Integrand size = 22, antiderivative size = 137 \[ \int \frac {1}{x \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {\arctan \left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log (x)}{2}+\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \] Output:
-1/3*arctan(1/3*(1+2*(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)+1/6*arctan(1/3*(1+2^ (2/3)*(-x^3+1)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)-1/2*ln(x)+1/12*ln(x^3+1)*2^ (1/3)+1/2*ln(1-(-x^3+1)^(1/3))-1/4*ln(2^(1/3)-(-x^3+1)^(1/3))*2^(1/3)
Time = 0.18 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.35 \[ \int \frac {1}{x \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{12} \left (-4 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )+2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )+4 \log \left (-1+\sqrt [3]{1-x^3}\right )-2 \sqrt [3]{2} \log \left (-2+2^{2/3} \sqrt [3]{1-x^3}\right )-2 \log \left (1+\sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )+\sqrt [3]{2} \log \left (2+2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \] Input:
Integrate[1/(x*(1 - x^3)^(2/3)*(1 + x^3)),x]
Output:
(-4*Sqrt[3]*ArcTan[(1 + 2*(1 - x^3)^(1/3))/Sqrt[3]] + 2*2^(1/3)*Sqrt[3]*Ar cTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] + 4*Log[-1 + (1 - x^3)^(1/3)] - 2*2^(1/3)*Log[-2 + 2^(2/3)*(1 - x^3)^(1/3)] - 2*Log[1 + (1 - x^3)^(1/3) + (1 - x^3)^(2/3)] + 2^(1/3)*Log[2 + 2^(2/3)*(1 - x^3)^(1/3) + 2^(1/3)*(1 - x^3)^(2/3)])/12
Time = 0.46 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {948, 97, 69, 16, 1082, 217, 1083, 217}
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.
| \(\displaystyle \int \frac {1}{x \left (1-x^3\right )^{2/3} \left (x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{x^3 \left (1-x^3\right )^{2/3} \left (x^3+1\right )}dx^3\) |
| \(\Big \downarrow \) 97 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {1}{x^3 \left (1-x^3\right )^{2/3}}dx^3-\int \frac {1}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}dx^3\right )\) |
| \(\Big \downarrow \) 69 |
| \(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{1-\sqrt [3]{1-x^3}}d\sqrt [3]{1-x^3}+\frac {3 \int \frac {1}{\sqrt [3]{2}-\sqrt [3]{1-x^3}}d\sqrt [3]{1-x^3}}{2\ 2^{2/3}}-\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{1-x^3}+1}d\sqrt [3]{1-x^3}+\frac {3 \int \frac {1}{x^6+\sqrt [3]{2} \sqrt [3]{1-x^3}+2^{2/3}}d\sqrt [3]{1-x^3}}{2 \sqrt [3]{2}}-\frac {1}{2} \log \left (x^3\right )+\frac {\log \left (x^3+1\right )}{2\ 2^{2/3}}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{1-x^3}+1}d\sqrt [3]{1-x^3}+\frac {3 \int \frac {1}{x^6+\sqrt [3]{2} \sqrt [3]{1-x^3}+2^{2/3}}d\sqrt [3]{1-x^3}}{2 \sqrt [3]{2}}-\frac {1}{2} \log \left (x^3\right )+\frac {\log \left (x^3+1\right )}{2\ 2^{2/3}}+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{3} \left (-\frac {3 \int \frac {1}{-x^6-3}d\left (2^{2/3} \sqrt [3]{1-x^3}+1\right )}{2^{2/3}}-\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{1-x^3}+1}d\sqrt [3]{1-x^3}-\frac {1}{2} \log \left (x^3\right )+\frac {\log \left (x^3+1\right )}{2\ 2^{2/3}}+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{1-x^3}+1}d\sqrt [3]{1-x^3}+\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2^{2/3}}-\frac {\log \left (x^3\right )}{2}+\frac {\log \left (x^3+1\right )}{2\ 2^{2/3}}+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{3} \left (3 \int \frac {1}{-x^6-3}d\left (2 \sqrt [3]{1-x^3}+1\right )+\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2^{2/3}}-\frac {\log \left (x^3\right )}{2}+\frac {\log \left (x^3+1\right )}{2\ 2^{2/3}}+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )+\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2^{2/3}}-\frac {\log \left (x^3\right )}{2}+\frac {\log \left (x^3+1\right )}{2\ 2^{2/3}}+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\right )\) |
Input:
Int[1/(x*(1 - x^3)^(2/3)*(1 + x^3)),x]
Output:
(-(Sqrt[3]*ArcTan[(1 + 2*(1 - x^3)^(1/3))/Sqrt[3]]) + (Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]])/2^(2/3) - Log[x^3]/2 + Log[1 + x^3]/(2 *2^(2/3)) + (3*Log[1 - (1 - x^3)^(1/3)])/2 - (3*Log[2^(1/3) - (1 - x^3)^(1 /3)])/(2*2^(2/3)))/3
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Simp[3/(2*b*q) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 /3)], x] - Simp[3/(2*b*q^2) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[b/(b*c - a*d) Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && !IntegerQ[p]
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])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
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]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Time = 2.34 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.06
| method | result | size |
| pseudoelliptic | \(-\frac {2^{\frac {1}{3}} \ln \left (\left (-x^{3}+1\right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right )}{6}+\frac {2^{\frac {1}{3}} \ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right )}{12}+\frac {\arctan \left (\frac {\left (1+2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) 2^{\frac {1}{3}} \sqrt {3}}{6}+\frac {\ln \left (-1+\left (-x^{3}+1\right )^{\frac {1}{3}}\right )}{3}-\frac {\ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+\left (-x^{3}+1\right )^{\frac {1}{3}}+1\right )}{6}-\frac {\arctan \left (\frac {\left (1+2 \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}\) | \(145\) |
Input:
int(1/x/(-x^3+1)^(2/3)/(x^3+1),x,method=_RETURNVERBOSE)
Output:
-1/6*2^(1/3)*ln((-x^3+1)^(1/3)-2^(1/3))+1/12*2^(1/3)*ln((-x^3+1)^(2/3)+2^( 1/3)*(-x^3+1)^(1/3)+2^(2/3))+1/6*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^( 1/2))*2^(1/3)*3^(1/2)+1/3*ln(-1+(-x^3+1)^(1/3))-1/6*ln((-x^3+1)^(2/3)+(-x^ 3+1)^(1/3)+1)-1/3*arctan(1/3*(1+2*(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)
Time = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{2} \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} \arctan \left (\frac {1}{2} \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}}\right )}\right ) + \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2 \cdot 4^{\frac {1}{3}}\right ) - \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (-4^{\frac {2}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \] Input:
integrate(1/x/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="fricas")
Output:
1/2*4^(1/6)*sqrt(1/3)*arctan(1/2*4^(1/6)*sqrt(1/3)*(4^(2/3)*(-x^3 + 1)^(1/ 3) + 4^(1/3))) + 1/24*4^(2/3)*log(4^(2/3)*(-x^3 + 1)^(1/3) + 2*(-x^3 + 1)^ (2/3) + 2*4^(1/3)) - 1/12*4^(2/3)*log(-4^(2/3) + 2*(-x^3 + 1)^(1/3)) - 1/3 *sqrt(3)*arctan(2/3*sqrt(3)*(-x^3 + 1)^(1/3) + 1/3*sqrt(3)) - 1/6*log((-x^ 3 + 1)^(2/3) + (-x^3 + 1)^(1/3) + 1) + 1/3*log((-x^3 + 1)^(1/3) - 1)
\[ \int \frac {1}{x \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {1}{x \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \] Input:
integrate(1/x/(-x**3+1)**(2/3)/(x**3+1),x)
Output:
Integral(1/(x*(-(x - 1)*(x**2 + x + 1))**(2/3)*(x + 1)*(x**2 - x + 1)), x)
\[ \int \frac {1}{x \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x} \,d x } \] Input:
integrate(1/x/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="maxima")
Output:
integrate(1/((x^3 + 1)*(-x^3 + 1)^(2/3)*x), x)
Time = 0.14 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{6} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left | {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \] Input:
integrate(1/x/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="giac")
Output:
1/6*sqrt(3)*2^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1/ 3))) - 1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-x^3 + 1)^(1/3) + 1)) + 1/12*2^( 1/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^(2/3)) - 1/6*2^(1 /3)*log(abs(-2^(1/3) + (-x^3 + 1)^(1/3))) - 1/6*log((-x^3 + 1)^(2/3) + (-x ^3 + 1)^(1/3) + 1) + 1/3*log(abs((-x^3 + 1)^(1/3) - 1))
Time = 3.81 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.51 \[ \int \frac {1}{x \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {\ln \left (5-5\,{\left (1-x^3\right )}^{1/3}\right )}{3}-\frac {2^{1/3}\,\ln \left (6\,{\left (1-x^3\right )}^{1/3}-\frac {2^{1/3}\,\left (\frac {2^{2/3}\,\left (243\,2^{1/3}+243\,{\left (1-x^3\right )}^{1/3}\right )}{36}+9\right )}{6}\right )}{6}+\ln \left (\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left ({\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\,\left (243\,{\left (1-x^3\right )}^{1/3}+243-\sqrt {3}\,243{}\mathrm {i}\right )+9\right )+6\,{\left (1-x^3\right )}^{1/3}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (6\,{\left (1-x^3\right )}^{1/3}-\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left ({\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\,\left (243\,{\left (1-x^3\right )}^{1/3}+243+\sqrt {3}\,243{}\mathrm {i}\right )+9\right )\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,\ln \left (6\,{\left (1-x^3\right )}^{1/3}-\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (\frac {{\left (-1\right )}^{2/3}\,2^{2/3}\,\left (243\,{\left (-1\right )}^{1/3}\,2^{1/3}-243\,{\left (1-x^3\right )}^{1/3}\right )}{36}-9\right )}{6}\right )}{6}-\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,\ln \left (6\,{\left (1-x^3\right )}^{1/3}-\frac {{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {{\left (-1\right )}^{2/3}\,2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (243\,{\left (1-x^3\right )}^{1/3}+\frac {243\,{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )}{144}+9\right )}{12}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \] Input:
int(1/(x*(1 - x^3)^(2/3)*(x^3 + 1)),x)
Output:
log(5 - 5*(1 - x^3)^(1/3))/3 - (2^(1/3)*log(6*(1 - x^3)^(1/3) - (2^(1/3)*( (2^(2/3)*(243*2^(1/3) + 243*(1 - x^3)^(1/3)))/36 + 9))/6))/6 + log(((3^(1/ 2)*1i)/6 - 1/6)*(((3^(1/2)*1i)/6 - 1/6)^2*(243*(1 - x^3)^(1/3) - 3^(1/2)*2 43i + 243) + 9) + 6*(1 - x^3)^(1/3))*((3^(1/2)*1i)/6 - 1/6) - log(6*(1 - x ^3)^(1/3) - ((3^(1/2)*1i)/6 + 1/6)*(((3^(1/2)*1i)/6 + 1/6)^2*(3^(1/2)*243i + 243*(1 - x^3)^(1/3) + 243) + 9))*((3^(1/2)*1i)/6 + 1/6) + ((-1)^(1/3)*2 ^(1/3)*log(6*(1 - x^3)^(1/3) - ((-1)^(1/3)*2^(1/3)*(((-1)^(2/3)*2^(2/3)*(2 43*(-1)^(1/3)*2^(1/3) - 243*(1 - x^3)^(1/3)))/36 - 9))/6))/6 - ((-1)^(1/3) *2^(1/3)*log(6*(1 - x^3)^(1/3) - ((-1)^(1/3)*2^(1/3)*(3^(1/2)*1i + 1)*(((- 1)^(2/3)*2^(2/3)*(3^(1/2)*1i + 1)^2*(243*(1 - x^3)^(1/3) + (243*(-1)^(1/3) *2^(1/3)*(3^(1/2)*1i + 1))/2))/144 + 9))/12)*(3^(1/2)*1i + 1))/12
\[ \int \frac {1}{x \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {1}{\left (-x^{3}+1\right )^{\frac {2}{3}} x^{4}+\left (-x^{3}+1\right )^{\frac {2}{3}} x}d x \] Input:
int(1/x/(-x^3+1)^(2/3)/(x^3+1),x)
Output:
int(1/(( - x**3 + 1)**(2/3)*x**4 + ( - x**3 + 1)**(2/3)*x),x)