Integrand size = 29, antiderivative size = 103 \[ \int \frac {1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3}}-\frac {\log \left (1+2 (1-x)^3-x^3\right )}{2\ 2^{2/3}}+\frac {3 \log \left (\sqrt [3]{2} (1-x)+\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \]
-1/4*ln(1+2*(1-x)^3-x^3)*2^(1/3)+3/4*ln(2^(1/3)*(1-x)+(-x^3+1)^(1/3))*2^(1 /3)+1/2*arctan(1/3*(1-2*2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)*2^( 1/3)
Time = 1.34 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.76 \[ \int \frac {1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \left (1-x^3\right )^{2/3}}{2^{2/3}+2^{2/3} x+2^{2/3} x^2-\left (1-x^3\right )^{2/3}}\right )-2 \log \left (2^{2/3}+2^{2/3} x+2^{2/3} x^2+2 \left (1-x^3\right )^{2/3}\right )+\log \left (-\left (\left (1+x+x^2\right ) \left (\sqrt [3]{2}+\sqrt [3]{2} x^2-\left (2-2 x^3\right )^{2/3}+2 \sqrt [3]{1-x^3}+x \left (\sqrt [3]{2}-2 \sqrt [3]{1-x^3}\right )\right )\right )\right )}{2\ 2^{2/3}} \]
-1/2*(2*Sqrt[3]*ArcTan[(Sqrt[3]*(1 - x^3)^(2/3))/(2^(2/3) + 2^(2/3)*x + 2^ (2/3)*x^2 - (1 - x^3)^(2/3))] - 2*Log[2^(2/3) + 2^(2/3)*x + 2^(2/3)*x^2 + 2*(1 - x^3)^(2/3)] + Log[-((1 + x + x^2)*(2^(1/3) + 2^(1/3)*x^2 - (2 - 2*x ^3)^(2/3) + 2*(1 - x^3)^(1/3) + x*(2^(1/3) - 2*(1 - x^3)^(1/3))))])/2^(2/3 )
Leaf count is larger than twice the leaf count of optimal. \(425\) vs. \(2(103)=206\).
Time = 0.96 (sec) , antiderivative size = 425, normalized size of antiderivative = 4.13, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, 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.
\(\displaystyle \int \frac {1-x^2}{\left (x^2-x+1\right ) \left (1-x^3\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 2583 |
\(\displaystyle \int \left (-\frac {x^3}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}+\frac {x}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}+\frac {1}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}-\frac {x^2}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (x^3+1\right )}{3\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}+\frac {1}{3} \sqrt [3]{2} \log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )-\frac {\log \left (\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{6\ 2^{2/3}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2\ 2^{2/3}}\) |
(2^(1/3)*ArcTan[(1 - (2*2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]])/Sqrt[3 ] + ArcTan[(1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[ 3]) - ArcTan[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3] ) + ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3]) + Log[ 1 + x^3]/(3*2^(2/3)) + Log[2^(2/3) - (1 - x)/(1 - x^3)^(1/3)]/(3*2^(2/3)) - Log[1 + (2^(2/3)*(1 - x)^2)/(1 - x^3)^(2/3) - (2^(1/3)*(1 - x))/(1 - x^3 )^(1/3)]/(3*2^(2/3)) + (2^(1/3)*Log[1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3)] )/3 - Log[2*2^(1/3) + (1 - x)^2/(1 - x^3)^(2/3) + (2^(2/3)*(1 - x))/(1 - x ^3)^(1/3)]/(6*2^(2/3)) - Log[2^(1/3) - (1 - x^3)^(1/3)]/(2*2^(2/3)) - Log[ -(2^(1/3)*x) - (1 - x^3)^(1/3)]/(2*2^(2/3))
3.11.17.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 10.01 (sec) , antiderivative size = 692, normalized size of antiderivative = 6.72
-1/2*ln(-(-2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z ^3-2)^4*x+2*(-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_ Z^2)*RootOf(_Z^3-2)^2-RootOf(_Z^3-2)^2*x^2+4*(-x^3+1)^(1/3)*RootOf(RootOf( _Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x+RootOf(_Z^3-2)^2*x-4*(-x^3+1)^(1/3 )*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-RootOf(_Z^3-2)^2)/(x ^2-x+1))*RootOf(_Z^3-2)-ln(-(-2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2 )+4*_Z^2)*RootOf(_Z^3-2)^4*x+2*(-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z *RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2-RootOf(_Z^3-2)^2*x^2+4*(-x^3+1)^( 1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x+RootOf(_Z^3-2)^ 2*x-4*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-R ootOf(_Z^3-2)^2)/(x^2-x+1))*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4* _Z^2)+RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*ln((-2*RootOf(Ro otOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^4*x+2*(-x^3+1)^( 2/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2+ RootOf(_Z^3-2)^2*x^2+2*(-x^3+1)^(1/3)*RootOf(_Z^3-2)*x+4*(-x^3+1)^(1/3)*Ro otOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x-3*RootOf(_Z^3-2)^2*x+2 *(-x^3+1)^(2/3)-2*(-x^3+1)^(1/3)*RootOf(_Z^3-2)-4*(-x^3+1)^(1/3)*RootOf(Ro otOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+RootOf(_Z^3-2)^2)/(x^2-x+1))
Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (78) = 156\).
Time = 2.92 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.81 \[ \int \frac {1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx=-\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (2 \cdot 4^{\frac {2}{3}} {\left (x^{5} - x^{4} - 3 \, x^{3} + 3 \, x^{2} + x - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 4 \, {\left (x^{4} - 4 \, x^{3} + 5 \, x^{2} - 4 \, x + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{6} - 7 \, x^{5} + 10 \, x^{4} - 7 \, x^{3} + 10 \, x^{2} - 7 \, x + 1\right )}\right )}}{6 \, {\left (3 \, x^{6} - 9 \, x^{5} + 6 \, x^{4} - x^{3} + 6 \, x^{2} - 9 \, x + 3\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {2 \cdot 4^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} - 3 \, x + 1\right )} - 4^{\frac {2}{3}} {\left (x^{4} - 3 \, x^{2} + 1\right )} - 8 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (-\frac {4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 4^{\frac {1}{3}} {\left (x^{2} - x + 1\right )} - 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} - x + 1}\right ) \]
-1/6*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*sqrt(3)*(2*4^(2/3)*(x^5 - x^4 - 3* x^3 + 3*x^2 + x - 1)*(-x^3 + 1)^(1/3) + 4*(x^4 - 4*x^3 + 5*x^2 - 4*x + 1)* (-x^3 + 1)^(2/3) + 4^(1/3)*(x^6 - 7*x^5 + 10*x^4 - 7*x^3 + 10*x^2 - 7*x + 1))/(3*x^6 - 9*x^5 + 6*x^4 - x^3 + 6*x^2 - 9*x + 3)) - 1/24*4^(2/3)*log((2 *4^(1/3)*(-x^3 + 1)^(2/3)*(x^2 - 3*x + 1) - 4^(2/3)*(x^4 - 3*x^2 + 1) - 8* (-x^3 + 1)^(1/3)*(x^2 - x))/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) + 1/12*4^(2/3 )*log(-(4^(2/3)*(-x^3 + 1)^(1/3)*(x - 1) - 4^(1/3)*(x^2 - x + 1) - 2*(-x^3 + 1)^(2/3))/(x^2 - x + 1))
\[ \int \frac {1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx=- \int \frac {x^{2}}{x^{2} \left (1 - x^{3}\right )^{\frac {2}{3}} - x \left (1 - x^{3}\right )^{\frac {2}{3}} + \left (1 - x^{3}\right )^{\frac {2}{3}}}\, dx - \int \left (- \frac {1}{x^{2} \left (1 - x^{3}\right )^{\frac {2}{3}} - x \left (1 - x^{3}\right )^{\frac {2}{3}} + \left (1 - x^{3}\right )^{\frac {2}{3}}}\right )\, dx \]
-Integral(x**2/(x**2*(1 - x**3)**(2/3) - x*(1 - x**3)**(2/3) + (1 - x**3)* *(2/3)), x) - Integral(-1/(x**2*(1 - x**3)**(2/3) - x*(1 - x**3)**(2/3) + (1 - x**3)**(2/3)), x)
\[ \int \frac {1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx=\int { -\frac {x^{2} - 1}{{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} - x + 1\right )}} \,d x } \]
\[ \int \frac {1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx=\int { -\frac {x^{2} - 1}{{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} - x + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx=-\int \frac {x^2-1}{{\left (1-x^3\right )}^{2/3}\,\left (x^2-x+1\right )} \,d x \]