Integrand size = 22, antiderivative size = 128 \[ \int \frac {x^{11}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {1}{5} \left (1-x^3\right )^{5/3}-\frac {1}{8} \left (1-x^3\right )^{8/3}-\frac {\arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \] Output:
-1/2*(-x^3+1)^(2/3)+1/5*(-x^3+1)^(5/3)-1/8*(-x^3+1)^(8/3)-1/6*arctan(1/3*( 1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)+1/12*ln(x^3+1)*2^(2/3)- 1/4*ln(2^(1/3)-(-x^3+1)^(1/3))*2^(2/3)
Time = 0.15 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.09 \[ \int \frac {x^{11}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {1}{120} \left (-3 \left (1-x^3\right )^{2/3} \left (17-2 x^3+5 x^6\right )-20\ 2^{2/3} \sqrt {3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )-20\ 2^{2/3} \log \left (-2+2^{2/3} \sqrt [3]{1-x^3}\right )+10\ 2^{2/3} \log \left (2+2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \] Input:
Integrate[x^11/((1 - x^3)^(1/3)*(1 + x^3)),x]
Output:
(-3*(1 - x^3)^(2/3)*(17 - 2*x^3 + 5*x^6) - 20*2^(2/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] - 20*2^(2/3)*Log[-2 + 2^(2/3)*(1 - x^3)^ (1/3)] + 10*2^(2/3)*Log[2 + 2^(2/3)*(1 - x^3)^(1/3) + 2^(1/3)*(1 - x^3)^(2 /3)])/120
Time = 0.44 (sec) , antiderivative size = 132, 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.
| \(\displaystyle \int \frac {x^{11}}{\sqrt [3]{1-x^3} \left (x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int \frac {x^9}{\sqrt [3]{1-x^3} \left (x^3+1\right )}dx^3\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {1}{3} \int \left (\left (1-x^3\right )^{5/3}-\left (1-x^3\right )^{2/3}-\frac {1}{\left (x^3+1\right ) \sqrt [3]{1-x^3}}+\frac {1}{\sqrt [3]{1-x^3}}\right )dx^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {3}{8} \left (1-x^3\right )^{8/3}+\frac {3}{5} \left (1-x^3\right )^{5/3}-\frac {3}{2} \left (1-x^3\right )^{2/3}+\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}-\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\right )\) |
Input:
Int[x^11/((1 - x^3)^(1/3)*(1 + x^3)),x]
Output:
((-3*(1 - x^3)^(2/3))/2 + (3*(1 - x^3)^(5/3))/5 - (3*(1 - x^3)^(8/3))/8 - (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
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 = 13.12 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.80
| method | result | size |
| pseudoelliptic | \(\frac {\left (-2 \arctan \left (\frac {\left (1+2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}-2 \ln \left (\left (-x^{3}+1\right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right )+\ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right )\right ) 2^{\frac {2}{3}}}{12}-\frac {\left (-x^{3}+1\right )^{\frac {2}{3}} \left (5 x^{6}-2 x^{3}+17\right )}{40}\) | \(102\) |
| trager | \(\left (-\frac {1}{8} x^{6}+\frac {1}{20} x^{3}-\frac {17}{40}\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 {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 (\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 ) x^{3}-15 \operatorname {RootOf}\left (\textit {\_Z}^{3}+4\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}}+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 )+35 \operatorname {RootOf}\left (\textit {\_Z}^{3}+4\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+4\right ) \ln \left (\frac {-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}-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}-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}-2 \operatorname {RootOf}\left (\textit {\_Z}^{3}+4\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}}+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 )+14 \operatorname {RootOf}\left (\textit {\_Z}^{3}+4\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{6}\) | \(497\) |
| risch | \(\text {Expression too large to display}\) | \(789\) |
Input:
int(x^11/(-x^3+1)^(1/3)/(x^3+1),x,method=_RETURNVERBOSE)
Output:
1/12*(-2*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)-2*ln((-x^3 +1)^(1/3)-2^(1/3))+ln((-x^3+1)^(2/3)+2^(1/3)*(-x^3+1)^(1/3)+2^(2/3)))*2^(2 /3)-1/40*(-x^3+1)^(2/3)*(5*x^6-2*x^3+17)
Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int \frac {x^{11}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {1}{40} \, {\left (5 \, x^{6} - 2 \, x^{3} + 17\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 2^{\frac {1}{6}} \sqrt {\frac {1}{6}} \arctan \left (2^{\frac {1}{6}} \sqrt {\frac {1}{6}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) \] Input:
integrate(x^11/(-x^3+1)^(1/3)/(x^3+1),x, algorithm="fricas")
Output:
-1/40*(5*x^6 - 2*x^3 + 17)*(-x^3 + 1)^(2/3) - 2^(1/6)*sqrt(1/6)*arctan(2^( 1/6)*sqrt(1/6)*(2^(1/3) + 2*(-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))
\[ \int \frac {x^{11}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int \frac {x^{11}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \] Input:
integrate(x**11/(-x**3+1)**(1/3)/(x**3+1),x)
Output:
Integral(x**11/((-(x - 1)*(x**2 + x + 1))**(1/3)*(x + 1)*(x**2 - x + 1)), x)
Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.93 \[ \int \frac {x^{11}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {1}{8} \, {\left (-x^{3} + 1\right )}^{\frac {8}{3}} - \frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{5} \, {\left (-x^{3} + 1\right )}^{\frac {5}{3}} + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} \] Input:
integrate(x^11/(-x^3+1)^(1/3)/(x^3+1),x, algorithm="maxima")
Output:
-1/8*(-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))) + 1/5*(-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)) - 1/2*(-x^3 + 1)^(2/3)
Time = 0.14 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.99 \[ \int \frac {x^{11}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {1}{8} \, {\left (x^{3} - 1\right )}^{2} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{5} \, {\left (-x^{3} + 1\right )}^{\frac {5}{3}} + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} \] Input:
integrate(x^11/(-x^3+1)^(1/3)/(x^3+1),x, algorithm="giac")
Output:
-1/8*(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))) + 1/5*(-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))) - 1/2*(-x^3 + 1)^(2/3)
Time = 3.42 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.04 \[ \int \frac {x^{11}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {{\left (1-x^3\right )}^{5/3}}{5}-\frac {{\left (1-x^3\right )}^{2/3}}{2}-\frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-2^{1/3}\right )}{6}-\frac {{\left (1-x^3\right )}^{8/3}}{8}-\frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-\frac {2^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12}+\frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-\frac {2^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \] Input:
int(x^11/((1 - x^3)^(1/3)*(x^3 + 1)),x)
Output:
(1 - x^3)^(5/3)/5 - (1 - x^3)^(2/3)/2 - (2^(2/3)*log((1 - x^3)^(1/3) - 2^( 1/3)))/6 - (1 - x^3)^(8/3)/8 - (2^(2/3)*log((1 - x^3)^(1/3) - (2^(1/3)*(3^ (1/2)*1i - 1)^2)/4)*(3^(1/2)*1i - 1))/12 + (2^(2/3)*log((1 - x^3)^(1/3) - (2^(1/3)*(3^(1/2)*1i + 1)^2)/4)*(3^(1/2)*1i + 1))/12
\[ \int \frac {x^{11}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int \frac {x^{11}}{\left (-x^{3}+1\right )^{\frac {1}{3}} x^{3}+\left (-x^{3}+1\right )^{\frac {1}{3}}}d x \] Input:
int(x^11/(-x^3+1)^(1/3)/(x^3+1),x)
Output:
int(x**11/(( - x**3 + 1)**(1/3)*x**3 + ( - x**3 + 1)**(1/3)),x)