\(\int \frac {1}{x^6 (1-3 x^4+x^8)} \, dx\) [135]

Optimal result

Integrand size = 16, antiderivative size = 173 \[ \int \frac {1}{x^6 \left (1-3 x^4+x^8\right )} \, dx=-\frac {1}{5 x^5}-\frac {3}{x}+\frac {\sqrt [4]{2889-1292 \sqrt {5}} \arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{2889+1292 \sqrt {5}} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{2889-1292 \sqrt {5}} \text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{2889+1292 \sqrt {5}} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}} \] Output:

-1/5/x^5-3/x+1/10*(2889-1292*5^(1/2))^(1/4)*arctan(2^(1/4)*(1/(3+5^(1/2))) 
^(1/4)*x)*5^(1/2)-1/10*(2889+1292*5^(1/2))^(1/4)*arctan((3/2+1/2*5^(1/2))^ 
(1/4)*x)*5^(1/2)-1/10*(2889-1292*5^(1/2))^(1/4)*arctanh(2^(1/4)*(1/(3+5^(1 
/2)))^(1/4)*x)*5^(1/2)+1/10*(2889+1292*5^(1/2))^(1/4)*arctanh((3/2+1/2*5^( 
1/2))^(1/4)*x)*5^(1/2)
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^6 \left (1-3 x^4+x^8\right )} \, dx=-\frac {1}{5 x^5}-\frac {3}{x}+\frac {\left (-7-3 \sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (-1+\sqrt {5}\right )}}+\frac {\left (7-3 \sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\left (-7-3 \sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (-1+\sqrt {5}\right )}}-\frac {\left (7-3 \sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}} \] Input:

Integrate[1/(x^6*(1 - 3*x^4 + x^8)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {1704, 27, 1828, 25, 1834, 27, 827, 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.

\(\displaystyle \int \frac {1}{x^6 \left (x^8-3 x^4+1\right )} \, dx\)

\(\Big \downarrow \) 1704

\(\displaystyle \frac {1}{5} \int \frac {5 \left (3-x^4\right )}{x^2 \left (x^8-3 x^4+1\right )}dx-\frac {1}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {3-x^4}{x^2 \left (x^8-3 x^4+1\right )}dx-\frac {1}{5 x^5}\)

\(\Big \downarrow \) 1828

\(\displaystyle -\int -\frac {x^2 \left (8-3 x^4\right )}{x^8-3 x^4+1}dx-\frac {1}{5 x^5}-\frac {3}{x}\)

\(\Big \downarrow \) 25

\(\displaystyle \int \frac {x^2 \left (8-3 x^4\right )}{x^8-3 x^4+1}dx-\frac {1}{5 x^5}-\frac {3}{x}\)

\(\Big \downarrow \) 1834

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 219

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

Input:

Int[1/(x^6*(1 - 3*x^4 + x^8)),x]
 

Output:

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

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 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 827

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

rule 1834

Int[(((f_.)*(x_))^(m_.)*((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[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 
 - (2*c*d - b*e)/(2*q))   Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ 
[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n 
, 0]
 

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 = 0.46

method	result	size
risch	\(\frac {-3 x^{4}-\frac {1}{5}}{x^{5}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}-380 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-55 \textit {\_R}^{3}+843 \textit {\_R} +34 x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}+380 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-55 \textit {\_R}^{3}-843 \textit {\_R} +34 x \right )\right )}{4}\)	\(79\)
default	\(-\frac {1}{5 x^{5}}-\frac {3}{x}+\frac {\left (-7+3 \sqrt {5}\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}-\frac {\sqrt {5}\, \left (7+3 \sqrt {5}\right ) \arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}+\frac {\sqrt {5}\, \left (7+3 \sqrt {5}\right ) \operatorname {arctanh}\left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}-\frac {\left (-7+3 \sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}\)	\(148\)

Input:

int(1/x^6/(x^8-3*x^4+1),x,method=_RETURNVERBOSE)
 

Output:

(-3*x^4-1/5)/x^5+1/4*sum(_R*ln(-55*_R^3+843*_R+34*x),_R=RootOf(25*_Z^4-380 
*_Z^2-1))+1/4*sum(_R*ln(-55*_R^3-843*_R+34*x),_R=RootOf(25*_Z^4+380*_Z^2-1 
))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.33 \[ \int \frac {1}{x^6 \left (1-3 x^4+x^8\right )} \, dx=-\frac {10 \, x^{5} \sqrt {\frac {17}{5} \, \sqrt {5} + \frac {38}{5}} \arctan \left (\frac {1}{2} \, {\left (7 \, \sqrt {5} x - 15 \, x\right )} \sqrt {\frac {17}{5} \, \sqrt {5} + \frac {38}{5}}\right ) - 10 \, x^{5} \sqrt {\frac {17}{5} \, \sqrt {5} - \frac {38}{5}} \arctan \left (\frac {1}{2} \, {\left (7 \, \sqrt {5} x + 15 \, x\right )} \sqrt {\frac {17}{5} \, \sqrt {5} - \frac {38}{5}}\right ) + 5 \, x^{5} \sqrt {\frac {17}{5} \, \sqrt {5} + \frac {38}{5}} \log \left ({\left (11 \, \sqrt {5} - 25\right )} \sqrt {\frac {17}{5} \, \sqrt {5} + \frac {38}{5}} + 2 \, x\right ) - 5 \, x^{5} \sqrt {\frac {17}{5} \, \sqrt {5} + \frac {38}{5}} \log \left (-{\left (11 \, \sqrt {5} - 25\right )} \sqrt {\frac {17}{5} \, \sqrt {5} + \frac {38}{5}} + 2 \, x\right ) + 5 \, x^{5} \sqrt {\frac {17}{5} \, \sqrt {5} - \frac {38}{5}} \log \left ({\left (11 \, \sqrt {5} + 25\right )} \sqrt {\frac {17}{5} \, \sqrt {5} - \frac {38}{5}} + 2 \, x\right ) - 5 \, x^{5} \sqrt {\frac {17}{5} \, \sqrt {5} - \frac {38}{5}} \log \left (-{\left (11 \, \sqrt {5} + 25\right )} \sqrt {\frac {17}{5} \, \sqrt {5} - \frac {38}{5}} + 2 \, x\right ) + 60 \, x^{4} + 4}{20 \, x^{5}} \] Input:

integrate(1/x^6/(x^8-3*x^4+1),x, algorithm="fricas")
 

Output:

-1/20*(10*x^5*sqrt(17/5*sqrt(5) + 38/5)*arctan(1/2*(7*sqrt(5)*x - 15*x)*sq 
rt(17/5*sqrt(5) + 38/5)) - 10*x^5*sqrt(17/5*sqrt(5) - 38/5)*arctan(1/2*(7* 
sqrt(5)*x + 15*x)*sqrt(17/5*sqrt(5) - 38/5)) + 5*x^5*sqrt(17/5*sqrt(5) + 3 
8/5)*log((11*sqrt(5) - 25)*sqrt(17/5*sqrt(5) + 38/5) + 2*x) - 5*x^5*sqrt(1 
7/5*sqrt(5) + 38/5)*log(-(11*sqrt(5) - 25)*sqrt(17/5*sqrt(5) + 38/5) + 2*x 
) + 5*x^5*sqrt(17/5*sqrt(5) - 38/5)*log((11*sqrt(5) + 25)*sqrt(17/5*sqrt(5 
) - 38/5) + 2*x) - 5*x^5*sqrt(17/5*sqrt(5) - 38/5)*log(-(11*sqrt(5) + 25)* 
sqrt(17/5*sqrt(5) - 38/5) + 2*x) + 60*x^4 + 4)/x^5
 

Sympy [A] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.42 \[ \int \frac {1}{x^6 \left (1-3 x^4+x^8\right )} \, dx=\operatorname {RootSum} {\left (6400 t^{4} - 6080 t^{2} - 1, \left ( t \mapsto t \log {\left (\frac {215808000 t^{7}}{323} - \frac {194833880 t^{3}}{323} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 6080 t^{2} - 1, \left ( t \mapsto t \log {\left (\frac {215808000 t^{7}}{323} - \frac {194833880 t^{3}}{323} + x \right )} \right )\right )} + \frac {- 15 x^{4} - 1}{5 x^{5}} \] Input:

integrate(1/x**6/(x**8-3*x**4+1),x)
 

Output:

RootSum(6400*_t**4 - 6080*_t**2 - 1, Lambda(_t, _t*log(215808000*_t**7/323 
 - 194833880*_t**3/323 + x))) + RootSum(6400*_t**4 + 6080*_t**2 - 1, Lambd 
a(_t, _t*log(215808000*_t**7/323 - 194833880*_t**3/323 + x))) + (-15*x**4 
- 1)/(5*x**5)
 

Maxima [F]

\[ \int \frac {1}{x^6 \left (1-3 x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - 3 \, x^{4} + 1\right )} x^{6}} \,d x } \] Input:

integrate(1/x^6/(x^8-3*x^4+1),x, algorithm="maxima")
 

Output:

-1/5*(15*x^4 + 1)/x^5 - 1/2*integrate((3*x^2 + 5)/(x^4 + x^2 - 1), x) - 1/ 
2*integrate((3*x^2 - 5)/(x^4 - x^2 - 1), x)
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^6 \left (1-3 x^4+x^8\right )} \, dx=\frac {1}{10} \, \sqrt {85 \, \sqrt {5} - 190} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) - \frac {1}{10} \, \sqrt {85 \, \sqrt {5} + 190} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{20} \, \sqrt {85 \, \sqrt {5} - 190} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {85 \, \sqrt {5} - 190} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {85 \, \sqrt {5} + 190} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{20} \, \sqrt {85 \, \sqrt {5} + 190} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {15 \, x^{4} + 1}{5 \, x^{5}} \] Input:

integrate(1/x^6/(x^8-3*x^4+1),x, algorithm="giac")
 

Output:

1/10*sqrt(85*sqrt(5) - 190)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) - 1/10*sqrt( 
85*sqrt(5) + 190)*arctan(x/sqrt(1/2*sqrt(5) - 1/2)) - 1/20*sqrt(85*sqrt(5) 
 - 190)*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/20*sqrt(85*sqrt(5) - 190 
)*log(abs(x - sqrt(1/2*sqrt(5) + 1/2))) + 1/20*sqrt(85*sqrt(5) + 190)*log( 
abs(x + sqrt(1/2*sqrt(5) - 1/2))) - 1/20*sqrt(85*sqrt(5) + 190)*log(abs(x 
- sqrt(1/2*sqrt(5) - 1/2))) - 1/5*(15*x^4 + 1)/x^5
 

Mupad [B] (verification not implemented)

Time = 19.36 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.49 \[ \int \frac {1}{x^6 \left (1-3 x^4+x^8\right )} \, dx=-\frac {3\,x^4+\frac {1}{5}}{x^5}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-85\,\sqrt {5}-190}\,372096{}\mathrm {i}}{2550408\,\sqrt {5}+5702888}+\frac {\sqrt {5}\,x\,\sqrt {-85\,\sqrt {5}-190}\,832048{}\mathrm {i}}{5\,\left (2550408\,\sqrt {5}+5702888\right )}\right )\,\sqrt {-85\,\sqrt {5}-190}\,1{}\mathrm {i}}{10}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {190-85\,\sqrt {5}}\,372096{}\mathrm {i}}{2550408\,\sqrt {5}-5702888}-\frac {\sqrt {5}\,x\,\sqrt {190-85\,\sqrt {5}}\,832048{}\mathrm {i}}{5\,\left (2550408\,\sqrt {5}-5702888\right )}\right )\,\sqrt {190-85\,\sqrt {5}}\,1{}\mathrm {i}}{10}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {85\,\sqrt {5}-190}\,372096{}\mathrm {i}}{2550408\,\sqrt {5}-5702888}-\frac {\sqrt {5}\,x\,\sqrt {85\,\sqrt {5}-190}\,832048{}\mathrm {i}}{5\,\left (2550408\,\sqrt {5}-5702888\right )}\right )\,\sqrt {85\,\sqrt {5}-190}\,1{}\mathrm {i}}{10}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {85\,\sqrt {5}+190}\,372096{}\mathrm {i}}{2550408\,\sqrt {5}+5702888}+\frac {\sqrt {5}\,x\,\sqrt {85\,\sqrt {5}+190}\,832048{}\mathrm {i}}{5\,\left (2550408\,\sqrt {5}+5702888\right )}\right )\,\sqrt {85\,\sqrt {5}+190}\,1{}\mathrm {i}}{10} \] Input:

int(1/(x^6*(x^8 - 3*x^4 + 1)),x)
 

Output:

(atan((x*(190 - 85*5^(1/2))^(1/2)*372096i)/(2550408*5^(1/2) - 5702888) - ( 
5^(1/2)*x*(190 - 85*5^(1/2))^(1/2)*832048i)/(5*(2550408*5^(1/2) - 5702888) 
))*(190 - 85*5^(1/2))^(1/2)*1i)/10 - (atan((x*(- 85*5^(1/2) - 190)^(1/2)*3 
72096i)/(2550408*5^(1/2) + 5702888) + (5^(1/2)*x*(- 85*5^(1/2) - 190)^(1/2 
)*832048i)/(5*(2550408*5^(1/2) + 5702888)))*(- 85*5^(1/2) - 190)^(1/2)*1i) 
/10 + (atan((x*(85*5^(1/2) - 190)^(1/2)*372096i)/(2550408*5^(1/2) - 570288 
8) - (5^(1/2)*x*(85*5^(1/2) - 190)^(1/2)*832048i)/(5*(2550408*5^(1/2) - 57 
02888)))*(85*5^(1/2) - 190)^(1/2)*1i)/10 - (atan((x*(85*5^(1/2) + 190)^(1/ 
2)*372096i)/(2550408*5^(1/2) + 5702888) + (5^(1/2)*x*(85*5^(1/2) + 190)^(1 
/2)*832048i)/(5*(2550408*5^(1/2) + 5702888)))*(85*5^(1/2) + 190)^(1/2)*1i) 
/10 - (3*x^4 + 1/5)/x^5
 

Reduce [F]

\[ \int \frac {1}{x^6 \left (1-3 x^4+x^8\right )} \, dx=\int \frac {1}{x^{14}-3 x^{10}+x^{6}}d x \] Input:

int(1/x^6/(x^8-3*x^4+1),x)
 

Output:

int(1/(x**14 - 3*x**10 + x**6),x)