Integrand size = 23, antiderivative size = 416 \[ \int \frac {1-x^3}{x^2 \left (1-x^3+x^6\right )} \, dx=-\frac {1}{x}-\frac {\left (i+\sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (i-\sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}+\frac {\left (3+i \sqrt {3}\right ) \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}} \] Output:
-1/x-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^(1/3)/(1-I*3^(1/2))^(1/3)+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^(1/3)/(1+I*3^(1/2))^(1/3)-1/18*(3+I*3^(1/2))*ln ((1-I*3^(1/2))^(1/3)-2^(1/3)*x)*2^(1/3)/(1-I*3^(1/2))^(1/3)-1/18*(3-I*3^(1 /2))*ln((1+I*3^(1/2))^(1/3)-2^(1/3)*x)*2^(1/3)/(1+I*3^(1/2))^(1/3)+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^ (1/3)/(1-I*3^(1/2))^(1/3)+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^(1/3)/(1+I*3^(1/2))^(1/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.11 \[ \int \frac {1-x^3}{x^2 \left (1-x^3+x^6\right )} \, dx=-\frac {1}{x}-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}^2}{-1+2 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(1 - x^3)/(x^2*(1 - x^3 + x^6)),x]
Output:
-x^(-1) - RootSum[1 - #1^3 + #1^6 & , (Log[x - #1]*#1^2)/(-1 + 2*#1^3) & ] /3
Time = 0.57 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.89, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1828, 1710, 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.
| \(\displaystyle \int \frac {1-x^3}{x^2 \left (x^6-x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 1828 |
| \(\displaystyle -\int \frac {x^4}{x^6-x^3+1}dx-\frac {1}{x}\) |
| \(\Big \downarrow \) 1710 |
| \(\displaystyle -\frac {1}{6} \left (3+i \sqrt {3}\right ) \int -\frac {2 x}{-2 x^3-i \sqrt {3}+1}dx-\frac {1}{6} \left (3-i \sqrt {3}\right ) \int -\frac {2 x}{-2 x^3+i \sqrt {3}+1}dx-\frac {1}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (3+i \sqrt {3}\right ) \int \frac {x}{-2 x^3-i \sqrt {3}+1}dx+\frac {1}{3} \left (3-i \sqrt {3}\right ) \int \frac {x}{-2 x^3+i \sqrt {3}+1}dx-\frac {1}{x}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {1}{3} \left (3+i \sqrt {3}\right ) \left (\frac {\int \frac {1}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x}dx}{3 \sqrt [3]{2 \left (1-i \sqrt {3}\right )}}-\frac {\int \frac {\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x}{2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}}dx}{3 \sqrt [3]{2 \left (1-i \sqrt {3}\right )}}\right )+\frac {1}{3} \left (3-i \sqrt {3}\right ) \left (\frac {\int \frac {1}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x}dx}{3 \sqrt [3]{2 \left (1+i \sqrt {3}\right )}}-\frac {\int \frac {\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x}{2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}}dx}{3 \sqrt [3]{2 \left (1+i \sqrt {3}\right )}}\right )-\frac {1}{x}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} \left (3+i \sqrt {3}\right ) \left (-\frac {\int \frac {\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x}{2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}}dx}{3 \sqrt [3]{2 \left (1-i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}\right )+\frac {1}{3} \left (3-i \sqrt {3}\right ) \left (-\frac {\int \frac {\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x}{2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}}dx}{3 \sqrt [3]{2 \left (1+i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\right )-\frac {1}{x}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{3} \left (3+i \sqrt {3}\right ) \left (-\frac {\frac {3}{2} \sqrt [3]{1-i \sqrt {3}} \int \frac {1}{2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}}dx-\frac {\int \frac {2\ 2^{2/3} x+\sqrt [3]{2 \left (1-i \sqrt {3}\right )}}{2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}}dx}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1-i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}\right )+\frac {1}{3} \left (3-i \sqrt {3}\right ) \left (-\frac {\frac {3}{2} \sqrt [3]{1+i \sqrt {3}} \int \frac {1}{2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}}dx-\frac {\int \frac {2\ 2^{2/3} x+\sqrt [3]{2 \left (1+i \sqrt {3}\right )}}{2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}}dx}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1+i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\right )-\frac {1}{x}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{3} \left (3+i \sqrt {3}\right ) \left (-\frac {-\frac {\int \frac {2\ 2^{2/3} x+\sqrt [3]{2 \left (1-i \sqrt {3}\right )}}{2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}}dx}{2 \sqrt [3]{2}}-\frac {3 \int \frac {1}{-\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}+1\right )^2-3}d\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}+1\right )}{\sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1-i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}\right )+\frac {1}{3} \left (3-i \sqrt {3}\right ) \left (-\frac {-\frac {\int \frac {2\ 2^{2/3} x+\sqrt [3]{2 \left (1+i \sqrt {3}\right )}}{2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}}dx}{2 \sqrt [3]{2}}-\frac {3 \int \frac {1}{-\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+1\right )^2-3}d\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+1\right )}{\sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1+i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\right )-\frac {1}{x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (3+i \sqrt {3}\right ) \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {\int \frac {2\ 2^{2/3} x+\sqrt [3]{2 \left (1-i \sqrt {3}\right )}}{2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}}dx}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1-i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}\right )+\frac {1}{3} \left (3-i \sqrt {3}\right ) \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {\int \frac {2\ 2^{2/3} x+\sqrt [3]{2 \left (1+i \sqrt {3}\right )}}{2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}}dx}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1+i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\right )-\frac {1}{x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{3} \left (3+i \sqrt {3}\right ) \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {\log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}\right )}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1-i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}\right )+\frac {1}{3} \left (3-i \sqrt {3}\right ) \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {\log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}\right )}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2 \left (1+i \sqrt {3}\right )}}-\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\right )-\frac {1}{x}\) |
Input:
Int[(1 - x^3)/(x^2*(1 - x^3 + x^6)),x]
Output:
-x^(-1) + ((3 + I*Sqrt[3])*(-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*Sqr t[3]))^(1/3)*x + 2^(2/3)*x^2]/(2*2^(1/3)))/(3*(2*(1 - I*Sqrt[3]))^(1/3)))) /3 + ((3 - I*Sqrt[3])*(-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))))/3
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_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]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*x^n + c*x^ (2*n))^(p + 1)/(a*f*(m + 1))), x] + Simp[1/(a*f^n*(m + 1)) Int[(f*x)^(m + n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1) - c*d*(m + 2*n*(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] && LtQ[m, -1] && Int egerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.10
| method | result | size |
| risch | \(-\frac {1}{x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )}{\sum }\textit {\_R} \ln \left (-27 \textit {\_R}^{5}+6 \textit {\_R}^{2}+x \right )\right )}{3}\) | \(40\) |
| default | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R}^{4} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{3}-\frac {1}{x}\) | \(46\) |
Input:
int((-x^3+1)/x^2/(x^6-x^3+1),x,method=_RETURNVERBOSE)
Output:
-1/x+1/3*sum(_R*ln(-27*_R^5+6*_R^2+x),_R=RootOf(27*_Z^6-9*_Z^3+1))
Time = 0.07 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.67 \[ \int \frac {1-x^3}{x^2 \left (1-x^3+x^6\right )} \, dx=\frac {{\left (\sqrt {-3} x - x\right )} {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} + \frac {1}{6}\right )}^{\frac {1}{3}} \log \left (3 \, {\left (3 \, \sqrt {-\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} - \sqrt {-3} - 1\right )} {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} + \frac {1}{6}\right )}^{\frac {2}{3}} + 4 \, x\right ) - {\left (\sqrt {-3} x + x\right )} {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} + \frac {1}{6}\right )}^{\frac {1}{3}} \log \left (-3 \, {\left (3 \, \sqrt {-\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} - \sqrt {-3} + 1\right )} {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} + \frac {1}{6}\right )}^{\frac {2}{3}} + 4 \, x\right ) + 2 \, x {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} + \frac {1}{6}\right )}^{\frac {1}{3}} \log \left (-3 \, {\left (3 \, \sqrt {-\frac {1}{3}} - 1\right )} {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} + \frac {1}{6}\right )}^{\frac {2}{3}} + 2 \, x\right ) + {\left (\sqrt {-3} x - x\right )} {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} + \frac {1}{6}\right )}^{\frac {1}{3}} \log \left (-3 \, {\left (3 \, \sqrt {-\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + \sqrt {-3} + 1\right )} {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} + \frac {1}{6}\right )}^{\frac {2}{3}} + 4 \, x\right ) - {\left (\sqrt {-3} x + x\right )} {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} + \frac {1}{6}\right )}^{\frac {1}{3}} \log \left (3 \, {\left (3 \, \sqrt {-\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + \sqrt {-3} - 1\right )} {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} + \frac {1}{6}\right )}^{\frac {2}{3}} + 4 \, x\right ) + 2 \, x {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} + \frac {1}{6}\right )}^{\frac {1}{3}} \log \left (3 \, {\left (3 \, \sqrt {-\frac {1}{3}} + 1\right )} {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} + \frac {1}{6}\right )}^{\frac {2}{3}} + 2 \, x\right ) - 6}{6 \, x} \] Input:
integrate((-x^3+1)/x^2/(x^6-x^3+1),x, algorithm="fricas")
Output:
1/6*((sqrt(-3)*x - x)*(1/6*sqrt(-1/3) + 1/6)^(1/3)*log(3*(3*sqrt(-1/3)*(sq rt(-3) + 1) - sqrt(-3) - 1)*(1/6*sqrt(-1/3) + 1/6)^(2/3) + 4*x) - (sqrt(-3 )*x + x)*(1/6*sqrt(-1/3) + 1/6)^(1/3)*log(-3*(3*sqrt(-1/3)*(sqrt(-3) - 1) - sqrt(-3) + 1)*(1/6*sqrt(-1/3) + 1/6)^(2/3) + 4*x) + 2*x*(1/6*sqrt(-1/3) + 1/6)^(1/3)*log(-3*(3*sqrt(-1/3) - 1)*(1/6*sqrt(-1/3) + 1/6)^(2/3) + 2*x) + (sqrt(-3)*x - x)*(-1/6*sqrt(-1/3) + 1/6)^(1/3)*log(-3*(3*sqrt(-1/3)*(sq rt(-3) + 1) + sqrt(-3) + 1)*(-1/6*sqrt(-1/3) + 1/6)^(2/3) + 4*x) - (sqrt(- 3)*x + x)*(-1/6*sqrt(-1/3) + 1/6)^(1/3)*log(3*(3*sqrt(-1/3)*(sqrt(-3) - 1) + sqrt(-3) - 1)*(-1/6*sqrt(-1/3) + 1/6)^(2/3) + 4*x) + 2*x*(-1/6*sqrt(-1/ 3) + 1/6)^(1/3)*log(3*(3*sqrt(-1/3) + 1)*(-1/6*sqrt(-1/3) + 1/6)^(2/3) + 2 *x) - 6)/x
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.07 \[ \int \frac {1-x^3}{x^2 \left (1-x^3+x^6\right )} \, dx=- \operatorname {RootSum} {\left (19683 t^{6} + 243 t^{3} + 1, \left ( t \mapsto t \log {\left (6561 t^{5} + 54 t^{2} + x \right )} \right )\right )} - \frac {1}{x} \] Input:
integrate((-x**3+1)/x**2/(x**6-x**3+1),x)
Output:
-RootSum(19683*_t**6 + 243*_t**3 + 1, Lambda(_t, _t*log(6561*_t**5 + 54*_t **2 + x))) - 1/x
\[ \int \frac {1-x^3}{x^2 \left (1-x^3+x^6\right )} \, dx=\int { -\frac {x^{3} - 1}{{\left (x^{6} - x^{3} + 1\right )} x^{2}} \,d x } \] Input:
integrate((-x^3+1)/x^2/(x^6-x^3+1),x, algorithm="maxima")
Output:
-1/x - integrate(x^4/(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. 832 vs. \(2 (260) = 520\).
Time = 0.14 (sec) , antiderivative size = 832, normalized size of antiderivative = 2.00 \[ \int \frac {1-x^3}{x^2 \left (1-x^3+x^6\right )} \, dx=\text {Too large to display} \] Input:
integrate((-x^3+1)/x^2/(x^6-x^3+1),x, algorithm="giac")
Output:
1/9*(2*sqrt(3)*cos(4/9*pi)^5 - 20*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi)^2 + 10 *sqrt(3)*cos(4/9*pi)*sin(4/9*pi)^4 - 10*cos(4/9*pi)^4*sin(4/9*pi) + 20*cos (4/9*pi)^2*sin(4/9*pi)^3 - 2*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*(2*sqrt(3)* cos(2/9*pi)^5 - 20*sqrt(3)*cos(2/9*pi)^3*sin(2/9*pi)^2 + 10*sqrt(3)*cos(2/ 9*pi)*sin(2/9*pi)^4 - 10*cos(2/9*pi)^4*sin(2/9*pi) + 20*cos(2/9*pi)^2*sin( 2/9*pi)^3 - 2*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*(2*sqrt(3)*cos(1/9*pi)^5 - 20*sqrt(3)*cos(1/9*pi)^3*sin(1/9*pi)^2 + 10*sqrt(3)*cos(1/9*pi)*sin(1/9*p i)^4 + 10*cos(1/9*pi)^4*sin(1/9*pi) - 20*cos(1/9*pi)^2*sin(1/9*pi)^3 + 2*s in(1/9*pi)^5 - sqrt(3)*cos(1/9*pi)^2 + sqrt(3)*sin(1/9*pi)^2 - 2*cos(1/9*p i)*sin(1/9*pi))*arctan(-1/2*((-I*sqrt(3) - 1)*cos(1/9*pi) - 2*x)/((1/2*I*s qrt(3) + 1/2)*sin(1/9*pi))) + 1/18*(10*sqrt(3)*cos(4/9*pi)^4*sin(4/9*pi) - 20*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi)^3 + 2*sqrt(3)*sin(4/9*pi)^5 + 2*cos( 4/9*pi)^5 - 20*cos(4/9*pi)^3*sin(4/9*pi)^2 + 10*cos(4/9*pi)*sin(4/9*pi)^4 + 2*sqrt(3)*cos(4/9*pi)*sin(4/9*pi) + cos(4/9*pi)^2 - sin(4/9*pi)^2)*log(( -I*sqrt(3)*cos(4/9*pi) - cos(4/9*pi))*x + x^2 + 1) + 1/18*(10*sqrt(3)*cos( 2/9*pi)^4*sin(2/9*pi) - 20*sqrt(3)*cos(2/9*pi)^2*sin(2/9*pi)^3 + 2*sqrt...
Time = 20.76 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.75 \[ \int \frac {1-x^3}{x^2 \left (1-x^3+x^6\right )} \, dx=\frac {\ln \left (-x+\left (162\,x+\frac {27\,{\left (36+\sqrt {3}\,12{}\mathrm {i}\right )}^{2/3}}{4}\right )\,\left (\frac {1}{162}+\frac {\sqrt {3}\,1{}\mathrm {i}}{486}\right )\right )\,{\left (36+\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}+\frac {\ln \left (-x-\left (162\,x+\frac {27\,{\left (36-\sqrt {3}\,12{}\mathrm {i}\right )}^{2/3}}{4}\right )\,\left (-\frac {1}{162}+\frac {\sqrt {3}\,1{}\mathrm {i}}{486}\right )\right )\,{\left (36-\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}-\frac {1}{x}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{1/3}\,3^{2/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{12}-\frac {2^{1/3}\,3^{1/6}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,1{}\mathrm {i}}{4}\right )\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{1/3}\,3^{2/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{12}+\frac {2^{1/3}\,3^{1/6}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,1{}\mathrm {i}}{4}\right )\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{1/3}\,3^{2/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{6}\right )\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{1/3}\,3^{2/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{6}\right )\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36} \] Input:
int(-(x^3 - 1)/(x^2*(x^6 - x^3 + 1)),x)
Output:
(log((162*x + (27*(3^(1/2)*12i + 36)^(2/3))/4)*((3^(1/2)*1i)/486 + 1/162) - x)*(3^(1/2)*12i + 36)^(1/3))/18 + (log(- x - (162*x + (27*(36 - 3^(1/2)* 12i)^(2/3))/4)*((3^(1/2)*1i)/486 - 1/162))*(36 - 3^(1/2)*12i)^(1/3))/18 - 1/x - (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
\[ \int \frac {1-x^3}{x^2 \left (1-x^3+x^6\right )} \, dx=\frac {-\left (\int \frac {x^{4}}{x^{6}-x^{3}+1}d x \right ) x -1}{x} \] Input:
int((-x^3+1)/x^2/(x^6-x^3+1),x)
Output:
( - (int(x**4/(x**6 - x**3 + 1),x)*x + 1))/x