Integrand size = 22, antiderivative size = 127 \[ \int \frac {x^{14}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {2}{5} \left (1-x^3\right )^{5/3}-\frac {1}{4} \left (1-x^3\right )^{8/3}+\frac {1}{11} \left (1-x^3\right )^{11/3}+\frac {\arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \]
2/5*(-x^3+1)^(5/3)-1/4*(-x^3+1)^(8/3)+1/11*(-x^3+1)^(11/3)-1/12*ln(x^3+1)* 2^(2/3)+1/4*ln(2^(1/3)-(-x^3+1)^(1/3))*2^(2/3)+1/6*arctan(1/3*(1+2^(2/3)*( -x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)
Time = 0.23 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.14 \[ \int \frac {x^{14}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {1}{220} \left (1-x^3\right )^{2/3} \left (53-38 x^3+5 x^6-20 x^9\right )+\frac {\arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2+2^{2/3} \sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2+2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}} \]
((1 - x^3)^(2/3)*(53 - 38*x^3 + 5*x^6 - 20*x^9))/220 + ArcTan[(1 + 2^(2/3) *(1 - x^3)^(1/3))/Sqrt[3]]/(2^(1/3)*Sqrt[3]) + Log[-2 + 2^(2/3)*(1 - x^3)^ (1/3)]/(3*2^(1/3)) - Log[2 + 2^(2/3)*(1 - x^3)^(1/3) + 2^(1/3)*(1 - x^3)^( 2/3)]/(6*2^(1/3))
Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {948, 99, 2009}
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.
| \(\displaystyle \int \frac {x^{14}}{\sqrt [3]{1-x^3} \left (x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int \frac {x^{12}}{\sqrt [3]{1-x^3} \left (x^3+1\right )}dx^3\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {1}{3} \int \left (-\left (1-x^3\right )^{8/3}+2 \left (1-x^3\right )^{5/3}-2 \left (1-x^3\right )^{2/3}+\frac {1}{\left (x^3+1\right ) \sqrt [3]{1-x^3}}\right )dx^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt [3]{2}}+\frac {3}{11} \left (1-x^3\right )^{11/3}-\frac {3}{4} \left (1-x^3\right )^{8/3}+\frac {6}{5} \left (1-x^3\right )^{5/3}-\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\right )\) |
((6*(1 - x^3)^(5/3))/5 - (3*(1 - x^3)^(8/3))/4 + (3*(1 - x^3)^(11/3))/11 + (Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]])/2^(1/3) - Log[1 + x^3]/(2*2^(1/3)) + (3*Log[2^(1/3) - (1 - x^3)^(1/3)])/(2*2^(1/3)))/3
3.7.5.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 9.74 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.86
| method | result | size |
| pseudoelliptic | \(\frac {\left (2 \arctan \left (\frac {\left (1+2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}-\ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right )+2 \ln \left (\left (-x^{3}+1\right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right )\right ) 2^{\frac {2}{3}}}{12}-\frac {\left (-x^{3}+1\right )^{\frac {2}{3}} \left (20 x^{9}-5 x^{6}+38 x^{3}-53\right )}{220}\) | \(109\) |
| trager | \(\left (-\frac {1}{11} x^{9}+\frac {1}{44} x^{6}-\frac {19}{110} x^{3}+\frac {53}{220}\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}+\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \ln \left (\frac {15 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{3} x^{3}+72 {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )}^{2} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2} x^{3}+15 \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right ) x^{3}+72 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{3}+126 \left (-x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+42 \left (-x^{3}+1\right )^{\frac {2}{3}}-35 \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )-168 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right ) \ln \left (-\frac {6 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{3} x^{3}+45 {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )}^{2} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right ) x^{3}-15 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-63 \left (-x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )-21 \left (-x^{3}+1\right )^{\frac {2}{3}}+14 \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+105 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{6}\) | \(503\) |
| risch | \(\frac {\left (20 x^{9}-5 x^{6}+38 x^{3}-53\right ) \left (x^{3}-1\right )}{220 \left (-x^{3}+1\right )^{\frac {1}{3}}}+\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \ln \left (\frac {15 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{3} x^{3}+72 {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )}^{2} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2} x^{3}+15 \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right ) x^{3}+72 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{3}+126 \left (-x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+42 \left (-x^{3}+1\right )^{\frac {2}{3}}-35 \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )-168 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right ) \ln \left (-\frac {6 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{3} x^{3}+45 {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )}^{2} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right ) x^{3}-15 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-63 \left (-x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )-21 \left (-x^{3}+1\right )^{\frac {2}{3}}+14 \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+105 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{6}\) | \(509\) |
1/12*(2*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)-ln((-x^3+1) ^(2/3)+2^(1/3)*(-x^3+1)^(1/3)+2^(2/3))+2*ln((-x^3+1)^(1/3)-2^(1/3)))*2^(2/ 3)-1/220*(-x^3+1)^(2/3)*(20*x^9-5*x^6+38*x^3-53)
Time = 0.32 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.93 \[ \int \frac {x^{14}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {1}{220} \, {\left (20 \, x^{9} - 5 \, x^{6} + 38 \, x^{3} - 53\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + \frac {1}{6} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (\frac {1}{6} \cdot 2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} + 2 \, \sqrt {6} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) \]
-1/220*(20*x^9 - 5*x^6 + 38*x^3 - 53)*(-x^3 + 1)^(2/3) + 1/6*sqrt(6)*2^(1/ 6)*arctan(1/6*2^(1/6)*(sqrt(6)*2^(1/3) + 2*sqrt(6)*(-x^3 + 1)^(1/3))) - 1/ 12*2^(2/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^(2/3)) + 1/ 6*2^(2/3)*log(-2^(1/3) + (-x^3 + 1)^(1/3))
\[ \int \frac {x^{14}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int \frac {x^{14}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.94 \[ \int \frac {x^{14}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {1}{11} \, {\left (-x^{3} + 1\right )}^{\frac {11}{3}} - \frac {1}{4} \, {\left (-x^{3} + 1\right )}^{\frac {8}{3}} + \frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {2}{5} \, {\left (-x^{3} + 1\right )}^{\frac {5}{3}} - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) \]
1/11*(-x^3 + 1)^(11/3) - 1/4*(-x^3 + 1)^(8/3) + 1/6*sqrt(3)*2^(2/3)*arctan (1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1/3))) + 2/5*(-x^3 + 1)^(5/3 ) - 1/12*2^(2/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^(2/3) ) + 1/6*2^(2/3)*log(-2^(1/3) + (-x^3 + 1)^(1/3))
Time = 0.31 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06 \[ \int \frac {x^{14}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {1}{11} \, {\left (x^{3} - 1\right )}^{3} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \frac {1}{4} \, {\left (x^{3} - 1\right )}^{2} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + \frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {2}{5} \, {\left (-x^{3} + 1\right )}^{\frac {5}{3}} - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) \]
-1/11*(x^3 - 1)^3*(-x^3 + 1)^(2/3) - 1/4*(x^3 - 1)^2*(-x^3 + 1)^(2/3) + 1/ 6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1/3) )) + 2/5*(-x^3 + 1)^(5/3) - 1/12*2^(2/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^ (1/3) + (-x^3 + 1)^(2/3)) + 1/6*2^(2/3)*log(abs(-2^(1/3) + (-x^3 + 1)^(1/3 )))
Time = 8.43 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.05 \[ \int \frac {x^{14}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-2^{1/3}\right )}{6}+\frac {2\,{\left (1-x^3\right )}^{5/3}}{5}-\frac {{\left (1-x^3\right )}^{8/3}}{4}+\frac {{\left (1-x^3\right )}^{11/3}}{11}+\frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-\frac {2^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12}-\frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-\frac {2^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \]