Integrand size = 27, antiderivative size = 199 \[ \int \frac {(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\frac {\left (1-x^3\right )^{2/3}}{1-x+x^2}-\frac {2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}+\log \left (x+\sqrt [3]{1-x^3}\right ) \]
(-x^3+1)^(2/3)/(x^2-x+1)+1/2*ln(2^(1/3)-(-x^3+1)^(1/3))*2^(2/3)-1/2*ln(-2^ (1/3)*x-(-x^3+1)^(1/3))*2^(2/3)+ln(x+(-x^3+1)^(1/3))-2/3*arctan(1/3*(1-2*x /(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)+1/3*arctan(1/3*(1-2*2^(1/3)*x/(-x^3+1)^( 1/3))*3^(1/2))*2^(2/3)*3^(1/2)+1/3*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3 ^(1/2))*2^(2/3)*3^(1/2)
\[ \int \frac {(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int \frac {(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx \]
Time = 0.46 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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.
| \(\displaystyle \int \frac {(1-2 x) \left (1-x^3\right )^{2/3}}{\left (x^2-x+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 2583 |
| \(\displaystyle \int \left (-\frac {2 \left (1-x^3\right )^{2/3} x^3}{\left (x^3+1\right )^2}+\frac {\left (1-x^3\right )^{2/3}}{\left (x^3+1\right )^2}-\frac {3 \left (1-x^3\right )^{2/3} x^2}{\left (x^3+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (1-x^3\right )^{2/3} x}{x^3+1}+\frac {\left (1-x^3\right )^{2/3}}{x^3+1}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}-\frac {2}{3} 2^{2/3} \log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}+\log \left (\sqrt [3]{1-x^3}+x\right )\) |
(1 - x^3)^(2/3)/(1 + x^3) + (x*(1 - x^3)^(2/3))/(1 + x^3) - (2*ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]])/Sqrt[3] + (2^(2/3)*ArcTan[(1 - (2*2^(1/3 )*x)/(1 - x^3)^(1/3))/Sqrt[3]])/Sqrt[3] + (2^(2/3)*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]])/Sqrt[3] + Log[2^(1/3) - (1 - x^3)^(1/3)]/2^(1/3) + Log[-(2^(1/3)*x) - (1 - x^3)^(1/3)]/(3*2^(1/3)) - (2*2^(2/3)*Log[-(2^(1/3 )*x) - (1 - x^3)^(1/3)])/3 + Log[x + (1 - x^3)^(1/3)]
3.2.10.3.1 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 = 26.80 (sec) , antiderivative size = 618, normalized size of antiderivative = 3.11
| method | result | size |
| trager | \(\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}-x +1}+\frac {2 \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{6} x^{3}-36 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3} \left (-x^{3}+1\right )^{\frac {2}{3}} x +36 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3} x^{3}-24 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3}+432 x \left (-x^{3}+1\right )^{\frac {2}{3}}+864 x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}\right )}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{5} \left (-x^{3}+1\right )^{\frac {1}{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4} x -\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4}+12 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}+36 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right ) x^{2}-36 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right ) x +144 \left (-x^{3}+1\right )^{\frac {2}{3}}-36 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )}{x^{2}-x +1}\right )}{3}+\frac {\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4} x -\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4}+12 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{2} \left (-x^{3}+1\right )^{\frac {1}{3}} x +72 \left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}-x +1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4}}{72}-\frac {\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4} x -\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4}+12 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{2} \left (-x^{3}+1\right )^{\frac {1}{3}} x +72 \left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}-x +1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )}{6}-\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{6} x^{3}+72 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3} \left (-x^{3}+1\right )^{\frac {2}{3}} x +72 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3} \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-24 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3}+864 x \left (-x^{3}+1\right )^{\frac {2}{3}}-864 x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}-1296 x^{3}+864\right ) \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3}}{36}-\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{6} x^{3}+72 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3} \left (-x^{3}+1\right )^{\frac {2}{3}} x +72 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3} \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-24 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3}+864 x \left (-x^{3}+1\right )^{\frac {2}{3}}-864 x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}-1296 x^{3}+864\right )}{3}\) | \(618\) |
| risch | \(\text {Expression too large to display}\) | \(1119\) |
(-x^3+1)^(2/3)/(x^2-x+1)+2/3*ln(-RootOf(_Z^6+432)^6*x^3-36*RootOf(_Z^6+432 )^3*(-x^3+1)^(2/3)*x+36*RootOf(_Z^6+432)^3*x^3-24*RootOf(_Z^6+432)^3+432*x *(-x^3+1)^(2/3)+864*x^2*(-x^3+1)^(1/3))+1/3*RootOf(_Z^6+432)*ln((RootOf(_Z ^6+432)^5*(-x^3+1)^(1/3)+RootOf(_Z^6+432)^4*x^2-RootOf(_Z^6+432)^4*x-RootO f(_Z^6+432)^4+12*RootOf(_Z^6+432)^2*(-x^3+1)^(1/3)+36*RootOf(_Z^6+432)*x^2 -36*RootOf(_Z^6+432)*x+144*(-x^3+1)^(2/3)-36*RootOf(_Z^6+432))/(x^2-x+1))+ 1/72*ln(-(RootOf(_Z^6+432)^4*x^2+RootOf(_Z^6+432)^4*x-RootOf(_Z^6+432)^4+1 2*RootOf(_Z^6+432)^2*(-x^3+1)^(1/3)*x+72*(-x^3+1)^(2/3))/(x^2-x+1))*RootOf (_Z^6+432)^4-1/6*ln(-(RootOf(_Z^6+432)^4*x^2+RootOf(_Z^6+432)^4*x-RootOf(_ Z^6+432)^4+12*RootOf(_Z^6+432)^2*(-x^3+1)^(1/3)*x+72*(-x^3+1)^(2/3))/(x^2- x+1))*RootOf(_Z^6+432)-1/36*ln(RootOf(_Z^6+432)^6*x^3+72*RootOf(_Z^6+432)^ 3*(-x^3+1)^(2/3)*x+72*RootOf(_Z^6+432)^3*(-x^3+1)^(1/3)*x^2-24*RootOf(_Z^6 +432)^3+864*x*(-x^3+1)^(2/3)-864*x^2*(-x^3+1)^(1/3)-1296*x^3+864)*RootOf(_ Z^6+432)^3-1/3*ln(RootOf(_Z^6+432)^6*x^3+72*RootOf(_Z^6+432)^3*(-x^3+1)^(2 /3)*x+72*RootOf(_Z^6+432)^3*(-x^3+1)^(1/3)*x^2-24*RootOf(_Z^6+432)^3+864*x *(-x^3+1)^(2/3)-864*x^2*(-x^3+1)^(1/3)-1296*x^3+864)
Result contains complex when optimal does not.
Time = 1.18 (sec) , antiderivative size = 2298, normalized size of antiderivative = 11.55 \[ \int \frac {(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\text {Too large to display} \]
1/36*(2*sqrt(3)*(-16)^(1/6)*(x^2 - x + 1)*log(9*(4*sqrt(3)*(-16)^(1/6)*(13 1*x^6 - 48*x^5 - 381*x^4 - 152*x^3 + 267*x^2 + 276*x - 112) + 12*2^(2/3)*( 15*x^6 - 200*x^5 + 5*x^4 + 216*x^3 + 157*x^2 - 124*x - 42) + 24*(31*x^4 + 107*x^3 - 243*x^2 - sqrt(3)*(-23*I*x^4 + 85*I*x^3 + 57*I*x^2 - 104*I*x + 4 *I) - 26*x + 50)*(-x^3 + 1)^(2/3) + 3*(-x^3 + 1)^(1/3)*(sqrt(3)*(-16)^(5/6 )*(4*x^5 + 69*x^4 - 58*x^3 - 77*x^2 + 12*x + 23) - 8*(-2)^(1/3)*(50*x^5 - 93*x^4 - 88*x^3 - 7*x^2 + 150*x - 31)))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x ^2 - 3*x + 1)) - 2*sqrt(3)*(-16)^(1/6)*(x^2 - x + 1)*log(-9*(4*sqrt(3)*(-1 6)^(1/6)*(131*x^6 - 48*x^5 - 381*x^4 - 152*x^3 + 267*x^2 + 276*x - 112) - 12*2^(2/3)*(15*x^6 - 200*x^5 + 5*x^4 + 216*x^3 + 157*x^2 - 124*x - 42) - 2 4*(31*x^4 + 107*x^3 - 243*x^2 - sqrt(3)*(23*I*x^4 - 85*I*x^3 - 57*I*x^2 + 104*I*x - 4*I) - 26*x + 50)*(-x^3 + 1)^(2/3) + 3*(-x^3 + 1)^(1/3)*(sqrt(3) *(-16)^(5/6)*(4*x^5 + 69*x^4 - 58*x^3 - 77*x^2 + 12*x + 23) + 8*(-2)^(1/3) *(50*x^5 - 93*x^4 - 88*x^3 - 7*x^2 + 150*x - 31)))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) - 24*sqrt(3)*(x^2 - x + 1)*arctan((4*sqrt(3)*(-x ^3 + 1)^(1/3)*x^2 + 2*sqrt(3)*(-x^3 + 1)^(2/3)*x - sqrt(3)*(x^3 - 1))/(9*x ^3 - 1)) + sqrt(3)*(sqrt(-3)*(-16)^(1/6)*(x^2 - x + 1) + (-16)^(1/6)*(x^2 - x + 1))*log(-9*(12*2^(2/3)*sqrt(-3)*(15*x^6 - 200*x^5 + 5*x^4 + 216*x^3 + 157*x^2 - 124*x - 42) + 12*2^(2/3)*(15*x^6 - 200*x^5 + 5*x^4 + 216*x^3 + 157*x^2 - 124*x - 42) - 48*(31*x^4 + 107*x^3 - 243*x^2 - sqrt(3)*(23*I...
\[ \int \frac {(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=- \int \left (- \frac {\left (1 - x^{3}\right )^{\frac {2}{3}}}{x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1}\right )\, dx - \int \frac {2 x \left (1 - x^{3}\right )^{\frac {2}{3}}}{x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1}\, dx \]
-Integral(-(1 - x**3)**(2/3)/(x**4 - 2*x**3 + 3*x**2 - 2*x + 1), x) - Inte gral(2*x*(1 - x**3)**(2/3)/(x**4 - 2*x**3 + 3*x**2 - 2*x + 1), x)
\[ \int \frac {(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int { -\frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (2 \, x - 1\right )}}{{\left (x^{2} - x + 1\right )}^{2}} \,d x } \]
\[ \int \frac {(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int { -\frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (2 \, x - 1\right )}}{{\left (x^{2} - x + 1\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=-\int \frac {\left (2\,x-1\right )\,{\left (1-x^3\right )}^{2/3}}{{\left (x^2-x+1\right )}^2} \,d x \]