Integrand size = 22, antiderivative size = 157 \[ \int \frac {1}{x^4 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {\left (1-x^3\right )^{2/3}}{3 x^3}-\frac {2 \arctan \left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log (x)}{3}-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac {1}{3} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \]
-1/3*(-x^3+1)^(2/3)/x^3+1/3*ln(x)-1/12*ln(x^3+1)*2^(2/3)-1/3*ln(1-(-x^3+1) ^(1/3))+1/4*ln(2^(1/3)-(-x^3+1)^(1/3))*2^(2/3)-2/9*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^(2/3)*3^(1/2)
Time = 0.39 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x^4 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {1}{36} \left (-\frac {12 \left (1-x^3\right )^{2/3}}{x^3}-8 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )+6\ 2^{2/3} \sqrt {3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )-8 \log \left (-1+\sqrt [3]{1-x^3}\right )+6\ 2^{2/3} \log \left (-2+2^{2/3} \sqrt [3]{1-x^3}\right )+4 \log \left (1+\sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )-3\ 2^{2/3} \log \left (2+2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \]
((-12*(1 - x^3)^(2/3))/x^3 - 8*Sqrt[3]*ArcTan[(1 + 2*(1 - x^3)^(1/3))/Sqrt [3]] + 6*2^(2/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] - 8 *Log[-1 + (1 - x^3)^(1/3)] + 6*2^(2/3)*Log[-2 + 2^(2/3)*(1 - x^3)^(1/3)] + 4*Log[1 + (1 - x^3)^(1/3) + (1 - x^3)^(2/3)] - 3*2^(2/3)*Log[2 + 2^(2/3)* (1 - x^3)^(1/3) + 2^(1/3)*(1 - x^3)^(2/3)])/36
Time = 0.31 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {948, 114, 27, 174, 67, 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.
| \(\displaystyle \int \frac {1}{x^4 \sqrt [3]{1-x^3} \left (x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{x^6 \sqrt [3]{1-x^3} \left (x^3+1\right )}dx^3\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{3} \left (-\int \frac {2-x^3}{3 x^3 \sqrt [3]{1-x^3} \left (x^3+1\right )}dx^3-\frac {\left (1-x^3\right )^{2/3}}{x^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (-\frac {1}{3} \int \frac {2-x^3}{x^3 \sqrt [3]{1-x^3} \left (x^3+1\right )}dx^3-\frac {\left (1-x^3\right )^{2/3}}{x^3}\right )\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (3 \int \frac {1}{\sqrt [3]{1-x^3} \left (x^3+1\right )}dx^3-2 \int \frac {1}{x^3 \sqrt [3]{1-x^3}}dx^3\right )-\frac {\left (1-x^3\right )^{2/3}}{x^3}\right )\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (3 \left (-\frac {3 \int \frac {1}{\sqrt [3]{2}-\sqrt [3]{1-x^3}}d\sqrt [3]{1-x^3}}{2 \sqrt [3]{2}}+\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{2} \sqrt [3]{1-x^3}+2^{2/3}}d\sqrt [3]{1-x^3}-\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}\right )-2 \left (-\frac {3}{2} \int \frac {1}{1-\sqrt [3]{1-x^3}}d\sqrt [3]{1-x^3}+\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{1-x^3}+1}d\sqrt [3]{1-x^3}-\frac {1}{2} \log \left (x^3\right )\right )\right )-\frac {\left (1-x^3\right )^{2/3}}{x^3}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (3 \left (\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{2} \sqrt [3]{1-x^3}+2^{2/3}}d\sqrt [3]{1-x^3}-\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\right )-2 \left (\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{1-x^3}+1}d\sqrt [3]{1-x^3}-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^3}\right )\right )\right )-\frac {\left (1-x^3\right )^{2/3}}{x^3}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (3 \left (-\frac {3 \int \frac {1}{-x^6-3}d\left (2^{2/3} \sqrt [3]{1-x^3}+1\right )}{\sqrt [3]{2}}-\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\right )-2 \left (\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{1-x^3}+1}d\sqrt [3]{1-x^3}-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^3}\right )\right )\right )-\frac {\left (1-x^3\right )^{2/3}}{x^3}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (3 \left (\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\right )-2 \left (\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{1-x^3}+1}d\sqrt [3]{1-x^3}-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^3}\right )\right )\right )-\frac {\left (1-x^3\right )^{2/3}}{x^3}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (3 \left (\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\right )-2 \left (-3 \int \frac {1}{-x^6-3}d\left (2 \sqrt [3]{1-x^3}+1\right )-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^3}\right )\right )\right )-\frac {\left (1-x^3\right )^{2/3}}{x^3}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (3 \left (\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\right )-2 \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )-\frac {\log \left (x^3\right )}{2}+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^3}\right )\right )\right )-\frac {\left (1-x^3\right )^{2/3}}{x^3}\right )\) |
(-((1 - x^3)^(2/3)/x^3) + (-2*(Sqrt[3]*ArcTan[(1 + 2*(1 - x^3)^(1/3))/Sqrt [3]] - Log[x^3]/2 + (3*Log[1 - (1 - x^3)^(1/3)])/2) + 3*((Sqrt[3]*ArcTan[( 1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]])/2^(1/3) - Log[1 + x^3]/(2*2^(1/3)) + (3*Log[2^(1/3) - (1 - x^3)^(1/3)])/(2*2^(1/3))))/3)/3
3.7.11.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
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 = 6.52 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.34
| method | result | size |
| pseudoelliptic | \(\frac {-6 \,2^{\frac {2}{3}} \sqrt {3}\, \arctan \left (\frac {\left (1+2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) x^{3}-6 \,2^{\frac {2}{3}} \ln \left (\left (-x^{3}+1\right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right ) x^{3}+3 \,2^{\frac {2}{3}} \ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right ) x^{3}+8 \sqrt {3}\, \arctan \left (\frac {\left (1+2 \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) x^{3}-4 \ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+\left (-x^{3}+1\right )^{\frac {1}{3}}+1\right ) x^{3}+8 \ln \left (-1+\left (-x^{3}+1\right )^{\frac {1}{3}}\right ) x^{3}+12 \left (-x^{3}+1\right )^{\frac {2}{3}}}{36 \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+\left (-x^{3}+1\right )^{\frac {1}{3}}+1\right ) \left (-1+\left (-x^{3}+1\right )^{\frac {1}{3}}\right )}\) | \(211\) |
1/36*(-6*2^(2/3)*3^(1/2)*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*x^ 3-6*2^(2/3)*ln((-x^3+1)^(1/3)-2^(1/3))*x^3+3*2^(2/3)*ln((-x^3+1)^(2/3)+2^( 1/3)*(-x^3+1)^(1/3)+2^(2/3))*x^3+8*3^(1/2)*arctan(1/3*(1+2*(-x^3+1)^(1/3)) *3^(1/2))*x^3-4*ln((-x^3+1)^(2/3)+(-x^3+1)^(1/3)+1)*x^3+8*ln(-1+(-x^3+1)^( 1/3))*x^3+12*(-x^3+1)^(2/3))/((-x^3+1)^(2/3)+(-x^3+1)^(1/3)+1)/(-1+(-x^3+1 )^(1/3))
Time = 0.29 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^4 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {6 \, \sqrt {6} 2^{\frac {1}{6}} x^{3} \arctan \left (\frac {1}{6} \cdot 2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} + 2 \, \sqrt {6} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - 3 \cdot 2^{\frac {2}{3}} x^{3} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + 6 \cdot 2^{\frac {2}{3}} x^{3} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - 8 \, \sqrt {3} x^{3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 4 \, x^{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - 8 \, x^{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) - 12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{36 \, x^{3}} \]
1/36*(6*sqrt(6)*2^(1/6)*x^3*arctan(1/6*2^(1/6)*(sqrt(6)*2^(1/3) + 2*sqrt(6 )*(-x^3 + 1)^(1/3))) - 3*2^(2/3)*x^3*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3 ) + (-x^3 + 1)^(2/3)) + 6*2^(2/3)*x^3*log(-2^(1/3) + (-x^3 + 1)^(1/3)) - 8 *sqrt(3)*x^3*arctan(2/3*sqrt(3)*(-x^3 + 1)^(1/3) + 1/3*sqrt(3)) + 4*x^3*lo g((-x^3 + 1)^(2/3) + (-x^3 + 1)^(1/3) + 1) - 8*x^3*log((-x^3 + 1)^(1/3) - 1) - 12*(-x^3 + 1)^(2/3))/x^3
\[ \int \frac {1}{x^4 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int \frac {1}{x^{4} \sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
\[ \int \frac {1}{x^4 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{4}} \,d x } \]
Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^4 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{3 \, x^{3}} + \frac {1}{9} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \log \left ({\left | {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1/ 3))) - 1/12*2^(2/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^(2 /3)) + 1/6*2^(2/3)*log(abs(-2^(1/3) + (-x^3 + 1)^(1/3))) - 2/9*sqrt(3)*arc tan(1/3*sqrt(3)*(2*(-x^3 + 1)^(1/3) + 1)) - 1/3*(-x^3 + 1)^(2/3)/x^3 + 1/9 *log((-x^3 + 1)^(2/3) + (-x^3 + 1)^(1/3) + 1) - 2/9*log(abs((-x^3 + 1)^(1/ 3) - 1))
Time = 8.48 (sec) , antiderivative size = 382, normalized size of antiderivative = 2.43 \[ \int \frac {1}{x^4 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {2^{2/3}\,\ln \left (\frac {2^{1/3}\,\left (\frac {2^{2/3}\,\left (81\,2^{1/3}-75\,{\left (1-x^3\right )}^{1/3}\right )}{6}-\frac {38}{3}\right )}{18}+\frac {16\,{\left (1-x^3\right )}^{1/3}}{27}\right )}{6}-\frac {{\left (1-x^3\right )}^{2/3}}{3\,x^3}-\frac {2\,\ln \left (\frac {344\,{\left (1-x^3\right )}^{1/3}}{243}-\frac {344}{243}\right )}{9}+\ln \left ({\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2\,\left (\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )\,\left (1458\,{\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2-75\,{\left (1-x^3\right )}^{1/3}\right )-\frac {38}{3}\right )+\frac {16\,{\left (1-x^3\right )}^{1/3}}{27}\right )\,\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )-\ln \left (\frac {16\,{\left (1-x^3\right )}^{1/3}}{27}-{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2\,\left (\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )\,\left (1458\,{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2-75\,{\left (1-x^3\right )}^{1/3}\right )+\frac {38}{3}\right )\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )+\frac {2^{2/3}\,\ln \left (\frac {16\,{\left (1-x^3\right )}^{1/3}}{27}+\frac {2^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {2^{2/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {81\,2^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}-75\,{\left (1-x^3\right )}^{1/3}\right )}{12}-\frac {38}{3}\right )}{72}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12}-\frac {2^{2/3}\,\ln \left (\frac {16\,{\left (1-x^3\right )}^{1/3}}{27}-\frac {2^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {2^{2/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {81\,2^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}-75\,{\left (1-x^3\right )}^{1/3}\right )}{12}+\frac {38}{3}\right )}{72}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \]
(2^(2/3)*log((2^(1/3)*((2^(2/3)*(81*2^(1/3) - 75*(1 - x^3)^(1/3)))/6 - 38/ 3))/18 + (16*(1 - x^3)^(1/3))/27))/6 - (1 - x^3)^(2/3)/(3*x^3) - (2*log((3 44*(1 - x^3)^(1/3))/243 - 344/243))/9 + log(((3^(1/2)*1i)/9 + 1/9)^2*(((3^ (1/2)*1i)/9 + 1/9)*(1458*((3^(1/2)*1i)/9 + 1/9)^2 - 75*(1 - x^3)^(1/3)) - 38/3) + (16*(1 - x^3)^(1/3))/27)*((3^(1/2)*1i)/9 + 1/9) - log((16*(1 - x^3 )^(1/3))/27 - ((3^(1/2)*1i)/9 - 1/9)^2*(((3^(1/2)*1i)/9 - 1/9)*(1458*((3^( 1/2)*1i)/9 - 1/9)^2 - 75*(1 - x^3)^(1/3)) + 38/3))*((3^(1/2)*1i)/9 - 1/9) + (2^(2/3)*log((16*(1 - x^3)^(1/3))/27 + (2^(1/3)*(3^(1/2)*1i - 1)^2*((2^( 2/3)*(3^(1/2)*1i - 1)*((81*2^(1/3)*(3^(1/2)*1i - 1)^2)/4 - 75*(1 - x^3)^(1 /3)))/12 - 38/3))/72)*(3^(1/2)*1i - 1))/12 - (2^(2/3)*log((16*(1 - x^3)^(1 /3))/27 - (2^(1/3)*(3^(1/2)*1i + 1)^2*((2^(2/3)*(3^(1/2)*1i + 1)*((81*2^(1 /3)*(3^(1/2)*1i + 1)^2)/4 - 75*(1 - x^3)^(1/3)))/12 + 38/3))/72)*(3^(1/2)* 1i + 1))/12