Integrand size = 13, antiderivative size = 88 \[ \int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\frac {1}{12} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \] Output:
-1/6*arctan(3^(1/2)*x^2/(x^2+2*(x^6-1)^(1/3)))*3^(1/2)-1/6*ln(-x^2+(x^6-1) ^(1/3))+1/12*ln(x^4+x^2*(x^6-1)^(1/3)+(x^6-1)^(2/3))
Time = 0.55 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.95 \[ \int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx=\frac {1}{12} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )-2 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \] Input:
Integrate[x^3/(-1 + x^6)^(2/3),x]
Output:
(-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)])/ 12
Time = 0.17 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.62, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {807, 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.
\(\displaystyle \int \frac {x^3}{\left (x^6-1\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int \frac {x^2}{\left (x^6-1\right )^{2/3}}dx^2\) |
\(\Big \downarrow \) 853 |
\(\displaystyle \frac {1}{2} \left (-\frac {\arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x^2-\sqrt [3]{x^6-1}\right )\right )\) |
Input:
Int[x^3/(-1 + x^6)^(2/3),x]
Output:
(-(ArcTan[(1 + (2*x^2)/(-1 + x^6)^(1/3))/Sqrt[3]]/Sqrt[3]) - Log[x^2 - (-1 + x^6)^(1/3)]/2)/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[(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 = 2.46 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.38
method | result | size |
meijerg | \(\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {2}{3}} x^{4} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{4 \operatorname {signum}\left (x^{6}-1\right )^{\frac {2}{3}}}\) | \(33\) |
pseudoelliptic | \(\frac {\ln \left (\frac {x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{4}}\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right )}{6}-\frac {\ln \left (\frac {-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}}{x^{2}}\right )}{6}\) | \(78\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right )}{6}-\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+4 x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{6}+\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+4 x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2\right )}{6}\) | \(338\) |
Input:
int(x^3/(x^6-1)^(2/3),x,method=_RETURNVERBOSE)
Output:
1/4/signum(x^6-1)^(2/3)*(-signum(x^6-1))^(2/3)*x^4*hypergeom([2/3,2/3],[5/ 3],x^6)
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) - \frac {1}{6} \, \log \left (-\frac {x^{2} - {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) + \frac {1}{12} \, \log \left (\frac {x^{4} + {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \] Input:
integrate(x^3/(x^6-1)^(2/3),x, algorithm="fricas")
Output:
1/6*sqrt(3)*arctan(1/3*(sqrt(3)*x^2 + 2*sqrt(3)*(x^6 - 1)^(1/3))/x^2) - 1/ 6*log(-(x^2 - (x^6 - 1)^(1/3))/x^2) + 1/12*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 = 0.50 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.36 \[ \int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx=\frac {x^{4} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {5}{3}\right )} \] Input:
integrate(x**3/(x**6-1)**(2/3),x)
Output:
x**4*exp(-2*I*pi/3)*gamma(2/3)*hyper((2/3, 2/3), (5/3,), x**6)/(6*gamma(5/ 3))
Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) + \frac {1}{12} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \] Input:
integrate(x^3/(x^6-1)^(2/3),x, algorithm="maxima")
Output:
1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3)/x^2 + 1)) + 1/12*log((x^ 6 - 1)^(1/3)/x^2 + (x^6 - 1)^(2/3)/x^4 + 1) - 1/6*log((x^6 - 1)^(1/3)/x^2 - 1)
\[ \int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx=\int { \frac {x^{3}}{{\left (x^{6} - 1\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate(x^3/(x^6-1)^(2/3),x, algorithm="giac")
Output:
integrate(x^3/(x^6 - 1)^(2/3), x)
Timed out. \[ \int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx=\int \frac {x^3}{{\left (x^6-1\right )}^{2/3}} \,d x \] Input:
int(x^3/(x^6 - 1)^(2/3),x)
Output:
int(x^3/(x^6 - 1)^(2/3), x)
\[ \int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx=\int \frac {x^{3}}{\left (x^{6}-1\right )^{\frac {2}{3}}}d x \] Input:
int(x^3/(x^6-1)^(2/3),x)
Output:
int(x**3/(x**6 - 1)**(2/3),x)