\(\int \frac {1}{x^4 (1+3 x^4+x^8)} \, dx\) [384]

Optimal result

Integrand size = 16, antiderivative size = 466 \[ \int \frac {1}{x^4 \left (1+3 x^4+x^8\right )} \, dx=-\frac {1}{3 x^3}+\frac {\sqrt [4]{843+377 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{843+377 \sqrt {5}} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{843-377 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{843-377 \sqrt {5}} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{843+377 \sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{843+377 \sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{843-377 \sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{843-377 \sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}} \]

[Out]

-1/3/x^3+1/20*arctan(-1+2^(3/4)*x/(3+5^(1/2))^(1/4))*(843-377*5^(1/2))^(1/4)*2^(1/4)*5^(1/2)+1/20*arctan(1+2^(
3/4)*x/(3+5^(1/2))^(1/4))*(843-377*5^(1/2))^(1/4)*2^(1/4)*5^(1/2)-1/40*ln(2*x^2-2*2^(1/4)*x*(3+5^(1/2))^(1/4)+
5^(1/2)+1)*(843-377*5^(1/2))^(1/4)*2^(1/4)*5^(1/2)+1/40*ln(2*x^2+2*2^(1/4)*x*(3+5^(1/2))^(1/4)+5^(1/2)+1)*(843
-377*5^(1/2))^(1/4)*2^(1/4)*5^(1/2)-1/20*arctan(-1+2^(3/4)*x/(3-5^(1/2))^(1/4))*(843+377*5^(1/2))^(1/4)*2^(1/4
)*5^(1/2)-1/20*arctan(1+2^(3/4)*x/(3-5^(1/2))^(1/4))*(843+377*5^(1/2))^(1/4)*2^(1/4)*5^(1/2)+1/40*ln(2*x^2-2*2
^(1/4)*x*(3-5^(1/2))^(1/4)+5^(1/2)-1)*(843+377*5^(1/2))^(1/4)*2^(1/4)*5^(1/2)-1/40*ln(2*x^2+2*2^(1/4)*x*(3-5^(
1/2))^(1/4)+5^(1/2)-1)*(843+377*5^(1/2))^(1/4)*2^(1/4)*5^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1382, 1436, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{x^4 \left (1+3 x^4+x^8\right )} \, dx=\frac {\sqrt [4]{843+377 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{843+377 \sqrt {5}} \arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{843-377 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{843-377 \sqrt {5}} \arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {1}{3 x^3}+\frac {\sqrt [4]{843+377 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{843+377 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{843-377 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{843-377 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}} \]

[In]

Int[1/(x^4*(1 + 3*x^4 + x^8)),x]

[Out]

-1/3*1/x^3 + ((843 + 377*Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) - ((8
43 + 377*Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) - ((843 - 377*Sqrt[5]
)^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) + ((843 - 377*Sqrt[5])^(1/4)*ArcTan[1
+ (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) + ((843 + 377*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])
] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)*Sqrt[5]) - ((843 + 377*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqr
t[5])] + 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)*Sqrt[5]) - ((843 - 377*Sqrt[5])^(1/4)*Log[Sqrt[2*(3
+ Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)*Sqrt[5]) + ((843 - 377*Sqrt[5])^(1/4)*Log[Sqrt[
2*(3 + Sqrt[5])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)*Sqrt[5])

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 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

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 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

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

Rule 1382

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 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}& = -\frac {1}{3 x^3}+\frac {1}{3} \int \frac {-9-3 x^4}{1+3 x^4+x^8} \, dx \\ & = -\frac {1}{3 x^3}+\frac {1}{10} \left (-5+3 \sqrt {5}\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx-\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx \\ & = -\frac {1}{3 x^3}-\frac {\left (3+\sqrt {5}\right )^{3/2} \int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{8 \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{3/2} \int \frac {\sqrt {3-\sqrt {5}}+\sqrt {2} x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{8 \sqrt {5}}+\frac {\left (-5+3 \sqrt {5}\right ) \int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{20 \sqrt {3+\sqrt {5}}}+\frac {\left (-5+3 \sqrt {5}\right ) \int \frac {\sqrt {3+\sqrt {5}}+\sqrt {2} x^2}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{20 \sqrt {3+\sqrt {5}}} \\ & = -\frac {1}{3 x^3}-\frac {\sqrt [4]{843-377 \sqrt {5}} \int \frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}+2 x}{-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x-x^2} \, dx}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{843-377 \sqrt {5}} \int \frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x-x^2} \, dx}{4\ 2^{3/4} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{3/2} \int \frac {1}{\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+x^2} \, dx}{8 \sqrt {10}}-\frac {\left (3+\sqrt {5}\right )^{3/2} \int \frac {1}{\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+x^2} \, dx}{8 \sqrt {10}}+\frac {\left (3+\sqrt {5}\right )^{7/4} \int \frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}+2 x}{-\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x-x^2} \, dx}{16 \sqrt [4]{2} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{7/4} \int \frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{-\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x-x^2} \, dx}{16 \sqrt [4]{2} \sqrt {5}}+\frac {\left (-5+3 \sqrt {5}\right ) \int \frac {1}{\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+x^2} \, dx}{20 \sqrt {2 \left (3+\sqrt {5}\right )}}+\frac {\left (-5+3 \sqrt {5}\right ) \int \frac {1}{\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+x^2} \, dx}{20 \sqrt {2 \left (3+\sqrt {5}\right )}} \\ & = -\frac {1}{3 x^3}+\frac {\left (3+\sqrt {5}\right )^{7/4} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{16 \sqrt [4]{2} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{7/4} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{16 \sqrt [4]{2} \sqrt {5}}-\frac {\sqrt [4]{843-377 \sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{843-377 \sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{843-377 \sqrt {5}} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{843-377 \sqrt {5}} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{7/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{8 \sqrt [4]{2} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{7/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{8 \sqrt [4]{2} \sqrt {5}} \\ & = -\frac {1}{3 x^3}+\frac {\left (3+\sqrt {5}\right )^{7/4} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{8 \sqrt [4]{2} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{7/4} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{8 \sqrt [4]{2} \sqrt {5}}-\frac {\sqrt [4]{843-377 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{843-377 \sqrt {5}} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{7/4} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{16 \sqrt [4]{2} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{7/4} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{16 \sqrt [4]{2} \sqrt {5}}-\frac {\sqrt [4]{843-377 \sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{843-377 \sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x^4 \left (1+3 x^4+x^8\right )} \, dx=-\frac {1}{3 x^3}-\frac {1}{4} \text {RootSum}\left [1+3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {3 \log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]

[In]

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

[Out]

-1/3*1/x^3 - RootSum[1 + 3*#1^4 + #1^8 & , (3*Log[x - #1] + Log[x - #1]*#1^4)/(3*#1^3 + 2*#1^7) & ]/4

Maple [C] (verified)

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

Time = 0.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.09

method	result	size
risch	\(-\frac {1}{3 x^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (625 \textit {\_Z}^{8}+21075 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (175 \textit {\_R}^{5}+5778 \textit {\_R} +377 x \right )\right )}{4}\)	\(40\)
default	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{4}-3\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}-\frac {1}{3 x^{3}}\)	\(50\)

[In]

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

[Out]

-1/3/x^3+1/4*sum(_R*ln(175*_R^5+5778*_R+377*x),_R=RootOf(625*_Z^8+21075*_Z^4+1))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 451, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^4 \left (1+3 x^4+x^8\right )} \, dx=\frac {3 \, \sqrt {10} x^{3} \sqrt {\sqrt {2} \sqrt {377 \, \sqrt {5} - 843}} \log \left (\sqrt {10} \sqrt {\sqrt {2} \sqrt {377 \, \sqrt {5} - 843}} {\left (7 \, \sqrt {5} + 15\right )} + 20 \, x\right ) - 3 \, \sqrt {10} x^{3} \sqrt {\sqrt {2} \sqrt {377 \, \sqrt {5} - 843}} \log \left (-\sqrt {10} \sqrt {\sqrt {2} \sqrt {377 \, \sqrt {5} - 843}} {\left (7 \, \sqrt {5} + 15\right )} + 20 \, x\right ) + 3 \, \sqrt {10} x^{3} \sqrt {-\sqrt {2} \sqrt {377 \, \sqrt {5} - 843}} \log \left (\sqrt {10} \sqrt {-\sqrt {2} \sqrt {377 \, \sqrt {5} - 843}} {\left (7 \, \sqrt {5} + 15\right )} + 20 \, x\right ) - 3 \, \sqrt {10} x^{3} \sqrt {-\sqrt {2} \sqrt {377 \, \sqrt {5} - 843}} \log \left (-\sqrt {10} \sqrt {-\sqrt {2} \sqrt {377 \, \sqrt {5} - 843}} {\left (7 \, \sqrt {5} + 15\right )} + 20 \, x\right ) - 3 \, \sqrt {10} x^{3} \sqrt {\sqrt {2} \sqrt {-377 \, \sqrt {5} - 843}} \log \left (\sqrt {10} \sqrt {\sqrt {2} \sqrt {-377 \, \sqrt {5} - 843}} {\left (7 \, \sqrt {5} - 15\right )} + 20 \, x\right ) + 3 \, \sqrt {10} x^{3} \sqrt {\sqrt {2} \sqrt {-377 \, \sqrt {5} - 843}} \log \left (-\sqrt {10} \sqrt {\sqrt {2} \sqrt {-377 \, \sqrt {5} - 843}} {\left (7 \, \sqrt {5} - 15\right )} + 20 \, x\right ) - 3 \, \sqrt {10} x^{3} \sqrt {-\sqrt {2} \sqrt {-377 \, \sqrt {5} - 843}} \log \left (\sqrt {10} \sqrt {-\sqrt {2} \sqrt {-377 \, \sqrt {5} - 843}} {\left (7 \, \sqrt {5} - 15\right )} + 20 \, x\right ) + 3 \, \sqrt {10} x^{3} \sqrt {-\sqrt {2} \sqrt {-377 \, \sqrt {5} - 843}} \log \left (-\sqrt {10} \sqrt {-\sqrt {2} \sqrt {-377 \, \sqrt {5} - 843}} {\left (7 \, \sqrt {5} - 15\right )} + 20 \, x\right ) - 40}{120 \, x^{3}} \]

[In]

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

[Out]

1/120*(3*sqrt(10)*x^3*sqrt(sqrt(2)*sqrt(377*sqrt(5) - 843))*log(sqrt(10)*sqrt(sqrt(2)*sqrt(377*sqrt(5) - 843))
*(7*sqrt(5) + 15) + 20*x) - 3*sqrt(10)*x^3*sqrt(sqrt(2)*sqrt(377*sqrt(5) - 843))*log(-sqrt(10)*sqrt(sqrt(2)*sq
rt(377*sqrt(5) - 843))*(7*sqrt(5) + 15) + 20*x) + 3*sqrt(10)*x^3*sqrt(-sqrt(2)*sqrt(377*sqrt(5) - 843))*log(sq
rt(10)*sqrt(-sqrt(2)*sqrt(377*sqrt(5) - 843))*(7*sqrt(5) + 15) + 20*x) - 3*sqrt(10)*x^3*sqrt(-sqrt(2)*sqrt(377
*sqrt(5) - 843))*log(-sqrt(10)*sqrt(-sqrt(2)*sqrt(377*sqrt(5) - 843))*(7*sqrt(5) + 15) + 20*x) - 3*sqrt(10)*x^
3*sqrt(sqrt(2)*sqrt(-377*sqrt(5) - 843))*log(sqrt(10)*sqrt(sqrt(2)*sqrt(-377*sqrt(5) - 843))*(7*sqrt(5) - 15)
+ 20*x) + 3*sqrt(10)*x^3*sqrt(sqrt(2)*sqrt(-377*sqrt(5) - 843))*log(-sqrt(10)*sqrt(sqrt(2)*sqrt(-377*sqrt(5) -
843))*(7*sqrt(5) - 15) + 20*x) - 3*sqrt(10)*x^3*sqrt(-sqrt(2)*sqrt(-377*sqrt(5) - 843))*log(sqrt(10)*sqrt(-sq
rt(2)*sqrt(-377*sqrt(5) - 843))*(7*sqrt(5) - 15) + 20*x) + 3*sqrt(10)*x^3*sqrt(-sqrt(2)*sqrt(-377*sqrt(5) - 84
3))*log(-sqrt(10)*sqrt(-sqrt(2)*sqrt(-377*sqrt(5) - 843))*(7*sqrt(5) - 15) + 20*x) - 40)/x^3

Sympy [A] (verification not implemented)

Time = 0.96 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.07 \[ \int \frac {1}{x^4 \left (1+3 x^4+x^8\right )} \, dx=\operatorname {RootSum} {\left (40960000 t^{8} + 5395200 t^{4} + 1, \left ( t \mapsto t \log {\left (\frac {179200 t^{5}}{377} + \frac {23112 t}{377} + x \right )} \right )\right )} - \frac {1}{3 x^{3}} \]

[In]

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

[Out]

RootSum(40960000*_t**8 + 5395200*_t**4 + 1, Lambda(_t, _t*log(179200*_t**5/377 + 23112*_t/377 + x))) - 1/(3*x*
*3)

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.52 \[ \int \frac {1}{x^4 \left (1+3 x^4+x^8\right )} \, dx=-\frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {65 \, \sqrt {5} + 145} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {65 \, \sqrt {5} + 145} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {65 \, \sqrt {5} - 145} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {65 \, \sqrt {5} - 145} + \frac {1}{40} \, \sqrt {65 \, \sqrt {5} - 145} \log \left (93122500 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 93122500 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {65 \, \sqrt {5} - 145} \log \left (93122500 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 93122500 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {65 \, \sqrt {5} + 145} \log \left (53728900 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 53728900 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {65 \, \sqrt {5} + 145} \log \left (53728900 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 53728900 \, x^{2}\right ) - \frac {1}{3 \, x^{3}} \]

[In]

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

[Out]

-1/80*(pi + 4*arctan(x*sqrt(sqrt(5) + 1) + 1))*sqrt(65*sqrt(5) + 145) + 1/80*(pi + 4*arctan(-x*sqrt(sqrt(5) +
1) + 1))*sqrt(65*sqrt(5) + 145) + 1/80*(pi + 4*arctan(x*sqrt(sqrt(5) - 1) - 1))*sqrt(65*sqrt(5) - 145) - 1/80*
(pi + 4*arctan(-x*sqrt(sqrt(5) - 1) - 1))*sqrt(65*sqrt(5) - 145) + 1/40*sqrt(65*sqrt(5) - 145)*log(93122500*(x
+ sqrt(sqrt(5) + 1))^2 + 93122500*x^2) - 1/40*sqrt(65*sqrt(5) - 145)*log(93122500*(x - sqrt(sqrt(5) + 1))^2 +
93122500*x^2) - 1/40*sqrt(65*sqrt(5) + 145)*log(53728900*(x + sqrt(sqrt(5) - 1))^2 + 53728900*x^2) + 1/40*sqr
t(65*sqrt(5) + 145)*log(53728900*(x - sqrt(sqrt(5) - 1))^2 + 53728900*x^2) - 1/3/x^3

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^4 \left (1+3 x^4+x^8\right )} \, dx=\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {46371\,2^{3/4}\,x\,{\left (377\,\sqrt {5}-843\right )}^{1/4}}{2\,\left (3393\,\sqrt {2}\,\sqrt {377\,\sqrt {5}-843}-1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {377\,\sqrt {5}-843}\right )}-\frac {20735\,2^{3/4}\,\sqrt {5}\,x\,{\left (377\,\sqrt {5}-843\right )}^{1/4}}{2\,\left (3393\,\sqrt {2}\,\sqrt {377\,\sqrt {5}-843}-1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {377\,\sqrt {5}-843}\right )}\right )\,{\left (377\,\sqrt {5}-843\right )}^{1/4}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {46371\,2^{3/4}\,x\,{\left (-377\,\sqrt {5}-843\right )}^{1/4}}{2\,\left (3393\,\sqrt {2}\,\sqrt {-377\,\sqrt {5}-843}+1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {-377\,\sqrt {5}-843}\right )}+\frac {20735\,2^{3/4}\,\sqrt {5}\,x\,{\left (-377\,\sqrt {5}-843\right )}^{1/4}}{2\,\left (3393\,\sqrt {2}\,\sqrt {-377\,\sqrt {5}-843}+1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {-377\,\sqrt {5}-843}\right )}\right )\,{\left (-377\,\sqrt {5}-843\right )}^{1/4}}{20}-\frac {1}{3\,x^3}+\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (-377\,\sqrt {5}-843\right )}^{1/4}\,46371{}\mathrm {i}}{2\,\left (3393\,\sqrt {2}\,\sqrt {-377\,\sqrt {5}-843}+1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {-377\,\sqrt {5}-843}\right )}+\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (-377\,\sqrt {5}-843\right )}^{1/4}\,20735{}\mathrm {i}}{2\,\left (3393\,\sqrt {2}\,\sqrt {-377\,\sqrt {5}-843}+1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {-377\,\sqrt {5}-843}\right )}\right )\,{\left (-377\,\sqrt {5}-843\right )}^{1/4}\,1{}\mathrm {i}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (377\,\sqrt {5}-843\right )}^{1/4}\,46371{}\mathrm {i}}{2\,\left (3393\,\sqrt {2}\,\sqrt {377\,\sqrt {5}-843}-1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {377\,\sqrt {5}-843}\right )}-\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (377\,\sqrt {5}-843\right )}^{1/4}\,20735{}\mathrm {i}}{2\,\left (3393\,\sqrt {2}\,\sqrt {377\,\sqrt {5}-843}-1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {377\,\sqrt {5}-843}\right )}\right )\,{\left (377\,\sqrt {5}-843\right )}^{1/4}\,1{}\mathrm {i}}{20} \]

[In]

int(1/(x^4*(3*x^4 + x^8 + 1)),x)

[Out]

(2^(3/4)*5^(1/2)*atan((46371*2^(3/4)*x*(377*5^(1/2) - 843)^(1/4))/(2*(3393*2^(1/2)*(377*5^(1/2) - 843)^(1/2) -
1508*2^(1/2)*5^(1/2)*(377*5^(1/2) - 843)^(1/2))) - (20735*2^(3/4)*5^(1/2)*x*(377*5^(1/2) - 843)^(1/4))/(2*(33
93*2^(1/2)*(377*5^(1/2) - 843)^(1/2) - 1508*2^(1/2)*5^(1/2)*(377*5^(1/2) - 843)^(1/2))))*(377*5^(1/2) - 843)^(
1/4))/20 - (2^(3/4)*5^(1/2)*atan((46371*2^(3/4)*x*(- 377*5^(1/2) - 843)^(1/4))/(2*(3393*2^(1/2)*(- 377*5^(1/2)
- 843)^(1/2) + 1508*2^(1/2)*5^(1/2)*(- 377*5^(1/2) - 843)^(1/2))) + (20735*2^(3/4)*5^(1/2)*x*(- 377*5^(1/2) -
843)^(1/4))/(2*(3393*2^(1/2)*(- 377*5^(1/2) - 843)^(1/2) + 1508*2^(1/2)*5^(1/2)*(- 377*5^(1/2) - 843)^(1/2)))
)*(- 377*5^(1/2) - 843)^(1/4))/20 - 1/(3*x^3) + (2^(3/4)*5^(1/2)*atan((2^(3/4)*x*(- 377*5^(1/2) - 843)^(1/4)*4
6371i)/(2*(3393*2^(1/2)*(- 377*5^(1/2) - 843)^(1/2) + 1508*2^(1/2)*5^(1/2)*(- 377*5^(1/2) - 843)^(1/2))) + (2^
(3/4)*5^(1/2)*x*(- 377*5^(1/2) - 843)^(1/4)*20735i)/(2*(3393*2^(1/2)*(- 377*5^(1/2) - 843)^(1/2) + 1508*2^(1/2
)*5^(1/2)*(- 377*5^(1/2) - 843)^(1/2))))*(- 377*5^(1/2) - 843)^(1/4)*1i)/20 - (2^(3/4)*5^(1/2)*atan((2^(3/4)*x
*(377*5^(1/2) - 843)^(1/4)*46371i)/(2*(3393*2^(1/2)*(377*5^(1/2) - 843)^(1/2) - 1508*2^(1/2)*5^(1/2)*(377*5^(1
/2) - 843)^(1/2))) - (2^(3/4)*5^(1/2)*x*(377*5^(1/2) - 843)^(1/4)*20735i)/(2*(3393*2^(1/2)*(377*5^(1/2) - 843)
^(1/2) - 1508*2^(1/2)*5^(1/2)*(377*5^(1/2) - 843)^(1/2))))*(377*5^(1/2) - 843)^(1/4)*1i)/20