\(\int x^{15} \sqrt [3]{-1+x^6} \, dx\) [1733]

Optimal result

Integrand size = 13, antiderivative size = 117 \[ \int x^{15} \sqrt [3]{-1+x^6} \, dx=\frac {1}{324} \sqrt [3]{-1+x^6} \left (-5 x^4-3 x^{10}+18 x^{16}\right )+\frac {5 \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{162 \sqrt {3}}+\frac {5}{486} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-\frac {5}{972} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \] Output:

1/324*(x^6-1)^(1/3)*(18*x^16-3*x^10-5*x^4)+5/486*arctan(3^(1/2)*x^2/(x^2+2 
*(x^6-1)^(1/3)))*3^(1/2)+5/486*ln(-x^2+(x^6-1)^(1/3))-5/972*ln(x^4+x^2*(x^ 
6-1)^(1/3)+(x^6-1)^(2/3))
 

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.96 \[ \int x^{15} \sqrt [3]{-1+x^6} \, dx=\frac {1}{972} \left (3 x^4 \sqrt [3]{-1+x^6} \left (-5-3 x^6+18 x^{12}\right )+10 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )+10 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-5 \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \] Input:

Integrate[x^15*(-1 + x^6)^(1/3),x]
 

Output:

(3*x^4*(-1 + x^6)^(1/3)*(-5 - 3*x^6 + 18*x^12) + 10*Sqrt[3]*ArcTan[(Sqrt[3 
]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))] + 10*Log[-x^2 + (-1 + x^6)^(1/3)] - 5*L 
og[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)])/972
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {807, 811, 843, 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.

\(\displaystyle \int x^{15} \sqrt [3]{x^6-1} \, dx\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {1}{2} \int x^{14} \sqrt [3]{x^6-1}dx^2\)

\(\Big \downarrow \) 811

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} x^{16} \sqrt [3]{x^6-1}-\frac {1}{9} \int \frac {x^{14}}{\left (x^6-1\right )^{2/3}}dx^2\right )\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (-\frac {5}{6} \int \frac {x^8}{\left (x^6-1\right )^{2/3}}dx^2-\frac {1}{6} \sqrt [3]{x^6-1} x^{10}\right )+\frac {1}{9} \sqrt [3]{x^6-1} x^{16}\right )\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (-\frac {5}{6} \left (\frac {2}{3} \int \frac {x^2}{\left (x^6-1\right )^{2/3}}dx^2+\frac {1}{3} \sqrt [3]{x^6-1} x^4\right )-\frac {1}{6} \sqrt [3]{x^6-1} x^{10}\right )+\frac {1}{9} \sqrt [3]{x^6-1} x^{16}\right )\)

\(\Big \downarrow \) 853

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (-\frac {5}{6} \left (\frac {2}{3} \left (-\frac {\arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x^2-\sqrt [3]{x^6-1}\right )\right )+\frac {1}{3} \sqrt [3]{x^6-1} x^4\right )-\frac {1}{6} \sqrt [3]{x^6-1} x^{10}\right )+\frac {1}{9} \sqrt [3]{x^6-1} x^{16}\right )\)

Input:

Int[x^15*(-1 + x^6)^(1/3),x]
 

Output:

((x^16*(-1 + x^6)^(1/3))/9 + (-1/6*(x^10*(-1 + x^6)^(1/3)) - (5*((x^4*(-1 
+ x^6)^(1/3))/3 + (2*(-(ArcTan[(1 + (2*x^2)/(-1 + x^6)^(1/3))/Sqrt[3]]/Sqr 
t[3]) - Log[x^2 - (-1 + x^6)^(1/3)]/2))/3))/6)/9)/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 = 2.43 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28

method	result	size
meijerg	\(\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} x^{16} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {8}{3}\right ], \left [\frac {11}{3}\right ], x^{6}\right )}{16 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}}}\)	\(33\)
risch	\(\frac {x^{4} \left (18 x^{12}-3 x^{6}-5\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{324}-\frac {5 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {2}{3}} x^{4} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{324 \operatorname {signum}\left (x^{6}-1\right )^{\frac {2}{3}}}\)	\(58\)
pseudoelliptic	\(\frac {-54 \left (x^{6}-1\right )^{\frac {1}{3}} x^{16}+9 \left (x^{6}-1\right )^{\frac {1}{3}} x^{10}+15 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right )+5 \ln \left (\frac {x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{4}}\right )-10 \ln \left (\frac {-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}}{x^{2}}\right )}{972 \left (x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right )^{3} \left (-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}\right )^{3}}\)	\(155\)
trager	\(\frac {x^{4} \left (18 x^{12}-3 x^{6}-5\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{324}+\frac {5 \ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}+x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{486}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}-2 x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1\right )}{486}\)	\(199\)

Input:

int(x^15*(x^6-1)^(1/3),x,method=_RETURNVERBOSE)
 

Output:

1/16*signum(x^6-1)^(1/3)/(-signum(x^6-1))^(1/3)*x^16*hypergeom([-1/3,8/3], 
[11/3],x^6)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int x^{15} \sqrt [3]{-1+x^6} \, dx=-\frac {5}{486} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) + \frac {1}{324} \, {\left (18 \, x^{16} - 3 \, x^{10} - 5 \, x^{4}\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}} + \frac {5}{486} \, \log \left (-\frac {x^{2} - {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) - \frac {5}{972} \, \log \left (\frac {x^{4} + {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \] Input:

integrate(x^15*(x^6-1)^(1/3),x, algorithm="fricas")
 

Output:

-5/486*sqrt(3)*arctan(1/3*(sqrt(3)*x^2 + 2*sqrt(3)*(x^6 - 1)^(1/3))/x^2) + 
 1/324*(18*x^16 - 3*x^10 - 5*x^4)*(x^6 - 1)^(1/3) + 5/486*log(-(x^2 - (x^6 
 - 1)^(1/3))/x^2) - 5/972*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 = 4.55 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.31 \[ \int x^{15} \sqrt [3]{-1+x^6} \, dx=- \frac {x^{16} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {11}{3}\right )} \] Input:

integrate(x**15*(x**6-1)**(1/3),x)
 

Output:

-x**16*exp(-2*I*pi/3)*gamma(8/3)*hyper((-1/3, 8/3), (11/3,), x**6)/(6*gamm 
a(11/3))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.24 \[ \int x^{15} \sqrt [3]{-1+x^6} \, dx=-\frac {5}{486} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {\frac {10 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {13 \, {\left (x^{6} - 1\right )}^{\frac {4}{3}}}{x^{8}} - \frac {5 \, {\left (x^{6} - 1\right )}^{\frac {7}{3}}}{x^{14}}}{324 \, {\left (\frac {3 \, {\left (x^{6} - 1\right )}}{x^{6}} - \frac {3 \, {\left (x^{6} - 1\right )}^{2}}{x^{12}} + \frac {{\left (x^{6} - 1\right )}^{3}}{x^{18}} - 1\right )}} - \frac {5}{972} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) + \frac {5}{486} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \] Input:

integrate(x^15*(x^6-1)^(1/3),x, algorithm="maxima")
 

Output:

-5/486*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3)/x^2 + 1)) - 1/324*(10 
*(x^6 - 1)^(1/3)/x^2 + 13*(x^6 - 1)^(4/3)/x^8 - 5*(x^6 - 1)^(7/3)/x^14)/(3 
*(x^6 - 1)/x^6 - 3*(x^6 - 1)^2/x^12 + (x^6 - 1)^3/x^18 - 1) - 5/972*log((x 
^6 - 1)^(1/3)/x^2 + (x^6 - 1)^(2/3)/x^4 + 1) + 5/486*log((x^6 - 1)^(1/3)/x 
^2 - 1)
 

Giac [F]

\[ \int x^{15} \sqrt [3]{-1+x^6} \, dx=\int { {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{15} \,d x } \] Input:

integrate(x^15*(x^6-1)^(1/3),x, algorithm="giac")
 

Output:

integrate((x^6 - 1)^(1/3)*x^15, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^{15} \sqrt [3]{-1+x^6} \, dx=\int x^{15}\,{\left (x^6-1\right )}^{1/3} \,d x \] Input:

int(x^15*(x^6 - 1)^(1/3),x)
 

Output:

int(x^15*(x^6 - 1)^(1/3), x)
 

Reduce [F]

\[ \int x^{15} \sqrt [3]{-1+x^6} \, dx=\frac {\left (x^{6}-1\right )^{\frac {1}{3}} x^{16}}{18}-\frac {\left (x^{6}-1\right )^{\frac {1}{3}} x^{10}}{108}-\frac {5 \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}}{324}-\frac {5 \left (\int \frac {x^{3}}{\left (x^{6}-1\right )^{\frac {2}{3}}}d x \right )}{81} \] Input:

int(x^15*(x^6-1)^(1/3),x)
 

Output:

(18*(x**6 - 1)**(1/3)*x**16 - 3*(x**6 - 1)**(1/3)*x**10 - 5*(x**6 - 1)**(1 
/3)*x**4 - 20*int(((x**6 - 1)**(1/3)*x**3)/(x**6 - 1),x))/324