Integrand size = 22, antiderivative size = 234 \[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=-\frac {\left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {x \left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {2 x^2 \left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}} \]
-1/3*(-x^3+1)^(2/3)/(x^3+1)+1/3*x*(-x^3+1)^(2/3)/(x^3+1)+2/3*x^2*(-x^3+1)^ (2/3)/(x^3+1)+1/3*x^2*hypergeom([1/3, 2/3],[5/3],x^3)-1/6*ln(2^(1/3)-(-x^3 +1)^(1/3))*2^(2/3)+1/6*ln(-2^(1/3)*x-(-x^3+1)^(1/3))*2^(2/3)-1/9*arctan(1/ 3*(1-2*2^(1/3)*x/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)-1/9*arctan(1/3*( 1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)
\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx \]
Time = 0.38 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2583, 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.
\(\displaystyle \int \frac {\left (1-x^3\right )^{2/3}}{\left (x^2-x+1\right )^2} \, dx\) |
\(\Big \downarrow \) 2583 |
\(\displaystyle \int \left (\frac {2 \left (1-x^3\right )^{2/3} x}{\left (x^3+1\right )^2}+\frac {\left (1-x^3\right )^{2/3}}{\left (x^3+1\right )^2}+\frac {\left (1-x^3\right )^{2/3} x^2}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {\left (1-x^3\right )^{2/3} x}{3 \left (x^3+1\right )}-\frac {\left (1-x^3\right )^{2/3}}{3 \left (x^3+1\right )}-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}+\frac {2 \left (1-x^3\right )^{2/3} x^2}{3 \left (x^3+1\right )}\) |
-1/3*(1 - x^3)^(2/3)/(1 + x^3) + (x*(1 - x^3)^(2/3))/(3*(1 + x^3)) + (2*x^ 2*(1 - x^3)^(2/3))/(3*(1 + x^3)) - (2^(2/3)*ArcTan[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) - (2^(2/3)*ArcTan[(1 + 2^(2/3)*(1 - x^3 )^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + (x^2*Hypergeometric2F1[1/3, 2/3, 5/3, x^3 ])/3 - Log[2^(1/3) - (1 - x^3)^(1/3)]/(3*2^(1/3)) + Log[-(2^(1/3)*x) - (1 - x^3)^(1/3)]/(3*2^(1/3))
3.2.9.3.1 Defintions of rubi rules used
Int[(Px_.)*((c_) + (d_.)*(x_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^3)^(p _.), x_Symbol] :> Simp[1/c^q Int[ExpandIntegrand[(c^3 - d^3*x^3)^q*(a + b *x^3)^p, Px/(c - d*x)^q, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && Poly Q[Px, x] && EqQ[d^2 - c*e, 0] && ILtQ[q, 0] && RationalQ[p] && EqQ[Denomina tor[p], 3]
\[\int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{\left (x^{2}-x +1\right )^{2}}d x\]
\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{2} - x + 1\right )}^{2}} \,d x } \]
\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}}}{\left (x^{2} - x + 1\right )^{2}}\, dx \]
\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{2} - x + 1\right )}^{2}} \,d x } \]
\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{2} - x + 1\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int \frac {{\left (1-x^3\right )}^{2/3}}{{\left (x^2-x+1\right )}^2} \,d x \]
\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{4}-2 x^{3}+3 x^{2}-2 x +1}d x \]