Integrand size = 23, antiderivative size = 382 \[ \int \frac {x^4 \left (1-x^3\right )}{1-x^3+x^6} \, dx=-\frac {x^2}{2}+\frac {i \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+\frac {i \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}-\frac {i \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-i \sqrt {3}}}+\frac {i \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+i \sqrt {3}}} \]
-1/2*x^2+1/3*I*2^(1/3)*arctan(1/3*(1+2*2^(1/3)*x/(1-I*3^(1/2))^(1/3))*3^(1 /2))/(1-I*3^(1/2))^(1/3)-1/3*I*2^(1/3)*arctan(1/3*(1+2*2^(1/3)*x/(1+I*3^(1 /2))^(1/3))*3^(1/2))/(1+I*3^(1/2))^(1/3)+1/9*I*2^(1/3)*ln(-2^(1/3)*x+(1-I* 3^(1/2))^(1/3))/(1-I*3^(1/2))^(1/3)*3^(1/2)-1/18*I*ln(2^(2/3)*x^2+2^(1/3)* x*(1-I*3^(1/2))^(1/3)+(1-I*3^(1/2))^(2/3))*2^(1/3)/(1-I*3^(1/2))^(1/3)*3^( 1/2)-1/9*I*2^(1/3)*ln(-2^(1/3)*x+(1+I*3^(1/2))^(1/3))/(1+I*3^(1/2))^(1/3)* 3^(1/2)+1/18*I*ln(2^(2/3)*x^2+2^(1/3)*x*(1+I*3^(1/2))^(1/3)+(1+I*3^(1/2))^ (2/3))*2^(1/3)/(1+I*3^(1/2))^(1/3)*3^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.13 \[ \int \frac {x^4 \left (1-x^3\right )}{1-x^3+x^6} \, dx=-\frac {x^2}{2}+\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]
Time = 0.59 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1826, 27, 1711, 27, 821, 16, 1142, 1082, 217, 1103}
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.
\(\displaystyle \int \frac {x^4 \left (1-x^3\right )}{x^6-x^3+1} \, dx\) |
\(\Big \downarrow \) 1826 |
\(\displaystyle -\frac {1}{2} \int -\frac {2 x}{x^6-x^3+1}dx-\frac {x^2}{2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {x}{x^6-x^3+1}dx-\frac {x^2}{2}\) |
\(\Big \downarrow \) 1711 |
\(\displaystyle \frac {i \int -\frac {2 x}{-2 x^3-i \sqrt {3}+1}dx}{\sqrt {3}}-\frac {i \int -\frac {2 x}{-2 x^3+i \sqrt {3}+1}dx}{\sqrt {3}}-\frac {x^2}{2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 i \int \frac {x}{-2 x^3-i \sqrt {3}+1}dx}{\sqrt {3}}+\frac {2 i \int \frac {x}{-2 x^3+i \sqrt {3}+1}dx}{\sqrt {3}}-\frac {x^2}{2}\) |
\(\Big \downarrow \) 821 |
\(\displaystyle -\frac {2 i \left (\frac {\int \frac {1}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x}dx}{3 \sqrt [3]{2 \left (1-i \sqrt {3}\right )}}-\frac {\int \frac {\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x}{2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}}dx}{3 \sqrt [3]{2 \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {3}}+\frac {2 i \left (\frac {\int \frac {1}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x}dx}{3 \sqrt [3]{2 \left (1+i \sqrt {3}\right )}}-\frac {\int \frac {\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x}{2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}}dx}{3 \sqrt [3]{2 \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {3}}-\frac {x^2}{2}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle -\frac {2 i \left (-\frac {\int \frac {\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x}{2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}}dx}{3 \sqrt [3]{2 \left (1-i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}\right )}{\sqrt {3}}+\frac {2 i \left (-\frac {\int \frac {\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x}{2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}}dx}{3 \sqrt [3]{2 \left (1+i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\right )}{\sqrt {3}}-\frac {x^2}{2}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle -\frac {2 i \left (-\frac {\frac {3}{2} \sqrt [3]{1-i \sqrt {3}} \int \frac {1}{2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}}dx-\frac {\int \frac {2\ 2^{2/3} x+\sqrt [3]{2 \left (1-i \sqrt {3}\right )}}{2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}}dx}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1-i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}\right )}{\sqrt {3}}+\frac {2 i \left (-\frac {\frac {3}{2} \sqrt [3]{1+i \sqrt {3}} \int \frac {1}{2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}}dx-\frac {\int \frac {2\ 2^{2/3} x+\sqrt [3]{2 \left (1+i \sqrt {3}\right )}}{2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}}dx}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1+i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\right )}{\sqrt {3}}-\frac {x^2}{2}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle -\frac {2 i \left (-\frac {-\frac {\int \frac {2\ 2^{2/3} x+\sqrt [3]{2 \left (1-i \sqrt {3}\right )}}{2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}}dx}{2 \sqrt [3]{2}}-\frac {3 \int \frac {1}{-\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}+1\right )^2-3}d\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}+1\right )}{\sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1-i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}\right )}{\sqrt {3}}+\frac {2 i \left (-\frac {-\frac {\int \frac {2\ 2^{2/3} x+\sqrt [3]{2 \left (1+i \sqrt {3}\right )}}{2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}}dx}{2 \sqrt [3]{2}}-\frac {3 \int \frac {1}{-\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+1\right )^2-3}d\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+1\right )}{\sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1+i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\right )}{\sqrt {3}}-\frac {x^2}{2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {2 i \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {\int \frac {2\ 2^{2/3} x+\sqrt [3]{2 \left (1-i \sqrt {3}\right )}}{2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}}dx}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1-i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}\right )}{\sqrt {3}}+\frac {2 i \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {\int \frac {2\ 2^{2/3} x+\sqrt [3]{2 \left (1+i \sqrt {3}\right )}}{2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}}dx}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1+i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\right )}{\sqrt {3}}-\frac {x^2}{2}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle -\frac {2 i \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {\log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}\right )}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1-i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}\right )}{\sqrt {3}}+\frac {2 i \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {\log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}\right )}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1+i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\right )}{\sqrt {3}}-\frac {x^2}{2}\) |
-1/2*x^2 - ((2*I)*(-1/3*Log[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x]/(2^(2/3)*(1 - I*Sqrt[3])^(1/3)) - ((Sqrt[3]*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/ 3))/Sqrt[3]])/2^(1/3) - Log[(1 - I*Sqrt[3])^(2/3) + (2*(1 - I*Sqrt[3]))^(1 /3)*x + 2^(2/3)*x^2]/(2*2^(1/3)))/(3*(2*(1 - I*Sqrt[3]))^(1/3))))/Sqrt[3] + ((2*I)*(-1/3*Log[(1 + I*Sqrt[3])^(1/3) - 2^(1/3)*x]/(2^(2/3)*(1 + I*Sqrt [3])^(1/3)) - ((Sqrt[3]*ArcTan[(1 + (2*x)/((1 + I*Sqrt[3])/2)^(1/3))/Sqrt[ 3]])/2^(1/3) - Log[(1 + I*Sqrt[3])^(2/3) + (2*(1 + I*Sqrt[3]))^(1/3)*x + 2 ^(2/3)*x^2]/(2*2^(1/3)))/(3*(2*(1 + I*Sqrt[3]))^(1/3))))/Sqrt[3]
3.1.26.3.1 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[((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_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
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[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_.)*(x_))^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symb ol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c/q Int[(d*x)^m/(b/2 - q/2 + c *x^n), x], x] - Simp[c/q Int[(d*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; Free Q[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[e*f^(n - 1)*(f*x)^(m - n + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + n*(2*p + 1) + 1))), x] - Simp[f^n/(c*( m + n*(2*p + 1) + 1)) Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^p*Simp[a* e*(m - n + 1) + (b*e*(m + n*p + 1) - c*d*(m + n*(2*p + 1) + 1))*x^n, x], x] , x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && Intege rQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.12
method | result | size |
default | \(-\frac {x^{2}}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{3}\) | \(44\) |
risch | \(-\frac {x^{2}}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{3}\) | \(44\) |
Time = 0.33 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.79 \[ \int \frac {x^4 \left (1-x^3\right )}{1-x^3+x^6} \, dx=-\frac {1}{108} \cdot 18^{\frac {2}{3}} {\left (i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (18^{\frac {1}{3}} {\left (\sqrt {3} {\left (i \, \sqrt {-3} - i\right )} + \sqrt {-3} - 1\right )} {\left (i \, \sqrt {3} + 3\right )}^{\frac {2}{3}} + 24 \, x\right ) + \frac {1}{108} \cdot 18^{\frac {2}{3}} {\left (i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (18^{\frac {1}{3}} {\left (\sqrt {3} {\left (-i \, \sqrt {-3} - i\right )} - \sqrt {-3} - 1\right )} {\left (i \, \sqrt {3} + 3\right )}^{\frac {2}{3}} + 24 \, x\right ) + \frac {1}{108} \cdot 18^{\frac {2}{3}} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (18^{\frac {1}{3}} {\left (\sqrt {3} {\left (i \, \sqrt {-3} + i\right )} - \sqrt {-3} - 1\right )} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {2}{3}} + 24 \, x\right ) - \frac {1}{108} \cdot 18^{\frac {2}{3}} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (18^{\frac {1}{3}} {\left (\sqrt {3} {\left (-i \, \sqrt {-3} + i\right )} + \sqrt {-3} - 1\right )} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {2}{3}} + 24 \, x\right ) - \frac {1}{2} \, x^{2} + \frac {1}{54} \cdot 18^{\frac {2}{3}} {\left (i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} \log \left (18^{\frac {1}{3}} {\left (i \, \sqrt {3} + 3\right )}^{\frac {2}{3}} {\left (i \, \sqrt {3} + 1\right )} + 12 \, x\right ) + \frac {1}{54} \cdot 18^{\frac {2}{3}} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} \log \left (18^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {2}{3}} {\left (-i \, \sqrt {3} + 1\right )} + 12 \, x\right ) \]
-1/108*18^(2/3)*(I*sqrt(3) + 3)^(1/3)*(sqrt(-3) + 1)*log(18^(1/3)*(sqrt(3) *(I*sqrt(-3) - I) + sqrt(-3) - 1)*(I*sqrt(3) + 3)^(2/3) + 24*x) + 1/108*18 ^(2/3)*(I*sqrt(3) + 3)^(1/3)*(sqrt(-3) - 1)*log(18^(1/3)*(sqrt(3)*(-I*sqrt (-3) - I) - sqrt(-3) - 1)*(I*sqrt(3) + 3)^(2/3) + 24*x) + 1/108*18^(2/3)*( -I*sqrt(3) + 3)^(1/3)*(sqrt(-3) - 1)*log(18^(1/3)*(sqrt(3)*(I*sqrt(-3) + I ) - sqrt(-3) - 1)*(-I*sqrt(3) + 3)^(2/3) + 24*x) - 1/108*18^(2/3)*(-I*sqrt (3) + 3)^(1/3)*(sqrt(-3) + 1)*log(18^(1/3)*(sqrt(3)*(-I*sqrt(-3) + I) + sq rt(-3) - 1)*(-I*sqrt(3) + 3)^(2/3) + 24*x) - 1/2*x^2 + 1/54*18^(2/3)*(I*sq rt(3) + 3)^(1/3)*log(18^(1/3)*(I*sqrt(3) + 3)^(2/3)*(I*sqrt(3) + 1) + 12*x ) + 1/54*18^(2/3)*(-I*sqrt(3) + 3)^(1/3)*log(18^(1/3)*(-I*sqrt(3) + 3)^(2/ 3)*(-I*sqrt(3) + 1) + 12*x)
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.08 \[ \int \frac {x^4 \left (1-x^3\right )}{1-x^3+x^6} \, dx=- \frac {x^{2}}{2} - \operatorname {RootSum} {\left (19683 t^{6} + 243 t^{3} + 1, \left ( t \mapsto t \log {\left (- 6561 t^{5} - 27 t^{2} + x \right )} \right )\right )} \]
-x**2/2 - RootSum(19683*_t**6 + 243*_t**3 + 1, Lambda(_t, _t*log(-6561*_t* *5 - 27*_t**2 + x)))
\[ \int \frac {x^4 \left (1-x^3\right )}{1-x^3+x^6} \, dx=\int { -\frac {{\left (x^{3} - 1\right )} x^{4}}{x^{6} - x^{3} + 1} \,d x } \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 820 vs. \(2 (246) = 492\).
Time = 0.35 (sec) , antiderivative size = 820, normalized size of antiderivative = 2.15 \[ \int \frac {x^4 \left (1-x^3\right )}{1-x^3+x^6} \, dx=\text {Too large to display} \]
-1/2*x^2 - 1/9*(sqrt(3)*cos(4/9*pi)^5 - 10*sqrt(3)*cos(4/9*pi)^3*sin(4/9*p i)^2 + 5*sqrt(3)*cos(4/9*pi)*sin(4/9*pi)^4 - 5*cos(4/9*pi)^4*sin(4/9*pi) + 10*cos(4/9*pi)^2*sin(4/9*pi)^3 - sin(4/9*pi)^5 - sqrt(3)*cos(4/9*pi)^2 + sqrt(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*x)/((1/2*I*sqrt(3) + 1/2)*sin(4/9*pi))) - 1/9*(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)*cos(2/9*pi) + 2 *x)/((1/2*I*sqrt(3) + 1/2)*sin(2/9*pi))) + 1/9*(sqrt(3)*cos(1/9*pi)^5 - 10 *sqrt(3)*cos(1/9*pi)^3*sin(1/9*pi)^2 + 5*sqrt(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*x)/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) - 1/18*(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)*c os(4/9*pi)*sin(4/9*pi) - cos(4/9*pi)^2 + sin(4/9*pi)^2)*log((-I*sqrt(3)*co s(4/9*pi) - cos(4/9*pi))*x + x^2 + 1) - 1/18*(5*sqrt(3)*cos(2/9*pi)^4*sin( 2/9*pi) - 10*sqrt(3)*cos(2/9*pi)^2*sin(2/9*pi)^3 + sqrt(3)*sin(2/9*pi)^...
Time = 10.57 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.81 \[ \int \frac {x^4 \left (1-x^3\right )}{1-x^3+x^6} \, dx=\frac {\ln \left (x+\left (81\,x-\frac {27\,{\left (36-\sqrt {3}\,12{}\mathrm {i}\right )}^{2/3}}{4}\right )\,\left (-\frac {1}{162}+\frac {\sqrt {3}\,1{}\mathrm {i}}{486}\right )\right )\,{\left (36-\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}+\frac {\ln \left (x-\left (81\,x-\frac {27\,{\left (36+\sqrt {3}\,12{}\mathrm {i}\right )}^{2/3}}{4}\right )\,\left (\frac {1}{162}+\frac {\sqrt {3}\,1{}\mathrm {i}}{486}\right )\right )\,{\left (36+\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}-\frac {x^2}{2}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{1/3}\,3^{2/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{12}+\frac {2^{1/3}\,3^{1/6}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,1{}\mathrm {i}}{4}\right )\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{1/3}\,3^{2/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{12}-\frac {2^{1/3}\,3^{1/6}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,1{}\mathrm {i}}{4}\right )\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{1/3}\,3^{2/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{6}\right )\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{1/3}\,3^{2/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{6}\right )\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36} \]
(log(x + (81*x - (27*(36 - 3^(1/2)*12i)^(2/3))/4)*((3^(1/2)*1i)/486 - 1/16 2))*(36 - 3^(1/2)*12i)^(1/3))/18 + (log(x - (81*x - (27*(3^(1/2)*12i + 36) ^(2/3))/4)*((3^(1/2)*1i)/486 + 1/162))*(3^(1/2)*12i + 36)^(1/3))/18 - x^2/ 2 - (2^(2/3)*log(x + (2^(1/3)*3^(2/3)*(3 - 3^(1/2)*1i)^(2/3))/12 + (2^(1/3 )*3^(1/6)*(3 - 3^(1/2)*1i)^(2/3)*1i)/4)*(3 - 3^(1/2)*1i)^(1/3)*(3^(1/3) + 3^(5/6)*1i))/36 - (2^(2/3)*log(x + (2^(1/3)*3^(2/3)*(3^(1/2)*1i + 3)^(2/3) )/12 - (2^(1/3)*3^(1/6)*(3^(1/2)*1i + 3)^(2/3)*1i)/4)*(3^(1/2)*1i + 3)^(1/ 3)*(3^(1/3) - 3^(5/6)*1i))/36 - (2^(2/3)*log(x - (2^(1/3)*3^(2/3)*(3 - 3^( 1/2)*1i)^(2/3))/6)*(3 - 3^(1/2)*1i)^(1/3)*(3^(1/3) - 3^(5/6)*1i))/36 - (2^ (2/3)*log(x - (2^(1/3)*3^(2/3)*(3^(1/2)*1i + 3)^(2/3))/6)*(3^(1/2)*1i + 3) ^(1/3)*(3^(1/3) + 3^(5/6)*1i))/36