\(\int \frac {1}{x^8 (1+x^4+x^8)} \, dx\) [81]

Optimal result

Integrand size = 14, antiderivative size = 123 \[ \int \frac {1}{x^8 \left (1+x^4+x^8\right )} \, dx=-\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{4} \arctan \left (\sqrt {3}-2 x\right )-\frac {\arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{4} \arctan \left (\sqrt {3}+2 x\right )+\frac {1}{4} \text {arctanh}\left (\frac {x}{1+x^2}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} x}{1+x^2}\right )}{4 \sqrt {3}} \] Output:

-1/7/x^7+1/3/x^3+1/12*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)+1/4*arctan(-3^(1 
/2)+2*x)-1/12*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)+1/4*arctan(3^(1/2)+2*x)+ 
1/4*arctanh(x/(x^2+1))-1/12*arctanh(3^(1/2)*x/(x^2+1))*3^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x^8 \left (1+x^4+x^8\right )} \, dx=-\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {\left (i+\sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) x\right )}{2 \sqrt {-6+6 i \sqrt {3}}}+\frac {\left (-i+\sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (1+i \sqrt {3}\right ) x\right )}{2 \sqrt {-6-6 i \sqrt {3}}}-\frac {\arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {\arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{8} \log \left (1-x+x^2\right )+\frac {1}{8} \log \left (1+x+x^2\right ) \] Input:

Integrate[1/(x^8*(1 + x^4 + x^8)),x]
 

Output:

-1/7*1/x^7 + 1/(3*x^3) + ((I + Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*x)/2])/(2* 
Sqrt[-6 + (6*I)*Sqrt[3]]) + ((-I + Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*x)/2]) 
/(2*Sqrt[-6 - (6*I)*Sqrt[3]]) - ArcTan[(-1 + 2*x)/Sqrt[3]]/(4*Sqrt[3]) - A 
rcTan[(1 + 2*x)/Sqrt[3]]/(4*Sqrt[3]) - Log[1 - x + x^2]/8 + Log[1 + x + x^ 
2]/8
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.41, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {1704, 27, 1828, 27, 1709, 1447, 1475, 1083, 217, 1478, 25, 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.

\(\displaystyle \int \frac {1}{x^8 \left (x^8+x^4+1\right )} \, dx\)

\(\Big \downarrow \) 1704

\(\displaystyle \frac {1}{7} \int -\frac {7 \left (x^4+1\right )}{x^4 \left (x^8+x^4+1\right )}dx-\frac {1}{7 x^7}\)

\(\Big \downarrow \) 27

\(\displaystyle -\int \frac {x^4+1}{x^4 \left (x^8+x^4+1\right )}dx-\frac {1}{7 x^7}\)

\(\Big \downarrow \) 1828

\(\displaystyle \frac {1}{3} \int \frac {3 x^4}{x^8+x^4+1}dx-\frac {1}{7 x^7}+\frac {1}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {x^4}{x^8+x^4+1}dx-\frac {1}{7 x^7}+\frac {1}{3 x^3}\)

\(\Big \downarrow \) 1709

\(\displaystyle \frac {1}{2} \int \frac {x^2}{x^4-x^2+1}dx-\frac {1}{2} \int \frac {x^2}{x^4+x^2+1}dx-\frac {1}{7 x^7}+\frac {1}{3 x^3}\)

\(\Big \downarrow \) 1447

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {x^2+1}{x^4-x^2+1}dx-\frac {1}{2} \int \frac {1-x^2}{x^4-x^2+1}dx\right )+\frac {1}{2} \left (\frac {1}{2} \int \frac {1-x^2}{x^4+x^2+1}dx-\frac {1}{2} \int \frac {x^2+1}{x^4+x^2+1}dx\right )-\frac {1}{7 x^7}+\frac {1}{3 x^3}\)

\(\Big \downarrow \) 1475

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^2-\sqrt {3} x+1}dx+\frac {1}{2} \int \frac {1}{x^2+\sqrt {3} x+1}dx\right )-\frac {1}{2} \int \frac {1-x^2}{x^4-x^2+1}dx\right )+\frac {1}{2} \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int \frac {1}{x^2+x+1}dx\right )+\frac {1}{2} \int \frac {1-x^2}{x^4+x^2+1}dx\right )-\frac {1}{7 x^7}+\frac {1}{3 x^3}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1-x^2}{x^4+x^2+1}dx+\frac {1}{2} \left (\int \frac {1}{-(2 x-1)^2-3}d(2 x-1)+\int \frac {1}{-(2 x+1)^2-3}d(2 x+1)\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (-\int \frac {1}{-\left (2 x-\sqrt {3}\right )^2-1}d\left (2 x-\sqrt {3}\right )-\int \frac {1}{-\left (2 x+\sqrt {3}\right )^2-1}d\left (2 x+\sqrt {3}\right )\right )-\frac {1}{2} \int \frac {1-x^2}{x^4-x^2+1}dx\right )-\frac {1}{7 x^7}+\frac {1}{3 x^3}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\arctan \left (2 x+\sqrt {3}\right )-\arctan \left (\sqrt {3}-2 x\right )\right )-\frac {1}{2} \int \frac {1-x^2}{x^4-x^2+1}dx\right )+\frac {1}{2} \left (\frac {1}{2} \int \frac {1-x^2}{x^4+x^2+1}dx+\frac {1}{2} \left (-\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}\right )\right )-\frac {1}{7 x^7}+\frac {1}{3 x^3}\)

\(\Big \downarrow \) 1478

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-\frac {1}{2} \int -\frac {1-2 x}{x^2-x+1}dx-\frac {1}{2} \int -\frac {2 x+1}{x^2+x+1}dx\right )+\frac {1}{2} \left (-\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {3}-2 x}{x^2-\sqrt {3} x+1}dx}{2 \sqrt {3}}+\frac {\int -\frac {2 x+\sqrt {3}}{x^2+\sqrt {3} x+1}dx}{2 \sqrt {3}}\right )+\frac {1}{2} \left (\arctan \left (2 x+\sqrt {3}\right )-\arctan \left (\sqrt {3}-2 x\right )\right )\right )-\frac {1}{7 x^7}+\frac {1}{3 x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx+\frac {1}{2} \int \frac {2 x+1}{x^2+x+1}dx\right )+\frac {1}{2} \left (-\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {3}-2 x}{x^2-\sqrt {3} x+1}dx}{2 \sqrt {3}}-\frac {\int \frac {2 x+\sqrt {3}}{x^2+\sqrt {3} x+1}dx}{2 \sqrt {3}}\right )+\frac {1}{2} \left (\arctan \left (2 x+\sqrt {3}\right )-\arctan \left (\sqrt {3}-2 x\right )\right )\right )-\frac {1}{7 x^7}+\frac {1}{3 x^3}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}\right )+\frac {1}{2} \left (\frac {1}{2} \log \left (x^2+x+1\right )-\frac {1}{2} \log \left (x^2-x+1\right )\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\arctan \left (2 x+\sqrt {3}\right )-\arctan \left (\sqrt {3}-2 x\right )\right )+\frac {1}{2} \left (\frac {\log \left (x^2-\sqrt {3} x+1\right )}{2 \sqrt {3}}-\frac {\log \left (x^2+\sqrt {3} x+1\right )}{2 \sqrt {3}}\right )\right )-\frac {1}{7 x^7}+\frac {1}{3 x^3}\)

Input:

Int[1/(x^8*(1 + x^4 + x^8)),x]
 

Output:

-1/7*1/x^7 + 1/(3*x^3) + ((-(ArcTan[(-1 + 2*x)/Sqrt[3]]/Sqrt[3]) - ArcTan[ 
(1 + 2*x)/Sqrt[3]]/Sqrt[3])/2 + (-1/2*Log[1 - x + x^2] + Log[1 + x + x^2]/ 
2)/2)/2 + ((-ArcTan[Sqrt[3] - 2*x] + ArcTan[Sqrt[3] + 2*x])/2 + (Log[1 - S 
qrt[3]*x + x^2]/(2*Sqrt[3]) - Log[1 + Sqrt[3]*x + x^2]/(2*Sqrt[3]))/2)/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

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]
 

rule 1447

Int[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
a/c, 2]}, Simp[1/2   Int[(q + x^2)/(a + b*x^2 + c*x^4), x], x] - Simp[1/2 
 Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && LtQ[b 
^2 - 4*a*c, 0] && PosQ[a*c]
 

rule 1475

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[2*(d/e) - b/c, 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^ 
2, x], x], x] + Simp[e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; F 
reeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && 
 (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2] 
, 0]))
 

rule 1478

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[-2*(d/e) - b/c, 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e 
 + q*x - x^2, x], x], x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - 
x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ 
[c*d^2 - a*e^2, 0] &&  !GtQ[b^2 - 4*a*c, 0]
 

rule 1704

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] - Simp[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 1709

Int[(x_)^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> W 
ith[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*r)   Int[x^( 
m - n/2)/(q - r*x^(n/2) + x^n), x], x] - Simp[1/(2*c*r)   Int[x^(m - n/2)/( 
q + r*x^(n/2) + x^n), x], x]]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && Ne 
Q[b^2 - 4*a*c, 0] && IGtQ[n/2, 0] && IGtQ[m, 0] && GeQ[m, n/2] && LtQ[m, 3* 
(n/2)] && NegQ[b^2 - 4*a*c]
 

rule 1828

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]
 

Maple [C] (verified)

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

Time = 0.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.76

method	result	size
risch	\(\frac {\frac {x^{4}}{3}-\frac {1}{7}}{x^{7}}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{12}-\frac {\ln \left (4 x^{2}-4 x +4\right )}{8}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (6 \textit {\_R}^{3}+\textit {\_R} +x \right )\right )}{4}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{12}+\frac {\ln \left (x^{2}+x +1\right )}{8}\)	\(94\)
default	\(-\frac {\ln \left (x^{2}-x +1\right )}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{12}+\frac {\ln \left (x^{2}+x +1\right )}{8}-\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}-\frac {\sqrt {3}\, \left (\frac {\ln \left (x^{2}+\sqrt {3}\, x +1\right )}{2}-\sqrt {3}\, \arctan \left (\sqrt {3}+2 x \right )\right )}{12}-\frac {\sqrt {3}\, \left (-\frac {\ln \left (x^{2}-\sqrt {3}\, x +1\right )}{2}-\sqrt {3}\, \arctan \left (-\sqrt {3}+2 x \right )\right )}{12}-\frac {1}{7 x^{7}}+\frac {1}{3 x^{3}}\)	\(131\)

Input:

int(1/x^8/(x^8+x^4+1),x,method=_RETURNVERBOSE)
 

Output:

(1/3*x^4-1/7)/x^7-1/12*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))-1/8*ln(4*x^2-4* 
x+4)+1/4*sum(_R*ln(6*_R^3+_R+x),_R=RootOf(9*_Z^4+3*_Z^2+1))-1/12*3^(1/2)*a 
rctan(2/3*3^(1/2)*(x+1/2))+1/8*ln(x^2+x+1)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^8 \left (1+x^4+x^8\right )} \, dx=-\frac {14 \, \sqrt {3} x^{7} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + 14 \, \sqrt {3} x^{7} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 7 \, \sqrt {3} x^{7} \log \left (x^{2} + \sqrt {3} x + 1\right ) - 7 \, \sqrt {3} x^{7} \log \left (x^{2} - \sqrt {3} x + 1\right ) - 42 \, x^{7} \arctan \left (2 \, x + \sqrt {3}\right ) + 42 \, x^{7} \arctan \left (-2 \, x + \sqrt {3}\right ) - 21 \, x^{7} \log \left (x^{2} + x + 1\right ) + 21 \, x^{7} \log \left (x^{2} - x + 1\right ) - 56 \, x^{4} + 24}{168 \, x^{7}} \] Input:

integrate(1/x^8/(x^8+x^4+1),x, algorithm="fricas")
 

Output:

-1/168*(14*sqrt(3)*x^7*arctan(1/3*sqrt(3)*(2*x + 1)) + 14*sqrt(3)*x^7*arct 
an(1/3*sqrt(3)*(2*x - 1)) + 7*sqrt(3)*x^7*log(x^2 + sqrt(3)*x + 1) - 7*sqr 
t(3)*x^7*log(x^2 - sqrt(3)*x + 1) - 42*x^7*arctan(2*x + sqrt(3)) + 42*x^7* 
arctan(-2*x + sqrt(3)) - 21*x^7*log(x^2 + x + 1) + 21*x^7*log(x^2 - x + 1) 
 - 56*x^4 + 24)/x^7
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.43 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.70 \[ \int \frac {1}{x^8 \left (1+x^4+x^8\right )} \, dx=\left (\frac {1}{8} - \frac {\sqrt {3} i}{24}\right ) \log {\left (x - \frac {1}{2} + \frac {\sqrt {3} i}{6} - 18432 \left (\frac {1}{8} - \frac {\sqrt {3} i}{24}\right )^{5} \right )} + \left (\frac {1}{8} + \frac {\sqrt {3} i}{24}\right ) \log {\left (x - \frac {1}{2} - 18432 \left (\frac {1}{8} + \frac {\sqrt {3} i}{24}\right )^{5} - \frac {\sqrt {3} i}{6} \right )} + \left (- \frac {1}{8} - \frac {\sqrt {3} i}{24}\right ) \log {\left (x + \frac {1}{2} + \frac {\sqrt {3} i}{6} - 18432 \left (- \frac {1}{8} - \frac {\sqrt {3} i}{24}\right )^{5} \right )} + \left (- \frac {1}{8} + \frac {\sqrt {3} i}{24}\right ) \log {\left (x + \frac {1}{2} - 18432 \left (- \frac {1}{8} + \frac {\sqrt {3} i}{24}\right )^{5} - \frac {\sqrt {3} i}{6} \right )} + \operatorname {RootSum} {\left (2304 t^{4} + 48 t^{2} + 1, \left ( t \mapsto t \log {\left (- 18432 t^{5} - 4 t + x \right )} \right )\right )} + \frac {7 x^{4} - 3}{21 x^{7}} \] Input:

integrate(1/x**8/(x**8+x**4+1),x)
 

Output:

(1/8 - sqrt(3)*I/24)*log(x - 1/2 + sqrt(3)*I/6 - 18432*(1/8 - sqrt(3)*I/24 
)**5) + (1/8 + sqrt(3)*I/24)*log(x - 1/2 - 18432*(1/8 + sqrt(3)*I/24)**5 - 
 sqrt(3)*I/6) + (-1/8 - sqrt(3)*I/24)*log(x + 1/2 + sqrt(3)*I/6 - 18432*(- 
1/8 - sqrt(3)*I/24)**5) + (-1/8 + sqrt(3)*I/24)*log(x + 1/2 - 18432*(-1/8 
+ sqrt(3)*I/24)**5 - sqrt(3)*I/6) + RootSum(2304*_t**4 + 48*_t**2 + 1, Lam 
bda(_t, _t*log(-18432*_t**5 - 4*_t + x))) + (7*x**4 - 3)/(21*x**7)
 

Maxima [F]

\[ \int \frac {1}{x^8 \left (1+x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} + x^{4} + 1\right )} x^{8}} \,d x } \] Input:

integrate(1/x^8/(x^8+x^4+1),x, algorithm="maxima")
 

Output:

-1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/12*sqrt(3)*arctan(1/3*sqrt 
(3)*(2*x - 1)) + 1/21*(7*x^4 - 3)/x^7 + 1/2*integrate(x^2/(x^4 - x^2 + 1), 
 x) + 1/8*log(x^2 + x + 1) - 1/8*log(x^2 - x + 1)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^8 \left (1+x^4+x^8\right )} \, dx=-\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{24} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) + \frac {1}{24} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) + \frac {7 \, x^{4} - 3}{21 \, x^{7}} + \frac {1}{4} \, \arctan \left (2 \, x + \sqrt {3}\right ) + \frac {1}{4} \, \arctan \left (2 \, x - \sqrt {3}\right ) + \frac {1}{8} \, \log \left (x^{2} + x + 1\right ) - \frac {1}{8} \, \log \left (x^{2} - x + 1\right ) \] Input:

integrate(1/x^8/(x^8+x^4+1),x, algorithm="giac")
 

Output:

-1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/12*sqrt(3)*arctan(1/3*sqrt 
(3)*(2*x - 1)) - 1/24*sqrt(3)*log(x^2 + sqrt(3)*x + 1) + 1/24*sqrt(3)*log( 
x^2 - sqrt(3)*x + 1) + 1/21*(7*x^4 - 3)/x^7 + 1/4*arctan(2*x + sqrt(3)) + 
1/4*arctan(2*x - sqrt(3)) + 1/8*log(x^2 + x + 1) - 1/8*log(x^2 - x + 1)
 

Mupad [B] (verification not implemented)

Time = 18.96 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^8 \left (1+x^4+x^8\right )} \, dx=\frac {\frac {x^4}{3}-\frac {1}{7}}{x^7}-\mathrm {atan}\left (\frac {2\,x}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\mathrm {atan}\left (\frac {x\,2{}\mathrm {i}}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{12}-\frac {1}{4}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {x\,2{}\mathrm {i}}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{12}+\frac {1}{4}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {2\,x}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \] Input:

int(1/(x^8*(x^4 + x^8 + 1)),x)
 

Output:

(x^4/3 - 1/7)/x^7 - atan((2*x)/(3^(1/2)*1i + 1))*((3^(1/2)*1i)/12 - 1/4) - 
 atan((x*2i)/(3^(1/2)*1i - 1))*(3^(1/2)/12 - 1i/4) - atan((x*2i)/(3^(1/2)* 
1i + 1))*(3^(1/2)/12 + 1i/4) - atan((2*x)/(3^(1/2)*1i - 1))*((3^(1/2)*1i)/ 
12 + 1/4)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^8 \left (1+x^4+x^8\right )} \, dx=\frac {-42 \mathit {atan} \left (\sqrt {3}-2 x \right ) x^{7}+42 \mathit {atan} \left (\sqrt {3}+2 x \right ) x^{7}-14 \sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right ) x^{7}-14 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) x^{7}+7 \sqrt {3}\, \mathrm {log}\left (-\sqrt {3}\, x +x^{2}+1\right ) x^{7}-7 \sqrt {3}\, \mathrm {log}\left (\sqrt {3}\, x +x^{2}+1\right ) x^{7}-21 \,\mathrm {log}\left (x^{2}-x +1\right ) x^{7}+21 \,\mathrm {log}\left (x^{2}+x +1\right ) x^{7}+56 x^{4}-24}{168 x^{7}} \] Input:

int(1/x^8/(x^8+x^4+1),x)
 

Output:

( - 42*atan(sqrt(3) - 2*x)*x**7 + 42*atan(sqrt(3) + 2*x)*x**7 - 14*sqrt(3) 
*atan((2*x - 1)/sqrt(3))*x**7 - 14*sqrt(3)*atan((2*x + 1)/sqrt(3))*x**7 + 
7*sqrt(3)*log( - sqrt(3)*x + x**2 + 1)*x**7 - 7*sqrt(3)*log(sqrt(3)*x + x* 
*2 + 1)*x**7 - 21*log(x**2 - x + 1)*x**7 + 21*log(x**2 + x + 1)*x**7 + 56* 
x**4 - 24)/(168*x**7)