Integrand size = 21, antiderivative size = 141 \[ \int \frac {-1+x}{\sqrt [3]{1-x-x^2+x^3}} \, dx=\frac {(-1+x)^{2/3} \sqrt [3]{1+x} \left (\sqrt [3]{-1+x} (1+x)^{2/3}+\frac {2 \arctan \left (\frac {\sqrt {3} \sqrt [3]{1+x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{1+x}}\right )}{\sqrt {3}}+\frac {2}{3} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{1+x}\right )-\frac {1}{3} \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )\right )}{\sqrt [3]{(-1+x)^2 (1+x)}} \]
(-1+x)^(2/3)*(1+x)^(1/3)*((-1+x)^(1/3)*(1+x)^(2/3)+2/3*3^(1/2)*arctan(3^(1 /2)*(1+x)^(1/3)/(2*(-1+x)^(1/3)+(1+x)^(1/3)))+2/3*ln((-1+x)^(1/3)-(1+x)^(1 /3))-1/3*ln((-1+x)^(2/3)+(-1+x)^(1/3)*(1+x)^(1/3)+(1+x)^(2/3)))/((-1+x)^2* (1+x))^(1/3)
Time = 0.11 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.13 \[ \int \frac {-1+x}{\sqrt [3]{1-x-x^2+x^3}} \, dx=\frac {-3+3 x^2+2 \sqrt {3} (-1+x)^{2/3} \sqrt [3]{1+x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1+x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{1+x}}\right )+2 (-1+x)^{2/3} \sqrt [3]{1+x} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{1+x}\right )-(-1+x)^{2/3} \sqrt [3]{1+x} \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )}{3 \sqrt [3]{(-1+x)^2 (1+x)}} \]
(-3 + 3*x^2 + 2*Sqrt[3]*(-1 + x)^(2/3)*(1 + x)^(1/3)*ArcTan[(Sqrt[3]*(1 + x)^(1/3))/(2*(-1 + x)^(1/3) + (1 + x)^(1/3))] + 2*(-1 + x)^(2/3)*(1 + x)^( 1/3)*Log[(-1 + x)^(1/3) - (1 + x)^(1/3)] - (-1 + x)^(2/3)*(1 + x)^(1/3)*Lo g[(-1 + x)^(2/3) + (-1 + x)^(1/3)*(1 + x)^(1/3) + (1 + x)^(2/3)])/(3*((-1 + x)^2*(1 + x))^(1/3))
Time = 0.23 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2489, 27, 60, 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.
| \(\displaystyle \int \frac {x-1}{\sqrt [3]{x^3-x^2-x+1}} \, dx\) |
| \(\Big \downarrow \) 2489 |
| \(\displaystyle \frac {4\ 2^{2/3} (x-1)^{2/3} \sqrt [3]{x+1} \int \frac {\sqrt [3]{x-1}}{4\ 2^{2/3} \sqrt [3]{x+1}}dx}{\sqrt [3]{x^3-x^2-x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(x-1)^{2/3} \sqrt [3]{x+1} \int \frac {\sqrt [3]{x-1}}{\sqrt [3]{x+1}}dx}{\sqrt [3]{x^3-x^2-x+1}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(x-1)^{2/3} \sqrt [3]{x+1} \left (\sqrt [3]{x-1} (x+1)^{2/3}-\frac {2}{3} \int \frac {1}{(x-1)^{2/3} \sqrt [3]{x+1}}dx\right )}{\sqrt [3]{x^3-x^2-x+1}}\) |
| \(\Big \downarrow \) 71 |
| \(\displaystyle \frac {(x-1)^{2/3} \sqrt [3]{x+1} \left (\sqrt [3]{x-1} (x+1)^{2/3}-\frac {2}{3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )-\frac {1}{2} \log (x-1)-\frac {3}{2} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{x-1}}-1\right )\right )\right )}{\sqrt [3]{x^3-x^2-x+1}}\) |
((-1 + x)^(2/3)*(1 + x)^(1/3)*((-1 + x)^(1/3)*(1 + x)^(2/3) - (2*(-(Sqrt[3 ]*ArcTan[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)
3.20.94.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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[((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 = 422, normalized size of antiderivative = 2.99
| method | result | size |
| trager | \(\frac {\left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}}{-1+x}+\frac {2 \ln \left (\frac {36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +12 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +x^{2}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-1}{-1+x}\right )}{3}+2 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +2 x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-4 x +2}{-1+x}\right )\) | \(422\) |
| risch | \(\frac {\left (1+x \right ) \left (-1+x \right )}{\left (\left (-1+x \right )^{2} \left (1+x \right )\right )^{\frac {1}{3}}}+\frac {2 \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 {2 \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 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +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 {2 \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 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +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}\) | \(557\) |
1/(-1+x)*(x^3-x^2-x+1)^(2/3)+2/3*ln((36*RootOf(9*_Z^2+3*_Z+1)^2*x^2-36*Roo tOf(9*_Z^2+3*_Z+1)^2*x-9*RootOf(9*_Z^2+3*_Z+1)*(x^3-x^2-x+1)^(2/3)+9*RootO f(9*_Z^2+3*_Z+1)*(x^3-x^2-x+1)^(1/3)*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)-6*RootOf(9*_Z^2+3*_Z+1)*x+3*(x^3-x^2 -x+1)^(1/3)*x+x^2-6*RootOf(9*_Z^2+3*_Z+1)-3*(x^3-x^2-x+1)^(1/3)-1)/(-1+x)) +2*RootOf(9*_Z^2+3*_Z+1)*ln(-(18*RootOf(9*_Z^2+3*_Z+1)^2*x^2-18*RootOf(9*_ Z^2+3*_Z+1)^2*x-9*RootOf(9*_Z^2+3*_Z+1)*(x^3-x^2-x+1)^(2/3)+15*RootOf(9*_Z ^2+3*_Z+1)*x^2-18*RootOf(9*_Z^2+3*_Z+1)*x-3*(x^3-x^2-x+1)^(2/3)+3*(x^3-x^2 -x+1)^(1/3)*x+2*x^2+3*RootOf(9*_Z^2+3*_Z+1)-3*(x^3-x^2-x+1)^(1/3)-4*x+2)/( -1+x))
Time = 0.32 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.14 \[ \int \frac {-1+x}{\sqrt [3]{1-x-x^2+x^3}} \, dx=\frac {2 \, \sqrt {3} {\left (x - 1\right )} \arctan \left (\frac {\sqrt {3} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}}}{3 \, {\left (x - 1\right )}}\right ) - {\left (x - 1\right )} \log \left (\frac {x^{2} + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 2 \, x + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {2}{3}} + 1}{x^{2} - 2 \, x + 1}\right ) + 2 \, {\left (x - 1\right )} \log \left (-\frac {x - {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} - 1}{x - 1}\right ) + 3 \, {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {2}{3}}}{3 \, {\left (x - 1\right )}} \]
1/3*(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)) + 3*(x^3 - x^2 - x + 1)^ (2/3))/(x - 1)
\[ \int \frac {-1+x}{\sqrt [3]{1-x-x^2+x^3}} \, dx=\int \frac {x - 1}{\sqrt [3]{\left (x - 1\right )^{2} \left (x + 1\right )}}\, dx \]
\[ \int \frac {-1+x}{\sqrt [3]{1-x-x^2+x^3}} \, dx=\int { \frac {x - 1}{{\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {-1+x}{\sqrt [3]{1-x-x^2+x^3}} \, dx=\int { \frac {x - 1}{{\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {-1+x}{\sqrt [3]{1-x-x^2+x^3}} \, dx=\int \frac {x-1}{{\left (x^3-x^2-x+1\right )}^{1/3}} \,d x \]