\(\int \frac {x^8}{1+3 x^4+x^8} \, dx\) [378]

Optimal result

Integrand size = 16, antiderivative size = 460 \[ \int \frac {x^8}{1+3 x^4+x^8} \, dx=x-\frac {\sqrt [4]{123-55 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{123-55 \sqrt {5}} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{123+55 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{123+55 \sqrt {5}} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{123-55 \sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{123-55 \sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{123+55 \sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{123+55 \sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}} \]

[Out]

x+1/20*arctan(-1+2^(3/4)*x/(3-5^(1/2))^(1/4))*(123-55*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))*(123-55*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
)*(123-55*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)*(123-55*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))*(123+55*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))*(123+55*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)*(123+55*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)*(123+55*5^(1/2))^(1/4)*2^(1/4)*5^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 440, normalized size of antiderivative = 0.96, number of steps used = 20, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1381, 1436, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^8}{1+3 x^4+x^8} \, dx=-\frac {\sqrt [4]{984-440 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{123+55 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{123+55 \sqrt {5}} \arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{984-440 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{8 \sqrt {10}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{8 \sqrt {10}}+\frac {\sqrt [4]{123+55 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{123+55 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}+x \]

[In]

Int[x^8/(1 + 3*x^4 + x^8),x]

[Out]

x - ((984 - 440*Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(4*Sqrt[10]) + ((984 - 440*Sqrt[5]
)^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(4*Sqrt[10]) + ((123 + 55*Sqrt[5])^(1/4)*ArcTan[1 - (2^(3
/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) - ((123 + 55*Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[
5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) - ((984 - 440*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(
1/4)*x + 2*x^2])/(8*Sqrt[10]) + ((984 - 440*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] + 2*(2*(3 - Sqrt[5]))^(1/
4)*x + 2*x^2])/(8*Sqrt[10]) + ((123 + 55*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]) - ((123 + 55*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 1381

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[d^(2*n - 1)*(d*x)^
(m - 2*n + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + 2*n*p + 1))), x] - Dist[d^(2*n)/(c*(m + 2*n*p + 1)), In
t[(d*x)^(m - 2*n)*Simp[a*(m - 2*n + 1) + b*(m + n*(p - 1) + 1)*x^n, x]*(a + b*x^n + c*x^(2*n))^p, x], x] /; Fr
eeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n
*p + 1, 0] && 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}& = x-\int \frac {1+3 x^4}{1+3 x^4+x^8} \, dx \\ & = x-\frac {1}{10} \left (15-7 \sqrt {5}\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx \\ & = x+\frac {1}{2} \sqrt {\frac {1}{10} \left (9-4 \sqrt {5}\right )} \int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx+\frac {1}{2} \sqrt {\frac {1}{10} \left (9-4 \sqrt {5}\right )} \int \frac {\sqrt {3-\sqrt {5}}+\sqrt {2} x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx-\frac {\left (15+7 \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 (15+7 \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}}} \\ & = x+\frac {1}{4} \sqrt {\frac {1}{5} \left (9-4 \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+\frac {1}{4} \sqrt {\frac {1}{5} \left (9-4 \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-\frac {\left (\sqrt {\frac {1}{5} \left (9-4 \sqrt {5}\right )} \sqrt [4]{3+\sqrt {5}}\right ) \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}}-\frac {\left (\sqrt {\frac {1}{5} \left (9-4 \sqrt {5}\right )} \sqrt [4]{3+\sqrt {5}}\right ) \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}}-\frac {1}{4} \sqrt {\frac {1}{5} \left (9+4 \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-\frac {1}{4} \sqrt {\frac {1}{5} \left (9+4 \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+\frac {\sqrt [4]{123+55 \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]{123+55 \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}} \\ & = x-\frac {1}{8} \sqrt [4]{\frac {246}{25}-\frac {22}{\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 )+\frac {1}{8} \sqrt [4]{\frac {246}{25}-\frac {22}{\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 )+\frac {\sqrt [4]{123+55 \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]{123+55 \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]{123-55 \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]{123-55 \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]{123+55 \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]{123+55 \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}} \\ & = x-\frac {\sqrt [4]{123-55 \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]{123-55 \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]{123+55 \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]{123+55 \sqrt {5}} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {1}{8} \sqrt [4]{\frac {246}{25}-\frac {22}{\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 )+\frac {1}{8} \sqrt [4]{\frac {246}{25}-\frac {22}{\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 )+\frac {\sqrt [4]{123+55 \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]{123+55 \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 = 58, normalized size of antiderivative = 0.13 \[ \int \frac {x^8}{1+3 x^4+x^8} \, dx=x-\frac {1}{4} \text {RootSum}\left [1+3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})+3 \log (x-\text {$\#$1}) \text {$\#$1}^4}{3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]

[In]

Integrate[x^8/(1 + 3*x^4 + x^8),x]

[Out]

x - RootSum[1 + 3*#1^4 + #1^8 & , (Log[x - #1] + 3*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 = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.10

method	result	size
default	\(x +\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-3 \textit {\_R}^{4}-1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}\)	\(46\)
risch	\(x +\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-3 \textit {\_R}^{4}-1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}\)	\(46\)

[In]

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

[Out]

x+1/4*sum((-3*_R^4-1)/(2*_R^7+3*_R^3)*ln(x-_R),_R=RootOf(_Z^8+3*_Z^4+1))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.92 \[ \int \frac {x^8}{1+3 x^4+x^8} \, dx=\frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {2} \sqrt {55 \, \sqrt {5} - 123}} \log \left (\sqrt {10} \sqrt {\sqrt {2} \sqrt {55 \, \sqrt {5} - 123}} {\left (3 \, \sqrt {5} + 5\right )} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {2} \sqrt {55 \, \sqrt {5} - 123}} \log \left (-\sqrt {10} \sqrt {\sqrt {2} \sqrt {55 \, \sqrt {5} - 123}} {\left (3 \, \sqrt {5} + 5\right )} + 20 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {2} \sqrt {55 \, \sqrt {5} - 123}} \log \left (\sqrt {10} \sqrt {-\sqrt {2} \sqrt {55 \, \sqrt {5} - 123}} {\left (3 \, \sqrt {5} + 5\right )} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {2} \sqrt {55 \, \sqrt {5} - 123}} \log \left (-\sqrt {10} \sqrt {-\sqrt {2} \sqrt {55 \, \sqrt {5} - 123}} {\left (3 \, \sqrt {5} + 5\right )} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {2} \sqrt {-55 \, \sqrt {5} - 123}} \log \left (\sqrt {10} \sqrt {\sqrt {2} \sqrt {-55 \, \sqrt {5} - 123}} {\left (3 \, \sqrt {5} - 5\right )} + 20 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {2} \sqrt {-55 \, \sqrt {5} - 123}} \log \left (-\sqrt {10} \sqrt {\sqrt {2} \sqrt {-55 \, \sqrt {5} - 123}} {\left (3 \, \sqrt {5} - 5\right )} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {2} \sqrt {-55 \, \sqrt {5} - 123}} \log \left (\sqrt {10} \sqrt {-\sqrt {2} \sqrt {-55 \, \sqrt {5} - 123}} {\left (3 \, \sqrt {5} - 5\right )} + 20 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {2} \sqrt {-55 \, \sqrt {5} - 123}} \log \left (-\sqrt {10} \sqrt {-\sqrt {2} \sqrt {-55 \, \sqrt {5} - 123}} {\left (3 \, \sqrt {5} - 5\right )} + 20 \, x\right ) + x \]

[In]

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

[Out]

1/40*sqrt(10)*sqrt(sqrt(2)*sqrt(55*sqrt(5) - 123))*log(sqrt(10)*sqrt(sqrt(2)*sqrt(55*sqrt(5) - 123))*(3*sqrt(5
) + 5) + 20*x) - 1/40*sqrt(10)*sqrt(sqrt(2)*sqrt(55*sqrt(5) - 123))*log(-sqrt(10)*sqrt(sqrt(2)*sqrt(55*sqrt(5)
- 123))*(3*sqrt(5) + 5) + 20*x) + 1/40*sqrt(10)*sqrt(-sqrt(2)*sqrt(55*sqrt(5) - 123))*log(sqrt(10)*sqrt(-sqrt
(2)*sqrt(55*sqrt(5) - 123))*(3*sqrt(5) + 5) + 20*x) - 1/40*sqrt(10)*sqrt(-sqrt(2)*sqrt(55*sqrt(5) - 123))*log(
-sqrt(10)*sqrt(-sqrt(2)*sqrt(55*sqrt(5) - 123))*(3*sqrt(5) + 5) + 20*x) - 1/40*sqrt(10)*sqrt(sqrt(2)*sqrt(-55*
sqrt(5) - 123))*log(sqrt(10)*sqrt(sqrt(2)*sqrt(-55*sqrt(5) - 123))*(3*sqrt(5) - 5) + 20*x) + 1/40*sqrt(10)*sqr
t(sqrt(2)*sqrt(-55*sqrt(5) - 123))*log(-sqrt(10)*sqrt(sqrt(2)*sqrt(-55*sqrt(5) - 123))*(3*sqrt(5) - 5) + 20*x)
- 1/40*sqrt(10)*sqrt(-sqrt(2)*sqrt(-55*sqrt(5) - 123))*log(sqrt(10)*sqrt(-sqrt(2)*sqrt(-55*sqrt(5) - 123))*(3
*sqrt(5) - 5) + 20*x) + 1/40*sqrt(10)*sqrt(-sqrt(2)*sqrt(-55*sqrt(5) - 123))*log(-sqrt(10)*sqrt(-sqrt(2)*sqrt(
-55*sqrt(5) - 123))*(3*sqrt(5) - 5) + 20*x) + x

Sympy [A] (verification not implemented)

Time = 0.93 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.06 \[ \int \frac {x^8}{1+3 x^4+x^8} \, dx=x + \operatorname {RootSum} {\left (40960000 t^{8} + 787200 t^{4} + 1, \left ( t \mapsto t \log {\left (\frac {15360 t^{5}}{11} + \frac {1288 t}{55} + x \right )} \right )\right )} \]

[In]

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

[Out]

x + RootSum(40960000*_t**8 + 787200*_t**4 + 1, Lambda(_t, _t*log(15360*_t**5/11 + 1288*_t/55 + x)))

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.52 \[ \int \frac {x^8}{1+3 x^4+x^8} \, dx=-\frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {25 \, \sqrt {5} + 55} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {25 \, \sqrt {5} + 55} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {25 \, \sqrt {5} - 55} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {25 \, \sqrt {5} - 55} - \frac {1}{40} \, \sqrt {25 \, \sqrt {5} + 55} \log \left (722500 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 722500 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {25 \, \sqrt {5} + 55} \log \left (722500 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 722500 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {25 \, \sqrt {5} - 55} \log \left (2992900 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 2992900 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {25 \, \sqrt {5} - 55} \log \left (2992900 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 2992900 \, x^{2}\right ) + x \]

[In]

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

[Out]

-1/80*(pi + 4*arctan(x*sqrt(sqrt(5) - 1) + 1))*sqrt(25*sqrt(5) + 55) + 1/80*(pi + 4*arctan(-x*sqrt(sqrt(5) - 1
) + 1))*sqrt(25*sqrt(5) + 55) + 1/80*(pi + 4*arctan(x*sqrt(sqrt(5) + 1) - 1))*sqrt(25*sqrt(5) - 55) - 1/80*(pi
+ 4*arctan(-x*sqrt(sqrt(5) + 1) - 1))*sqrt(25*sqrt(5) - 55) - 1/40*sqrt(25*sqrt(5) + 55)*log(722500*(x + sqrt
(sqrt(5) + 1))^2 + 722500*x^2) + 1/40*sqrt(25*sqrt(5) + 55)*log(722500*(x - sqrt(sqrt(5) + 1))^2 + 722500*x^2)
+ 1/40*sqrt(25*sqrt(5) - 55)*log(2992900*(x + sqrt(sqrt(5) - 1))^2 + 2992900*x^2) - 1/40*sqrt(25*sqrt(5) - 55
)*log(2992900*(x - sqrt(sqrt(5) - 1))^2 + 2992900*x^2) + x

Mupad [B] (verification not implemented)

Time = 8.31 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.47 \[ \int \frac {x^8}{1+3 x^4+x^8} \, dx=x-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {3\,2^{1/4}\,x}{2\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}+\frac {2^{1/4}\,\sqrt {5}\,x}{2\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}\right )\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}{20}+\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {3\,2^{1/4}\,x}{2\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}-\frac {2^{1/4}\,\sqrt {5}\,x}{2\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}\right )\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}{20}+\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{1/4}\,x\,3{}\mathrm {i}}{2\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}+\frac {2^{1/4}\,\sqrt {5}\,x\,1{}\mathrm {i}}{2\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}\right )\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}\,1{}\mathrm {i}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{1/4}\,x\,3{}\mathrm {i}}{2\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}-\frac {2^{1/4}\,\sqrt {5}\,x\,1{}\mathrm {i}}{2\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}\right )\,{\left (55\,\sqrt {5}-123\right )}^{1/4}\,1{}\mathrm {i}}{20} \]

[In]

int(x^8/(3*x^4 + x^8 + 1),x)

[Out]

x - (2^(3/4)*5^(1/2)*atan((3*2^(1/4)*x)/(2*(- 55*5^(1/2) - 123)^(1/4)) + (2^(1/4)*5^(1/2)*x)/(2*(- 55*5^(1/2)
- 123)^(1/4)))*(- 55*5^(1/2) - 123)^(1/4))/20 + (2^(3/4)*5^(1/2)*atan((3*2^(1/4)*x)/(2*(55*5^(1/2) - 123)^(1/4
)) - (2^(1/4)*5^(1/2)*x)/(2*(55*5^(1/2) - 123)^(1/4)))*(55*5^(1/2) - 123)^(1/4))/20 + (2^(3/4)*5^(1/2)*atan((2
^(1/4)*x*3i)/(2*(- 55*5^(1/2) - 123)^(1/4)) + (2^(1/4)*5^(1/2)*x*1i)/(2*(- 55*5^(1/2) - 123)^(1/4)))*(- 55*5^(
1/2) - 123)^(1/4)*1i)/20 - (2^(3/4)*5^(1/2)*atan((2^(1/4)*x*3i)/(2*(55*5^(1/2) - 123)^(1/4)) - (2^(1/4)*5^(1/2
)*x*1i)/(2*(55*5^(1/2) - 123)^(1/4)))*(55*5^(1/2) - 123)^(1/4)*1i)/20