Integrand size = 16, antiderivative size = 411 \[ \int \frac {x^3}{1-x^3+x^6} \, dx=-\frac {\left (i+\sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {\left (i-\sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {\left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}} \] Output:
-1/6*(3^(1/2)+I)*arctan(1/3*(1+2*x/(1/2-1/2*I*3^(1/2))^(1/3))*3^(1/2))*2^( 2/3)/(1-I*3^(1/2))^(2/3)+1/6*(I-3^(1/2))*arctan(1/3*(1+2*x/(1/2+1/2*I*3^(1 /2))^(1/3))*3^(1/2))*2^(2/3)/(1+I*3^(1/2))^(2/3)+1/18*(3+I*3^(1/2))*ln((1- I*3^(1/2))^(1/3)-2^(1/3)*x)*2^(2/3)/(1-I*3^(1/2))^(2/3)+1/18*(3-I*3^(1/2)) *ln((1+I*3^(1/2))^(1/3)-2^(1/3)*x)*2^(2/3)/(1+I*3^(1/2))^(2/3)-1/36*(3+I*3 ^(1/2))*ln((1-I*3^(1/2))^(2/3)+(2-2*I*3^(1/2))^(1/3)*x+2^(2/3)*x^2)*2^(2/3 )/(1-I*3^(1/2))^(2/3)-1/36*(3-I*3^(1/2))*ln((1+I*3^(1/2))^(2/3)+(2+2*I*3^( 1/2))^(1/3)*x+2^(2/3)*x^2)*2^(2/3)/(1+I*3^(1/2))^(2/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.09 \[ \int \frac {x^3}{1-x^3+x^6} \, dx=\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{-1+2 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[x^3/(1 - x^3 + x^6),x]
Output:
RootSum[1 - #1^3 + #1^6 & , (Log[x - #1]*#1)/(-1 + 2*#1^3) & ]/3
Time = 0.52 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.85, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1710, 750, 16, 25, 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.
| \(\displaystyle \int \frac {x^3}{x^6-x^3+1} \, dx\) |
| \(\Big \downarrow \) 1710 |
| \(\displaystyle \frac {1}{6} \left (3-i \sqrt {3}\right ) \int \frac {1}{x^3+\frac {1}{2} \left (-1-i \sqrt {3}\right )}dx+\frac {1}{6} \left (3+i \sqrt {3}\right ) \int \frac {1}{x^3+\frac {1}{2} \left (-1+i \sqrt {3}\right )}dx\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {1}{6} \left (3+i \sqrt {3}\right ) \left (\frac {\int -\frac {x+2^{2/3} \sqrt [3]{1-i \sqrt {3}}}{x^2+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}+\frac {\int \frac {1}{x-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}\right )+\frac {1}{6} \left (3-i \sqrt {3}\right ) \left (\frac {\int -\frac {x+2^{2/3} \sqrt [3]{1+i \sqrt {3}}}{x^2+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}+\frac {\int \frac {1}{x-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{6} \left (3+i \sqrt {3}\right ) \left (\frac {\int -\frac {x+2^{2/3} \sqrt [3]{1-i \sqrt {3}}}{x^2+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}+\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}\right )+\frac {1}{6} \left (3-i \sqrt {3}\right ) \left (\frac {\int -\frac {x+2^{2/3} \sqrt [3]{1+i \sqrt {3}}}{x^2+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}+\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (3+i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}-\frac {\int \frac {x+2^{2/3} \sqrt [3]{1-i \sqrt {3}}}{x^2+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}\right )+\frac {1}{6} \left (3-i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}-\frac {\int \frac {x+2^{2/3} \sqrt [3]{1+i \sqrt {3}}}{x^2+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{6} \left (3+i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}-\frac {\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} \int \frac {1}{x^2+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}dx+\frac {1}{2} \int \frac {2 x+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}{x^2+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}\right )+\frac {1}{6} \left (3-i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}-\frac {\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} \int \frac {1}{x^2+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}dx+\frac {1}{2} \int \frac {2 x+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}{x^2+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{6} \left (3+i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}-\frac {\frac {1}{2} \int \frac {2 x+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}{x^2+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}dx-3 \int \frac {1}{-\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}+1\right )^2-3}d\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}+1\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}\right )+\frac {1}{6} \left (3-i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}-\frac {\frac {1}{2} \int \frac {2 x+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}{x^2+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}dx-3 \int \frac {1}{-\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+1\right )^2-3}d\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+1\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{6} \left (3+i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}-\frac {\frac {1}{2} \int \frac {2 x+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}{x^2+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}dx+\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}\right )+\frac {1}{6} \left (3-i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}-\frac {\frac {1}{2} \int \frac {2 x+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}{x^2+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}dx+\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{6} \left (3+i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )+\frac {1}{2} \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}\right )+\frac {1}{6} \left (3-i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )+\frac {1}{2} \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}\right )\) |
Input:
Int[x^3/(1 - x^3 + x^6),x]
Output:
((3 + I*Sqrt[3])*(Log[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x]/(3*((1 - I*Sqrt[3 ])/2)^(2/3)) - (Sqrt[3]*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/3))/Sqrt[ 3]] + Log[(1 - I*Sqrt[3])^(2/3) + (2*(1 - I*Sqrt[3]))^(1/3)*x + 2^(2/3)*x^ 2]/2)/(3*((1 - I*Sqrt[3])/2)^(2/3))))/6 + ((3 - I*Sqrt[3])*(Log[(1 + I*Sqr t[3])^(1/3) - 2^(1/3)*x]/(3*((1 + I*Sqrt[3])/2)^(2/3)) - (Sqrt[3]*ArcTan[( 1 + (2*x)/((1 + I*Sqrt[3])/2)^(1/3))/Sqrt[3]] + Log[(1 + I*Sqrt[3])^(2/3) + (2*(1 + I*Sqrt[3]))^(1/3)*x + 2^(2/3)*x^2]/2)/(3*((1 + I*Sqrt[3])/2)^(2/ 3))))/6
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, 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[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*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_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(d^n/2)*(b/q + 1) Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Simp[(d^n/2)*(b/q - 1) Int[(d*x)^(m - n)/(b/2 - q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] & & NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[m, n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.10
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{3}\) | \(40\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{3}\) | \(40\) |
Input:
int(x^3/(x^6-x^3+1),x,method=_RETURNVERBOSE)
Output:
1/3*sum(_R^3/(2*_R^5-_R^2)*ln(x-_R),_R=RootOf(_Z^6-_Z^3+1))
Time = 0.07 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.53 \[ \int \frac {x^3}{1-x^3+x^6} \, dx=-\frac {1}{6} \, {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (3 \, \sqrt {-\frac {1}{3}} {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + 2 \, x\right ) - \frac {1}{6} \, {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (-3 \, \sqrt {-\frac {1}{3}} {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + 2 \, x\right ) + \frac {1}{6} \, {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (-3 \, \sqrt {-\frac {1}{3}} {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + 2 \, x\right ) + \frac {1}{6} \, {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (3 \, \sqrt {-\frac {1}{3}} {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + 2 \, x\right ) + \frac {1}{3} \, {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} \log \left (x - 3 \, \sqrt {-\frac {1}{3}} {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}}\right ) + \frac {1}{3} \, {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} \log \left (x + 3 \, \sqrt {-\frac {1}{3}} {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}}\right ) \] Input:
integrate(x^3/(x^6-x^3+1),x, algorithm="fricas")
Output:
-1/6*(1/6*sqrt(-1/3) - 1/6)^(1/3)*(sqrt(-3) + 1)*log(3*sqrt(-1/3)*(1/6*sqr t(-1/3) - 1/6)^(1/3)*(sqrt(-3) + 1) + 2*x) - 1/6*(-1/6*sqrt(-1/3) - 1/6)^( 1/3)*(sqrt(-3) + 1)*log(-3*sqrt(-1/3)*(-1/6*sqrt(-1/3) - 1/6)^(1/3)*(sqrt( -3) + 1) + 2*x) + 1/6*(1/6*sqrt(-1/3) - 1/6)^(1/3)*(sqrt(-3) - 1)*log(-3*s qrt(-1/3)*(1/6*sqrt(-1/3) - 1/6)^(1/3)*(sqrt(-3) - 1) + 2*x) + 1/6*(-1/6*s qrt(-1/3) - 1/6)^(1/3)*(sqrt(-3) - 1)*log(3*sqrt(-1/3)*(-1/6*sqrt(-1/3) - 1/6)^(1/3)*(sqrt(-3) - 1) + 2*x) + 1/3*(1/6*sqrt(-1/3) - 1/6)^(1/3)*log(x - 3*sqrt(-1/3)*(1/6*sqrt(-1/3) - 1/6)^(1/3)) + 1/3*(-1/6*sqrt(-1/3) - 1/6) ^(1/3)*log(x + 3*sqrt(-1/3)*(-1/6*sqrt(-1/3) - 1/6)^(1/3))
Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.06 \[ \int \frac {x^3}{1-x^3+x^6} \, dx=\operatorname {RootSum} {\left (19683 t^{6} + 243 t^{3} + 1, \left ( t \mapsto t \log {\left (- 1458 t^{4} - 9 t + x \right )} \right )\right )} \] Input:
integrate(x**3/(x**6-x**3+1),x)
Output:
RootSum(19683*_t**6 + 243*_t**3 + 1, Lambda(_t, _t*log(-1458*_t**4 - 9*_t + x)))
\[ \int \frac {x^3}{1-x^3+x^6} \, dx=\int { \frac {x^{3}}{x^{6} - x^{3} + 1} \,d x } \] Input:
integrate(x^3/(x^6-x^3+1),x, algorithm="maxima")
Output:
integrate(x^3/(x^6 - x^3 + 1), x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (255) = 510\).
Time = 0.14 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.56 \[ \int \frac {x^3}{1-x^3+x^6} \, dx=\text {Too large to display} \] Input:
integrate(x^3/(x^6-x^3+1),x, algorithm="giac")
Output:
-1/9*(2*sqrt(3)*cos(4/9*pi)^4 - 12*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi)^2 + 2 *sqrt(3)*sin(4/9*pi)^4 + 8*cos(4/9*pi)^3*sin(4/9*pi) - 8*cos(4/9*pi)*sin(4 /9*pi)^3 + sqrt(3)*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*(2*sqrt(3)* cos(2/9*pi)^4 - 12*sqrt(3)*cos(2/9*pi)^2*sin(2/9*pi)^2 + 2*sqrt(3)*sin(2/9 *pi)^4 + 8*cos(2/9*pi)^3*sin(2/9*pi) - 8*cos(2/9*pi)*sin(2/9*pi)^3 + sqrt( 3)*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*(2*sqrt(3)*cos(1/9*pi)^4 - 12*sqrt(3)*cos(1/9*pi)^2*sin(1/9*pi)^2 + 2*sqrt(3)*sin(1/9*pi)^4 - 8*cos(1 /9*pi)^3*sin(1/9*pi) + 8*cos(1/9*pi)*sin(1/9*pi)^3 - sqrt(3)*cos(1/9*pi) + sin(1/9*pi))*arctan(-1/2*((-I*sqrt(3) - 1)*cos(1/9*pi) - 2*x)/((1/2*I*sqr t(3) + 1/2)*sin(1/9*pi))) - 1/18*(8*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi) - 8* sqrt(3)*cos(4/9*pi)*sin(4/9*pi)^3 - 2*cos(4/9*pi)^4 + 12*cos(4/9*pi)^2*sin (4/9*pi)^2 - 2*sin(4/9*pi)^4 + sqrt(3)*sin(4/9*pi) - cos(4/9*pi))*log((-I* sqrt(3)*cos(4/9*pi) - cos(4/9*pi))*x + x^2 + 1) - 1/18*(8*sqrt(3)*cos(2/9* pi)^3*sin(2/9*pi) - 8*sqrt(3)*cos(2/9*pi)*sin(2/9*pi)^3 - 2*cos(2/9*pi)^4 + 12*cos(2/9*pi)^2*sin(2/9*pi)^2 - 2*sin(2/9*pi)^4 + sqrt(3)*sin(2/9*pi) - cos(2/9*pi))*log((-I*sqrt(3)*cos(2/9*pi) - cos(2/9*pi))*x + x^2 + 1) + 1/ 18*(8*sqrt(3)*cos(1/9*pi)^3*sin(1/9*pi) - 8*sqrt(3)*cos(1/9*pi)*sin(1/9*pi )^3 + 2*cos(1/9*pi)^4 - 12*cos(1/9*pi)^2*sin(1/9*pi)^2 + 2*sin(1/9*pi)^...
Time = 20.15 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.80 \[ \int \frac {x^3}{1-x^3+x^6} \, dx=\frac {\ln \left (x+\frac {2^{2/3}\,3^{5/6}\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{6}\right )\,{\left (-36-\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}+\frac {\ln \left (x-\frac {2^{2/3}\,3^{5/6}\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{6}\right )\,{\left (-36+\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{2/3}\,3^{1/3}\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}}{2}+\frac {2^{2/3}\,3^{1/3}\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{4/3}}{12}\right )\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{2/3}\,3^{1/3}\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}}{2}+\frac {2^{2/3}\,3^{1/3}\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{4/3}}{12}\right )\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{2/3}\,3^{1/3}\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}}{4}-\frac {2^{2/3}\,3^{5/6}\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{12}\right )\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{2/3}\,3^{1/3}\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}}{4}+\frac {2^{2/3}\,3^{5/6}\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{12}\right )\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36} \] Input:
int(x^3/(x^6 - x^3 + 1),x)
Output:
(log(x + (2^(2/3)*3^(5/6)*(- 3^(1/2)*1i - 3)^(1/3)*1i)/6)*(- 3^(1/2)*12i - 36)^(1/3))/18 + (log(x - (2^(2/3)*3^(5/6)*(3^(1/2)*1i - 3)^(1/3)*1i)/6)*( 3^(1/2)*12i - 36)^(1/3))/18 - (2^(2/3)*log(x + (2^(2/3)*3^(1/3)*(- 3^(1/2) *1i - 3)^(1/3))/2 + (2^(2/3)*3^(1/3)*(- 3^(1/2)*1i - 3)^(4/3))/12)*(- 3^(1 /2)*1i - 3)^(1/3)*(3^(1/3) + 3^(5/6)*1i))/36 - (2^(2/3)*log(x + (2^(2/3)*3 ^(1/3)*(3^(1/2)*1i - 3)^(1/3))/2 + (2^(2/3)*3^(1/3)*(3^(1/2)*1i - 3)^(4/3) )/12)*(3^(1/2)*1i - 3)^(1/3)*(3^(1/3) - 3^(5/6)*1i))/36 - (2^(2/3)*log(x - (2^(2/3)*3^(1/3)*(- 3^(1/2)*1i - 3)^(1/3))/4 - (2^(2/3)*3^(5/6)*(- 3^(1/2 )*1i - 3)^(1/3)*1i)/12)*(- 3^(1/2)*1i - 3)^(1/3)*(3^(1/3) - 3^(5/6)*1i))/3 6 - (2^(2/3)*log(x - (2^(2/3)*3^(1/3)*(3^(1/2)*1i - 3)^(1/3))/4 + (2^(2/3) *3^(5/6)*(3^(1/2)*1i - 3)^(1/3)*1i)/12)*(3^(1/2)*1i - 3)^(1/3)*(3^(1/3) + 3^(5/6)*1i))/36
\[ \int \frac {x^3}{1-x^3+x^6} \, dx=\int \frac {x^{3}}{x^{6}-x^{3}+1}d x \] Input:
int(x^3/(x^6-x^3+1),x)
Output:
int(x**3/(x**6 - x**3 + 1),x)