Integrand size = 13, antiderivative size = 104 \[ \int x^3 \sqrt [3]{-1+x^6} \, dx=\frac {1}{6} x^4 \sqrt [3]{-1+x^6}+\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{6 \sqrt {3}}+\frac {1}{18} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-\frac {1}{36} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \] Output:
1/6*x^4*(x^6-1)^(1/3)+1/18*arctan(3^(1/2)*x^2/(x^2+2*(x^6-1)^(1/3)))*3^(1/ 2)+1/18*ln(-x^2+(x^6-1)^(1/3))-1/36*ln(x^4+x^2*(x^6-1)^(1/3)+(x^6-1)^(2/3) )
Time = 0.48 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.96 \[ \int x^3 \sqrt [3]{-1+x^6} \, dx=\frac {1}{36} \left (6 x^4 \sqrt [3]{-1+x^6}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )+2 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-\log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \] Input:
Integrate[x^3*(-1 + x^6)^(1/3),x]
Output:
(6*x^4*(-1 + x^6)^(1/3) + 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)])/36
Time = 0.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.72, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {807, 811, 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.
\(\displaystyle \int x^3 \sqrt [3]{x^6-1} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int x^2 \sqrt [3]{x^6-1}dx^2\) |
\(\Big \downarrow \) 811 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^4 \sqrt [3]{x^6-1}-\frac {1}{3} \int \frac {x^2}{\left (x^6-1\right )^{2/3}}dx^2\right )\) |
\(\Big \downarrow \) 853 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {\arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^6-1}\right )\right )+\frac {1}{3} \sqrt [3]{x^6-1} x^4\right )\) |
Input:
Int[x^3*(-1 + x^6)^(1/3),x]
Output:
((x^4*(-1 + x^6)^(1/3))/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)/2
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]
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.32
method | result | size |
meijerg | \(\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} x^{4} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{4 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}}}\) | \(33\) |
risch | \(\frac {x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}}{6}-\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {2}{3}} x^{4} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{12 \operatorname {signum}\left (x^{6}-1\right )^{\frac {2}{3}}}\) | \(46\) |
pseudoelliptic | \(\frac {-6 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right )+\ln \left (\frac {x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{4}}\right )-2 \ln \left (\frac {-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}}{x^{2}}\right )}{36 \left (x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right ) \left (-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}\right )}\) | \(129\) |
trager | \(\frac {x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{18}-\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+4 x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{18}-\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+4 x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right )}{18}\) | \(318\) |
Input:
int(x^3*(x^6-1)^(1/3),x,method=_RETURNVERBOSE)
Output:
1/4*signum(x^6-1)^(1/3)/(-signum(x^6-1))^(1/3)*x^4*hypergeom([-1/3,2/3],[5 /3],x^6)
Time = 0.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int x^3 \sqrt [3]{-1+x^6} \, dx=\frac {1}{6} \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{4} - \frac {1}{18} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) + \frac {1}{18} \, \log \left (-\frac {x^{2} - {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) - \frac {1}{36} \, \log \left (\frac {x^{4} + {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \] Input:
integrate(x^3*(x^6-1)^(1/3),x, algorithm="fricas")
Output:
1/6*(x^6 - 1)^(1/3)*x^4 - 1/18*sqrt(3)*arctan(1/3*(sqrt(3)*x^2 + 2*sqrt(3) *(x^6 - 1)^(1/3))/x^2) + 1/18*log(-(x^2 - (x^6 - 1)^(1/3))/x^2) - 1/36*log ((x^4 + (x^6 - 1)^(1/3)*x^2 + (x^6 - 1)^(2/3))/x^4)
Result contains complex when optimal does not.
Time = 0.81 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.35 \[ \int x^3 \sqrt [3]{-1+x^6} \, dx=- \frac {x^{4} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {5}{3}\right )} \] Input:
integrate(x**3*(x**6-1)**(1/3),x)
Output:
-x**4*exp(-2*I*pi/3)*gamma(2/3)*hyper((-1/3, 2/3), (5/3,), x**6)/(6*gamma( 5/3))
Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int x^3 \sqrt [3]{-1+x^6} \, dx=-\frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{6 \, x^{2} {\left (\frac {x^{6} - 1}{x^{6}} - 1\right )}} - \frac {1}{36} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) + \frac {1}{18} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \] Input:
integrate(x^3*(x^6-1)^(1/3),x, algorithm="maxima")
Output:
-1/18*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3)/x^2 + 1)) - 1/6*(x^6 - 1)^(1/3)/(x^2*((x^6 - 1)/x^6 - 1)) - 1/36*log((x^6 - 1)^(1/3)/x^2 + (x^6 - 1)^(2/3)/x^4 + 1) + 1/18*log((x^6 - 1)^(1/3)/x^2 - 1)
\[ \int x^3 \sqrt [3]{-1+x^6} \, dx=\int { {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{3} \,d x } \] Input:
integrate(x^3*(x^6-1)^(1/3),x, algorithm="giac")
Output:
integrate((x^6 - 1)^(1/3)*x^3, x)
Timed out. \[ \int x^3 \sqrt [3]{-1+x^6} \, dx=\int x^3\,{\left (x^6-1\right )}^{1/3} \,d x \] Input:
int(x^3*(x^6 - 1)^(1/3),x)
Output:
int(x^3*(x^6 - 1)^(1/3), x)
\[ \int x^3 \sqrt [3]{-1+x^6} \, dx=\frac {\left (x^{6}-1\right )^{\frac {1}{3}} x^{4}}{6}-\frac {\left (\int \frac {x^{3}}{\left (x^{6}-1\right )^{\frac {2}{3}}}d x \right )}{3} \] Input:
int(x^3*(x^6-1)^(1/3),x)
Output:
((x**6 - 1)**(1/3)*x**4 - 2*int(((x**6 - 1)**(1/3)*x**3)/(x**6 - 1),x))/6