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

Optimal. Leaf size=460 \[ -\frac{\sqrt [4]{123-55 \sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{123-55 \sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{123+55 \sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{123+55 \sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}+x-\frac{\sqrt [4]{123-55 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{123-55 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{123+55 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{123+55 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}} \]

[Out]

x - ((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*Sq
rt[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)*ArcT
an[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]) - ((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]) + ((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])

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Rubi [A]  time = 0.416256, antiderivative size = 440, normalized size of antiderivative = 0.96, number of steps used = 20, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1367, 1422, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\sqrt [4]{984-440 \sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{8 \sqrt{10}}+\frac{\sqrt [4]{984-440 \sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{8 \sqrt{10}}+\frac{\sqrt [4]{123+55 \sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{123+55 \sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}+x-\frac{\sqrt [4]{984-440 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{4 \sqrt{10}}+\frac{\sqrt [4]{984-440 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{4 \sqrt{10}}+\frac{\sqrt [4]{123+55 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{123+55 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}} \]

Antiderivative was successfully verified.

[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 1367

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 1422

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])

Rule 211

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 1165

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 628

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 1162

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 617

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 204

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

Rubi steps

\begin{align*} \int \frac{x^8}{1+3 x^4+x^8} \, dx &=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}} \operatorname{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}} \operatorname{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}} \operatorname{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}} \operatorname{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]  time = 0.0140273, size = 58, normalized size = 0.13 \[ x-\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\& ,\frac{3 \text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{2 \text{$\#$1}^7+3 \text{$\#$1}^3}\& \right ] \]

Antiderivative was successfully verified.

[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

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Maple [C]  time = 0.006, size = 46, normalized size = 0.1 \begin{align*} x+{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+3\,{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ( -3\,{{\it \_R}}^{4}-1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}+3\,{{\it \_R}}^{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} x - \int \frac{3 \, x^{4} + 1}{x^{8} + 3 \, x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.91333, size = 3565, normalized size = 7.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80*sqrt(10)*(110*sqrt(5) + 246)^(3/4)*sqrt(55*sqrt(5) + 123)*(55*sqrt(5) - 123)*arctan(1/80*sqrt(10)*sqrt(20
*x^2 + sqrt(10)*(3*sqrt(5)*sqrt(2)*x - 5*sqrt(2)*x)*(110*sqrt(5) + 246)^(1/4) - 5*sqrt(110*sqrt(5) + 246)*(3*s
qrt(5) - 7))*(1292*sqrt(5) - 2889)*(110*sqrt(5) + 246)^(5/4)*sqrt(55*sqrt(5) + 123) + 1/40*sqrt(10)*(2889*sqrt
(5)*x - 6460*x)*(110*sqrt(5) + 246)^(5/4)*sqrt(55*sqrt(5) + 123) - 1/8*(55*sqrt(5)*sqrt(2) - 123*sqrt(2))*sqrt
(110*sqrt(5) + 246)*sqrt(55*sqrt(5) + 123)) + 1/80*sqrt(10)*(110*sqrt(5) + 246)^(3/4)*sqrt(55*sqrt(5) + 123)*(
55*sqrt(5) - 123)*arctan(1/80*sqrt(10)*sqrt(20*x^2 - sqrt(10)*(3*sqrt(5)*sqrt(2)*x - 5*sqrt(2)*x)*(110*sqrt(5)
+ 246)^(1/4) - 5*sqrt(110*sqrt(5) + 246)*(3*sqrt(5) - 7))*(1292*sqrt(5) - 2889)*(110*sqrt(5) + 246)^(5/4)*sqr
t(55*sqrt(5) + 123) + 1/40*sqrt(10)*(2889*sqrt(5)*x - 6460*x)*(110*sqrt(5) + 246)^(5/4)*sqrt(55*sqrt(5) + 123)
+ 1/8*(55*sqrt(5)*sqrt(2) - 123*sqrt(2))*sqrt(110*sqrt(5) + 246)*sqrt(55*sqrt(5) + 123)) - 1/80*sqrt(10)*(55*
sqrt(5) + 123)*sqrt(-55*sqrt(5) + 123)*(-110*sqrt(5) + 246)^(3/4)*arctan(1/80*sqrt(10)*sqrt(20*x^2 + sqrt(10)*
(3*sqrt(5)*sqrt(2)*x + 5*sqrt(2)*x)*(-110*sqrt(5) + 246)^(1/4) + 5*(3*sqrt(5) + 7)*sqrt(-110*sqrt(5) + 246))*(
1292*sqrt(5) + 2889)*sqrt(-55*sqrt(5) + 123)*(-110*sqrt(5) + 246)^(5/4) - 1/40*(sqrt(10)*(2889*sqrt(5)*x + 646
0*x)*(-110*sqrt(5) + 246)^(5/4) + 5*(55*sqrt(5)*sqrt(2) + 123*sqrt(2))*sqrt(-110*sqrt(5) + 246))*sqrt(-55*sqrt
(5) + 123)) - 1/80*sqrt(10)*(55*sqrt(5) + 123)*sqrt(-55*sqrt(5) + 123)*(-110*sqrt(5) + 246)^(3/4)*arctan(1/80*
sqrt(10)*sqrt(20*x^2 - sqrt(10)*(3*sqrt(5)*sqrt(2)*x + 5*sqrt(2)*x)*(-110*sqrt(5) + 246)^(1/4) + 5*(3*sqrt(5)
+ 7)*sqrt(-110*sqrt(5) + 246))*(1292*sqrt(5) + 2889)*sqrt(-55*sqrt(5) + 123)*(-110*sqrt(5) + 246)^(5/4) - 1/40
*(sqrt(10)*(2889*sqrt(5)*x + 6460*x)*(-110*sqrt(5) + 246)^(5/4) - 5*(55*sqrt(5)*sqrt(2) + 123*sqrt(2))*sqrt(-1
10*sqrt(5) + 246))*sqrt(-55*sqrt(5) + 123)) - 1/80*sqrt(10)*sqrt(2)*(110*sqrt(5) + 246)^(1/4)*log(20*x^2 + sqr
t(10)*(3*sqrt(5)*sqrt(2)*x - 5*sqrt(2)*x)*(110*sqrt(5) + 246)^(1/4) - 5*sqrt(110*sqrt(5) + 246)*(3*sqrt(5) - 7
)) + 1/80*sqrt(10)*sqrt(2)*(110*sqrt(5) + 246)^(1/4)*log(20*x^2 - sqrt(10)*(3*sqrt(5)*sqrt(2)*x - 5*sqrt(2)*x)
*(110*sqrt(5) + 246)^(1/4) - 5*sqrt(110*sqrt(5) + 246)*(3*sqrt(5) - 7)) + 1/80*sqrt(10)*sqrt(2)*(-110*sqrt(5)
+ 246)^(1/4)*log(20*x^2 + sqrt(10)*(3*sqrt(5)*sqrt(2)*x + 5*sqrt(2)*x)*(-110*sqrt(5) + 246)^(1/4) + 5*(3*sqrt(
5) + 7)*sqrt(-110*sqrt(5) + 246)) - 1/80*sqrt(10)*sqrt(2)*(-110*sqrt(5) + 246)^(1/4)*log(20*x^2 - sqrt(10)*(3*
sqrt(5)*sqrt(2)*x + 5*sqrt(2)*x)*(-110*sqrt(5) + 246)^(1/4) + 5*(3*sqrt(5) + 7)*sqrt(-110*sqrt(5) + 246)) + x

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Sympy [A]  time = 1.17545, size = 29, normalized size = 0.06 \begin{align*} 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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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Giac [A]  time = 1.35593, size = 343, normalized size = 0.75 \begin{align*} \frac{1}{40} \,{\left (i + 1\right )} \sqrt{25 \, \sqrt{5} - 55} \log \left (1730 \,{\left (i + 1\right )} x + 1730 \, i \sqrt{\sqrt{5} - 1}\right ) - \frac{1}{40} \,{\left (i + 1\right )} \sqrt{25 \, \sqrt{5} - 55} \log \left (1730 \,{\left (i + 1\right )} x - 1730 \, i \sqrt{\sqrt{5} - 1}\right ) - \frac{1}{40} \,{\left (i - 1\right )} \sqrt{25 \, \sqrt{5} - 55} \log \left (1730 \,{\left (i + 1\right )} x + 1730 \, \sqrt{\sqrt{5} - 1}\right ) + \frac{1}{40} \,{\left (i - 1\right )} \sqrt{25 \, \sqrt{5} - 55} \log \left (1730 \,{\left (i + 1\right )} x - 1730 \, \sqrt{\sqrt{5} - 1}\right ) - \frac{1}{40} \,{\left (i + 1\right )} \sqrt{25 \, \sqrt{5} + 55} \log \left (850 \,{\left (i + 1\right )} x + 850 \, i \sqrt{\sqrt{5} + 1}\right ) + \frac{1}{40} \,{\left (i + 1\right )} \sqrt{25 \, \sqrt{5} + 55} \log \left (850 \,{\left (i + 1\right )} x - 850 \, i \sqrt{\sqrt{5} + 1}\right ) + \frac{1}{40} \,{\left (i - 1\right )} \sqrt{25 \, \sqrt{5} + 55} \log \left (850 \,{\left (i + 1\right )} x + 850 \, \sqrt{\sqrt{5} + 1}\right ) - \frac{1}{40} \,{\left (i - 1\right )} \sqrt{25 \, \sqrt{5} + 55} \log \left (850 \,{\left (i + 1\right )} x - 850 \, \sqrt{\sqrt{5} + 1}\right ) + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(i + 1)*sqrt(25*sqrt(5) - 55)*log(1730*(i + 1)*x + 1730*i*sqrt(sqrt(5) - 1)) - 1/40*(i + 1)*sqrt(25*sqrt(
5) - 55)*log(1730*(i + 1)*x - 1730*i*sqrt(sqrt(5) - 1)) - 1/40*(i - 1)*sqrt(25*sqrt(5) - 55)*log(1730*(i + 1)*
x + 1730*sqrt(sqrt(5) - 1)) + 1/40*(i - 1)*sqrt(25*sqrt(5) - 55)*log(1730*(i + 1)*x - 1730*sqrt(sqrt(5) - 1))
- 1/40*(i + 1)*sqrt(25*sqrt(5) + 55)*log(850*(i + 1)*x + 850*i*sqrt(sqrt(5) + 1)) + 1/40*(i + 1)*sqrt(25*sqrt(
5) + 55)*log(850*(i + 1)*x - 850*i*sqrt(sqrt(5) + 1)) + 1/40*(i - 1)*sqrt(25*sqrt(5) + 55)*log(850*(i + 1)*x +
850*sqrt(sqrt(5) + 1)) - 1/40*(i - 1)*sqrt(25*sqrt(5) + 55)*log(850*(i + 1)*x - 850*sqrt(sqrt(5) + 1)) + x