3.181 \(\int \frac{1}{x^5 (1-x^3+x^6)} \, dx\)
Optimal. Leaf size=423 \[ -\frac{1}{4 x^4}+\frac{\left (3+i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}}-\frac{1}{x}-\frac{\left (3+i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}-\frac{\left (3-i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}}-\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}+\frac{\left (-\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}} \]
[Out]
-1/(4*x^4) - x^(-1) - ((I + Sqrt[3])*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/(3*2^(2/3)*(1 - I*
Sqrt[3])^(1/3)) + ((I - Sqrt[3])*ArcTan[(1 + (2*x)/((1 + I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/(3*2^(2/3)*(1 + I*Sqrt
[3])^(1/3)) - ((3 + I*Sqrt[3])*Log[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(2/3)*(1 - I*Sqrt[3])^(1/3)) - ((3
- I*Sqrt[3])*Log[(1 + I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(2/3)*(1 + I*Sqrt[3])^(1/3)) + ((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^(2/3)*(1 - I*Sqrt[3])^(1/3)) + ((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^(2/3)*(1 + I*Sqrt[3
])^(1/3))
________________________________________________________________________________________
Rubi [A] time = 0.369171, antiderivative size = 423, normalized size of antiderivative = 1.,
number of steps used = 16, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used
= {1368, 1504, 12, 1374, 292, 31, 634, 617, 204, 628} \[ -\frac{1}{4 x^4}+\frac{\left (3+i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}}-\frac{1}{x}-\frac{\left (3+i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}-\frac{\left (3-i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}}-\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}+\frac{\left (-\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}} \]
Antiderivative was successfully verified.
[In]
Int[1/(x^5*(1 - x^3 + x^6)),x]
[Out]
-1/(4*x^4) - x^(-1) - ((I + Sqrt[3])*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/(3*2^(2/3)*(1 - I*
Sqrt[3])^(1/3)) + ((I - Sqrt[3])*ArcTan[(1 + (2*x)/((1 + I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/(3*2^(2/3)*(1 + I*Sqrt
[3])^(1/3)) - ((3 + I*Sqrt[3])*Log[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(2/3)*(1 - I*Sqrt[3])^(1/3)) - ((3
- I*Sqrt[3])*Log[(1 + I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(2/3)*(1 + I*Sqrt[3])^(1/3)) + ((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^(2/3)*(1 - I*Sqrt[3])^(1/3)) + ((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^(2/3)*(1 + I*Sqrt[3
])^(1/3))
Rule 1368
Int[((d_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((d*x)^(m + 1)*(a +
b*x^n + c*x^(2*n))^(p + 1))/(a*d*(m + 1)), x] - Dist[1/(a*d^n*(m + 1)), Int[(d*x)^(m + n)*(b*(m + n*(p + 1) +
1) + c*(m + 2*n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2
*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntegerQ[p]
Rule 1504
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] + Dist[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] && IntegerQ[p]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 1374
Int[((d_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[(d^n*(b/q + 1))/2, Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Dist[(d^n*(b/q - 1))/2, 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]
Rule 292
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[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]
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rule 634
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 617
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 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 628
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]
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (1-x^3+x^6\right )} \, dx &=-\frac{1}{4 x^4}+\frac{1}{4} \int \frac{4-4 x^3}{x^2 \left (1-x^3+x^6\right )} \, dx\\ &=-\frac{1}{4 x^4}-\frac{1}{x}-\frac{1}{4} \int \frac{4 x^4}{1-x^3+x^6} \, dx\\ &=-\frac{1}{4 x^4}-\frac{1}{x}-\int \frac{x^4}{1-x^3+x^6} \, dx\\ &=-\frac{1}{4 x^4}-\frac{1}{x}+\frac{1}{6} \left (-3+i \sqrt{3}\right ) \int \frac{x}{-\frac{1}{2}-\frac{i \sqrt{3}}{2}+x^3} \, dx-\frac{1}{6} \left (3+i \sqrt{3}\right ) \int \frac{x}{-\frac{1}{2}+\frac{i \sqrt{3}}{2}+x^3} \, dx\\ &=-\frac{1}{4 x^4}-\frac{1}{x}+-\frac{\left (-3-i \sqrt{3}\right ) \int \frac{-\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}+\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )} x+x^2} \, dx}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt{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\ 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 )}+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\ 2^{2/3} \sqrt [3]{1+i \sqrt{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\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}\\ &=-\frac{1}{4 x^4}-\frac{1}{x}-\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 ) \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\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}+\frac{1}{12} \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+\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\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}}-\frac{1}{12} \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\\ &=-\frac{1}{4 x^4}-\frac{1}{x}-\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}}}+-\frac{\left (-3-i \sqrt{3}\right ) \operatorname{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\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}+\frac{\left (3-i \sqrt{3}\right ) \operatorname{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\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}}\\ &=-\frac{1}{4 x^4}-\frac{1}{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\ 2^{2/3} \sqrt [3]{1-i \sqrt{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\ 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}}}\\ \end{align*}
Mathematica [C] time = 0.0148857, size = 54, normalized size = 0.13 \[ -\frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\& ,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\& \right ]-\frac{1}{4 x^4}-\frac{1}{x} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(x^5*(1 - x^3 + x^6)),x]
[Out]
-1/(4*x^4) - x^(-1) - RootSum[1 - #1^3 + #1^6 & , (Log[x - #1]*#1^2)/(-1 + 2*#1^3) & ]/3
________________________________________________________________________________________
Maple [C] time = 0.007, size = 51, normalized size = 0.1 \begin{align*} -{\frac{1}{4\,{x}^{4}}}-{x}^{-1}-{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{3}+1 \right ) }{\frac{{{\it \_R}}^{4}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}-{{\it \_R}}^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^5/(x^6-x^3+1),x)
[Out]
-1/4/x^4-1/x-1/3*sum(_R^4/(2*_R^5-_R^2)*ln(x-_R),_R=RootOf(_Z^6-_Z^3+1))
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{4 \, x^{3} + 1}{4 \, x^{4}} - \int \frac{x^{4}}{x^{6} - x^{3} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^5/(x^6-x^3+1),x, algorithm="maxima")
[Out]
-1/4*(4*x^3 + 1)/x^4 - integrate(x^4/(x^6 - x^3 + 1), x)
________________________________________________________________________________________
Fricas [B] time = 2.29423, size = 5994, normalized size = 14.17 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^5/(x^6-x^3+1),x, algorithm="fricas")
[Out]
1/108*(2*18^(2/3)*12^(1/6)*x^4*cos(2/3*arctan(sqrt(3) - 2))*log(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2))
^4 + 18^(2/3)*12^(2/3)*sin(2/3*arctan(sqrt(3) - 2))^4 - 12*18^(1/3)*12^(1/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3)
- 2))*sin(2/3*arctan(sqrt(3) - 2)) + 6*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 2*(18^(2/3)*12^(2/
3)*cos(2/3*arctan(sqrt(3) - 2))^2 - 3*18^(1/3)*12^(1/3)*x)*sin(2/3*arctan(sqrt(3) - 2))^2 + 36*x^2) + 8*18^(2/
3)*12^(1/6)*x^4*arctan(1/108*(6*18^(2/3)*12^(2/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 108*sqrt(3)*cos(2
/3*arctan(sqrt(3) - 2))^4 + 108*sqrt(3)*sin(2/3*arctan(sqrt(3) - 2))^4 + 864*cos(2/3*arctan(sqrt(3) - 2))*sin(
2/3*arctan(sqrt(3) - 2))^3 - 6*(18^(2/3)*12^(2/3)*sqrt(3)*x - 36*sqrt(3)*cos(2/3*arctan(sqrt(3) - 2))^2)*sin(2
/3*arctan(sqrt(3) - 2))^2 - 12*(18^(2/3)*12^(2/3)*x*cos(2/3*arctan(sqrt(3) - 2)) + 72*cos(2/3*arctan(sqrt(3) -
2))^3)*sin(2/3*arctan(sqrt(3) - 2)) - sqrt(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2))^4 + 18^(2/3)*12^(2/
3)*sin(2/3*arctan(sqrt(3) - 2))^4 - 12*18^(1/3)*12^(1/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan
(sqrt(3) - 2)) + 6*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 2*(18^(2/3)*12^(2/3)*cos(2/3*arctan(sq
rt(3) - 2))^2 - 3*18^(1/3)*12^(1/3)*x)*sin(2/3*arctan(sqrt(3) - 2))^2 + 36*x^2)*(18^(2/3)*12^(2/3)*sqrt(3)*cos
(2/3*arctan(sqrt(3) - 2))^2 - 18^(2/3)*12^(2/3)*sqrt(3)*sin(2/3*arctan(sqrt(3) - 2))^2 - 2*18^(2/3)*12^(2/3)*c
os(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2))))/(3*cos(2/3*arctan(sqrt(3) - 2))^4 - 10*cos(2/3*arct
an(sqrt(3) - 2))^2*sin(2/3*arctan(sqrt(3) - 2))^2 + 3*sin(2/3*arctan(sqrt(3) - 2))^4))*sin(2/3*arctan(sqrt(3)
- 2)) - 108*x^3 - 4*(18^(2/3)*12^(1/6)*sqrt(3)*x^4*cos(2/3*arctan(sqrt(3) - 2)) - 18^(2/3)*12^(1/6)*x^4*sin(2/
3*arctan(sqrt(3) - 2)))*arctan(1/108*(6*18^(2/3)*12^(2/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 108*sqrt(
3)*cos(2/3*arctan(sqrt(3) - 2))^4 + 108*sqrt(3)*sin(2/3*arctan(sqrt(3) - 2))^4 - 864*cos(2/3*arctan(sqrt(3) -
2))*sin(2/3*arctan(sqrt(3) - 2))^3 - 6*(18^(2/3)*12^(2/3)*sqrt(3)*x - 36*sqrt(3)*cos(2/3*arctan(sqrt(3) - 2))^
2)*sin(2/3*arctan(sqrt(3) - 2))^2 + 12*(18^(2/3)*12^(2/3)*x*cos(2/3*arctan(sqrt(3) - 2)) + 72*cos(2/3*arctan(s
qrt(3) - 2))^3)*sin(2/3*arctan(sqrt(3) - 2)) - sqrt(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2))^4 + 18^(2/3
)*12^(2/3)*sin(2/3*arctan(sqrt(3) - 2))^4 + 12*18^(1/3)*12^(1/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2/
3*arctan(sqrt(3) - 2)) + 6*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 2*(18^(2/3)*12^(2/3)*cos(2/3*a
rctan(sqrt(3) - 2))^2 - 3*18^(1/3)*12^(1/3)*x)*sin(2/3*arctan(sqrt(3) - 2))^2 + 36*x^2)*(18^(2/3)*12^(2/3)*sqr
t(3)*cos(2/3*arctan(sqrt(3) - 2))^2 - 18^(2/3)*12^(2/3)*sqrt(3)*sin(2/3*arctan(sqrt(3) - 2))^2 + 2*18^(2/3)*12
^(2/3)*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2))))/(3*cos(2/3*arctan(sqrt(3) - 2))^4 - 10*cos(
2/3*arctan(sqrt(3) - 2))^2*sin(2/3*arctan(sqrt(3) - 2))^2 + 3*sin(2/3*arctan(sqrt(3) - 2))^4)) - 4*(18^(2/3)*1
2^(1/6)*sqrt(3)*x^4*cos(2/3*arctan(sqrt(3) - 2)) + 18^(2/3)*12^(1/6)*x^4*sin(2/3*arctan(sqrt(3) - 2)))*arctan(
-1/432*(6*18^(2/3)*12^(2/3)*x - 216*cos(2/3*arctan(sqrt(3) - 2))^2 + 216*sin(2/3*arctan(sqrt(3) - 2))^2 - 18^(
2/3)*12^(2/3)*sqrt(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2))^4 + 18^(2/3)*12^(2/3)*sin(2/3*arctan(sqrt(3)
- 2))^4 - 12*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 2*(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3)
- 2))^2 + 6*18^(1/3)*12^(1/3)*x)*sin(2/3*arctan(sqrt(3) - 2))^2 + 36*x^2))/(cos(2/3*arctan(sqrt(3) - 2))*sin(
2/3*arctan(sqrt(3) - 2)))) - (18^(2/3)*12^(1/6)*sqrt(3)*x^4*sin(2/3*arctan(sqrt(3) - 2)) + 18^(2/3)*12^(1/6)*x
^4*cos(2/3*arctan(sqrt(3) - 2)))*log(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2))^4 + 18^(2/3)*12^(2/3)*sin(
2/3*arctan(sqrt(3) - 2))^4 + 12*18^(1/3)*12^(1/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3
) - 2)) + 6*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 2*(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) -
2))^2 - 3*18^(1/3)*12^(1/3)*x)*sin(2/3*arctan(sqrt(3) - 2))^2 + 36*x^2) + (18^(2/3)*12^(1/6)*sqrt(3)*x^4*sin(
2/3*arctan(sqrt(3) - 2)) - 18^(2/3)*12^(1/6)*x^4*cos(2/3*arctan(sqrt(3) - 2)))*log(18^(2/3)*12^(2/3)*cos(2/3*a
rctan(sqrt(3) - 2))^4 + 18^(2/3)*12^(2/3)*sin(2/3*arctan(sqrt(3) - 2))^4 - 12*18^(1/3)*12^(1/3)*x*cos(2/3*arct
an(sqrt(3) - 2))^2 + 2*(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2))^2 + 6*18^(1/3)*12^(1/3)*x)*sin(2/3*arct
an(sqrt(3) - 2))^2 + 36*x^2) - 27)/x^4
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Sympy [A] time = 0.215088, size = 37, normalized size = 0.09 \begin{align*} \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{4 x^{3} + 1}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**5/(x**6-x**3+1),x)
[Out]
RootSum(19683*_t**6 - 243*_t**3 + 1, Lambda(_t, _t*log(-6561*_t**5 + 54*_t**2 + x))) - (4*x**3 + 1)/(4*x**4)
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Giac [B] time = 1.15665, size = 1129, normalized size = 2.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^5/(x^6-x^3+1),x, algorithm="giac")
[Out]
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 - sqr
t(3)*sin(4/9*pi)^2 - 2*cos(4/9*pi)*sin(4/9*pi))*arctan(-((sqrt(3)*i + 1)*cos(4/9*pi) - 2*x)/((sqrt(3)*i + 1)*s
in(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(-((sqrt(3)*i + 1)*cos(2/9*pi) - 2*x)/((s
qrt(3)*i + 1)*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*pi)^4 + 10*cos(1/9*pi)^4*sin(1/9*pi) - 20*cos(1/9*pi)^2*sin(1/9*pi)^3 + 2*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(((sqrt(3)*i + 1)*cos(1/9*
pi) + 2*x)/((sqrt(3)*i + 1)*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(-(sqrt(3)*i*cos(4/9*p
i) + 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(3)*sin(2/9*pi)^5 + 2*cos(2/9*pi)^5 - 20*cos(2/9*pi)^3*sin(2/9*pi)^2 + 10*cos(2/9*pi)*sin(2/9*p
i)^4 + 2*sqrt(3)*cos(2/9*pi)*sin(2/9*pi) + cos(2/9*pi)^2 - sin(2/9*pi)^2)*log(-(sqrt(3)*i*cos(2/9*pi) + cos(2/
9*pi))*x + x^2 + 1) + 1/18*(10*sqrt(3)*cos(1/9*pi)^4*sin(1/9*pi) - 20*sqrt(3)*cos(1/9*pi)^2*sin(1/9*pi)^3 + 2*
sqrt(3)*sin(1/9*pi)^5 - 2*cos(1/9*pi)^5 + 20*cos(1/9*pi)^3*sin(1/9*pi)^2 - 10*cos(1/9*pi)*sin(1/9*pi)^4 - 2*sq
rt(3)*cos(1/9*pi)*sin(1/9*pi) + cos(1/9*pi)^2 - sin(1/9*pi)^2)*log((sqrt(3)*i*cos(1/9*pi) + cos(1/9*pi))*x + x
^2 + 1) - 1/4*(4*x^3 + 1)/x^4