\(\int x^9 \sqrt [3]{-1+x^6} \, dx\) [1662]

Optimal result

Integrand size = 13, antiderivative size = 112 \[ \int x^9 \sqrt [3]{-1+x^6} \, dx=\frac {1}{36} \sqrt [3]{-1+x^6} \left (-x^4+3 x^{10}\right )+\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{18 \sqrt {3}}+\frac {1}{54} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-\frac {1}{108} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \] Output:

1/36*(x^6-1)^(1/3)*(3*x^10-x^4)+1/54*arctan(3^(1/2)*x^2/(x^2+2*(x^6-1)^(1/ 
3)))*3^(1/2)+1/54*ln(-x^2+(x^6-1)^(1/3))-1/108*ln(x^4+x^2*(x^6-1)^(1/3)+(x 
^6-1)^(2/3))
 

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int x^9 \sqrt [3]{-1+x^6} \, dx=\frac {1}{108} \left (3 x^4 \sqrt [3]{-1+x^6} \left (-1+3 x^6\right )+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )+2 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-\log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \] Input:

Integrate[x^9*(-1 + x^6)^(1/3),x]
 

Output:

(3*x^4*(-1 + x^6)^(1/3)*(-1 + 3*x^6) + 2*Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 
 + 2*(-1 + x^6)^(1/3))] + 2*Log[-x^2 + (-1 + x^6)^(1/3)] - Log[x^4 + x^2*( 
-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)])/108
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {807, 811, 843, 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.

\(\displaystyle \int x^9 \sqrt [3]{x^6-1} \, dx\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {1}{2} \int x^8 \sqrt [3]{x^6-1}dx^2\)

\(\Big \downarrow \) 811

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} x^{10} \sqrt [3]{x^6-1}-\frac {1}{6} \int \frac {x^8}{\left (x^6-1\right )^{2/3}}dx^2\right )\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} \left (-\frac {2}{3} \int \frac {x^2}{\left (x^6-1\right )^{2/3}}dx^2-\frac {1}{3} \sqrt [3]{x^6-1} x^4\right )+\frac {1}{6} \sqrt [3]{x^6-1} x^{10}\right )\)

\(\Big \downarrow \) 853

\(\displaystyle \frac {1}{2} \left (\frac {1}{6} \left (-\frac {2}{3} \left (-\frac {\arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x^2-\sqrt [3]{x^6-1}\right )\right )-\frac {1}{3} \sqrt [3]{x^6-1} x^4\right )+\frac {1}{6} \sqrt [3]{x^6-1} x^{10}\right )\)

Input:

Int[x^9*(-1 + x^6)^(1/3),x]
 

Output:

((x^10*(-1 + x^6)^(1/3))/6 + (-1/3*(x^4*(-1 + x^6)^(1/3)) - (2*(-(ArcTan[( 
1 + (2*x^2)/(-1 + x^6)^(1/3))/Sqrt[3]]/Sqrt[3]) - Log[x^2 - (-1 + x^6)^(1/ 
3)]/2))/3)/6)/2
 

Defintions of rubi rules used

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 811

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

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]
 

rule 853

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]
 

Maple [C] (warning: unable to verify)

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

Time = 2.14 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.29

method	result	size
meijerg	\(\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} x^{10} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], x^{6}\right )}{10 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}}}\)	\(33\)
risch	\(\frac {x^{4} \left (3 x^{6}-1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{36}-\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {2}{3}} x^{4} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{36 \operatorname {signum}\left (x^{6}-1\right )^{\frac {2}{3}}}\)	\(53\)
pseudoelliptic	\(\frac {9 \left (x^{6}-1\right )^{\frac {1}{3}} x^{10}-3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right )-\ln \left (\frac {x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{4}}\right )+2 \ln \left (\frac {-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}}{x^{2}}\right )}{108 \left (x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right )^{2} \left (-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}\right )^{2}}\)	\(143\)
trager	\(\frac {x^{4} \left (3 x^{6}-1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{36}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{54}-\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+4 x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{54}-\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+4 x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right )}{54}\)	\(325\)

Input:

int(x^9*(x^6-1)^(1/3),x,method=_RETURNVERBOSE)
 

Output:

1/10*signum(x^6-1)^(1/3)/(-signum(x^6-1))^(1/3)*x^10*hypergeom([-1/3,5/3], 
[8/3],x^6)
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.91 \[ \int x^9 \sqrt [3]{-1+x^6} \, dx=-\frac {1}{54} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) + \frac {1}{36} \, {\left (3 \, x^{10} - x^{4}\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}} + \frac {1}{54} \, \log \left (-\frac {x^{2} - {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) - \frac {1}{108} \, \log \left (\frac {x^{4} + {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \] Input:

integrate(x^9*(x^6-1)^(1/3),x, algorithm="fricas")
 

Output:

-1/54*sqrt(3)*arctan(1/3*(sqrt(3)*x^2 + 2*sqrt(3)*(x^6 - 1)^(1/3))/x^2) + 
1/36*(3*x^10 - x^4)*(x^6 - 1)^(1/3) + 1/54*log(-(x^2 - (x^6 - 1)^(1/3))/x^ 
2) - 1/108*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 = 1.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.29 \[ \int x^9 \sqrt [3]{-1+x^6} \, dx=\frac {x^{10} e^{\frac {i \pi }{3}} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {8}{3}\right )} \] Input:

integrate(x**9*(x**6-1)**(1/3),x)
 

Output:

x**10*exp(I*pi/3)*gamma(5/3)*hyper((-1/3, 5/3), (8/3,), x**6)/(6*gamma(8/3 
))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.08 \[ \int x^9 \sqrt [3]{-1+x^6} \, dx=-\frac {1}{54} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {4}{3}}}{x^{8}}}{36 \, {\left (\frac {2 \, {\left (x^{6} - 1\right )}}{x^{6}} - \frac {{\left (x^{6} - 1\right )}^{2}}{x^{12}} - 1\right )}} - \frac {1}{108} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) + \frac {1}{54} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \] Input:

integrate(x^9*(x^6-1)^(1/3),x, algorithm="maxima")
 

Output:

-1/54*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3)/x^2 + 1)) - 1/36*(2*(x 
^6 - 1)^(1/3)/x^2 + (x^6 - 1)^(4/3)/x^8)/(2*(x^6 - 1)/x^6 - (x^6 - 1)^2/x^ 
12 - 1) - 1/108*log((x^6 - 1)^(1/3)/x^2 + (x^6 - 1)^(2/3)/x^4 + 1) + 1/54* 
log((x^6 - 1)^(1/3)/x^2 - 1)
 

Giac [F]

\[ \int x^9 \sqrt [3]{-1+x^6} \, dx=\int { {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{9} \,d x } \] Input:

integrate(x^9*(x^6-1)^(1/3),x, algorithm="giac")
 

Output:

integrate((x^6 - 1)^(1/3)*x^9, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^9 \sqrt [3]{-1+x^6} \, dx=\int x^9\,{\left (x^6-1\right )}^{1/3} \,d x \] Input:

int(x^9*(x^6 - 1)^(1/3),x)
 

Output:

int(x^9*(x^6 - 1)^(1/3), x)
 

Reduce [F]

\[ \int x^9 \sqrt [3]{-1+x^6} \, dx=\frac {\left (x^{6}-1\right )^{\frac {1}{3}} x^{10}}{12}-\frac {\left (x^{6}-1\right )^{\frac {1}{3}} x^{4}}{36}-\frac {\left (\int \frac {x^{3}}{\left (x^{6}-1\right )^{\frac {2}{3}}}d x \right )}{9} \] Input:

int(x^9*(x^6-1)^(1/3),x)
 

Output:

(3*(x**6 - 1)**(1/3)*x**10 - (x**6 - 1)**(1/3)*x**4 - 4*int(((x**6 - 1)**( 
1/3)*x**3)/(x**6 - 1),x))/36