Integrand size = 25, antiderivative size = 135 \[ \int \frac {1+x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}+\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{\sqrt [3]{2}} \]
1/4*ln(1+2^(2/3)*(1-x)^2/(-x^3+1)^(2/3)-2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(2 /3)-1/2*ln(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(2/3)+1/2*arctan(1/3*(1-2*2^( 1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)
Time = 1.16 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.07 \[ \int \frac {1+x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{-2 \sqrt [3]{2}+2 \sqrt [3]{2} x+\sqrt [3]{1-x^3}}\right )-2 \log \left (-\sqrt [3]{2}+\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )+\log \left (2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2+(-1+x) \sqrt [3]{2-2 x^3}+\left (1-x^3\right )^{2/3}\right )}{2 \sqrt [3]{2}} \]
(-2*Sqrt[3]*ArcTan[(Sqrt[3]*(1 - x^3)^(1/3))/(-2*2^(1/3) + 2*2^(1/3)*x + ( 1 - x^3)^(1/3))] - 2*Log[-2^(1/3) + 2^(1/3)*x - (1 - x^3)^(1/3)] + Log[2^( 2/3) - 2*2^(2/3)*x + 2^(2/3)*x^2 + (-1 + x)*(2 - 2*x^3)^(1/3) + (1 - x^3)^ (2/3)])/(2*2^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(383\) vs. \(2(135)=270\).
Time = 0.54 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.84, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
| \(\displaystyle \int \frac {x+1}{\left (x^2-x+1\right ) \sqrt [3]{1-x^3}} \, dx\) |
| \(\Big \downarrow \) 2583 |
| \(\displaystyle \int \left (\frac {2 x}{\sqrt [3]{1-x^3} \left (x^3+1\right )}+\frac {1}{\sqrt [3]{1-x^3} \left (x^3+1\right )}+\frac {x^2}{\sqrt [3]{1-x^3} \left (x^3+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (x^3+1\right )}{3 \sqrt [3]{2}}+\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}-\frac {1}{3} 2^{2/3} \log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2 \sqrt [3]{2}}-\frac {\log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{2 \sqrt [3]{2}}+\frac {\log \left ((1-x) (x+1)^2\right )}{6 \sqrt [3]{2}}\) |
(2^(2/3)*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^(1/3)*Sqrt[ 3]) - ArcTan[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(1/3)*Sqrt[3] ) + ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]]/(2^(1/3)*Sqrt[3]) + Log[ (1 - x)*(1 + x)^2]/(6*2^(1/3)) - Log[1 + x^3]/(3*2^(1/3)) + 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^(1 /3)) - (2^(2/3)*Log[1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3)])/3 + Log[2^(1/3 ) - (1 - x^3)^(1/3)]/(2*2^(1/3)) + Log[-(2^(1/3)*x) - (1 - x^3)^(1/3)]/(2* 2^(1/3)) - Log[-1 + x + 2^(2/3)*(1 - x^3)^(1/3)]/(2*2^(1/3))
3.1.100.3.1 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 = 6.46 (sec) , antiderivative size = 677, normalized size of antiderivative = 5.01
1/2*RootOf(_Z^3+4)*ln((RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2) *RootOf(_Z^3+4)^2*(-x^3+1)^(2/3)-RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+ 4)+4*_Z^2)*RootOf(_Z^3+4)^3*x-2*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4 )+4*_Z^2)^2*RootOf(_Z^3+4)^2*x+(-x^3+1)^(1/3)*RootOf(_Z^3+4)^2*x+2*(-x^3+1 )^(1/3)*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4) *x-(-x^3+1)^(1/3)*RootOf(_Z^3+4)^2-2*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3+4)^ 2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)+RootOf(_Z^3+4)*x^2+2*RootOf(R ootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x^2-RootOf(_Z^3+4)*x-2*RootOf( RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x+RootOf(_Z^3+4)+2*RootOf(Roo tOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2))/(x^2-x+1))+RootOf(RootOf(_Z^3+4 )^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*ln(-(RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_ Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)^2*(-x^3+1)^(2/3)-RootOf(RootOf(_Z^3+4)^2+2*_ Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)^3*x-2*RootOf(RootOf(_Z^3+4)^2+2*_Z *RootOf(_Z^3+4)+4*_Z^2)^2*RootOf(_Z^3+4)^2*x+(-x^3+1)^(1/3)*RootOf(_Z^3+4) ^2*x-(-x^3+1)^(1/3)*RootOf(_Z^3+4)^2-RootOf(_Z^3+4)*x^2-2*RootOf(RootOf(_Z ^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x^2-2*(-x^3+1)^(2/3)+3*RootOf(_Z^3+4)* x+6*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x-RootOf(_Z^3+4)-2 *RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2))/(x^2-x+1))
Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (101) = 202\).
Time = 5.54 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.36 \[ \int \frac {1+x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (4 \cdot 2^{\frac {1}{6}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - 4 \, x^{3} + 5 \, x^{2} - 4 \, x + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4 \, \sqrt {2} \left (-1\right )^{\frac {1}{3}} {\left (x^{5} - x^{4} - 3 \, x^{3} + 3 \, x^{2} + x - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 2^{\frac {5}{6}} {\left (x^{6} - 7 \, x^{5} + 10 \, x^{4} - 7 \, x^{3} + 10 \, x^{2} - 7 \, x + 1\right )}\right )}}{6 \, {\left (3 \, x^{6} - 9 \, x^{5} + 6 \, x^{4} - x^{3} + 6 \, x^{2} - 9 \, x + 3\right )}}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} - 3 \, x + 1\right )} + 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - 3 \, x^{2} + 1\right )} + 4 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {2 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} - x + 1\right )} - 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} - x + 1}\right ) \]
1/6*sqrt(3)*2^(2/3)*(-1)^(1/3)*arctan(1/6*sqrt(3)*2^(1/6)*(4*2^(1/6)*(-1)^ (2/3)*(x^4 - 4*x^3 + 5*x^2 - 4*x + 1)*(-x^3 + 1)^(2/3) - 4*sqrt(2)*(-1)^(1 /3)*(x^5 - x^4 - 3*x^3 + 3*x^2 + x - 1)*(-x^3 + 1)^(1/3) + 2^(5/6)*(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)) - 1/12*2^(2/3)*(-1)^(1/3)*log(-(2^(2/3)*(-1)^(1/3)*(-x^ 3 + 1)^(2/3)*(x^2 - 3*x + 1) + 2^(1/3)*(-1)^(2/3)*(x^4 - 3*x^2 + 1) + 4*(- x^3 + 1)^(1/3)*(x^2 - x))/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) + 1/6*2^(2/3)*( -1)^(1/3)*log(-(2*2^(1/3)*(-1)^(2/3)*(-x^3 + 1)^(1/3)*(x - 1) + 2^(2/3)*(- 1)^(1/3)*(x^2 - x + 1) - 2*(-x^3 + 1)^(2/3))/(x^2 - x + 1))
\[ \int \frac {1+x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int \frac {x + 1}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} - x + 1\right )}\, dx \]
\[ \int \frac {1+x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int { \frac {x + 1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x + 1\right )}} \,d x } \]
\[ \int \frac {1+x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int { \frac {x + 1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {1+x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int \frac {x+1}{{\left (1-x^3\right )}^{1/3}\,\left (x^2-x+1\right )} \,d x \]
\[ \int \frac {1+x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx=\int \frac {x}{\left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-\left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}d x +\int \frac {1}{\left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-\left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}d x \]