\(\int \frac {\sqrt [3]{1-x^3} (-2+x^3)}{x^3 (-1+x^3+x^6)} \, dx\) [8]

Optimal result

Integrand size = 30, antiderivative size = 110 \[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=-\frac {\sqrt [3]{1-x^3}}{x^2}-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{1-x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x^2+\sqrt [3]{1-x^3}\right )+\frac {1}{6} \log \left (x^4+x^2 \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right ) \] Output:

-(-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))
 

Mathematica [A] (verified)

Time = 1.22 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=-\frac {\sqrt [3]{1-x^3}}{x^2}-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{1-x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x^2+\sqrt [3]{1-x^3}\right )+\frac {1}{6} \log \left (x^4+x^2 \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right ) \] Input:

Integrate[((1 - x^3)^(1/3)*(-2 + x^3))/(x^3*(-1 + x^3 + x^6)),x]
 

Output:

-((1 - x^3)^(1/3)/x^2) - ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(1 - x^3)^(1/3))]/S 
qrt[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
 

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.62 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {7279, 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.

\(\displaystyle \int \frac {\sqrt [3]{1-x^3} \left (x^3-2\right )}{x^3 \left (x^6+x^3-1\right )} \, dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {2 \sqrt [3]{1-x^3}}{x^3}+\frac {\sqrt [3]{1-x^3} \left (-2 x^3-1\right )}{x^6+x^3-1}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 x \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},1,\frac {4}{3},x^3,-\frac {2 x^3}{1-\sqrt {5}}\right )}{1-\sqrt {5}}-\frac {2 x \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},1,\frac {4}{3},x^3,-\frac {2 x^3}{1+\sqrt {5}}\right )}{1+\sqrt {5}}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {1}{3},x^3\right )}{x^2}\)

Input:

Int[((1 - x^3)^(1/3)*(-2 + x^3))/(x^3*(-1 + x^3 + x^6)),x]
 

Output:

(-2*x*AppellF1[1/3, -1/3, 1, 4/3, x^3, (-2*x^3)/(1 - Sqrt[5])])/(1 - Sqrt[ 
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
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 6.09 (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	\(\text {Expression too large to display}\)	\(638\)

Input:

int((-x^3+1)^(1/3)*(x^3-2)/x^3/(x^6+x^3-1),x,method=_RETURNVERBOSE)
 

Output:

(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*Ro 
otOf(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)
 

Fricas [A] (verification not implemented)

Time = 1.50 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=-\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {\sqrt {3} x^{6} - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{4} + 4 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x^{2}}{x^{6} - 8 \, x^{3} + 8}\right ) + x^{2} \log \left (\frac {x^{6} - 3 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{4} + x^{3} + 3 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x^{2} - 1}{x^{6} + x^{3} - 1}\right ) + 6 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{6 \, x^{2}} \] Input:

integrate((-x^3+1)^(1/3)*(x^3-2)/x^3/(x^6+x^3-1),x, algorithm="fricas")
 

Output:

-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
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=\text {Timed out} \] Input:

integrate((-x**3+1)**(1/3)*(x**3-2)/x**3/(x**6+x**3-1),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} - 2\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{{\left (x^{6} + x^{3} - 1\right )} x^{3}} \,d x } \] Input:

integrate((-x^3+1)^(1/3)*(x^3-2)/x^3/(x^6+x^3-1),x, algorithm="maxima")
 

Output:

integrate((x^3 - 2)*(-x^3 + 1)^(1/3)/((x^6 + x^3 - 1)*x^3), x)
 

Giac [F]

\[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} - 2\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{{\left (x^{6} + x^{3} - 1\right )} x^{3}} \,d x } \] Input:

integrate((-x^3+1)^(1/3)*(x^3-2)/x^3/(x^6+x^3-1),x, algorithm="giac")
 

Output:

integrate((x^3 - 2)*(-x^3 + 1)^(1/3)/((x^6 + x^3 - 1)*x^3), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=\int \frac {{\left (1-x^3\right )}^{1/3}\,\left (x^3-2\right )}{x^3\,\left (x^6+x^3-1\right )} \,d x \] Input:

int(((1 - x^3)^(1/3)*(x^3 - 2))/(x^3*(x^3 + x^6 - 1)),x)
 

Output:

int(((1 - x^3)^(1/3)*(x^3 - 2))/(x^3*(x^3 + x^6 - 1)), x)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{1-x^3} \left (-2+x^3\right )}{x^3 \left (-1+x^3+x^6\right )} \, dx=-2 \left (\int \frac {\left (-x^{3}+1\right )^{\frac {1}{3}}}{x^{9}+x^{6}-x^{3}}d x \right )+\int \frac {\left (-x^{3}+1\right )^{\frac {1}{3}}}{x^{6}+x^{3}-1}d x \] Input:

int((-x^3+1)^(1/3)*(x^3-2)/x^3/(x^6+x^3-1),x)
 

Output:

 - 2*int(( - x**3 + 1)**(1/3)/(x**9 + x**6 - x**3),x) + int(( - x**3 + 1)* 
*(1/3)/(x**6 + x**3 - 1),x)