\(\int \frac {x^3}{\sqrt [3]{-x^2+x^3} (-1+x^6)} \, dx\) [2752]

Optimal result

Integrand size = 24, antiderivative size = 257 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx=-\frac {\left (-x^2+x^3\right )^{2/3}}{2 (-1+x) x}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{6 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{12 \sqrt [3]{2}}+\frac {1}{6} \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]-\frac {1}{6} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \] Output:

Unintegrable
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.09 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx=\frac {x^{2/3} \left (-12 \sqrt [3]{x}+2\ 2^{2/3} \sqrt {3} \sqrt [3]{-1+x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2^{2/3} \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2\ 2^{2/3} \sqrt [3]{-1+x} \log \left (2^{2/3} \sqrt [3]{-1+x}-2 \sqrt [3]{x}\right )+2^{2/3} \sqrt [3]{-1+x} \log \left (\sqrt [3]{2} (-1+x)^{2/3}+2^{2/3} \sqrt [3]{-1+x} \sqrt [3]{x}+2 x^{2/3}\right )+4 \sqrt [3]{-1+x} \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]-4 \sqrt [3]{-1+x} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{24 \sqrt [3]{(-1+x) x^2}} \] Input:

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

Output:

(x^(2/3)*(-12*x^(1/3) + 2*2^(2/3)*Sqrt[3]*(-1 + x)^(1/3)*ArcTan[(Sqrt[3]*x 
^(1/3))/(2^(2/3)*(-1 + x)^(1/3) + x^(1/3))] - 2*2^(2/3)*(-1 + x)^(1/3)*Log 
[2^(2/3)*(-1 + x)^(1/3) - 2*x^(1/3)] + 2^(2/3)*(-1 + x)^(1/3)*Log[2^(1/3)* 
(-1 + x)^(2/3) + 2^(2/3)*(-1 + x)^(1/3)*x^(1/3) + 2*x^(2/3)] + 4*(-1 + x)^ 
(1/3)*RootSum[3 - 3*#1^3 + #1^6 & , (-Log[x^(1/3)] + Log[(-1 + x)^(1/3) - 
x^(1/3)*#1])/#1 & ] - 4*(-1 + x)^(1/3)*RootSum[1 - #1^3 + #1^6 & , (-Log[x 
^(1/3)] + Log[(-1 + x)^(1/3) - x^(1/3)*#1])/#1 & ]))/(24*((-1 + x)*x^2)^(1 
/3))
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.36 (sec) , antiderivative size = 1274, normalized size of antiderivative = 4.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2467, 25, 2035, 7276, 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.

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

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [3]{x-1} x^{2/3} \int -\frac {x^{7/3}}{\sqrt [3]{x-1} \left (1-x^6\right )}dx}{\sqrt [3]{x^3-x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt [3]{x-1} x^{2/3} \int \frac {x^{7/3}}{\sqrt [3]{x-1} \left (1-x^6\right )}dx}{\sqrt [3]{x^3-x^2}}\)

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {3 \sqrt [3]{x-1} x^{2/3} \int \left (\frac {\sqrt [3]{x}-2}{18 \left (x^{2/3}-\sqrt [3]{x}+1\right ) \sqrt [3]{x-1}}-\frac {\sqrt [3]{x}}{9 \left (x^{2/3}-1\right ) \sqrt [3]{x-1}}+\frac {\sqrt [3]{x}+2}{18 \left (x^{2/3}+\sqrt [3]{x}+1\right ) \sqrt [3]{x-1}}+\frac {x-2}{6 \sqrt [3]{x-1} \left (x^2-x+1\right )}+\frac {x+2}{6 \sqrt [3]{x-1} \left (x^2+x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^3-x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt [3]{x-1} x^{2/3} \left (-\frac {\arctan \left (\frac {1-\frac {\sqrt [3]{2} \left (1-\sqrt [3]{x}\right )}{\sqrt [3]{x-1}}}{\sqrt {3}}\right )}{18 \sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} \left (1-\sqrt [3]{x}\right )}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{18 \sqrt [3]{2} \sqrt {3}}-\frac {\arctan \left (\frac {1-2^{2/3} \sqrt [3]{x-1}}{\sqrt {3}}\right )}{18 \sqrt [3]{2} \sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{9 \sqrt [3]{2} \sqrt {3}}+\frac {\sqrt [3]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}}}+\frac {\sqrt [3]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {\log \left (1-\frac {\sqrt [3]{2} \left (1-\sqrt [3]{x}\right )}{\sqrt [3]{x-1}}\right )}{54 \sqrt [3]{2}}+\frac {\log \left (\frac {2^{2/3} \left (1-\sqrt [3]{x}\right )^2}{(x-1)^{2/3}}+\frac {\sqrt [3]{2} \left (1-\sqrt [3]{x}\right )}{\sqrt [3]{x-1}}+1\right )}{108 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{x-1}+\sqrt [3]{2}\right )}{36 \sqrt [3]{2}}+\frac {\log \left (2^{2/3} \sqrt [3]{x-1}-\sqrt [3]{x}+1\right )}{36 \sqrt [3]{2}}-\frac {\log \left (\left (1-\sqrt [3]{x}\right ) \left (\sqrt [3]{x}+1\right )^2\right )}{108 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{18 \sqrt [3]{2}}-\frac {1}{12} \sqrt [3]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}}}-\sqrt [3]{x-1}\right )+\frac {1}{12} \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}}}-\sqrt [3]{x-1}\right )+\frac {\log \left (\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{12 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}}}-\frac {1}{12} \sqrt [3]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}}}-\sqrt [3]{x-1}\right )-\frac {\log (x+1)}{108 \sqrt [3]{2}}-\frac {1}{36} \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \log \left (2 x-i \sqrt {3}-1\right )+\frac {1}{36} \sqrt [3]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \log \left (2 x-i \sqrt {3}+1\right )-\frac {\log \left (2 x+i \sqrt {3}-1\right )}{36 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}}}+\frac {1}{36} \sqrt [3]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \log \left (2 x+i \sqrt {3}+1\right )+\frac {\sqrt [3]{x}}{6 \sqrt [3]{x-1}}\right )}{\sqrt [3]{x^3-x^2}}\)

Input:

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

Output:

(-3*(-1 + x)^(1/3)*x^(2/3)*(x^(1/3)/(6*(-1 + x)^(1/3)) - ArcTan[(1 - (2^(1 
/3)*(1 - x^(1/3)))/(-1 + x)^(1/3))/Sqrt[3]]/(18*2^(1/3)*Sqrt[3]) + ArcTan[ 
(1 + (2*2^(1/3)*(1 - x^(1/3)))/(-1 + x)^(1/3))/Sqrt[3]]/(18*2^(1/3)*Sqrt[3 
]) - ArcTan[(1 - 2^(2/3)*(-1 + x)^(1/3))/Sqrt[3]]/(18*2^(1/3)*Sqrt[3]) - A 
rcTan[(1 + (2*2^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]]/(9*2^(1/3)*Sqrt[3] 
) + (((I - Sqrt[3])/(3*I - Sqrt[3]))^(1/3)*ArcTan[(1 + (2*x^(1/3))/(((I - 
Sqrt[3])/(3*I - Sqrt[3]))^(1/3)*(-1 + x)^(1/3)))/Sqrt[3]])/(6*Sqrt[3]) - ( 
(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*ArcTan[(1 + (2*x^(1/3))/((-((I - Sq 
rt[3])/(I + Sqrt[3])))^(1/3)*(-1 + x)^(1/3)))/Sqrt[3]])/(6*Sqrt[3]) - ArcT 
an[(1 + (2*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*x^(1/3))/(-1 + x)^(1/3)) 
/Sqrt[3]]/(6*Sqrt[3]*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)) + (((I + Sqrt 
[3])/(3*I + Sqrt[3]))^(1/3)*ArcTan[(1 + (2*x^(1/3))/(((I + Sqrt[3])/(3*I + 
 Sqrt[3]))^(1/3)*(-1 + x)^(1/3)))/Sqrt[3]])/(6*Sqrt[3]) - Log[1 - (2^(1/3) 
*(1 - x^(1/3)))/(-1 + x)^(1/3)]/(54*2^(1/3)) + Log[1 + (2^(2/3)*(1 - x^(1/ 
3))^2)/(-1 + x)^(2/3) + (2^(1/3)*(1 - x^(1/3)))/(-1 + x)^(1/3)]/(108*2^(1/ 
3)) - Log[2^(1/3) + (-1 + x)^(1/3)]/(36*2^(1/3)) + Log[1 + 2^(2/3)*(-1 + x 
)^(1/3) - x^(1/3)]/(36*2^(1/3)) - Log[(1 - x^(1/3))*(1 + x^(1/3))^2]/(108* 
2^(1/3)) + Log[-(-1 + x)^(1/3) + 2^(1/3)*x^(1/3)]/(18*2^(1/3)) - (((I - Sq 
rt[3])/(3*I - Sqrt[3]))^(1/3)*Log[-(-1 + x)^(1/3) + x^(1/3)/((I - Sqrt[3]) 
/(3*I - Sqrt[3]))^(1/3)])/12 + ((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*...
 

Defintions of rubi rules used

rule 25

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

rule 2009

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

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [N/A] (verified)

Time = 14.84 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.92

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-2 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-3 \textit {\_Z}^{3}+3\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-12 x}{24 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}\)	\(237\)
trager	\(\text {Expression too large to display}\)	\(14316\)

Input:

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

Output:

1/24*(-2*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(2^(2/3)*((-1+x)*x^2)^(1/3)+x) 
/x)*((-1+x)*x^2)^(1/3)-2*2^(2/3)*ln((-2^(1/3)*x+((-1+x)*x^2)^(1/3))/x)*((- 
1+x)*x^2)^(1/3)+2^(2/3)*ln((2^(2/3)*x^2+2^(1/3)*((-1+x)*x^2)^(1/3)*x+((-1+ 
x)*x^2)^(2/3))/x^2)*((-1+x)*x^2)^(1/3)+4*sum(ln((-_R*x+((-1+x)*x^2)^(1/3)) 
/x)/_R,_R=RootOf(_Z^6-3*_Z^3+3))*((-1+x)*x^2)^(1/3)-4*sum(ln((-_R*x+((-1+x 
)*x^2)^(1/3))/x)/_R,_R=RootOf(_Z^6-_Z^3+1))*((-1+x)*x^2)^(1/3)-12*x)/((-1+ 
x)*x^2)^(1/3)
 

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.10 (sec) , antiderivative size = 1029, normalized size of antiderivative = 4.00 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/24*(12*2^(1/6)*sqrt(1/6)*(x^2 - x)*arctan(2^(1/6)*sqrt(1/6)*(2^(1/3)*x 
+ 2*(x^3 - x^2)^(1/3))/x) + 2*2^(2/3)*(x^2 - x)*log(-(2^(1/3)*x - (x^3 - x 
^2)^(1/3))/x) - 2^(2/3)*(x^2 - x)*log((2^(2/3)*x^2 + 2^(1/3)*(x^3 - x^2)^( 
1/3)*x + (x^3 - x^2)^(2/3))/x^2) + 2*(x^2 - sqrt(-3)*(x^2 - x) - x)*(3/2*s 
qrt(-1/3) - 1/2)^(1/3)*log(((3*sqrt(-1/3)*(sqrt(-3)*x + x) + sqrt(-3)*x + 
x)*(3/2*sqrt(-1/3) - 1/2)^(2/3) + 4*(x^3 - x^2)^(1/3))/x) + 2*(x^2 + sqrt( 
-3)*(x^2 - x) - x)*(3/2*sqrt(-1/3) - 1/2)^(1/3)*log(-((3*sqrt(-1/3)*(sqrt( 
-3)*x - x) + sqrt(-3)*x - x)*(3/2*sqrt(-1/3) - 1/2)^(2/3) - 4*(x^3 - x^2)^ 
(1/3))/x) - 4*(x^2 - x)*(3/2*sqrt(-1/3) - 1/2)^(1/3)*log(-((3*sqrt(-1/3)*x 
 + x)*(3/2*sqrt(-1/3) - 1/2)^(2/3) - 2*(x^3 - x^2)^(1/3))/x) + 2*(x^2 - sq 
rt(-3)*(x^2 - x) - x)*(1/2*sqrt(-1/3) + 1/2)^(1/3)*log(-(3*(sqrt(-1/3)*(sq 
rt(-3)*x + x) - sqrt(-3)*x - x)*(1/2*sqrt(-1/3) + 1/2)^(2/3) - 4*(x^3 - x^ 
2)^(1/3))/x) + 2*(x^2 + sqrt(-3)*(x^2 - x) - x)*(1/2*sqrt(-1/3) + 1/2)^(1/ 
3)*log((3*(sqrt(-1/3)*(sqrt(-3)*x - x) - sqrt(-3)*x + x)*(1/2*sqrt(-1/3) + 
 1/2)^(2/3) + 4*(x^3 - x^2)^(1/3))/x) - 4*(x^2 - x)*(1/2*sqrt(-1/3) + 1/2) 
^(1/3)*log((3*(sqrt(-1/3)*x - x)*(1/2*sqrt(-1/3) + 1/2)^(2/3) + 2*(x^3 - x 
^2)^(1/3))/x) + 2*(x^2 - sqrt(-3)*(x^2 - x) - x)*(-1/2*sqrt(-1/3) + 1/2)^( 
1/3)*log((3*(sqrt(-1/3)*(sqrt(-3)*x + x) + sqrt(-3)*x + x)*(-1/2*sqrt(-1/3 
) + 1/2)^(2/3) + 4*(x^3 - x^2)^(1/3))/x) + 2*(x^2 + sqrt(-3)*(x^2 - x) - x 
)*(-1/2*sqrt(-1/3) + 1/2)^(1/3)*log(-(3*(sqrt(-1/3)*(sqrt(-3)*x - x) + ...
 

Sympy [N/A]

Not integrable

Time = 0.77 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.13 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx=\int \frac {x^{3}}{\sqrt [3]{x^{2} \left (x - 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \] Input:

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

Output:

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

Maxima [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.09 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx=\int { \frac {x^{3}}{{\left (x^{6} - 1\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:proot error [1,0,0,1,0,0,1]proot er 
ror [1,0,0,-1,0,0,1]Invalid _EXT in replace_ext Error: Bad Argument Valuep 
root erro
 

Mupad [N/A]

Not integrable

Time = 9.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.09 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx=\int \frac {x^3}{\left (x^6-1\right )\,{\left (x^3-x^2\right )}^{1/3}} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.11 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx=\int \frac {x^{3}}{x^{\frac {20}{3}} \left (x -1\right )^{\frac {1}{3}}-x^{\frac {2}{3}} \left (x -1\right )^{\frac {1}{3}}}d x \] Input:

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

Output:

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