3.381 \(\int \frac{x^2}{1+3 x^4+x^8} \, dx\)

Optimal. Leaf size=427 \[ \frac{\log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4 \sqrt{5} \sqrt [4]{2 \left (3-\sqrt{5}\right )}}-\frac{\log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4 \sqrt{5} \sqrt [4]{2 \left (3-\sqrt{5}\right )}}-\frac{\log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}+\frac{\log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}-\frac{\tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3-\sqrt{5}\right )}}+\frac{\tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3-\sqrt{5}\right )}}+\frac{\tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}-\frac{\tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}} \]

[Out]

-ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)]/(2*Sqrt[5]*(2*(3 - Sqrt[5]))^(1/4)) + ArcTan[1 + (2^(3/4)*x)/(3 -
Sqrt[5])^(1/4)]/(2*Sqrt[5]*(2*(3 - Sqrt[5]))^(1/4)) + ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]/(2*Sqrt[5]*
(2*(3 + Sqrt[5]))^(1/4)) - ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]/(2*Sqrt[5]*(2*(3 + Sqrt[5]))^(1/4)) + L
og[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2]/(4*Sqrt[5]*(2*(3 - Sqrt[5]))^(1/4)) - Log[Sqrt
[2*(3 - Sqrt[5])] + 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2]/(4*Sqrt[5]*(2*(3 - Sqrt[5]))^(1/4)) - Log[Sqrt[2*(3 +
Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2]/(4*Sqrt[5]*(2*(3 + Sqrt[5]))^(1/4)) + Log[Sqrt[2*(3 + Sqrt[5
])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2]/(4*Sqrt[5]*(2*(3 + Sqrt[5]))^(1/4))

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Rubi [A]  time = 0.25911, antiderivative size = 431, normalized size of antiderivative = 1.01, 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 = {1375, 297, 1162, 617, 204, 1165, 628} \[ \frac{\sqrt [4]{3+\sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3+\sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}+\frac{\log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}-\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}-\frac{\tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(1 + 3*x^4 + x^8),x]

[Out]

-((3 + Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) + ((3 + Sqrt[5])^(1/4)*
ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) + ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]
/(2*Sqrt[5]*(2*(3 + Sqrt[5]))^(1/4)) - ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]/(2*Sqrt[5]*(2*(3 + Sqrt[5])
)^(1/4)) + ((3 + 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)*S
qrt[5]) - ((3 + 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)*Sq
rt[5]) - Log[Sqrt[2*(3 + Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2]/(4*Sqrt[5]*(2*(3 + Sqrt[5]))^(1/4))
+ Log[Sqrt[2*(3 + Sqrt[5])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2]/(4*Sqrt[5]*(2*(3 + Sqrt[5]))^(1/4))

Rule 1375

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

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

Mathematica [C]  time = 0.0104187, size = 40, normalized size = 0.09 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\& ,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5+3 \text{$\#$1}}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(1 + 3*x^4 + x^8),x]

[Out]

RootSum[1 + 3*#1^4 + #1^8 & , Log[x - #1]/(3*#1 + 2*#1^5) & ]/4

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Maple [C]  time = 0.005, 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}}^{2}\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^2/(x^8+3*x^4+1),x)

[Out]

1/4*sum(_R^2/(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^{2}}{x^{8} + 3 \, x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^8+3*x^4+1),x, algorithm="maxima")

[Out]

integrate(x^2/(x^8 + 3*x^4 + 1), x)

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Fricas [B]  time = 1.85963, size = 3101, normalized size = 7.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^8+3*x^4+1),x, algorithm="fricas")

[Out]

1/80*sqrt(10)*(2*sqrt(5) + 6)^(3/4)*sqrt(sqrt(5) + 3)*(sqrt(5) - 3)*arctan(-1/40*sqrt(10)*(3*sqrt(5)*x - 5*x)*
(2*sqrt(5) + 6)^(3/4)*sqrt(sqrt(5) + 3) + 1/80*sqrt(sqrt(10)*(3*sqrt(5)*sqrt(2)*x - 5*sqrt(2)*x)*(2*sqrt(5) +
6)^(3/4) + 40*x^2 - 10*sqrt(2*sqrt(5) + 6)*(sqrt(5) - 3))*(3*sqrt(5) - 5)*(2*sqrt(5) + 6)^(3/4)*sqrt(sqrt(5) +
3) + 1/8*(sqrt(5)*sqrt(2) - 3*sqrt(2))*sqrt(2*sqrt(5) + 6)*sqrt(sqrt(5) + 3)) + 1/80*sqrt(10)*(2*sqrt(5) + 6)
^(3/4)*sqrt(sqrt(5) + 3)*(sqrt(5) - 3)*arctan(-1/40*sqrt(10)*(3*sqrt(5)*x - 5*x)*(2*sqrt(5) + 6)^(3/4)*sqrt(sq
rt(5) + 3) + 1/80*sqrt(-sqrt(10)*(3*sqrt(5)*sqrt(2)*x - 5*sqrt(2)*x)*(2*sqrt(5) + 6)^(3/4) + 40*x^2 - 10*sqrt(
2*sqrt(5) + 6)*(sqrt(5) - 3))*(3*sqrt(5) - 5)*(2*sqrt(5) + 6)^(3/4)*sqrt(sqrt(5) + 3) - 1/8*(sqrt(5)*sqrt(2) -
3*sqrt(2))*sqrt(2*sqrt(5) + 6)*sqrt(sqrt(5) + 3)) + 1/80*sqrt(10)*(sqrt(5) + 3)*sqrt(-sqrt(5) + 3)*(-2*sqrt(5
) + 6)^(3/4)*arctan(1/80*sqrt(sqrt(10)*(3*sqrt(5)*sqrt(2)*x + 5*sqrt(2)*x)*(-2*sqrt(5) + 6)^(3/4) + 40*x^2 + 1
0*(sqrt(5) + 3)*sqrt(-2*sqrt(5) + 6))*(3*sqrt(5) + 5)*sqrt(-sqrt(5) + 3)*(-2*sqrt(5) + 6)^(3/4) - 1/40*(sqrt(1
0)*(3*sqrt(5)*x + 5*x)*(-2*sqrt(5) + 6)^(3/4) + 5*(sqrt(5)*sqrt(2) + 3*sqrt(2))*sqrt(-2*sqrt(5) + 6))*sqrt(-sq
rt(5) + 3)) + 1/80*sqrt(10)*(sqrt(5) + 3)*sqrt(-sqrt(5) + 3)*(-2*sqrt(5) + 6)^(3/4)*arctan(1/80*sqrt(-sqrt(10)
*(3*sqrt(5)*sqrt(2)*x + 5*sqrt(2)*x)*(-2*sqrt(5) + 6)^(3/4) + 40*x^2 + 10*(sqrt(5) + 3)*sqrt(-2*sqrt(5) + 6))*
(3*sqrt(5) + 5)*sqrt(-sqrt(5) + 3)*(-2*sqrt(5) + 6)^(3/4) - 1/40*(sqrt(10)*(3*sqrt(5)*x + 5*x)*(-2*sqrt(5) + 6
)^(3/4) - 5*(sqrt(5)*sqrt(2) + 3*sqrt(2))*sqrt(-2*sqrt(5) + 6))*sqrt(-sqrt(5) + 3)) - 1/80*sqrt(10)*sqrt(2)*(2
*sqrt(5) + 6)^(1/4)*log(sqrt(10)*(3*sqrt(5)*sqrt(2)*x - 5*sqrt(2)*x)*(2*sqrt(5) + 6)^(3/4) + 40*x^2 - 10*sqrt(
2*sqrt(5) + 6)*(sqrt(5) - 3)) + 1/80*sqrt(10)*sqrt(2)*(2*sqrt(5) + 6)^(1/4)*log(-sqrt(10)*(3*sqrt(5)*sqrt(2)*x
- 5*sqrt(2)*x)*(2*sqrt(5) + 6)^(3/4) + 40*x^2 - 10*sqrt(2*sqrt(5) + 6)*(sqrt(5) - 3)) + 1/80*sqrt(10)*sqrt(2)
*(-2*sqrt(5) + 6)^(1/4)*log(sqrt(10)*(3*sqrt(5)*sqrt(2)*x + 5*sqrt(2)*x)*(-2*sqrt(5) + 6)^(3/4) + 40*x^2 + 10*
(sqrt(5) + 3)*sqrt(-2*sqrt(5) + 6)) - 1/80*sqrt(10)*sqrt(2)*(-2*sqrt(5) + 6)^(1/4)*log(-sqrt(10)*(3*sqrt(5)*sq
rt(2)*x + 5*sqrt(2)*x)*(-2*sqrt(5) + 6)^(3/4) + 40*x^2 + 10*(sqrt(5) + 3)*sqrt(-2*sqrt(5) + 6))

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Sympy [A]  time = 1.14028, size = 26, normalized size = 0.06 \begin{align*} \operatorname{RootSum}{\left (40960000 t^{8} + 19200 t^{4} + 1, \left ( t \mapsto t \log{\left (- 6144000 t^{7} - 2240 t^{3} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(x**8+3*x**4+1),x)

[Out]

RootSum(40960000*_t**8 + 19200*_t**4 + 1, Lambda(_t, _t*log(-6144000*_t**7 - 2240*_t**3 + x)))

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Giac [A]  time = 1.36071, size = 342, normalized size = 0.8 \begin{align*} -\frac{1}{40} \,{\left (i - 1\right )} \sqrt{5 \, \sqrt{5} - 5} \log \left (130 \,{\left (i + 1\right )} x + 130 \, i \sqrt{\sqrt{5} + 1}\right ) + \frac{1}{40} \,{\left (i - 1\right )} \sqrt{5 \, \sqrt{5} - 5} \log \left (130 \,{\left (i + 1\right )} x - 130 \, i \sqrt{\sqrt{5} + 1}\right ) + \frac{1}{40} \,{\left (i + 1\right )} \sqrt{5 \, \sqrt{5} - 5} \log \left (130 \,{\left (i + 1\right )} x + 130 \, \sqrt{\sqrt{5} + 1}\right ) - \frac{1}{40} \,{\left (i + 1\right )} \sqrt{5 \, \sqrt{5} - 5} \log \left (130 \,{\left (i + 1\right )} x - 130 \, \sqrt{\sqrt{5} + 1}\right ) + \frac{1}{40} \,{\left (i - 1\right )} \sqrt{5 \, \sqrt{5} + 5} \log \left (50 \,{\left (i + 1\right )} x + 50 \, i \sqrt{\sqrt{5} - 1}\right ) - \frac{1}{40} \,{\left (i - 1\right )} \sqrt{5 \, \sqrt{5} + 5} \log \left (50 \,{\left (i + 1\right )} x - 50 \, i \sqrt{\sqrt{5} - 1}\right ) - \frac{1}{40} \,{\left (i + 1\right )} \sqrt{5 \, \sqrt{5} + 5} \log \left (50 \,{\left (i + 1\right )} x + 50 \, \sqrt{\sqrt{5} - 1}\right ) + \frac{1}{40} \,{\left (i + 1\right )} \sqrt{5 \, \sqrt{5} + 5} \log \left (50 \,{\left (i + 1\right )} x - 50 \, \sqrt{\sqrt{5} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^8+3*x^4+1),x, algorithm="giac")

[Out]

-1/40*(i - 1)*sqrt(5*sqrt(5) - 5)*log(130*(i + 1)*x + 130*i*sqrt(sqrt(5) + 1)) + 1/40*(i - 1)*sqrt(5*sqrt(5) -
5)*log(130*(i + 1)*x - 130*i*sqrt(sqrt(5) + 1)) + 1/40*(i + 1)*sqrt(5*sqrt(5) - 5)*log(130*(i + 1)*x + 130*sq
rt(sqrt(5) + 1)) - 1/40*(i + 1)*sqrt(5*sqrt(5) - 5)*log(130*(i + 1)*x - 130*sqrt(sqrt(5) + 1)) + 1/40*(i - 1)*
sqrt(5*sqrt(5) + 5)*log(50*(i + 1)*x + 50*i*sqrt(sqrt(5) - 1)) - 1/40*(i - 1)*sqrt(5*sqrt(5) + 5)*log(50*(i +
1)*x - 50*i*sqrt(sqrt(5) - 1)) - 1/40*(i + 1)*sqrt(5*sqrt(5) + 5)*log(50*(i + 1)*x + 50*sqrt(sqrt(5) - 1)) + 1
/40*(i + 1)*sqrt(5*sqrt(5) + 5)*log(50*(i + 1)*x - 50*sqrt(sqrt(5) - 1))