\(\int \frac {1}{x^8 (1-3 x^4+x^8)} \, dx\) [136]

Optimal result

Integrand size = 16, antiderivative size = 189 \[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=-\frac {1}{7 x^7}-\frac {1}{x^3}-\frac {\sqrt [4]{\frac {1}{2} \left (39603-17711 \sqrt {5}\right )} \arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (39603+17711 \sqrt {5}\right )} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (39603-17711 \sqrt {5}\right )} \text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (39603+17711 \sqrt {5}\right )} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}} \] Output:

-1/7/x^7-1/x^3-1/10*(39603/2-17711/2*5^(1/2))^(1/4)*arctan(2^(1/4)*(1/(3+5 
^(1/2)))^(1/4)*x)*5^(1/2)+1/10*(39603/2+17711/2*5^(1/2))^(1/4)*arctan((3/2 
+1/2*5^(1/2))^(1/4)*x)*5^(1/2)-1/10*(39603/2-17711/2*5^(1/2))^(1/4)*arctan 
h(2^(1/4)*(1/(3+5^(1/2)))^(1/4)*x)*5^(1/2)+1/10*(39603/2+17711/2*5^(1/2))^ 
(1/4)*arctanh((3/2+1/2*5^(1/2))^(1/4)*x)*5^(1/2)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=-\frac {1}{7 x^7}-\frac {1}{x^3}+\frac {\left (11+5 \sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (-1+\sqrt {5}\right )}}+\frac {\left (11-5 \sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\left (-11-5 \sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (-1+\sqrt {5}\right )}}-\frac {\left (-11+5 \sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}} \] Input:

Integrate[1/(x^8*(1 - 3*x^4 + x^8)),x]
 

Output:

-1/7*1/x^7 - x^(-3) + ((11 + 5*Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/ 
(2*Sqrt[10*(-1 + Sqrt[5])]) + ((11 - 5*Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5] 
)]*x])/(2*Sqrt[10*(1 + Sqrt[5])]) - ((-11 - 5*Sqrt[5])*ArcTanh[Sqrt[2/(-1 
+ Sqrt[5])]*x])/(2*Sqrt[10*(-1 + Sqrt[5])]) - ((-11 + 5*Sqrt[5])*ArcTanh[S 
qrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[10*(1 + Sqrt[5])])
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1704, 27, 1828, 27, 1752, 756, 216, 219}

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.

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

\(\Big \downarrow \) 1704

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1828

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1752

\(\displaystyle -\frac {1}{10} \left (15-7 \sqrt {5}\right ) \int \frac {1}{x^4+\frac {1}{2} \left (-3-\sqrt {5}\right )}dx-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \int \frac {1}{x^4+\frac {1}{2} \left (-3+\sqrt {5}\right )}dx-\frac {1}{7 x^7}-\frac {1}{x^3}\)

\(\Big \downarrow \) 756

\(\displaystyle -\frac {1}{10} \left (15+7 \sqrt {5}\right ) \left (-\frac {\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}dx}{\sqrt {3-\sqrt {5}}}-\frac {\int \frac {1}{\sqrt {2} x^2+\sqrt {3-\sqrt {5}}}dx}{\sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (15-7 \sqrt {5}\right ) \left (-\frac {\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}dx}{\sqrt {3+\sqrt {5}}}-\frac {\int \frac {1}{\sqrt {2} x^2+\sqrt {3+\sqrt {5}}}dx}{\sqrt {3+\sqrt {5}}}\right )-\frac {1}{7 x^7}-\frac {1}{x^3}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {1}{10} \left (15+7 \sqrt {5}\right ) \left (-\frac {\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}dx}{\sqrt {3-\sqrt {5}}}-\frac {\arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{\sqrt [4]{2} \left (3-\sqrt {5}\right )^{3/4}}\right )-\frac {1}{10} \left (15-7 \sqrt {5}\right ) \left (-\frac {\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}dx}{\sqrt {3+\sqrt {5}}}-\frac {\arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \left (3+\sqrt {5}\right )^{3/4}}\right )-\frac {1}{7 x^7}-\frac {1}{x^3}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {1}{10} \left (15-7 \sqrt {5}\right ) \left (-\frac {\arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \left (3+\sqrt {5}\right )^{3/4}}-\frac {\text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \left (3+\sqrt {5}\right )^{3/4}}\right )-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \left (-\frac {\arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{\sqrt [4]{2} \left (3-\sqrt {5}\right )^{3/4}}-\frac {\text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{\sqrt [4]{2} \left (3-\sqrt {5}\right )^{3/4}}\right )-\frac {1}{7 x^7}-\frac {1}{x^3}\)

Input:

Int[1/(x^8*(1 - 3*x^4 + x^8)),x]
 

Output:

-1/7*1/x^7 - x^(-3) - ((15 - 7*Sqrt[5])*(-(ArcTan[(2/(3 + Sqrt[5]))^(1/4)* 
x]/(2^(1/4)*(3 + Sqrt[5])^(3/4))) - ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^ 
(1/4)*(3 + Sqrt[5])^(3/4))))/10 - ((15 + 7*Sqrt[5])*(-(ArcTan[((3 + Sqrt[5 
])/2)^(1/4)*x]/(2^(1/4)*(3 - Sqrt[5])^(3/4))) - ArcTanh[((3 + Sqrt[5])/2)^ 
(1/4)*x]/(2^(1/4)*(3 - Sqrt[5])^(3/4))))/10
 

Defintions of rubi rules used

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 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 1752

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x 
_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) 
   Int[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   I 
nt[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])
 

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 = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.42

method	result	size
risch	\(\frac {-x^{4}-\frac {1}{7}}{x^{7}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}-995 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (90 \textit {\_R}^{3}-3571 \textit {\_R} +89 x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}+995 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-90 \textit {\_R}^{3}-3571 \textit {\_R} +89 x \right )\right )}{4}\)	\(79\)
default	\(-\frac {1}{7 x^{7}}-\frac {1}{x^{3}}-\frac {\left (-11+5 \sqrt {5}\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}+\frac {\sqrt {5}\, \left (11+5 \sqrt {5}\right ) \arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}+\frac {\sqrt {5}\, \left (11+5 \sqrt {5}\right ) \operatorname {arctanh}\left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}-\frac {\left (-11+5 \sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}\)	\(148\)

Input:

int(1/x^8/(x^8-3*x^4+1),x,method=_RETURNVERBOSE)
 

Output:

(-x^4-1/7)/x^7+1/4*sum(_R*ln(90*_R^3-3571*_R+89*x),_R=RootOf(25*_Z^4-995*_ 
Z^2-1))+1/4*sum(_R*ln(-90*_R^3-3571*_R+89*x),_R=RootOf(25*_Z^4+995*_Z^2-1) 
)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=-\frac {14 \, x^{7} \sqrt {\frac {89}{10} \, \sqrt {5} + \frac {199}{10}} \arctan \left (\frac {1}{2} \, {\left (11 \, \sqrt {5} x - 25 \, x\right )} \sqrt {\frac {89}{10} \, \sqrt {5} + \frac {199}{10}}\right ) + 14 \, x^{7} \sqrt {\frac {89}{10} \, \sqrt {5} - \frac {199}{10}} \arctan \left (\frac {1}{2} \, {\left (11 \, \sqrt {5} x + 25 \, x\right )} \sqrt {\frac {89}{10} \, \sqrt {5} - \frac {199}{10}}\right ) - 7 \, x^{7} \sqrt {\frac {89}{10} \, \sqrt {5} + \frac {199}{10}} \log \left ({\left (9 \, \sqrt {5} - 20\right )} \sqrt {\frac {89}{10} \, \sqrt {5} + \frac {199}{10}} + x\right ) + 7 \, x^{7} \sqrt {\frac {89}{10} \, \sqrt {5} + \frac {199}{10}} \log \left (-{\left (9 \, \sqrt {5} - 20\right )} \sqrt {\frac {89}{10} \, \sqrt {5} + \frac {199}{10}} + x\right ) + 7 \, x^{7} \sqrt {\frac {89}{10} \, \sqrt {5} - \frac {199}{10}} \log \left ({\left (9 \, \sqrt {5} + 20\right )} \sqrt {\frac {89}{10} \, \sqrt {5} - \frac {199}{10}} + x\right ) - 7 \, x^{7} \sqrt {\frac {89}{10} \, \sqrt {5} - \frac {199}{10}} \log \left (-{\left (9 \, \sqrt {5} + 20\right )} \sqrt {\frac {89}{10} \, \sqrt {5} - \frac {199}{10}} + x\right ) + 28 \, x^{4} + 4}{28 \, x^{7}} \] Input:

integrate(1/x^8/(x^8-3*x^4+1),x, algorithm="fricas")
 

Output:

-1/28*(14*x^7*sqrt(89/10*sqrt(5) + 199/10)*arctan(1/2*(11*sqrt(5)*x - 25*x 
)*sqrt(89/10*sqrt(5) + 199/10)) + 14*x^7*sqrt(89/10*sqrt(5) - 199/10)*arct 
an(1/2*(11*sqrt(5)*x + 25*x)*sqrt(89/10*sqrt(5) - 199/10)) - 7*x^7*sqrt(89 
/10*sqrt(5) + 199/10)*log((9*sqrt(5) - 20)*sqrt(89/10*sqrt(5) + 199/10) + 
x) + 7*x^7*sqrt(89/10*sqrt(5) + 199/10)*log(-(9*sqrt(5) - 20)*sqrt(89/10*s 
qrt(5) + 199/10) + x) + 7*x^7*sqrt(89/10*sqrt(5) - 199/10)*log((9*sqrt(5) 
+ 20)*sqrt(89/10*sqrt(5) - 199/10) + x) - 7*x^7*sqrt(89/10*sqrt(5) - 199/1 
0)*log(-(9*sqrt(5) + 20)*sqrt(89/10*sqrt(5) - 199/10) + x) + 28*x^4 + 4)/x 
^7
 

Sympy [A] (verification not implemented)

Time = 1.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.37 \[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=\operatorname {RootSum} {\left (6400 t^{4} - 15920 t^{2} - 1, \left ( t \mapsto t \log {\left (\frac {460800 t^{5}}{17711} - \frac {2842588 t}{17711} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 15920 t^{2} - 1, \left ( t \mapsto t \log {\left (\frac {460800 t^{5}}{17711} - \frac {2842588 t}{17711} + x \right )} \right )\right )} + \frac {- 7 x^{4} - 1}{7 x^{7}} \] Input:

integrate(1/x**8/(x**8-3*x**4+1),x)
 

Output:

RootSum(6400*_t**4 - 15920*_t**2 - 1, Lambda(_t, _t*log(460800*_t**5/17711 
 - 2842588*_t/17711 + x))) + RootSum(6400*_t**4 + 15920*_t**2 - 1, Lambda( 
_t, _t*log(460800*_t**5/17711 - 2842588*_t/17711 + x))) + (-7*x**4 - 1)/(7 
*x**7)
 

Maxima [F]

\[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - 3 \, x^{4} + 1\right )} x^{8}} \,d x } \] Input:

integrate(1/x^8/(x^8-3*x^4+1),x, algorithm="maxima")
 

Output:

-1/7*(7*x^4 + 1)/x^7 - 1/2*integrate((5*x^2 + 8)/(x^4 + x^2 - 1), x) + 1/2 
*integrate((5*x^2 - 8)/(x^4 - x^2 - 1), x)
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=-\frac {1}{20} \, \sqrt {890 \, \sqrt {5} - 1990} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{20} \, \sqrt {890 \, \sqrt {5} + 1990} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{40} \, \sqrt {890 \, \sqrt {5} - 1990} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {890 \, \sqrt {5} - 1990} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {890 \, \sqrt {5} + 1990} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{40} \, \sqrt {890 \, \sqrt {5} + 1990} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {7 \, x^{4} + 1}{7 \, x^{7}} \] Input:

integrate(1/x^8/(x^8-3*x^4+1),x, algorithm="giac")
 

Output:

-1/20*sqrt(890*sqrt(5) - 1990)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/20*sq 
rt(890*sqrt(5) + 1990)*arctan(x/sqrt(1/2*sqrt(5) - 1/2)) - 1/40*sqrt(890*s 
qrt(5) - 1990)*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(890*sqrt( 
5) - 1990)*log(abs(x - sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(890*sqrt(5) + 
 1990)*log(abs(x + sqrt(1/2*sqrt(5) - 1/2))) - 1/40*sqrt(890*sqrt(5) + 199 
0)*log(abs(x - sqrt(1/2*sqrt(5) - 1/2))) - 1/7*(7*x^4 + 1)/x^7
 

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.54 \[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=-\frac {x^4+\frac {1}{7}}{x^7}+\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {-89\,\sqrt {5}-199}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}[phone]\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {-89\,\sqrt {5}-199}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}[phone]\right )}\right )\,\sqrt {-89\,\sqrt {5}-199}\,1{}\mathrm {i}}{20}+\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {199-89\,\sqrt {5}}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}-165580139\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {199-89\,\sqrt {5}}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}-165580139\right )}\right )\,\sqrt {199-89\,\sqrt {5}}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}-199}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}-165580139\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}-199}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}-165580139\right )}\right )\,\sqrt {89\,\sqrt {5}-199}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}+199}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}[phone]\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}+199}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}[phone]\right )}\right )\,\sqrt {89\,\sqrt {5}+199}\,1{}\mathrm {i}}{20} \] Input:

int(1/(x^8*(x^8 - 3*x^4 + 1)),x)
                                                                                    
                                                                                    
 

Output:

(10^(1/2)*atan((10^(1/2)*x*(- 89*5^(1/2) - 199)^(1/2)*6677047i)/(2*(740496 
91*5^(1/2) + 165580139)) + (5^(1/2)*10^(1/2)*x*(- 89*5^(1/2) - 199)^(1/2)* 
14930373i)/(10*(74049691*5^(1/2) + 165580139)))*(- 89*5^(1/2) - 199)^(1/2) 
*1i)/20 - (x^4 + 1/7)/x^7 + (10^(1/2)*atan((10^(1/2)*x*(199 - 89*5^(1/2))^ 
(1/2)*6677047i)/(2*(74049691*5^(1/2) - 165580139)) - (5^(1/2)*10^(1/2)*x*( 
199 - 89*5^(1/2))^(1/2)*14930373i)/(10*(74049691*5^(1/2) - 165580139)))*(1 
99 - 89*5^(1/2))^(1/2)*1i)/20 - (10^(1/2)*atan((10^(1/2)*x*(89*5^(1/2) - 1 
99)^(1/2)*6677047i)/(2*(74049691*5^(1/2) - 165580139)) - (5^(1/2)*10^(1/2) 
*x*(89*5^(1/2) - 199)^(1/2)*14930373i)/(10*(74049691*5^(1/2) - 165580139)) 
)*(89*5^(1/2) - 199)^(1/2)*1i)/20 - (10^(1/2)*atan((10^(1/2)*x*(89*5^(1/2) 
 + 199)^(1/2)*6677047i)/(2*(74049691*5^(1/2) + 165580139)) + (5^(1/2)*10^( 
1/2)*x*(89*5^(1/2) + 199)^(1/2)*14930373i)/(10*(74049691*5^(1/2) + 1655801 
39)))*(89*5^(1/2) + 199)^(1/2)*1i)/20
 

Reduce [F]

\[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=\int \frac {1}{x^{16}-3 x^{12}+x^{8}}d x \] Input:

int(1/x^8/(x^8-3*x^4+1),x)
 

Output:

int(1/(x**16 - 3*x**12 + x**8),x)