {28 \left (\frac {197}{84}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+6 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{243 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{18}}+\frac {\pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{12}}+\frac {4 \pi \sqrt {3}}{27 \Gamma \left (\frac {2}{3}\right ) x^{6}}\right )}{12 \pi \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\) | \(123\) |
| trager | \(\text {Expression too large to display}\) | \(453\) |
Input:
int(1/x^19/(x^6-1)^(1/3),x,method=_RETURNVERBOSE)
Output:
1/324*(28*x^18-7*x^12-3*x^6-18)/x^18/(x^6-1)^(1/3)+7/486/Pi*3^(1/2)*GAMMA( 2/3)/signum(x^6-1)^(1/3)*(-signum(x^6-1))^(1/3)*(2/9*Pi*3^(1/2)/GAMMA(2/3) *x^6*hypergeom([1,1,4/3],[2,2],x^6)+2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)+6*ln(x) +I*Pi)*Pi*3^(1/2)/GAMMA(2/3))
Time = 0.11 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=\frac {28 \, \sqrt {3} x^{18} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 14 \, x^{18} \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - 28 \, x^{18} \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (28 \, x^{12} + 21 \, x^{6} + 18\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{972 \, x^{18}} \] Input:
integrate(1/x^19/(x^6-1)^(1/3),x, algorithm="fricas")
Output:
1/972*(28*sqrt(3)*x^18*arctan(2/3*sqrt(3)*(x^6 - 1)^(1/3) - 1/3*sqrt(3)) + 14*x^18*log((x^6 - 1)^(2/3) - (x^6 - 1)^(1/3) + 1) - 28*x^18*log((x^6 - 1 )^(1/3) + 1) + 3*(28*x^12 + 21*x^6 + 18)*(x^6 - 1)^(2/3))/x^18
Result contains complex when optimal does not.
Time = 7.37 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.31 \[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=- \frac {\Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{6}}} \right )}}{6 x^{20} \Gamma \left (\frac {13}{3}\right )} \] Input:
integrate(1/x**19/(x**6-1)**(1/3),x)
Output:
-gamma(10/3)*hyper((1/3, 10/3), (13/3,), exp_polar(2*I*pi)/x**6)/(6*x**20* gamma(13/3))
Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=\frac {7}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {28 \, {\left (x^{6} - 1\right )}^{\frac {8}{3}} + 77 \, {\left (x^{6} - 1\right )}^{\frac {5}{3}} + 67 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{324 \, {\left (3 \, x^{6} + {\left (x^{6} - 1\right )}^{3} + 3 \, {\left (x^{6} - 1\right )}^{2} - 2\right )}} + \frac {7}{486} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {7}{243} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) \] Input:
integrate(1/x^19/(x^6-1)^(1/3),x, algorithm="maxima")
Output:
7/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3) - 1)) + 1/324*(28*(x^6 - 1)^(8/3) + 77*(x^6 - 1)^(5/3) + 67*(x^6 - 1)^(2/3))/(3*x^6 + (x^6 - 1)^ 3 + 3*(x^6 - 1)^2 - 2) + 7/486*log((x^6 - 1)^(2/3) - (x^6 - 1)^(1/3) + 1) - 7/243*log((x^6 - 1)^(1/3) + 1)
Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=\frac {7}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {28 \, {\left (x^{6} - 1\right )}^{\frac {8}{3}} + 77 \, {\left (x^{6} - 1\right )}^{\frac {5}{3}} + 67 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{324 \, x^{18}} + \frac {7}{486} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {7}{243} \, \log \left ({\left | {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \] Input:
integrate(1/x^19/(x^6-1)^(1/3),x, algorithm="giac")
Output:
7/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3) - 1)) + 1/324*(28*(x^6 - 1)^(8/3) + 77*(x^6 - 1)^(5/3) + 67*(x^6 - 1)^(2/3))/x^18 + 7/486*log((x ^6 - 1)^(2/3) - (x^6 - 1)^(1/3) + 1) - 7/243*log(abs((x^6 - 1)^(1/3) + 1))
Time = 10.04 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=\frac {\frac {67\,{\left (x^6-1\right )}^{2/3}}{324}+\frac {77\,{\left (x^6-1\right )}^{5/3}}{324}+\frac {7\,{\left (x^6-1\right )}^{8/3}}{81}}{3\,{\left (x^6-1\right )}^2+{\left (x^6-1\right )}^3+3\,x^6-2}-\ln \left (9\,{\left (-\frac {7}{486}+\frac {\sqrt {3}\,7{}\mathrm {i}}{486}\right )}^2+\frac {49\,{\left (x^6-1\right )}^{1/3}}{6561}\right )\,\left (-\frac {7}{486}+\frac {\sqrt {3}\,7{}\mathrm {i}}{486}\right )+\ln \left (9\,{\left (\frac {7}{486}+\frac {\sqrt {3}\,7{}\mathrm {i}}{486}\right )}^2+\frac {49\,{\left (x^6-1\right )}^{1/3}}{6561}\right )\,\left (\frac {7}{486}+\frac {\sqrt {3}\,7{}\mathrm {i}}{486}\right )-\frac {7\,\ln \left (\frac {49\,{\left (x^6-1\right )}^{1/3}}{6561}+\frac {49}{6561}\right )}{243} \] Input:
int(1/(x^19*(x^6 - 1)^(1/3)),x)
Output:
log(9*((3^(1/2)*7i)/486 + 7/486)^2 + (49*(x^6 - 1)^(1/3))/6561)*((3^(1/2)* 7i)/486 + 7/486) - log(9*((3^(1/2)*7i)/486 - 7/486)^2 + (49*(x^6 - 1)^(1/3 ))/6561)*((3^(1/2)*7i)/486 - 7/486) - (7*log((49*(x^6 - 1)^(1/3))/6561 + 4 9/6561))/243 + ((67*(x^6 - 1)^(2/3))/324 + (77*(x^6 - 1)^(5/3))/324 + (7*( x^6 - 1)^(8/3))/81)/(3*(x^6 - 1)^2 + (x^6 - 1)^3 + 3*x^6 - 2)
\[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=\int \frac {1}{\left (x^{6}-1\right )^{\frac {1}{3}} x^{19}}d x \] Input:
int(1/x^19/(x^6-1)^(1/3),x)
Output:
int(1/((x**6 - 1)**(1/3)*x**19),x)