Integrand size = 22, antiderivative size = 139 \[ \int \frac {x^4}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}-\frac {1}{2} \log \left (-x-\sqrt [3]{1-x^3}\right )+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \]
-1/12*ln(x^3+1)*2^(1/3)-1/2*ln(-x-(-x^3+1)^(1/3))+1/4*ln(-2^(1/3)*x-(-x^3+ 1)^(1/3))*2^(1/3)-1/3*arctan(1/3*(1-2*x/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)+1 /6*arctan(1/3*(1-2*2^(1/3)*x/(-x^3+1)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)
Time = 0.49 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.48 \[ \int \frac {x^4}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{12} \left (-4 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1-x^3}}\right )+2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )-4 \log \left (x+\sqrt [3]{1-x^3}\right )+2 \sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )+2 \log \left (x^2-x \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )-\sqrt [3]{2} \log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3}-\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \]
(-4*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(1 - x^3)^(1/3))] + 2*2^(1/3)*Sqrt[3 ]*ArcTan[(Sqrt[3]*x)/(x - 2^(2/3)*(1 - x^3)^(1/3))] - 4*Log[x + (1 - x^3)^ (1/3)] + 2*2^(1/3)*Log[2*x + 2^(2/3)*(1 - x^3)^(1/3)] + 2*Log[x^2 - x*(1 - x^3)^(1/3) + (1 - x^3)^(2/3)] - 2^(1/3)*Log[-2*x^2 + 2^(2/3)*x*(1 - x^3)^ (1/3) - 2^(1/3)*(1 - x^3)^(2/3)])/12
Time = 0.25 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {983, 853, 992}
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.
| \(\displaystyle \int \frac {x^4}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 983 |
| \(\displaystyle \int \frac {x}{\left (1-x^3\right )^{2/3}}dx-\int \frac {x}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}dx\) |
| \(\Big \downarrow \) 853 |
| \(\displaystyle -\int \frac {x}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}dx-\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-\sqrt [3]{1-x^3}-x\right )\) |
| \(\Big \downarrow \) 992 |
| \(\displaystyle -\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (x^3+1\right )}{6\ 2^{2/3}}-\frac {1}{2} \log \left (-\sqrt [3]{1-x^3}-x\right )+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2\ 2^{2/3}}\) |
-(ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]]/Sqrt[3]) + ArcTan[(1 - (2*2^ (1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3]) - Log[1 + x^3]/(6*2^( 2/3)) - Log[-x - (1 - x^3)^(1/3)]/2 + Log[-(2^(1/3)*x) - (1 - x^3)^(1/3)]/ (2*2^(2/3))
3.7.30.3.1 Defintions of rubi rules used
Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Sim p[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp [Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]
Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^( n_)), x_Symbol] :> Simp[e^n/b Int[(e*x)^(m - n)*(c + d*x^n)^q, x], x] - S imp[a*(e^n/b) Int[(e*x)^(m - n)*((c + d*x^n)^q/(a + b*x^n)), x], x] /; Fr eeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b, c, d, e, m, n, -1, q, x]
Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[-ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3 ))/Sqrt[3]]/(Sqrt[3]*c*q^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c* q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Time = 4.67 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.29
| method | result | size |
| pseudoelliptic | \(-\frac {2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )}{12}+\frac {\ln \left (\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}-\left (-x^{3}+1\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{6}+\frac {2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )}{6}-\frac {\ln \left (\frac {x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )}{3}+\frac {\left (-\arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) 2^{\frac {1}{3}}+2 \arctan \left (\frac {\left (-2 \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )\right ) \sqrt {3}}{6}\) | \(179\) |
-1/12*2^(1/3)*ln((2^(2/3)*x^2-2^(1/3)*(-x^3+1)^(1/3)*x+(-x^3+1)^(2/3))/x^2 )+1/6*ln(((-x^3+1)^(2/3)-(-x^3+1)^(1/3)*x+x^2)/x^2)+1/6*2^(1/3)*ln((2^(1/3 )*x+(-x^3+1)^(1/3))/x)-1/3*ln((x+(-x^3+1)^(1/3))/x)+1/6*(-arctan(1/3*3^(1/ 2)*(-2^(2/3)*(-x^3+1)^(1/3)+x)/x)*2^(1/3)+2*arctan(1/3*(-2*(-x^3+1)^(1/3)+ x)*3^(1/2)/x))*3^(1/2)
Time = 0.27 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.42 \[ \int \frac {x^4}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (-\frac {4^{\frac {1}{6}} {\left (4^{\frac {1}{3}} \sqrt {3} x - 4^{\frac {2}{3}} \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, x}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} x + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {2 \cdot 4^{\frac {1}{3}} x^{2} - 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \frac {1}{3} \, \log \left (\frac {x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{6} \, \log \left (\frac {x^{2} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
1/6*4^(1/6)*sqrt(3)*arctan(-1/6*4^(1/6)*(4^(1/3)*sqrt(3)*x - 4^(2/3)*sqrt( 3)*(-x^3 + 1)^(1/3))/x) + 1/12*4^(2/3)*log((4^(2/3)*x + 2*(-x^3 + 1)^(1/3) )/x) - 1/24*4^(2/3)*log((2*4^(1/3)*x^2 - 4^(2/3)*(-x^3 + 1)^(1/3)*x + 2*(- x^3 + 1)^(2/3))/x^2) - 1/3*sqrt(3)*arctan(-1/3*(sqrt(3)*x - 2*sqrt(3)*(-x^ 3 + 1)^(1/3))/x) - 1/3*log((x + (-x^3 + 1)^(1/3))/x) + 1/6*log((x^2 - (-x^ 3 + 1)^(1/3)*x + (-x^3 + 1)^(2/3))/x^2)
\[ \int \frac {x^4}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x^{4}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
\[ \int \frac {x^4}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {x^{4}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {x^4}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {x^{4}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}} \,d x } \]
Timed out. \[ \int \frac {x^4}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x^4}{{\left (1-x^3\right )}^{2/3}\,\left (x^3+1\right )} \,d x \]