Integrand size = 22, antiderivative size = 98 \[ \int \frac {x^8}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{4} \left (1-x^3\right )^{4/3}-\frac {\arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \]
1/4*(-x^3+1)^(4/3)-1/12*ln(x^3+1)*2^(1/3)+1/4*ln(2^(1/3)-(-x^3+1)^(1/3))*2 ^(1/3)-1/6*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)
Time = 0.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.30 \[ \int \frac {x^8}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{12} \left (3 \left (1-x^3\right )^{4/3}-2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )+2 \sqrt [3]{2} \log \left (-2+2^{2/3} \sqrt [3]{1-x^3}\right )-\sqrt [3]{2} \log \left (2+2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \]
(3*(1 - x^3)^(4/3) - 2*2^(1/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3) )/Sqrt[3]] + 2*2^(1/3)*Log[-2 + 2^(2/3)*(1 - x^3)^(1/3)] - 2^(1/3)*Log[2 + 2^(2/3)*(1 - x^3)^(1/3) + 2^(1/3)*(1 - x^3)^(2/3)])/12
Time = 0.24 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.04, 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.
\(\displaystyle \int \frac {x^8}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {x^6}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}dx^3\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{3} \int \left (\frac {1}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}-\sqrt [3]{1-x^3}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2^{2/3}}+\frac {3}{4} \left (1-x^3\right )^{4/3}-\frac {\log \left (x^3+1\right )}{2\ 2^{2/3}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\right )\) |
((3*(1 - x^3)^(4/3))/4 - (Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqr t[3]])/2^(2/3) - Log[1 + x^3]/(2*2^(2/3)) + (3*Log[2^(1/3) - (1 - x^3)^(1/ 3)])/(2*2^(2/3)))/3
3.7.24.3.1 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 = 9.51 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.11
method | result | size |
pseudoelliptic | \(-\frac {\left (-x^{3}+1\right )^{\frac {1}{3}} x^{3}}{4}+\frac {\left (-x^{3}+1\right )^{\frac {1}{3}}}{4}+\frac {2^{\frac {1}{3}} \ln \left (\left (-x^{3}+1\right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right )}{6}-\frac {2^{\frac {1}{3}} \ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right )}{12}-\frac {\arctan \left (\frac {\left (1+2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) 2^{\frac {1}{3}} \sqrt {3}}{6}\) | \(109\) |
trager | \(\text {Expression too large to display}\) | \(545\) |
risch | \(\text {Expression too large to display}\) | \(1064\) |
-1/4*(-x^3+1)^(1/3)*x^3+1/4*(-x^3+1)^(1/3)+1/6*2^(1/3)*ln((-x^3+1)^(1/3)-2 ^(1/3))-1/12*2^(1/3)*ln((-x^3+1)^(2/3)+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^(1/3)*3^(1/2)
Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.16 \[ \int \frac {x^8}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} {\left (4^{\frac {2}{3}} \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} \sqrt {3}\right )}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2 \cdot 4^{\frac {1}{3}}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (-4^{\frac {2}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - \frac {1}{4} \, {\left (x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \]
-1/6*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*(4^(2/3)*sqrt(3)*(-x^3 + 1)^(1/3) + 4^(1/3)*sqrt(3))) - 1/24*4^(2/3)*log(4^(2/3)*(-x^3 + 1)^(1/3) + 2*(-x^3 + 1)^(2/3) + 2*4^(1/3)) + 1/12*4^(2/3)*log(-4^(2/3) + 2*(-x^3 + 1)^(1/3)) - 1/4*(x^3 - 1)*(-x^3 + 1)^(1/3)
\[ \int \frac {x^8}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x^{8}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.99 \[ \int \frac {x^8}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {1}{6} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{4} \, {\left (-x^{3} + 1\right )}^{\frac {4}{3}} - \frac {1}{12} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 2^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) \]
-1/6*sqrt(3)*2^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1 /3))) + 1/4*(-x^3 + 1)^(4/3) - 1/12*2^(1/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^(2/3)) + 1/6*2^(1/3)*log(-2^(1/3) + (-x^3 + 1)^(1/3) )
Time = 0.36 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \frac {x^8}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {1}{6} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{4} \, {\left (-x^{3} + 1\right )}^{\frac {4}{3}} - \frac {1}{12} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) \]
-1/6*sqrt(3)*2^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1 /3))) + 1/4*(-x^3 + 1)^(4/3) - 1/12*2^(1/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^(2/3)) + 1/6*2^(1/3)*log(abs(-2^(1/3) + (-x^3 + 1)^( 1/3)))
Time = 8.44 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.15 \[ \int \frac {x^8}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^3\right )}^{1/3}}{2}-\frac {2^{1/3}}{2}\right )}{6}+\frac {{\left (1-x^3\right )}^{4/3}}{4}+\frac {2^{1/3}\,\ln \left (3\,{\left (1-x^3\right )}^{1/3}-\frac {3\,2^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12}-\frac {2^{1/3}\,\ln \left (\frac {3\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+3\,{\left (1-x^3\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \]