Integrand size = 17, antiderivative size = 163 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x+x^3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x+x^3}\right )}{6 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x+x^3}+\sqrt [3]{2} \left (x+x^3\right )^{2/3}\right )}{12 \sqrt [3]{2}}+\frac {1}{6} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \left (-2^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2^{2/3} \sqrt [3]{1+x^2}}\right )-2 \log \left (-2 x^{2/3}+2^{2/3} \sqrt [3]{1+x^2}\right )+\log \left (2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{1+x^2}+\sqrt [3]{2} \left (1+x^2\right )^{2/3}\right )\right )+4 \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{1+x^2}-x^{2/3} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{24 \sqrt [3]{x+x^3}} \]
(x^(1/3)*(1 + x^2)^(1/3)*(-(2^(2/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x ^(2/3) + 2^(2/3)*(1 + x^2)^(1/3))] - 2*Log[-2*x^(2/3) + 2^(2/3)*(1 + x^2)^ (1/3)] + Log[2*x^(4/3) + 2^(2/3)*x^(2/3)*(1 + x^2)^(1/3) + 2^(1/3)*(1 + x^ 2)^(2/3)])) + 4*RootSum[1 - #1^3 + #1^6 & , (-2*Log[x^(1/3)] + Log[(1 + x^ 2)^(1/3) - x^(2/3)*#1])/#1 & ]))/(24*(x + x^3)^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(1643\) vs. \(2(163)=326\).
Time = 2.39 (sec) , antiderivative size = 1643, normalized size of antiderivative = 10.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2467, 25, 2035, 7266, 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.
| \(\displaystyle \int \frac {1}{\sqrt [3]{x^3+x} \left (x^6-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \int -\frac {1}{\sqrt [3]{x} \sqrt [3]{x^2+1} \left (1-x^6\right )}dx}{\sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \int \frac {1}{\sqrt [3]{x} \sqrt [3]{x^2+1} \left (1-x^6\right )}dx}{\sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \int \frac {\sqrt [3]{x}}{\sqrt [3]{x^2+1} \left (1-x^6\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \int \frac {1}{\sqrt [3]{x+1} \left (1-x^3\right )}dx^{2/3}}{2 \sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \int \left (\frac {1}{9 \left (1-x^{2/3}\right ) \sqrt [3]{x+1}}+\frac {1}{9 \left (\sqrt [9]{-1} x^{2/3}+1\right ) \sqrt [3]{x+1}}+\frac {1}{9 \left (1-(-1)^{2/9} x^{2/3}\right ) \sqrt [3]{x+1}}+\frac {1}{9 \left (\sqrt [3]{-1} x^{2/3}+1\right ) \sqrt [3]{x+1}}+\frac {1}{9 \left (1-(-1)^{4/9} x^{2/3}\right ) \sqrt [3]{x+1}}+\frac {1}{9 \left ((-1)^{5/9} x^{2/3}+1\right ) \sqrt [3]{x+1}}+\frac {1}{9 \left (1-(-1)^{2/3} x^{2/3}\right ) \sqrt [3]{x+1}}+\frac {1}{9 \left ((-1)^{7/9} x^{2/3}+1\right ) \sqrt [3]{x+1}}+\frac {1}{9 \left (1-(-1)^{8/9} x^{2/3}\right ) \sqrt [3]{x+1}}\right )dx^{2/3}}{2 \sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \left (-\frac {1}{18} (-1)^{7/9} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\sqrt [3]{-1} x\right )+\frac {1}{18} (-1)^{4/9} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\sqrt [3]{-1} x\right )-\frac {1}{18} \sqrt [9]{-1} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\sqrt [3]{-1} x\right )+\frac {1}{18} (-1)^{8/9} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,(-1)^{2/3} x\right )-\frac {1}{18} (-1)^{5/9} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,(-1)^{2/3} x\right )+\frac {1}{18} (-1)^{2/9} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,(-1)^{2/3} x\right )+\frac {\arctan \left (\frac {\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {\frac {(-1)^{2/3} \sqrt [3]{2} \left (1-\sqrt [3]{-1} x^{2/3}\right )}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {1-\frac {\sqrt [3]{-2} \left ((-1)^{2/3} x^{2/3}+1\right )}{\sqrt [3]{x+1}}}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{1-\sqrt [3]{-1}} x^{2/3}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{1+(-1)^{2/3}} x^{2/3}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{1+(-1)^{2/3}}}+\frac {(-1)^{7/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x+1}}{\sqrt [3]{1-\sqrt [3]{-1}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {(-1)^{4/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x+1}}{\sqrt [3]{1-\sqrt [3]{-1}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}}}+\frac {\sqrt [9]{-1} \arctan \left (\frac {\frac {2 \sqrt [3]{x+1}}{\sqrt [3]{1-\sqrt [3]{-1}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {(-1)^{8/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x+1}}{\sqrt [3]{1+(-1)^{2/3}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{1+(-1)^{2/3}}}+\frac {(-1)^{5/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x+1}}{\sqrt [3]{1+(-1)^{2/3}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{1+(-1)^{2/3}}}-\frac {(-1)^{2/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x+1}}{\sqrt [3]{1+(-1)^{2/3}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{1+(-1)^{2/3}}}+\frac {\log \left (\left (1-x^{2/3}\right )^2 \left (x^{2/3}+1\right )\right )}{36 \sqrt [3]{2}}+\frac {\log \left (\left (1-\sqrt [3]{-1} x^{2/3}\right ) \left (\sqrt [3]{-1} x^{2/3}+1\right )^2\right )}{36 \sqrt [3]{2}}+\frac {\log \left (\left (1-(-1)^{2/3} x^{2/3}\right )^2 \left ((-1)^{2/3} x^{2/3}+1\right )\right )}{36 \sqrt [3]{2}}-\frac {(-1)^{7/9} \log \left (x+\sqrt [3]{-1}\right )}{54 \sqrt [3]{1-\sqrt [3]{-1}}}+\frac {(-1)^{4/9} \log \left (x+\sqrt [3]{-1}\right )}{54 \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {\sqrt [9]{-1} \log \left (x+\sqrt [3]{-1}\right )}{54 \sqrt [3]{1-\sqrt [3]{-1}}}+\frac {(-1)^{8/9} \log \left (x-(-1)^{2/3}\right )}{54 \sqrt [3]{1+(-1)^{2/3}}}-\frac {(-1)^{5/9} \log \left (x-(-1)^{2/3}\right )}{54 \sqrt [3]{1+(-1)^{2/3}}}+\frac {(-1)^{2/9} \log \left (x-(-1)^{2/3}\right )}{54 \sqrt [3]{1+(-1)^{2/3}}}+\frac {\log \left (\sqrt [3]{-1} x+1\right )}{18 \sqrt [3]{1-\sqrt [3]{-1}}}+\frac {\log \left (1-(-1)^{2/3} x\right )}{18 \sqrt [3]{1+(-1)^{2/3}}}+\frac {(-1)^{7/9} \log \left (\sqrt [3]{1-\sqrt [3]{-1}}-\sqrt [3]{x+1}\right )}{18 \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {(-1)^{4/9} \log \left (\sqrt [3]{1-\sqrt [3]{-1}}-\sqrt [3]{x+1}\right )}{18 \sqrt [3]{1-\sqrt [3]{-1}}}+\frac {\sqrt [9]{-1} \log \left (\sqrt [3]{1-\sqrt [3]{-1}}-\sqrt [3]{x+1}\right )}{18 \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {(-1)^{8/9} \log \left (\sqrt [3]{1+(-1)^{2/3}}-\sqrt [3]{x+1}\right )}{18 \sqrt [3]{1+(-1)^{2/3}}}+\frac {(-1)^{5/9} \log \left (\sqrt [3]{1+(-1)^{2/3}}-\sqrt [3]{x+1}\right )}{18 \sqrt [3]{1+(-1)^{2/3}}}-\frac {(-1)^{2/9} \log \left (\sqrt [3]{1+(-1)^{2/3}}-\sqrt [3]{x+1}\right )}{18 \sqrt [3]{1+(-1)^{2/3}}}-\frac {\log \left (\sqrt [3]{1-\sqrt [3]{-1}} x^{2/3}-\sqrt [3]{x+1}\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {\log \left (\sqrt [3]{1+(-1)^{2/3}} x^{2/3}-\sqrt [3]{x+1}\right )}{6 \sqrt [3]{1+(-1)^{2/3}}}-\frac {\log \left ((-1)^{2/3} x^{2/3}-(-2)^{2/3} \sqrt [3]{x+1}+1\right )}{12 \sqrt [3]{2}}-\frac {\log \left (x^{2/3}-2^{2/3} \sqrt [3]{x+1}+1\right )}{12 \sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{-1} x^{2/3}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{x+1}+1\right )}{12 \sqrt [3]{2}}\right )}{2 \sqrt [3]{x^3+x}}\) |
(-3*x^(1/3)*(1 + x^2)^(1/3)*(-1/18*((-1)^(1/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x, -((-1)^(1/3)*x)]) + ((-1)^(4/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x, -((-1)^(1/3)*x)])/18 - ((-1)^(7/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x, -((-1)^(1/3)*x)])/18 + ((-1)^(2/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x, (-1)^(2/3)*x])/18 - ((-1)^(5/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3 , -x, (-1)^(2/3)*x])/18 + ((-1)^(8/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, - x, (-1)^(2/3)*x])/18 + ArcTan[(1 + (2^(1/3)*(1 + x^(2/3)))/(1 + x)^(1/3))/ Sqrt[3]]/(6*2^(1/3)*Sqrt[3]) + ArcTan[(1 + ((-1)^(2/3)*2^(1/3)*(1 - (-1)^( 1/3)*x^(2/3)))/(1 + x)^(1/3))/Sqrt[3]]/(6*2^(1/3)*Sqrt[3]) + ArcTan[(1 - ( (-2)^(1/3)*(1 + (-1)^(2/3)*x^(2/3)))/(1 + x)^(1/3))/Sqrt[3]]/(6*2^(1/3)*Sq rt[3]) + ArcTan[(1 + (2*(1 - (-1)^(1/3))^(1/3)*x^(2/3))/(1 + x)^(1/3))/Sqr t[3]]/(3*Sqrt[3]*(1 - (-1)^(1/3))^(1/3)) + ArcTan[(1 + (2*(1 + (-1)^(2/3)) ^(1/3)*x^(2/3))/(1 + x)^(1/3))/Sqrt[3]]/(3*Sqrt[3]*(1 + (-1)^(2/3))^(1/3)) + ((-1)^(1/9)*ArcTan[(1 + (2*(1 + x)^(1/3))/(1 - (-1)^(1/3))^(1/3))/Sqrt[ 3]])/(9*Sqrt[3]*(1 - (-1)^(1/3))^(1/3)) - ((-1)^(4/9)*ArcTan[(1 + (2*(1 + x)^(1/3))/(1 - (-1)^(1/3))^(1/3))/Sqrt[3]])/(9*Sqrt[3]*(1 - (-1)^(1/3))^(1 /3)) + ((-1)^(7/9)*ArcTan[(1 + (2*(1 + x)^(1/3))/(1 - (-1)^(1/3))^(1/3))/S qrt[3]])/(9*Sqrt[3]*(1 - (-1)^(1/3))^(1/3)) - ((-1)^(2/9)*ArcTan[(1 + (2*( 1 + x)^(1/3))/(1 + (-1)^(2/3))^(1/3))/Sqrt[3]])/(9*Sqrt[3]*(1 + (-1)^(2/3) )^(1/3)) + ((-1)^(5/9)*ArcTan[(1 + (2*(1 + x)^(1/3))/(1 + (-1)^(2/3))^(...
3.23.2.3.1 Defintions of rubi rules used
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]
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]
Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), u, x]
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]
Time = 57.87 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.86
| method | result | size |
| pseudoelliptic | \(\frac {2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )}{12}-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{24}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right )}{3 x}\right )}{12}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )}{6}\) | \(140\) |
| trager | \(\text {Expression too large to display}\) | \(5368\) |
1/12*2^(2/3)*ln((-2^(1/3)*x+(x*(x^2+1))^(1/3))/x)-1/24*2^(2/3)*ln((2^(2/3) *x^2+2^(1/3)*(x*(x^2+1))^(1/3)*x+(x*(x^2+1))^(2/3))/x^2)+1/12*3^(1/2)*2^(2 /3)*arctan(1/3*3^(1/2)*(2^(2/3)*(x*(x^2+1))^(1/3)+x)/x)+1/6*sum(ln((-_R*x+ (x*(x^2+1))^(1/3))/x)/_R,_R=RootOf(_Z^6-_Z^3+1))
Exception generated. \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (trace 0)
Not integrable
Time = 0.79 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.20 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\int \frac {1}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
Not integrable
Time = 0.33 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\int { \frac {1}{{\left (x^{6} - 1\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}} \,d x } \]
-3/80*(9*x^7 + 3*x^5 - x^3 + 5*x)/((x^(19/3) - x^(1/3))*(x^2 + 1)^(1/3)) - integrate(9/40*(9*x^6 + 3*x^4 - x^2 + 5)/((x^(37/3) - 2*x^(19/3) + x^(1/3 ))*(x^2 + 1)^(1/3)), x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.38 (sec) , antiderivative size = 941, normalized size of antiderivative = 5.77 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\text {Too large to display} \]
1/12*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(1/x^2 + 1)^( 1/3))) - 1/6*(sqrt(3)*cos(4/9*pi)^5 - 10*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi) ^2 + 5*sqrt(3)*cos(4/9*pi)*sin(4/9*pi)^4 - 5*cos(4/9*pi)^4*sin(4/9*pi) + 1 0*cos(4/9*pi)^2*sin(4/9*pi)^3 - sin(4/9*pi)^5 + sqrt(3)*cos(4/9*pi)^2 - sq rt(3)*sin(4/9*pi)^2 - 2*cos(4/9*pi)*sin(4/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(4/9*pi) + 2*(1/x^2 + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(4/9*pi)) ) - 1/6*(sqrt(3)*cos(2/9*pi)^5 - 10*sqrt(3)*cos(2/9*pi)^3*sin(2/9*pi)^2 + 5*sqrt(3)*cos(2/9*pi)*sin(2/9*pi)^4 - 5*cos(2/9*pi)^4*sin(2/9*pi) + 10*cos (2/9*pi)^2*sin(2/9*pi)^3 - sin(2/9*pi)^5 + sqrt(3)*cos(2/9*pi)^2 - sqrt(3) *sin(2/9*pi)^2 - 2*cos(2/9*pi)*sin(2/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*c os(2/9*pi) + 2*(1/x^2 + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(2/9*pi))) + 1 /6*(sqrt(3)*cos(1/9*pi)^5 - 10*sqrt(3)*cos(1/9*pi)^3*sin(1/9*pi)^2 + 5*sqr t(3)*cos(1/9*pi)*sin(1/9*pi)^4 + 5*cos(1/9*pi)^4*sin(1/9*pi) - 10*cos(1/9* pi)^2*sin(1/9*pi)^3 + sin(1/9*pi)^5 - sqrt(3)*cos(1/9*pi)^2 + sqrt(3)*sin( 1/9*pi)^2 - 2*cos(1/9*pi)*sin(1/9*pi))*arctan(-1/2*((-I*sqrt(3) - 1)*cos(1 /9*pi) - 2*(1/x^2 + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) - 1/12* (5*sqrt(3)*cos(4/9*pi)^4*sin(4/9*pi) - 10*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi )^3 + sqrt(3)*sin(4/9*pi)^5 + cos(4/9*pi)^5 - 10*cos(4/9*pi)^3*sin(4/9*pi) ^2 + 5*cos(4/9*pi)*sin(4/9*pi)^4 + 2*sqrt(3)*cos(4/9*pi)*sin(4/9*pi) + cos (4/9*pi)^2 - sin(4/9*pi)^2)*log((-I*sqrt(3)*cos(4/9*pi) - cos(4/9*pi))*...
Not integrable
Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.10 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\int \frac {1}{\left (x^6-1\right )\,{\left (x^3+x\right )}^{1/3}} \,d x \]