\(\int \frac {x^6}{1-x^3+x^6} \, dx\) [170]

Optimal result

Integrand size = 16, antiderivative size = 412 \[ \int \frac {x^6}{1-x^3+x^6} \, dx=x+\frac {\left (i-\sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (i+\sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {\left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}} \]

[Out]

x+1/6*arctan(1/3*(1+2*2^(1/3)*x/(1-I*3^(1/2))^(1/3))*3^(1/2))*(I-3^(1/2))*2^(2/3)/(1-I*3^(1/2))^(2/3)+1/18*ln(
-2^(1/3)*x+(1-I*3^(1/2))^(1/3))*(3-I*3^(1/2))*2^(2/3)/(1-I*3^(1/2))^(2/3)-1/36*ln(2^(2/3)*x^2+2^(1/3)*x*(1-I*3
^(1/2))^(1/3)+(1-I*3^(1/2))^(2/3))*(3-I*3^(1/2))*2^(2/3)/(1-I*3^(1/2))^(2/3)+1/18*ln(-2^(1/3)*x+(1+I*3^(1/2))^
(1/3))*(3+I*3^(1/2))*2^(2/3)/(1+I*3^(1/2))^(2/3)-1/36*ln(2^(2/3)*x^2+2^(1/3)*x*(1+I*3^(1/2))^(1/3)+(1+I*3^(1/2
))^(2/3))*(3+I*3^(1/2))*2^(2/3)/(1+I*3^(1/2))^(2/3)-1/6*arctan(1/3*(1+2*2^(1/3)*x/(1+I*3^(1/2))^(1/3))*3^(1/2)
)*(3^(1/2)+I)*2^(2/3)/(1+I*3^(1/2))^(2/3)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1381, 1436, 206, 31, 648, 631, 210, 642} \[ \int \frac {x^6}{1-x^3+x^6} \, dx=\frac {\left (-\sqrt {3}+i\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (\sqrt {3}+i\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}+x+\frac {\left (3-i \sqrt {3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{9 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {\left (3+i \sqrt {3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{9 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}} \]

[In]

Int[x^6/(1 - x^3 + x^6),x]

[Out]

x + ((I - Sqrt[3])*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/(3*2^(1/3)*(1 - I*Sqrt[3])^(2/3)) -
((I + Sqrt[3])*ArcTan[(1 + (2*x)/((1 + I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/(3*2^(1/3)*(1 + I*Sqrt[3])^(2/3)) + ((3
- I*Sqrt[3])*Log[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(1/3)*(1 - I*Sqrt[3])^(2/3)) + ((3 + I*Sqrt[3])*Log[
(1 + I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(1/3)*(1 + I*Sqrt[3])^(2/3)) - ((3 - I*Sqrt[3])*Log[(1 - I*Sqrt[3])^(
2/3) + (2*(1 - I*Sqrt[3]))^(1/3)*x + 2^(2/3)*x^2])/(18*2^(1/3)*(1 - I*Sqrt[3])^(2/3)) - ((3 + I*Sqrt[3])*Log[(
1 + I*Sqrt[3])^(2/3) + (2*(1 + I*Sqrt[3]))^(1/3)*x + 2^(2/3)*x^2])/(18*2^(1/3)*(1 + I*Sqrt[3])^(2/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[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]

Rule 210

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])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1381

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[d^(2*n - 1)*(d*x)^
(m - 2*n + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + 2*n*p + 1))), x] - Dist[d^(2*n)/(c*(m + 2*n*p + 1)), In
t[(d*x)^(m - 2*n)*Simp[a*(m - 2*n + 1) + b*(m + n*(p - 1) + 1)*x^n, x]*(a + b*x^n + c*x^(2*n))^p, x], x] /; Fr
eeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n
*p + 1, 0] && IntegerQ[p]

Rule 1436

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] ||  !IGtQ[n/2, 0])

Rubi steps \begin{align*} \text {integral}& = x-\int \frac {1-x^3}{1-x^3+x^6} \, dx \\ & = x-\frac {1}{6} \left (-3+i \sqrt {3}\right ) \int \frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}+x^3} \, dx+\frac {1}{6} \left (3+i \sqrt {3}\right ) \int \frac {1}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}+x^3} \, dx \\ & = x+\frac {\left (3-i \sqrt {3}\right ) \int \frac {1}{-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}+x} \, dx}{9 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {\left (3-i \sqrt {3}\right ) \int \frac {-2^{2/3} \sqrt [3]{1-i \sqrt {3}}-x}{\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+x^2} \, dx}{9 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {\left (3+i \sqrt {3}\right ) \int \frac {1}{-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}+x} \, dx}{9 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {\left (3+i \sqrt {3}\right ) \int \frac {-2^{2/3} \sqrt [3]{1+i \sqrt {3}}-x}{\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+x^2} \, dx}{9 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}} \\ & = x+\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 ) \int \frac {\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}+2 x}{\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+x^2} \, dx}{18 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (3-i \sqrt {3}\right ) \int \frac {1}{\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+x^2} \, dx}{6\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (3+i \sqrt {3}\right ) \int \frac {\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}+2 x}{\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+x^2} \, dx}{18 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {\left (3+i \sqrt {3}\right ) \int \frac {1}{\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+x^2} \, dx}{6\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}} \\ & = x+\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}}+\frac {\left (3-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {\left (3+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}} \\ & = x+\frac {\left (i-\sqrt {3}\right ) \tan ^{-1}\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 ) \tan ^{-1}\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}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.14 \[ \int \frac {x^6}{1-x^3+x^6} \, dx=x+\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^3}{-\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ] \]

[In]

Integrate[x^6/(1 - x^3 + x^6),x]

[Out]

x + RootSum[1 - #1^3 + #1^6 & , (-Log[x - #1] + Log[x - #1]*#1^3)/(-#1^2 + 2*#1^5) & ]/3

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.11

method	result	size
default	\(x +\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (\textit {\_R}^{3}-1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{3}\)	\(44\)
risch	\(x +\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (\textit {\_R}^{3}-1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{3}\)	\(44\)

[In]

int(x^6/(x^6-x^3+1),x,method=_RETURNVERBOSE)

[Out]

x+1/3*sum((_R^3-1)/(2*_R^5-_R^2)*ln(x-_R),_R=RootOf(_Z^6-_Z^3+1))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.73 \[ \int \frac {x^6}{1-x^3+x^6} \, dx=\frac {1}{108} \cdot 18^{\frac {2}{3}} {\left (i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (18^{\frac {2}{3}} {\left (\sqrt {3} {\left (i \, \sqrt {-3} - i\right )} - 3 \, \sqrt {-3} + 3\right )} {\left (i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} + 72 \, x\right ) - \frac {1}{108} \cdot 18^{\frac {2}{3}} {\left (i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (18^{\frac {2}{3}} {\left (\sqrt {3} {\left (-i \, \sqrt {-3} - i\right )} + 3 \, \sqrt {-3} + 3\right )} {\left (i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} + 72 \, x\right ) - \frac {1}{108} \cdot 18^{\frac {2}{3}} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (18^{\frac {2}{3}} {\left (\sqrt {3} {\left (i \, \sqrt {-3} + i\right )} + 3 \, \sqrt {-3} + 3\right )} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} + 72 \, x\right ) + \frac {1}{108} \cdot 18^{\frac {2}{3}} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (18^{\frac {2}{3}} {\left (\sqrt {3} {\left (-i \, \sqrt {-3} + i\right )} - 3 \, \sqrt {-3} + 3\right )} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} + 72 \, x\right ) + \frac {1}{54} \cdot 18^{\frac {2}{3}} {\left (i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} \log \left (18^{\frac {2}{3}} {\left (i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} {\left (i \, \sqrt {3} - 3\right )} + 36 \, x\right ) + \frac {1}{54} \cdot 18^{\frac {2}{3}} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} \log \left (18^{\frac {2}{3}} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 3\right )} + 36 \, x\right ) + x \]

[In]

integrate(x^6/(x^6-x^3+1),x, algorithm="fricas")

[Out]

1/108*18^(2/3)*(I*sqrt(3) + 3)^(1/3)*(sqrt(-3) - 1)*log(18^(2/3)*(sqrt(3)*(I*sqrt(-3) - I) - 3*sqrt(-3) + 3)*(
I*sqrt(3) + 3)^(1/3) + 72*x) - 1/108*18^(2/3)*(I*sqrt(3) + 3)^(1/3)*(sqrt(-3) + 1)*log(18^(2/3)*(sqrt(3)*(-I*s
qrt(-3) - I) + 3*sqrt(-3) + 3)*(I*sqrt(3) + 3)^(1/3) + 72*x) - 1/108*18^(2/3)*(-I*sqrt(3) + 3)^(1/3)*(sqrt(-3)
+ 1)*log(18^(2/3)*(sqrt(3)*(I*sqrt(-3) + I) + 3*sqrt(-3) + 3)*(-I*sqrt(3) + 3)^(1/3) + 72*x) + 1/108*18^(2/3)
*(-I*sqrt(3) + 3)^(1/3)*(sqrt(-3) - 1)*log(18^(2/3)*(sqrt(3)*(-I*sqrt(-3) + I) - 3*sqrt(-3) + 3)*(-I*sqrt(3) +
3)^(1/3) + 72*x) + 1/54*18^(2/3)*(I*sqrt(3) + 3)^(1/3)*log(18^(2/3)*(I*sqrt(3) + 3)^(1/3)*(I*sqrt(3) - 3) + 3
6*x) + 1/54*18^(2/3)*(-I*sqrt(3) + 3)^(1/3)*log(18^(2/3)*(-I*sqrt(3) + 3)^(1/3)*(-I*sqrt(3) - 3) + 36*x) + x

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.06 \[ \int \frac {x^6}{1-x^3+x^6} \, dx=x + \operatorname {RootSum} {\left (19683 t^{6} - 243 t^{3} + 1, \left ( t \mapsto t \log {\left (729 t^{4} - 9 t + x \right )} \right )\right )} \]

[In]

integrate(x**6/(x**6-x**3+1),x)

[Out]

x + RootSum(19683*_t**6 - 243*_t**3 + 1, Lambda(_t, _t*log(729*_t**4 - 9*_t + x)))

Maxima [F]

\[ \int \frac {x^6}{1-x^3+x^6} \, dx=\int { \frac {x^{6}}{x^{6} - x^{3} + 1} \,d x } \]

[In]

integrate(x^6/(x^6-x^3+1),x, algorithm="maxima")

[Out]

x + integrate((x^3 - 1)/(x^6 - x^3 + 1), x)

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (268) = 536\).

Time = 0.33 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.56 \[ \int \frac {x^6}{1-x^3+x^6} \, dx=\text {Too large to display} \]

[In]

integrate(x^6/(x^6-x^3+1),x, algorithm="giac")

[Out]

-1/9*(sqrt(3)*cos(4/9*pi)^4 - 6*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi)^2 + sqrt(3)*sin(4/9*pi)^4 + 4*cos(4/9*pi)^3*
sin(4/9*pi) - 4*cos(4/9*pi)*sin(4/9*pi)^3 + 2*sqrt(3)*cos(4/9*pi) + 2*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)^4 - 6*sqrt(3)*cos(2/9*pi)
^2*sin(2/9*pi)^2 + sqrt(3)*sin(2/9*pi)^4 + 4*cos(2/9*pi)^3*sin(2/9*pi) - 4*cos(2/9*pi)*sin(2/9*pi)^3 + 2*sqrt(
3)*cos(2/9*pi) + 2*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)^4 - 6*sqrt(3)*cos(1/9*pi)^2*sin(1/9*pi)^2 + sqrt(3)*sin(1/9*pi)^4 - 4*cos(1/
9*pi)^3*sin(1/9*pi) + 4*cos(1/9*pi)*sin(1/9*pi)^3 - 2*sqrt(3)*cos(1/9*pi) + 2*sin(1/9*pi))*arctan(-1/2*((-I*sq
rt(3) - 1)*cos(1/9*pi) - 2*x)/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) - 1/18*(4*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi)
- 4*sqrt(3)*cos(4/9*pi)*sin(4/9*pi)^3 - cos(4/9*pi)^4 + 6*cos(4/9*pi)^2*sin(4/9*pi)^2 - sin(4/9*pi)^4 + 2*sqr
t(3)*sin(4/9*pi) - 2*cos(4/9*pi))*log((-I*sqrt(3)*cos(4/9*pi) - cos(4/9*pi))*x + x^2 + 1) - 1/18*(4*sqrt(3)*co
s(2/9*pi)^3*sin(2/9*pi) - 4*sqrt(3)*cos(2/9*pi)*sin(2/9*pi)^3 - cos(2/9*pi)^4 + 6*cos(2/9*pi)^2*sin(2/9*pi)^2
- sin(2/9*pi)^4 + 2*sqrt(3)*sin(2/9*pi) - 2*cos(2/9*pi))*log((-I*sqrt(3)*cos(2/9*pi) - cos(2/9*pi))*x + x^2 +
1) + 1/18*(4*sqrt(3)*cos(1/9*pi)^3*sin(1/9*pi) - 4*sqrt(3)*cos(1/9*pi)*sin(1/9*pi)^3 + cos(1/9*pi)^4 - 6*cos(1
/9*pi)^2*sin(1/9*pi)^2 + sin(1/9*pi)^4 - 2*sqrt(3)*sin(1/9*pi) - 2*cos(1/9*pi))*log((I*sqrt(3)*cos(1/9*pi) + c
os(1/9*pi))*x + x^2 + 1) + x

Mupad [B] (verification not implemented)

Time = 8.73 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.78 \[ \int \frac {x^6}{1-x^3+x^6} \, dx=x+\frac {\ln \left (x+\frac {\left (-\frac {27}{2}+\frac {\sqrt {3}\,9{}\mathrm {i}}{2}\right )\,{\left (36+\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{54}\right )\,{\left (36+\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}+\frac {\ln \left (x-\frac {\left (\frac {27}{2}+\frac {\sqrt {3}\,9{}\mathrm {i}}{2}\right )\,{\left (36-\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{54}\right )\,{\left (36-\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{2/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )\,\left (\frac {3\,\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}^3}{16}-27\right )}{108}\right )\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{2/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )\,\left (\frac {3\,\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}^3}{16}+27\right )}{108}\right )\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{2/3}\,3^{5/6}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{6}\right )\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{2/3}\,3^{5/6}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{6}\right )\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36} \]

[In]

int(x^6/(x^6 - x^3 + 1),x)

[Out]

x + (log(x + (((3^(1/2)*9i)/2 - 27/2)*(3^(1/2)*12i + 36)^(1/3))/54)*(3^(1/2)*12i + 36)^(1/3))/18 + (log(x - ((
(3^(1/2)*9i)/2 + 27/2)*(36 - 3^(1/2)*12i)^(1/3))/54)*(36 - 3^(1/2)*12i)^(1/3))/18 - (2^(2/3)*log(x - (2^(2/3)*
(3 - 3^(1/2)*1i)^(1/3)*(3^(1/3) - 3^(5/6)*1i)*((3*(3^(1/2)*1i - 3)*(3^(1/3) - 3^(5/6)*1i)^3)/16 - 27))/108)*(3
- 3^(1/2)*1i)^(1/3)*(3^(1/3) - 3^(5/6)*1i))/36 - (2^(2/3)*log(x + (2^(2/3)*(3^(1/2)*1i + 3)^(1/3)*(3^(1/3) +
3^(5/6)*1i)*((3*(3^(1/2)*1i + 3)*(3^(1/3) + 3^(5/6)*1i)^3)/16 + 27))/108)*(3^(1/2)*1i + 3)^(1/3)*(3^(1/3) + 3^
(5/6)*1i))/36 - (2^(2/3)*log(x + (2^(2/3)*3^(5/6)*(3 - 3^(1/2)*1i)^(1/3)*1i)/6)*(3 - 3^(1/2)*1i)^(1/3)*(3^(1/3
) + 3^(5/6)*1i))/36 - (2^(2/3)*log(x - (2^(2/3)*3^(5/6)*(3^(1/2)*1i + 3)^(1/3)*1i)/6)*(3^(1/2)*1i + 3)^(1/3)*(
3^(1/3) - 3^(5/6)*1i))/36