\(\int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx\) [1732]

Optimal result

Integrand size = 13, antiderivative size = 117 \[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=\frac {1}{324} \left (-1+x^6\right )^{2/3} \left (28 x^2+21 x^8+18 x^{14}\right )+\frac {7 \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{81 \sqrt {3}}-\frac {7}{243} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\frac {7}{486} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \] Output:

1/324*(x^6-1)^(2/3)*(18*x^14+21*x^8+28*x^2)+7/243*arctan(3^(1/2)*x^2/(x^2+ 
2*(x^6-1)^(1/3)))*3^(1/2)-7/243*ln(-x^2+(x^6-1)^(1/3))+7/486*ln(x^4+x^2*(x 
^6-1)^(1/3)+(x^6-1)^(2/3))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 4.45 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.96 \[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=\frac {1}{972} \left (3 x^2 \left (-1+x^6\right )^{2/3} \left (28+21 x^6+18 x^{12}\right )+28 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )-28 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+14 \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \] Input:

Integrate[x^19/(-1 + x^6)^(1/3),x]
 

Output:

(3*x^2*(-1 + x^6)^(2/3)*(28 + 21*x^6 + 18*x^12) + 28*Sqrt[3]*ArcTan[(Sqrt[ 
3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))] - 28*Log[-x^2 + (-1 + x^6)^(1/3)] + 14 
*Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)])/972
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {807, 843, 843, 843, 769}

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.

\(\displaystyle \int \frac {x^{19}}{\sqrt [3]{x^6-1}} \, dx\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {1}{2} \int \frac {x^{18}}{\sqrt [3]{x^6-1}}dx^2\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {1}{2} \left (\frac {7}{9} \int \frac {x^{12}}{\sqrt [3]{x^6-1}}dx^2+\frac {1}{9} \left (x^6-1\right )^{2/3} x^{14}\right )\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {1}{2} \left (\frac {7}{9} \left (\frac {2}{3} \int \frac {x^6}{\sqrt [3]{x^6-1}}dx^2+\frac {1}{6} \left (x^6-1\right )^{2/3} x^8\right )+\frac {1}{9} \left (x^6-1\right )^{2/3} x^{14}\right )\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {1}{2} \left (\frac {7}{9} \left (\frac {2}{3} \left (\frac {1}{3} \int \frac {1}{\sqrt [3]{x^6-1}}dx^2+\frac {1}{3} \left (x^6-1\right )^{2/3} x^2\right )+\frac {1}{6} \left (x^6-1\right )^{2/3} x^8\right )+\frac {1}{9} \left (x^6-1\right )^{2/3} x^{14}\right )\)

\(\Big \downarrow \) 769

\(\displaystyle \frac {1}{2} \left (\frac {7}{9} \left (\frac {2}{3} \left (\frac {1}{3} \left (\frac {\arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^6-1}-x^2\right )\right )+\frac {1}{3} \left (x^6-1\right )^{2/3} x^2\right )+\frac {1}{6} \left (x^6-1\right )^{2/3} x^8\right )+\frac {1}{9} \left (x^6-1\right )^{2/3} x^{14}\right )\)

Input:

Int[x^19/(-1 + x^6)^(1/3),x]
 

Output:

((x^14*(-1 + x^6)^(2/3))/9 + (7*((x^8*(-1 + x^6)^(2/3))/6 + (2*((x^2*(-1 + 
 x^6)^(2/3))/3 + (ArcTan[(1 + (2*x^2)/(-1 + x^6)^(1/3))/Sqrt[3]]/Sqrt[3] - 
 Log[-x^2 + (-1 + x^6)^(1/3)]/2)/3))/3))/9)/2
 

Defintions of rubi rules used

rule 769

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* 
(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 
3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
 

rule 807

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]
 

rule 843

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n 
 - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ 
a*c^n*((m - n + 1)/(b*(m + n*p + 1)))   Int[(c*x)^(m - n)*(a + b*x^n)^p, x] 
, x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* 
p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.67 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28

method	result	size
meijerg	\(\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{20} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {10}{3}\right ], \left [\frac {13}{3}\right ], x^{6}\right )}{20 \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\)	\(33\)
risch	\(\frac {x^{2} \left (18 x^{12}+21 x^{6}+28\right ) \left (x^{6}-1\right )^{\frac {2}{3}}}{324}+\frac {7 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right )}{81 \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\)	\(58\)
pseudoelliptic	\(\frac {-54 \left (x^{6}-1\right )^{\frac {2}{3}} x^{14}-63 x^{8} \left (x^{6}-1\right )^{\frac {2}{3}}-84 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+28 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right )-14 \ln \left (\frac {x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{4}}\right )+28 \ln \left (\frac {-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}}{x^{2}}\right )}{972 \left (x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right )^{3} \left (-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}\right )^{3}}\)	\(155\)
trager	\(\frac {x^{2} \left (18 x^{12}+21 x^{6}+28\right ) \left (x^{6}-1\right )^{\frac {2}{3}}}{324}+\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+4 x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2\right )}{243}-\frac {7 \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{243}+\frac {7 \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right )}{243}\)	\(338\)

Input:

int(x^19/(x^6-1)^(1/3),x,method=_RETURNVERBOSE)
 

Output:

1/20/signum(x^6-1)^(1/3)*(-signum(x^6-1))^(1/3)*x^20*hypergeom([1/3,10/3], 
[13/3],x^6)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=\frac {1}{324} \, {\left (18 \, x^{14} + 21 \, x^{8} + 28 \, x^{2}\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}} - \frac {7}{243} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) - \frac {7}{243} \, \log \left (-\frac {x^{2} - {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) + \frac {7}{486} \, \log \left (\frac {x^{4} + {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \] Input:

integrate(x^19/(x^6-1)^(1/3),x, algorithm="fricas")
 

Output:

1/324*(18*x^14 + 21*x^8 + 28*x^2)*(x^6 - 1)^(2/3) - 7/243*sqrt(3)*arctan(1 
/3*(sqrt(3)*x^2 + 2*sqrt(3)*(x^6 - 1)^(1/3))/x^2) - 7/243*log(-(x^2 - (x^6 
 - 1)^(1/3))/x^2) + 7/486*log((x^4 + (x^6 - 1)^(1/3)*x^2 + (x^6 - 1)^(2/3) 
)/x^4)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.98 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.29 \[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=- \frac {x^{20} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {13}{3}\right )} \] Input:

integrate(x**19/(x**6-1)**(1/3),x)
 

Output:

-x**20*exp(2*I*pi/3)*gamma(10/3)*hyper((1/3, 10/3), (13/3,), x**6)/(6*gamm 
a(13/3))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.24 \[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=-\frac {7}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {\frac {67 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} - \frac {77 \, {\left (x^{6} - 1\right )}^{\frac {5}{3}}}{x^{10}} + \frac {28 \, {\left (x^{6} - 1\right )}^{\frac {8}{3}}}{x^{16}}}{324 \, {\left (\frac {3 \, {\left (x^{6} - 1\right )}}{x^{6}} - \frac {3 \, {\left (x^{6} - 1\right )}^{2}}{x^{12}} + \frac {{\left (x^{6} - 1\right )}^{3}}{x^{18}} - 1\right )}} + \frac {7}{486} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) - \frac {7}{243} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \] Input:

integrate(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)/x^2 + 1)) - 1/324*(67 
*(x^6 - 1)^(2/3)/x^4 - 77*(x^6 - 1)^(5/3)/x^10 + 28*(x^6 - 1)^(8/3)/x^16)/ 
(3*(x^6 - 1)/x^6 - 3*(x^6 - 1)^2/x^12 + (x^6 - 1)^3/x^18 - 1) + 7/486*log( 
(x^6 - 1)^(1/3)/x^2 + (x^6 - 1)^(2/3)/x^4 + 1) - 7/243*log((x^6 - 1)^(1/3) 
/x^2 - 1)
 

Giac [F]

\[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=\int { \frac {x^{19}}{{\left (x^{6} - 1\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(x^19/(x^6-1)^(1/3),x, algorithm="giac")
 

Output:

integrate(x^19/(x^6 - 1)^(1/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=\int \frac {x^{19}}{{\left (x^6-1\right )}^{1/3}} \,d x \] Input:

int(x^19/(x^6 - 1)^(1/3),x)
 

Output:

int(x^19/(x^6 - 1)^(1/3), x)
 

Reduce [F]

\[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=\int \frac {x^{19}}{\left (x^{6}-1\right )^{\frac {1}{3}}}d x \] Input:

int(x^19/(x^6-1)^(1/3),x)
 

Output:

int(x**19/(x**6 - 1)**(1/3),x)