Integrand size = 17, antiderivative size = 141 \[ \int \frac {x}{\sqrt [3]{-1-x+x^2+x^3}} \, dx=\frac {\sqrt [3]{-1+x} (1+x)^{2/3} \left ((-1+x)^{2/3} \sqrt [3]{1+x}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{1+x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{1+x}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{1+x}\right )-\frac {1}{6} \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )\right )}{\sqrt [3]{(-1+x) (1+x)^2}} \]
(-1+x)^(1/3)*(1+x)^(2/3)*((-1+x)^(2/3)*(1+x)^(1/3)-1/3*3^(1/2)*arctan(3^(1 /2)*(1+x)^(1/3)/(2*(-1+x)^(1/3)+(1+x)^(1/3)))+1/3*ln((-1+x)^(1/3)-(1+x)^(1 /3))-1/6*ln((-1+x)^(2/3)+(-1+x)^(1/3)*(1+x)^(1/3)+(1+x)^(2/3)))/((-1+x)*(1 +x)^2)^(1/3)
Time = 0.19 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.04 \[ \int \frac {x}{\sqrt [3]{-1-x+x^2+x^3}} \, dx=\frac {-6+6 x^2-2 \sqrt {3} \sqrt [3]{-1+x} (1+x)^{2/3} \arctan \left (\frac {1+\frac {2}{\sqrt [3]{\frac {-1+x}{1+x}}}}{\sqrt {3}}\right )+2 \sqrt [3]{-1+x} (1+x)^{2/3} \log \left (-1+\frac {1}{\sqrt [3]{\frac {-1+x}{1+x}}}\right )-\sqrt [3]{-1+x} (1+x)^{2/3} \log \left (1+\frac {1}{\left (\frac {-1+x}{1+x}\right )^{2/3}}+\frac {1}{\sqrt [3]{\frac {-1+x}{1+x}}}\right )}{6 \sqrt [3]{(-1+x) (1+x)^2}} \]
(-6 + 6*x^2 - 2*Sqrt[3]*(-1 + x)^(1/3)*(1 + x)^(2/3)*ArcTan[(1 + 2/((-1 + x)/(1 + x))^(1/3))/Sqrt[3]] + 2*(-1 + x)^(1/3)*(1 + x)^(2/3)*Log[-1 + ((-1 + x)/(1 + x))^(-1/3)] - (-1 + x)^(1/3)*(1 + x)^(2/3)*Log[1 + ((-1 + x)/(1 + x))^(-2/3) + ((-1 + x)/(1 + x))^(-1/3)])/(6*((-1 + x)*(1 + x)^2)^(1/3))
Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2489, 27, 90, 71}
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.
| \(\displaystyle \int \frac {x}{\sqrt [3]{x^3+x^2-x-1}} \, dx\) |
| \(\Big \downarrow \) 2489 |
| \(\displaystyle \frac {4\ 2^{2/3} \sqrt [3]{x-1} (x+1)^{2/3} \int \frac {x}{4\ 2^{2/3} \sqrt [3]{x-1} (x+1)^{2/3}}dx}{\sqrt [3]{x^3+x^2-x-1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt [3]{x-1} (x+1)^{2/3} \int \frac {x}{\sqrt [3]{x-1} (x+1)^{2/3}}dx}{\sqrt [3]{x^3+x^2-x-1}}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {\sqrt [3]{x-1} (x+1)^{2/3} \left ((x-1)^{2/3} \sqrt [3]{x+1}-\frac {1}{3} \int \frac {1}{\sqrt [3]{x-1} (x+1)^{2/3}}dx\right )}{\sqrt [3]{x^3+x^2-x-1}}\) |
| \(\Big \downarrow \) 71 |
| \(\displaystyle \frac {\sqrt [3]{x-1} (x+1)^{2/3} \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )+\frac {1}{2} \log (x+1)+\frac {3}{2} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x+1}}-1\right )\right )+(x-1)^{2/3} \sqrt [3]{x+1}\right )}{\sqrt [3]{x^3+x^2-x-1}}\) |
((-1 + x)^(1/3)*(1 + x)^(2/3)*((-1 + x)^(2/3)*(1 + x)^(1/3) + (Sqrt[3]*Arc Tan[1/Sqrt[3] + (2*(-1 + x)^(1/3))/(Sqrt[3]*(1 + x)^(1/3))] + Log[1 + x]/2 + (3*Log[-1 + (-1 + x)^(1/3)/(1 + x)^(1/3)])/2)/3))/(-1 - x + x^2 + x^3)^ (1/3)
3.20.95.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, Simp[(-Sqrt[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/( Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*((a + b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[d/b]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((e_.) + (f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*( x_)^3)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2 + d*x^3)^p/((c^3 - 4*b*c*d + 9*a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3*b*d)*x)^(2*p)) Int[(e + f*x)^m*(c^3 - 4*b*c*d + 9*a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a *d + 2*(c^2 - 3*b*d)*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && NeQ[c^2 - 3*b*d, 0] && EqQ[b^2*c^2 - 4*a*c^3 - 4*b^3*d + 18*a*b*c*d - 2 7*a^2*d^2, 0] && !IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.54 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.86
| method | result | size |
| trager | \(\frac {\left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}}{1+x}+\frac {\ln \left (\frac {-36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x -12 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-3 x \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{1+x}\right )}{3}+\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x -15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-3 x \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -2 x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4 x -2}{1+x}\right )\) | \(403\) |
| risch | \(\frac {\left (1+x \right ) \left (-1+x \right )}{\left (\left (-1+x \right ) \left (1+x \right )^{2}\right )^{\frac {1}{3}}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +2 x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2}{1+x}\right )}{3}-\frac {\ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+3 x \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+x^{2}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2 x +1}{1+x}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{3}-\frac {\ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+3 x \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+x^{2}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2 x +1}{1+x}\right )}{3}\) | \(528\) |
1/(1+x)*(x^3+x^2-x-1)^(2/3)+1/3*ln((-36*RootOf(9*_Z^2+3*_Z+1)^2*x^2+9*Root Of(9*_Z^2+3*_Z+1)*(x^3+x^2-x-1)^(2/3)-9*RootOf(9*_Z^2+3*_Z+1)*(x^3+x^2-x-1 )^(1/3)*x-36*RootOf(9*_Z^2+3*_Z+1)^2*x-12*RootOf(9*_Z^2+3*_Z+1)*x^2-9*Root Of(9*_Z^2+3*_Z+1)*(x^3+x^2-x-1)^(1/3)-3*x*(x^3+x^2-x-1)^(1/3)-6*RootOf(9*_ Z^2+3*_Z+1)*x-x^2-3*(x^3+x^2-x-1)^(1/3)+6*RootOf(9*_Z^2+3*_Z+1)+1)/(1+x))+ RootOf(9*_Z^2+3*_Z+1)*ln(-(-18*RootOf(9*_Z^2+3*_Z+1)^2*x^2+9*RootOf(9*_Z^2 +3*_Z+1)*(x^3+x^2-x-1)^(2/3)-18*RootOf(9*_Z^2+3*_Z+1)^2*x-15*RootOf(9*_Z^2 +3*_Z+1)*x^2+3*(x^3+x^2-x-1)^(2/3)-3*x*(x^3+x^2-x-1)^(1/3)-18*RootOf(9*_Z^ 2+3*_Z+1)*x-2*x^2-3*(x^3+x^2-x-1)^(1/3)-3*RootOf(9*_Z^2+3*_Z+1)-4*x-2)/(1+ x))
Time = 0.32 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.07 \[ \int \frac {x}{\sqrt [3]{-1-x+x^2+x^3}} \, dx=\frac {2 \, \sqrt {3} {\left (x + 1\right )} \arctan \left (\frac {\sqrt {3} {\left (x + 1\right )} + 2 \, \sqrt {3} {\left (x^{3} + x^{2} - x - 1\right )}^{\frac {1}{3}}}{3 \, {\left (x + 1\right )}}\right ) - {\left (x + 1\right )} \log \left (\frac {x^{2} + {\left (x^{3} + x^{2} - x - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + 2 \, x + {\left (x^{3} + x^{2} - x - 1\right )}^{\frac {2}{3}} + 1}{x^{2} + 2 \, x + 1}\right ) + 2 \, {\left (x + 1\right )} \log \left (-\frac {x - {\left (x^{3} + x^{2} - x - 1\right )}^{\frac {1}{3}} + 1}{x + 1}\right ) + 6 \, {\left (x^{3} + x^{2} - x - 1\right )}^{\frac {2}{3}}}{6 \, {\left (x + 1\right )}} \]
1/6*(2*sqrt(3)*(x + 1)*arctan(1/3*(sqrt(3)*(x + 1) + 2*sqrt(3)*(x^3 + x^2 - x - 1)^(1/3))/(x + 1)) - (x + 1)*log((x^2 + (x^3 + x^2 - x - 1)^(1/3)*(x + 1) + 2*x + (x^3 + x^2 - x - 1)^(2/3) + 1)/(x^2 + 2*x + 1)) + 2*(x + 1)* log(-(x - (x^3 + x^2 - x - 1)^(1/3) + 1)/(x + 1)) + 6*(x^3 + x^2 - x - 1)^ (2/3))/(x + 1)
\[ \int \frac {x}{\sqrt [3]{-1-x+x^2+x^3}} \, dx=\int \frac {x}{\sqrt [3]{\left (x - 1\right ) \left (x + 1\right )^{2}}}\, dx \]
\[ \int \frac {x}{\sqrt [3]{-1-x+x^2+x^3}} \, dx=\int { \frac {x}{{\left (x^{3} + x^{2} - x - 1\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {x}{\sqrt [3]{-1-x+x^2+x^3}} \, dx=\int { \frac {x}{{\left (x^{3} + x^{2} - x - 1\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {x}{\sqrt [3]{-1-x+x^2+x^3}} \, dx=\int \frac {x}{{\left (x^3+x^2-x-1\right )}^{1/3}} \,d x \]