\(\int x^7 (-1+x^6)^{2/3} \, dx\) [1663]

Optimal result

Integrand size = 13, antiderivative size = 112 \[ \int x^7 \left (-1+x^6\right )^{2/3} \, dx=\frac {1}{36} \left (-1+x^6\right )^{2/3} \left (-2 x^2+3 x^8\right )-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{18 \sqrt {3}}+\frac {1}{54} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-\frac {1}{108} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]

[Out]

1/36*(x^6-1)^(2/3)*(3*x^8-2*x^2)-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))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {281, 285, 327, 245} \[ \int x^7 \left (-1+x^6\right )^{2/3} \, dx=-\frac {\arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{18 \sqrt {3}}+\frac {1}{12} \left (x^6-1\right )^{2/3} x^8-\frac {1}{18} \left (x^6-1\right )^{2/3} x^2+\frac {1}{36} \log \left (x^2-\sqrt [3]{x^6-1}\right ) \]

[In]

Int[x^7*(-1 + x^6)^(2/3),x]

[Out]

-1/18*(x^2*(-1 + x^6)^(2/3)) + (x^8*(-1 + x^6)^(2/3))/12 - ArcTan[(1 + (2*x^2)/(-1 + x^6)^(1/3))/Sqrt[3]]/(18*
Sqrt[3]) + Log[x^2 - (-1 + x^6)^(1/3)]/36

Rule 245

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]]/(Sq
rt[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 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[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 285

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] + Dist[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]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x^3 \left (-1+x^3\right )^{2/3} \, dx,x,x^2\right ) \\ & = \frac {1}{12} x^8 \left (-1+x^6\right )^{2/3}-\frac {1}{6} \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^3}} \, dx,x,x^2\right ) \\ & = -\frac {1}{18} x^2 \left (-1+x^6\right )^{2/3}+\frac {1}{12} x^8 \left (-1+x^6\right )^{2/3}-\frac {1}{18} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,x^2\right ) \\ & = -\frac {1}{18} x^2 \left (-1+x^6\right )^{2/3}+\frac {1}{12} x^8 \left (-1+x^6\right )^{2/3}-\frac {\arctan \left (\frac {1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}}{\sqrt {3}}\right )}{18 \sqrt {3}}+\frac {1}{36} \log \left (x^2-\sqrt [3]{-1+x^6}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int x^7 \left (-1+x^6\right )^{2/3} \, dx=\frac {1}{108} \left (3 x^2 \left (-1+x^6\right )^{2/3} \left (-2+3 x^6\right )-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )+2 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-\log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \]

[In]

Integrate[x^7*(-1 + x^6)^(2/3),x]

[Out]

(3*x^2*(-1 + x^6)^(2/3)*(-2 + 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

Maple [C] (warning: unable to verify)

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

Time = 2.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.29

method	result	size
meijerg	\(\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {2}{3}} x^{8} \operatorname {hypergeom}\left (\left [-\frac {2}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], x^{6}\right )}{8 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {2}{3}}}\)	\(33\)
risch	\(\frac {x^{2} \left (3 x^{6}-2\right ) \left (x^{6}-1\right )^{\frac {2}{3}}}{36}-\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right )}{18 \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\)	\(53\)
pseudoelliptic	\(\frac {9 x^{8} \left (x^{6}-1\right )^{\frac {2}{3}}-6 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right )-\ln \left (\frac {x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{4}}\right )+2 \ln \left (\frac {-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}}{x^{2}}\right )}{108 \left (x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right )^{2} \left (-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}\right )^{2}}\)	\(143\)
trager	\(\frac {x^{2} \left (3 x^{6}-2\right ) \left (x^{6}-1\right )^{\frac {2}{3}}}{36}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+4 x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right )}{54}-\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{54}-\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{54}\)	\(297\)

[In]

int(x^7*(x^6-1)^(2/3),x,method=_RETURNVERBOSE)

[Out]

1/8*signum(x^6-1)^(2/3)/(-signum(x^6-1))^(2/3)*x^8*hypergeom([-2/3,4/3],[7/3],x^6)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.91 \[ \int x^7 \left (-1+x^6\right )^{2/3} \, dx=\frac {1}{36} \, {\left (3 \, x^{8} - 2 \, x^{2}\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}} + \frac {1}{54} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) + \frac {1}{54} \, \log \left (-\frac {x^{2} - {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) - \frac {1}{108} \, \log \left (\frac {x^{4} + {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \]

[In]

integrate(x^7*(x^6-1)^(2/3),x, algorithm="fricas")

[Out]

1/36*(3*x^8 - 2*x^2)*(x^6 - 1)^(2/3) + 1/54*sqrt(3)*arctan(1/3*(sqrt(3)*x^2 + 2*sqrt(3)*(x^6 - 1)^(1/3))/x^2)
+ 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 = 1.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.30 \[ \int x^7 \left (-1+x^6\right )^{2/3} \, dx=\frac {x^{8} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {7}{3}\right )} \]

[In]

integrate(x**7*(x**6-1)**(2/3),x)

[Out]

x**8*exp(2*I*pi/3)*gamma(4/3)*hyper((-2/3, 4/3), (7/3,), x**6)/(6*gamma(7/3))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.08 \[ \int x^7 \left (-1+x^6\right )^{2/3} \, dx=\frac {1}{54} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {\frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + \frac {2 \, {\left (x^{6} - 1\right )}^{\frac {5}{3}}}{x^{10}}}{36 \, {\left (\frac {2 \, {\left (x^{6} - 1\right )}}{x^{6}} - \frac {{\left (x^{6} - 1\right )}^{2}}{x^{12}} - 1\right )}} - \frac {1}{108} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) + \frac {1}{54} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \]

[In]

integrate(x^7*(x^6-1)^(2/3),x, algorithm="maxima")

[Out]

1/54*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3)/x^2 + 1)) - 1/36*((x^6 - 1)^(2/3)/x^4 + 2*(x^6 - 1)^(5/3)/x
^10)/(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) + 1/5
4*log((x^6 - 1)^(1/3)/x^2 - 1)

Giac [F]

\[ \int x^7 \left (-1+x^6\right )^{2/3} \, dx=\int { {\left (x^{6} - 1\right )}^{\frac {2}{3}} x^{7} \,d x } \]

[In]

integrate(x^7*(x^6-1)^(2/3),x, algorithm="giac")

[Out]

integrate((x^6 - 1)^(2/3)*x^7, x)

Mupad [F(-1)]

Timed out. \[ \int x^7 \left (-1+x^6\right )^{2/3} \, dx=\int x^7\,{\left (x^6-1\right )}^{2/3} \,d x \]

[In]

int(x^7*(x^6 - 1)^(2/3),x)

[Out]

int(x^7*(x^6 - 1)^(2/3), x)