Integrand size = 30, antiderivative size = 110 \[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=-\frac {\sqrt [3]{1-x^3}}{x^2}-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{1-x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x^2+\sqrt [3]{1-x^3}\right )+\frac {1}{6} \log \left (x^4+x^2 \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right ) \]
[Out] -(-x^3+1)^(1/3)/x^2-1/3*arctan(3^(1/2)*x^2/(x^2+2*(-x^3+1)^(1/3)))*3^( 1/2)-1/3*ln(-x^2+(-x^3+1)^(1/3))+1/6*ln(x^4+x^2*(-x^3+1)^(1/3)+(-x^3+1 )^(2/3))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.37 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {7281, 888, 936} \[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=-\frac {2 x \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},1,\frac {4}{3},x^3,-\frac {2 x^3}{1-\sqrt {5}}\right )}{1-\sqrt {5}}-\frac {2 x \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},1,\frac {4}{3},x^3,-\frac {2 x^3}{1+\sqrt {5}}\right )}{1+\sqrt {5}}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {1}{3},x^3\right )}{x^2} \]
[In] Int[((1 - x^3)^(1/3)*(-2 + x^3))/(x^3*(-1 + x^3 + x^6)),x]
[Out] (-2*x*AppellF1[1/3, -1/3, 1, 4/3, x^3, (-2*x^3)/(1 - Sqrt[5])])/(1 - S qrt[5]) - (2*x*AppellF1[1/3, -1/3, 1, 4/3, x^3, (-2*x^3)/(1 + Sqrt[5]) ])/(1 + Sqrt[5]) - Hypergeometric2F1[-2/3, -1/3, 1/3, x^3]/x^2
Rule 888
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Rule 936
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) ], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Rule 7281
Int[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /; !FalseQ[lst]]
Rubi steps \begin{align*} \text {integral}= \int \left (\frac {2 \sqrt [3]{1-x^3}}{x^3}+\frac {\left (-1-2 x^3\right ) \sqrt [3]{1-x^3}}{-1+x^3+x^6}\right ) \, dx \\ = 2 \int \frac {\sqrt [3]{1-x^3}}{x^3} \, dx+\int \frac {\left (-1-2 x^3\right ) \sqrt [3]{1-x^3}}{-1+x^3+x^6} \, dx \\ = -\frac {\, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {1}{3};x^3\right )}{x^2}+\int \left (-\frac {2 \sqrt [3]{1-x^3}}{1-\sqrt {5}+2 x^3}-\frac {2 \sqrt [3]{1-x^3}}{1+\sqrt {5}+2 x^3}\right ) \, dx \\ = -\frac {\, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {1}{3};x^3\right )}{x^2}-2 \int \frac {\sqrt [3]{1-x^3}}{1-\sqrt {5}+2 x^3} \, dx-2 \int \frac {\sqrt [3]{1-x^3}}{1+\sqrt {5}+2 x^3} \, dx \\ = -\frac {2 x F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};x^3,-\frac {2 x^3}{1-\sqrt {5}}\right )}{1-\sqrt {5}}-\frac {2 x F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};x^3,-\frac {2 x^3}{1+\sqrt {5}}\right )}{1+\sqrt {5}}-\frac {\, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {1}{3};x^3\right )}{x^2} \\ \end{align*}
Time = 1.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=-\frac {\sqrt [3]{1-x^3}}{x^2}-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{1-x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x^2+\sqrt [3]{1-x^3}\right )+\frac {1}{6} \log \left (x^4+x^2 \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right ) \]
[In] Integrate[((1 - x^3)^(1/3)*(-2 + x^3))/(x^3*(-1 + x^3 + x^6)),x]
[Out] -((1 - x^3)^(1/3)/x^2) - ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(1 - x^3)^(1/3) )]/Sqrt[3] - Log[-x^2 + (1 - x^3)^(1/3)]/3 + Log[x^4 + x^2*(1 - x^3)^( 1/3) + (1 - x^3)^(2/3)]/6
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 5.34 (sec) , antiderivative size = 600, normalized size of antiderivative = 5.45
| method | result | size |
| risch | \(\frac {x^{3}-1}{x^{2} \left (-x^{3}+1\right )^{\frac {2}{3}}}+\frac {\left (\frac {\ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{9}-x^{9}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{6}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}+3 \left (x^{6}-2 x^{3}+1\right )^{\frac {2}{3}} x^{4}-3 \left (x^{6}-2 x^{3}+1\right )^{\frac {1}{3}} x^{5}+2 x^{6}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+3 \left (x^{6}-2 x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+1}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (x^{6}+x^{3}-1\right )}\right )}{3}-\ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{9}-x^{9}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{6}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}+3 \left (x^{6}-2 x^{3}+1\right )^{\frac {2}{3}} x^{4}-3 \left (x^{6}-2 x^{3}+1\right )^{\frac {1}{3}} x^{5}+2 x^{6}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+3 \left (x^{6}-2 x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+1}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (x^{6}+x^{3}-1\right )}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+\operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{9}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{9}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{6}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}+3 \left (x^{6}-2 x^{3}+1\right )^{\frac {2}{3}} x^{4}-3 \left (x^{6}-2 x^{3}+1\right )^{\frac {1}{3}} x^{5}+2 x^{6}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+3 \left (x^{6}-2 x^{3}+1\right )^{\frac {1}{3}} x^{2}-4 x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+2}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (x^{6}+x^{3}-1\right )}\right )\right ) {\left (\left (x^{3}-1\right )^{2}\right )}^{\frac {1}{3}}}{\left (-x^{3}+1\right )^{\frac {2}{3}}}\) | \(600\) |
| trager | \(\text {Expression too large to display}\) | \(639\) |
[In] int((-x^3+1)^(1/3)*(x^3-2)/x^3/(x^6+x^3-1),x,method=_RETURNVERBOSE)
[Out] (x^3-1)/x^2/(-x^3+1)^(2/3)+(1/3*ln((9*RootOf(9*_Z^2-3*_Z+1)^2*x^9-x^9- 9*RootOf(9*_Z^2-3*_Z+1)^2*x^6+3*RootOf(9*_Z^2-3*_Z+1)*x^6+3*(x^6-2*x^3 +1)^(2/3)*x^4-3*(x^6-2*x^3+1)^(1/3)*x^5+2*x^6-6*RootOf(9*_Z^2-3*_Z+1)* x^3+3*(x^6-2*x^3+1)^(1/3)*x^2-2*x^3+3*RootOf(9*_Z^2-3*_Z+1)+1)/(-1+x)/ (x^2+x+1)/(x^6+x^3-1))-ln((9*RootOf(9*_Z^2-3*_Z+1)^2*x^9-x^9-9*RootOf( 9*_Z^2-3*_Z+1)^2*x^6+3*RootOf(9*_Z^2-3*_Z+1)*x^6+3*(x^6-2*x^3+1)^(2/3) *x^4-3*(x^6-2*x^3+1)^(1/3)*x^5+2*x^6-6*RootOf(9*_Z^2-3*_Z+1)*x^3+3*(x^ 6-2*x^3+1)^(1/3)*x^2-2*x^3+3*RootOf(9*_Z^2-3*_Z+1)+1)/(-1+x)/(x^2+x+1) /(x^6+x^3-1))*RootOf(9*_Z^2-3*_Z+1)+RootOf(9*_Z^2-3*_Z+1)*ln((9*RootOf (9*_Z^2-3*_Z+1)^2*x^9-6*RootOf(9*_Z^2-3*_Z+1)*x^9-9*RootOf(9*_Z^2-3*_Z +1)^2*x^6+3*RootOf(9*_Z^2-3*_Z+1)*x^6+3*(x^6-2*x^3+1)^(2/3)*x^4-3*(x^6 -2*x^3+1)^(1/3)*x^5+2*x^6+6*RootOf(9*_Z^2-3*_Z+1)*x^3+3*(x^6-2*x^3+1)^ (1/3)*x^2-4*x^3-3*RootOf(9*_Z^2-3*_Z+1)+2)/(-1+x)/(x^2+x+1)/(x^6+x^3-1 )))/(-x^3+1)^(2/3)*((x^3-1)^2)^(1/3)
none
Time = 1.50 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=-\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {\sqrt {3} x^{6} - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{4} + 4 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x^{2}}{x^{6} - 8 \, x^{3} + 8}\right ) + x^{2} \log \left (\frac {x^{6} - 3 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{4} + x^{3} + 3 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x^{2} - 1}{x^{6} + x^{3} - 1}\right ) + 6 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{6 \, x^{2}} \]
[In] integrate((-x^3+1)^(1/3)*(x^3-2)/x^3/(x^6+x^3-1),x, algorithm="fricas" )
[Out] -1/6*(2*sqrt(3)*x^2*arctan((sqrt(3)*x^6 - 2*sqrt(3)*(-x^3 + 1)^(1/3)*x ^4 + 4*sqrt(3)*(-x^3 + 1)^(2/3)*x^2)/(x^6 - 8*x^3 + 8)) + x^2*log((x^6 - 3*(-x^3 + 1)^(1/3)*x^4 + x^3 + 3*(-x^3 + 1)^(2/3)*x^2 - 1)/(x^6 + x ^3 - 1)) + 6*(-x^3 + 1)^(1/3))/x^2
Timed out. \[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=\text {Timed out} \]
[In] integrate((-x**3+1)**(1/3)*(x**3-2)/x**3/(x**6+x**3-1),x)
[Out] Timed out
\[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} - 2\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{{\left (x^{6} + x^{3} - 1\right )} x^{3}} \,d x } \]
[In] integrate((-x^3+1)^(1/3)*(x^3-2)/x^3/(x^6+x^3-1),x, algorithm="maxima" )
[Out] integrate((x^3 - 2)*(-x^3 + 1)^(1/3)/((x^6 + x^3 - 1)*x^3), x)
\[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} - 2\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{{\left (x^{6} + x^{3} - 1\right )} x^{3}} \,d x } \]
[In] integrate((-x^3+1)^(1/3)*(x^3-2)/x^3/(x^6+x^3-1),x, algorithm="giac")
[Out] integrate((x^3 - 2)*(-x^3 + 1)^(1/3)/((x^6 + x^3 - 1)*x^3), x)
Timed out. \[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=\int \frac {{\left (1-x^3\right )}^{1/3}\,\left (x^3-2\right )}{x^3\,\left (x^6+x^3-1\right )} \,d x \]
[In] int(((1 - x^3)^(1/3)*(x^3 - 2))/(x^3*(x^3 + x^6 - 1)),x)
[Out] int(((1 - x^3)^(1/3)*(x^3 - 2))/(x^3*(x^3 + x^6 - 1)), x)