\(\int \frac {x}{\sqrt [3]{1-x^3} (1+x^3)} \, dx\) [98]

Optimal result

Integrand size = 20, antiderivative size = 233 \[ \int \frac {x}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left ((1-x) (1+x)^2\right )}{12 \sqrt [3]{2}}+\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{6 \sqrt [3]{2}}-\frac {\log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}} \] Output:

1/24*ln((1-x)*(1+x)^2)*2^(2/3)+1/12*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)-1/6*ln(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3)) 
*2^(2/3)-1/8*ln(-1+x+2^(2/3)*(-x^3+1)^(1/3))*2^(2/3)+1/6*arctan(1/3*(1-2*2 
^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)+1/12*arctan(1/3*(1+2 
^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)*2^(2/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.21 \[ \int \frac {x}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{\sqrt [3]{2}-\sqrt [3]{2} x+\sqrt [3]{1-x^3}}\right )-4 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{-2 \sqrt [3]{2}+2 \sqrt [3]{2} x+\sqrt [3]{1-x^3}}\right )-4 \log \left (-\sqrt [3]{2}+\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )-2 \log \left (-\sqrt [3]{2}+\sqrt [3]{2} x+2 \sqrt [3]{1-x^3}\right )+2 \log \left (2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2+(-1+x) \sqrt [3]{2-2 x^3}+\left (1-x^3\right )^{2/3}\right )+\log \left (2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2-2 (-1+x) \sqrt [3]{2-2 x^3}+4 \left (1-x^3\right )^{2/3}\right )}{12 \sqrt [3]{2}} \] Input:

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

Output:

(-2*Sqrt[3]*ArcTan[(Sqrt[3]*(1 - x^3)^(1/3))/(2^(1/3) - 2^(1/3)*x + (1 - x 
^3)^(1/3))] - 4*Sqrt[3]*ArcTan[(Sqrt[3]*(1 - x^3)^(1/3))/(-2*2^(1/3) + 2*2 
^(1/3)*x + (1 - x^3)^(1/3))] - 4*Log[-2^(1/3) + 2^(1/3)*x - (1 - x^3)^(1/3 
)] - 2*Log[-2^(1/3) + 2^(1/3)*x + 2*(1 - x^3)^(1/3)] + 2*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)] + L 
og[2^(2/3) - 2*2^(2/3)*x + 2^(2/3)*x^2 - 2*(-1 + x)*(2 - 2*x^3)^(1/3) + 4* 
(1 - x^3)^(2/3)])/(12*2^(1/3))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {991, 750, 16, 27, 1142, 25, 27, 1082, 217, 1103, 2574}

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.

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

\(\Big \downarrow \) 991

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

\(\Big \downarrow \) 750

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

\(\Big \downarrow \) 2574

\(\displaystyle -\frac {1}{3} \sqrt [3]{2} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3}}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{2\ 2^{2/3}}\right )+\frac {1}{3} \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}-\frac {3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}+\frac {\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}}\right )-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}\)

Input:

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

Output:

-1/3*(2^(1/3)*(-((Sqrt[3]*ArcTan[(1 - (2*2^(1/3)*(1 - x))/(1 - x^3)^(1/3)) 
/Sqrt[3]])/2^(2/3)) - Log[1 + (2^(2/3)*(1 - x)^2)/(1 - x^3)^(2/3) - (2^(1/ 
3)*(1 - x))/(1 - x^3)^(1/3)]/(2*2^(2/3)))) - Log[1 + (2^(1/3)*(1 - x))/(1 
- x^3)^(1/3)]/(3*2^(1/3)) + ((Sqrt[3]*ArcTan[(1 + (2^(1/3)*(1 - x))/(1 - x 
^3)^(1/3))/Sqrt[3]])/(2*2^(1/3)) + Log[(1 - x)*(1 + x)^2]/(4*2^(1/3)) - (3 
*Log[-1 + x + 2^(2/3)*(1 - x^3)^(1/3)])/(4*2^(1/3)))/3
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 750

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2)   Int[1/ 
(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2)   Int[(2*Rt[a, 3] - 
 Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; 
 FreeQ[{a, b}, x]
 

rule 991

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> 
With[{q = Rt[b/a, 3]}, Simp[-q^2/(3*d)   Int[1/((1 - q*x)*(a + b*x^3)^(1/3) 
), x], x] + Simp[q/d   Subst[Int[1/(1 + 2*a*x^3), x], x, (1 + q*x)/(a + b*x 
^3)^(1/3)], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 
 a*d, 0]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 2574

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[ 
Sqrt[3]*(ArcTan[(1 - 2^(1/3)*Rt[b, 3]*((c - d*x)/(d*(a + b*x^3)^(1/3))))/Sq 
rt[3]]/(2^(4/3)*Rt[b, 3]*c)), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2^(7/3 
)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^ 
(1/3)])/(2^(7/3)*Rt[b, 3]*c), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + 
a*d^3, 0]
 

Maple [F]

\[\int \frac {x}{\left (-x^{3}+1\right )^{\frac {1}{3}} \left (x^{3}+1\right )}d x\]

Input:

int(x/(-x^3+1)^(1/3)/(x^3+1),x)
 

Output:

int(x/(-x^3+1)^(1/3)/(x^3+1),x)
 

Fricas [A] (verification not implemented)

Time = 1.57 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.44 \[ \int \frac {x}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {1}{6} \cdot 2^{\frac {1}{6}} \sqrt {\frac {1}{6}} \arctan \left (\frac {2^{\frac {1}{6}} \sqrt {\frac {1}{6}} {\left (24 \cdot 2^{\frac {2}{3}} {\left (x^{14} - 2 \, x^{11} - 6 \, x^{8} - 2 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{18} + 42 \, x^{15} - 417 \, x^{12} + 812 \, x^{9} - 417 \, x^{6} + 42 \, x^{3} + 1\right )} - 12 \, {\left (x^{16} - 33 \, x^{13} + 110 \, x^{10} - 110 \, x^{7} + 33 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{x^{18} - 102 \, x^{15} + 447 \, x^{12} - 628 \, x^{9} + 447 \, x^{6} - 102 \, x^{3} + 1}\right ) - \frac {1}{36} \cdot 2^{\frac {2}{3}} \log \left (-\frac {12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x^{2} + 2^{\frac {2}{3}} {\left (x^{6} + 2 \, x^{3} + 1\right )} - 6 \cdot 2^{\frac {1}{3}} {\left (x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + \frac {1}{72} \cdot 2^{\frac {2}{3}} \log \left (\frac {12 \cdot 2^{\frac {2}{3}} {\left (x^{8} - 4 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{12} - 32 \, x^{9} + 78 \, x^{6} - 32 \, x^{3} + 1\right )} + 6 \, {\left (x^{10} - 11 \, x^{7} + 11 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) \] Input:

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

Output:

1/6*2^(1/6)*sqrt(1/6)*arctan(2^(1/6)*sqrt(1/6)*(24*2^(2/3)*(x^14 - 2*x^11 
- 6*x^8 - 2*x^5 + x^2)*(-x^3 + 1)^(2/3) + 2^(1/3)*(x^18 + 42*x^15 - 417*x^ 
12 + 812*x^9 - 417*x^6 + 42*x^3 + 1) - 12*(x^16 - 33*x^13 + 110*x^10 - 110 
*x^7 + 33*x^4 - x)*(-x^3 + 1)^(1/3))/(x^18 - 102*x^15 + 447*x^12 - 628*x^9 
 + 447*x^6 - 102*x^3 + 1)) - 1/36*2^(2/3)*log(-(12*(-x^3 + 1)^(2/3)*x^2 + 
2^(2/3)*(x^6 + 2*x^3 + 1) - 6*2^(1/3)*(x^4 - x)*(-x^3 + 1)^(1/3))/(x^6 + 2 
*x^3 + 1)) + 1/72*2^(2/3)*log((12*2^(2/3)*(x^8 - 4*x^5 + x^2)*(-x^3 + 1)^( 
2/3) + 2^(1/3)*(x^12 - 32*x^9 + 78*x^6 - 32*x^3 + 1) + 6*(x^10 - 11*x^7 + 
11*x^4 - x)*(-x^3 + 1)^(1/3))/(x^12 + 4*x^9 + 6*x^6 + 4*x^3 + 1))
 

Sympy [F]

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

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

Output:

Integral(x/((-(x - 1)*(x**2 + x + 1))**(1/3)*(x + 1)*(x**2 - x + 1)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int(x/(-x^3+1)^(1/3)/(x^3+1),x)
 

Output:

int(x/(( - x**3 + 1)**(1/3)*x**3 + ( - x**3 + 1)**(1/3)),x)