3.379 \(\int \frac{x^6}{1+3 x^4+x^8} \, dx\)

Optimal. Leaf size=431 \[ -\frac{\sqrt [4]{9-4 \sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4 \sqrt{10}}+\frac{\sqrt [4]{9-4 \sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4 \sqrt{10}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{8 \sqrt [4]{2} \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{8 \sqrt [4]{2} \sqrt{5}}+\frac{\sqrt [4]{9-4 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2 \sqrt{10}}-\frac{\sqrt [4]{9-4 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2 \sqrt{10}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{4 \sqrt [4]{2} \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{4 \sqrt [4]{2} \sqrt{5}} \]

[Out]

((9 - 4*Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*Sqrt[10]) - ((9 - 4*Sqrt[5])^(1/4)*ArcT
an[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*Sqrt[10]) - ((3 + Sqrt[5])^(3/4)*ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[
5])^(1/4)])/(4*2^(1/4)*Sqrt[5]) + ((3 + Sqrt[5])^(3/4)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(4*2^(1/4)
*Sqrt[5]) - ((9 - 4*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*Sqrt[1
0]) + ((9 - 4*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] + 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*Sqrt[10]) +
((3 + Sqrt[5])^(3/4)*Log[Sqrt[2*(3 + Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(8*2^(1/4)*Sqrt[5]) - (
(3 + Sqrt[5])^(3/4)*Log[Sqrt[2*(3 + Sqrt[5])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(8*2^(1/4)*Sqrt[5])

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Rubi [A]  time = 0.294608, antiderivative size = 431, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {1374, 297, 1162, 617, 204, 1165, 628} \[ -\frac{\sqrt [4]{9-4 \sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4 \sqrt{10}}+\frac{\sqrt [4]{9-4 \sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4 \sqrt{10}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{8 \sqrt [4]{2} \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{8 \sqrt [4]{2} \sqrt{5}}+\frac{\sqrt [4]{9-4 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2 \sqrt{10}}-\frac{\sqrt [4]{9-4 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2 \sqrt{10}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{4 \sqrt [4]{2} \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{4 \sqrt [4]{2} \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(1 + 3*x^4 + x^8),x]

[Out]

((9 - 4*Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*Sqrt[10]) - ((9 - 4*Sqrt[5])^(1/4)*ArcT
an[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*Sqrt[10]) - ((3 + Sqrt[5])^(3/4)*ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[
5])^(1/4)])/(4*2^(1/4)*Sqrt[5]) + ((3 + Sqrt[5])^(3/4)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(4*2^(1/4)
*Sqrt[5]) - ((9 - 4*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*Sqrt[1
0]) + ((9 - 4*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] + 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*Sqrt[10]) +
((3 + Sqrt[5])^(3/4)*Log[Sqrt[2*(3 + Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(8*2^(1/4)*Sqrt[5]) - (
(3 + Sqrt[5])^(3/4)*Log[Sqrt[2*(3 + Sqrt[5])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(8*2^(1/4)*Sqrt[5])

Rule 1374

Int[((d_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[(d^n*(b/q + 1))/2, Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Dist[(d^n*(b/q - 1))/2, Int[(d*x)^(m
- n)/(b/2 - q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n,
0] && GeQ[m, n]

Rule 297

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

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])

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]

Rubi steps

\begin{align*} \int \frac{x^6}{1+3 x^4+x^8} \, dx &=-\left (\frac{1}{10} \left (-5+3 \sqrt{5}\right ) \int \frac{x^2}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x^4} \, dx\right )+\frac{1}{10} \left (5+3 \sqrt{5}\right ) \int \frac{x^2}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x^4} \, dx\\ &=\frac{\left (3-\sqrt{5}\right ) \int \frac{\sqrt{3-\sqrt{5}}-\sqrt{2} x^2}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x^4} \, dx}{4 \sqrt{10}}-\frac{\left (3-\sqrt{5}\right ) \int \frac{\sqrt{3-\sqrt{5}}+\sqrt{2} x^2}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x^4} \, dx}{4 \sqrt{10}}-\frac{\left (3+\sqrt{5}\right ) \int \frac{\sqrt{3+\sqrt{5}}-\sqrt{2} x^2}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x^4} \, dx}{4 \sqrt{10}}+\frac{\left (3+\sqrt{5}\right ) \int \frac{\sqrt{3+\sqrt{5}}+\sqrt{2} x^2}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x^4} \, dx}{4 \sqrt{10}}\\ &=-\frac{\sqrt [4]{9-4 \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 \sqrt{10}}-\frac{\sqrt [4]{9-4 \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 \sqrt{10}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \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}{8 \sqrt [4]{2} \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \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}{8 \sqrt [4]{2} \sqrt{5}}-\frac{1}{40} \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-\frac{1}{40} \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+\frac{1}{40} \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+\frac{1}{40} \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\\ &=-\frac{\sqrt [4]{9-4 \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 \sqrt{10}}+\frac{\sqrt [4]{9-4 \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 \sqrt{10}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \log \left (\sqrt{2 \left (3+\sqrt{5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+2 x^2\right )}{8 \sqrt [4]{2} \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \log \left (\sqrt{2 \left (3+\sqrt{5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+2 x^2\right )}{8 \sqrt [4]{2} \sqrt{5}}-\frac{\left (3-\sqrt{5}\right )^{3/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{4 \sqrt [4]{2} \sqrt{5}}-\frac{\left (5-3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{20 \sqrt [4]{2 \left (3-\sqrt{5}\right )}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{4 \sqrt [4]{2} \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{4 \sqrt [4]{2} \sqrt{5}}\\ &=\frac{\left (3-\sqrt{5}\right )^{3/4} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{4 \sqrt [4]{2} \sqrt{5}}-\frac{\sqrt [4]{36-16 \sqrt{5}} \tan ^{-1}\left (1+\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{4 \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{4 \sqrt [4]{2} \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (1+\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{4 \sqrt [4]{2} \sqrt{5}}-\frac{\sqrt [4]{9-4 \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 \sqrt{10}}+\frac{\sqrt [4]{9-4 \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 \sqrt{10}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \log \left (\sqrt{2 \left (3+\sqrt{5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+2 x^2\right )}{8 \sqrt [4]{2} \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \log \left (\sqrt{2 \left (3+\sqrt{5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+2 x^2\right )}{8 \sqrt [4]{2} \sqrt{5}}\\ \end{align*}

Mathematica [C]  time = 0.0121073, size = 41, normalized size = 0.1 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\& ,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4+3}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(1 + 3*x^4 + x^8),x]

[Out]

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

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Maple [C]  time = 0.007, size = 40, normalized size = 0.1 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+3\,{{\it \_Z}}^{4}+1 \right ) }{\frac{{{\it \_R}}^{6}\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^6/(x^8+3*x^4+1),x)

[Out]

1/4*sum(_R^6/(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*} \int \frac{x^{6}}{x^{8} + 3 \, x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^6/(x^8 + 3*x^4 + 1), x)

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Fricas [B]  time = 1.80635, size = 2326, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*sqrt(5)*sqrt(2)*(4*sqrt(5) + 9)^(1/4)*arctan(1/2*sqrt(2*x^2 + (3*sqrt(5)*sqrt(2)*x - 7*sqrt(2)*x)*(4*sqrt
(5) + 9)^(3/4) - sqrt(4*sqrt(5) + 9)*(sqrt(5) - 3))*(21*sqrt(5) - 47)*(4*sqrt(5) + 9)^(5/4) - 1/2*(21*sqrt(5)*
sqrt(2)*x - 47*sqrt(2)*x)*(4*sqrt(5) + 9)^(5/4) - 1) + 1/10*sqrt(5)*sqrt(2)*(4*sqrt(5) + 9)^(1/4)*arctan(1/2*s
qrt(2*x^2 - (3*sqrt(5)*sqrt(2)*x - 7*sqrt(2)*x)*(4*sqrt(5) + 9)^(3/4) - sqrt(4*sqrt(5) + 9)*(sqrt(5) - 3))*(21
*sqrt(5) - 47)*(4*sqrt(5) + 9)^(5/4) - 1/2*(21*sqrt(5)*sqrt(2)*x - 47*sqrt(2)*x)*(4*sqrt(5) + 9)^(5/4) + 1) +
1/10*sqrt(5)*sqrt(2)*(-4*sqrt(5) + 9)^(1/4)*arctan(1/2*sqrt(2*x^2 + (3*sqrt(5)*sqrt(2)*x + 7*sqrt(2)*x)*(-4*sq
rt(5) + 9)^(3/4) + (sqrt(5) + 3)*sqrt(-4*sqrt(5) + 9))*(21*sqrt(5) + 47)*(-4*sqrt(5) + 9)^(5/4) - 1/2*(21*sqrt
(5)*sqrt(2)*x + 47*sqrt(2)*x)*(-4*sqrt(5) + 9)^(5/4) - 1) + 1/10*sqrt(5)*sqrt(2)*(-4*sqrt(5) + 9)^(1/4)*arctan
(1/2*sqrt(2*x^2 - (3*sqrt(5)*sqrt(2)*x + 7*sqrt(2)*x)*(-4*sqrt(5) + 9)^(3/4) + (sqrt(5) + 3)*sqrt(-4*sqrt(5) +
9))*(21*sqrt(5) + 47)*(-4*sqrt(5) + 9)^(5/4) - 1/2*(21*sqrt(5)*sqrt(2)*x + 47*sqrt(2)*x)*(-4*sqrt(5) + 9)^(5/
4) + 1) + 1/40*sqrt(5)*sqrt(2)*(4*sqrt(5) + 9)^(1/4)*log(2*x^2 + (3*sqrt(5)*sqrt(2)*x - 7*sqrt(2)*x)*(4*sqrt(5
) + 9)^(3/4) - sqrt(4*sqrt(5) + 9)*(sqrt(5) - 3)) - 1/40*sqrt(5)*sqrt(2)*(4*sqrt(5) + 9)^(1/4)*log(2*x^2 - (3*
sqrt(5)*sqrt(2)*x - 7*sqrt(2)*x)*(4*sqrt(5) + 9)^(3/4) - sqrt(4*sqrt(5) + 9)*(sqrt(5) - 3)) + 1/40*sqrt(5)*sqr
t(2)*(-4*sqrt(5) + 9)^(1/4)*log(2*x^2 + (3*sqrt(5)*sqrt(2)*x + 7*sqrt(2)*x)*(-4*sqrt(5) + 9)^(3/4) + (sqrt(5)
+ 3)*sqrt(-4*sqrt(5) + 9)) - 1/40*sqrt(5)*sqrt(2)*(-4*sqrt(5) + 9)^(1/4)*log(2*x^2 - (3*sqrt(5)*sqrt(2)*x + 7*
sqrt(2)*x)*(-4*sqrt(5) + 9)^(3/4) + (sqrt(5) + 3)*sqrt(-4*sqrt(5) + 9))

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Sympy [A]  time = 1.14604, size = 26, normalized size = 0.06 \begin{align*} \operatorname{RootSum}{\left (40960000 t^{8} + 115200 t^{4} + 1, \left ( t \mapsto t \log{\left (- 1792000 t^{7} - 4920 t^{3} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(40960000*_t**8 + 115200*_t**4 + 1, Lambda(_t, _t*log(-1792000*_t**7 - 4920*_t**3 + x)))

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Giac [A]  time = 1.36248, size = 342, normalized size = 0.79 \begin{align*} -\frac{1}{40} \,{\left (i - 1\right )} \sqrt{10 \, \sqrt{5} - 20} \log \left (100 \,{\left (i + 1\right )} x + 100 \, i \sqrt{\sqrt{5} - 1}\right ) + \frac{1}{40} \,{\left (i - 1\right )} \sqrt{10 \, \sqrt{5} - 20} \log \left (100 \,{\left (i + 1\right )} x - 100 \, i \sqrt{\sqrt{5} - 1}\right ) + \frac{1}{40} \,{\left (i + 1\right )} \sqrt{10 \, \sqrt{5} - 20} \log \left (100 \,{\left (i + 1\right )} x + 100 \, \sqrt{\sqrt{5} - 1}\right ) - \frac{1}{40} \,{\left (i + 1\right )} \sqrt{10 \, \sqrt{5} - 20} \log \left (100 \,{\left (i + 1\right )} x - 100 \, \sqrt{\sqrt{5} - 1}\right ) + \frac{1}{40} \,{\left (i - 1\right )} \sqrt{10 \, \sqrt{5} + 20} \log \left (20 \,{\left (i + 1\right )} x + 20 \, i \sqrt{\sqrt{5} + 1}\right ) - \frac{1}{40} \,{\left (i - 1\right )} \sqrt{10 \, \sqrt{5} + 20} \log \left (20 \,{\left (i + 1\right )} x - 20 \, i \sqrt{\sqrt{5} + 1}\right ) - \frac{1}{40} \,{\left (i + 1\right )} \sqrt{10 \, \sqrt{5} + 20} \log \left (20 \,{\left (i + 1\right )} x + 20 \, \sqrt{\sqrt{5} + 1}\right ) + \frac{1}{40} \,{\left (i + 1\right )} \sqrt{10 \, \sqrt{5} + 20} \log \left (20 \,{\left (i + 1\right )} x - 20 \, \sqrt{\sqrt{5} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*(i - 1)*sqrt(10*sqrt(5) - 20)*log(100*(i + 1)*x + 100*i*sqrt(sqrt(5) - 1)) + 1/40*(i - 1)*sqrt(10*sqrt(5
) - 20)*log(100*(i + 1)*x - 100*i*sqrt(sqrt(5) - 1)) + 1/40*(i + 1)*sqrt(10*sqrt(5) - 20)*log(100*(i + 1)*x +
100*sqrt(sqrt(5) - 1)) - 1/40*(i + 1)*sqrt(10*sqrt(5) - 20)*log(100*(i + 1)*x - 100*sqrt(sqrt(5) - 1)) + 1/40*
(i - 1)*sqrt(10*sqrt(5) + 20)*log(20*(i + 1)*x + 20*i*sqrt(sqrt(5) + 1)) - 1/40*(i - 1)*sqrt(10*sqrt(5) + 20)*
log(20*(i + 1)*x - 20*i*sqrt(sqrt(5) + 1)) - 1/40*(i + 1)*sqrt(10*sqrt(5) + 20)*log(20*(i + 1)*x + 20*sqrt(sqr
t(5) + 1)) + 1/40*(i + 1)*sqrt(10*sqrt(5) + 20)*log(20*(i + 1)*x - 20*sqrt(sqrt(5) + 1))