3.4.80 \(\int \frac {x^4}{1+3 x^4+x^8} \, dx\) [380]

Optimal. Leaf size=451 \[ \frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3-\sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 451, normalized size of antiderivative = 1.00, 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 = {1388, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {\sqrt [4]{3-\sqrt {5}} \text {ArcTan}\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}} \text {ArcTan}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\sqrt {5}} \text {ArcTan}\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}} \text {ArcTan}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2\ 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 {\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 {\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(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)*A
rcTan[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]) + ((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)*Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(
4*2^(3/4)*Sqrt[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)*Sqrt[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)*Sqrt[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)*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 1388

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/2)*(b/q + 1), Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Dist[(d^n/2)*(b/q - 1), 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]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.01, size = 39, normalized size = 0.09 \begin {gather*} \frac {1}{4} \text {RootSum}\left [1+3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{3+2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(1 + 3*x^4 + x^8),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.02, size = 40, normalized size = 0.09

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/(x^8 + 3*x^4 + 1), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (293) = 586\).
time = 0.40, size = 859, normalized size = 1.90 \begin {gather*} \frac {1}{80} \, \sqrt {10} {\left (2 \, \sqrt {5} + 6\right )}^{\frac {3}{4}} \sqrt {\sqrt {5} + 3} {\left (\sqrt {5} - 3\right )} \arctan \left (\frac {1}{800} \, \sqrt {10} \sqrt {10 \, \sqrt {10} \sqrt {5} \sqrt {2} x {\left (2 \, \sqrt {5} + 6\right )}^{\frac {1}{4}} + 100 \, x^{2} + 50 \, \sqrt {2 \, \sqrt {5} + 6}} {\left (7 \, \sqrt {5} - 15\right )} {\left (2 \, \sqrt {5} + 6\right )}^{\frac {5}{4}} \sqrt {\sqrt {5} + 3} - \frac {1}{80} \, \sqrt {10} {\left (7 \, \sqrt {5} x - 15 \, x\right )} {\left (2 \, \sqrt {5} + 6\right )}^{\frac {5}{4}} \sqrt {\sqrt {5} + 3} + \frac {1}{8} \, {\left (\sqrt {5} \sqrt {2} - 3 \, \sqrt {2}\right )} \sqrt {2 \, \sqrt {5} + 6} \sqrt {\sqrt {5} + 3}\right ) + \frac {1}{80} \, \sqrt {10} {\left (2 \, \sqrt {5} + 6\right )}^{\frac {3}{4}} \sqrt {\sqrt {5} + 3} {\left (\sqrt {5} - 3\right )} \arctan \left (\frac {1}{800} \, \sqrt {10} \sqrt {-10 \, \sqrt {10} \sqrt {5} \sqrt {2} x {\left (2 \, \sqrt {5} + 6\right )}^{\frac {1}{4}} + 100 \, x^{2} + 50 \, \sqrt {2 \, \sqrt {5} + 6}} {\left (7 \, \sqrt {5} - 15\right )} {\left (2 \, \sqrt {5} + 6\right )}^{\frac {5}{4}} \sqrt {\sqrt {5} + 3} - \frac {1}{80} \, \sqrt {10} {\left (7 \, \sqrt {5} x - 15 \, x\right )} {\left (2 \, \sqrt {5} + 6\right )}^{\frac {5}{4}} \sqrt {\sqrt {5} + 3} - \frac {1}{8} \, {\left (\sqrt {5} \sqrt {2} - 3 \, \sqrt {2}\right )} \sqrt {2 \, \sqrt {5} + 6} \sqrt {\sqrt {5} + 3}\right ) + \frac {1}{80} \, \sqrt {10} {\left (\sqrt {5} + 3\right )} \sqrt {-\sqrt {5} + 3} {\left (-2 \, \sqrt {5} + 6\right )}^{\frac {3}{4}} \arctan \left (\frac {1}{800} \, \sqrt {10} \sqrt {10 \, \sqrt {10} \sqrt {5} \sqrt {2} x {\left (-2 \, \sqrt {5} + 6\right )}^{\frac {1}{4}} + 100 \, x^{2} + 50 \, \sqrt {-2 \, \sqrt {5} + 6}} {\left (7 \, \sqrt {5} + 15\right )} \sqrt {-\sqrt {5} + 3} {\left (-2 \, \sqrt {5} + 6\right )}^{\frac {5}{4}} - \frac {1}{80} \, {\left (\sqrt {10} {\left (7 \, \sqrt {5} x + 15 \, x\right )} {\left (-2 \, \sqrt {5} + 6\right )}^{\frac {5}{4}} + 10 \, {\left (\sqrt {5} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {-2 \, \sqrt {5} + 6}\right )} \sqrt {-\sqrt {5} + 3}\right ) + \frac {1}{80} \, \sqrt {10} {\left (\sqrt {5} + 3\right )} \sqrt {-\sqrt {5} + 3} {\left (-2 \, \sqrt {5} + 6\right )}^{\frac {3}{4}} \arctan \left (\frac {1}{800} \, \sqrt {10} \sqrt {-10 \, \sqrt {10} \sqrt {5} \sqrt {2} x {\left (-2 \, \sqrt {5} + 6\right )}^{\frac {1}{4}} + 100 \, x^{2} + 50 \, \sqrt {-2 \, \sqrt {5} + 6}} {\left (7 \, \sqrt {5} + 15\right )} \sqrt {-\sqrt {5} + 3} {\left (-2 \, \sqrt {5} + 6\right )}^{\frac {5}{4}} - \frac {1}{80} \, {\left (\sqrt {10} {\left (7 \, \sqrt {5} x + 15 \, x\right )} {\left (-2 \, \sqrt {5} + 6\right )}^{\frac {5}{4}} - 10 \, {\left (\sqrt {5} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {-2 \, \sqrt {5} + 6}\right )} \sqrt {-\sqrt {5} + 3}\right ) + \frac {1}{80} \, \sqrt {10} \sqrt {2} {\left (2 \, \sqrt {5} + 6\right )}^{\frac {1}{4}} \log \left (10 \, \sqrt {10} \sqrt {5} \sqrt {2} x {\left (2 \, \sqrt {5} + 6\right )}^{\frac {1}{4}} + 100 \, x^{2} + 50 \, \sqrt {2 \, \sqrt {5} + 6}\right ) - \frac {1}{80} \, \sqrt {10} \sqrt {2} {\left (2 \, \sqrt {5} + 6\right )}^{\frac {1}{4}} \log \left (-10 \, \sqrt {10} \sqrt {5} \sqrt {2} x {\left (2 \, \sqrt {5} + 6\right )}^{\frac {1}{4}} + 100 \, x^{2} + 50 \, \sqrt {2 \, \sqrt {5} + 6}\right ) - \frac {1}{80} \, \sqrt {10} \sqrt {2} {\left (-2 \, \sqrt {5} + 6\right )}^{\frac {1}{4}} \log \left (10 \, \sqrt {10} \sqrt {5} \sqrt {2} x {\left (-2 \, \sqrt {5} + 6\right )}^{\frac {1}{4}} + 100 \, x^{2} + 50 \, \sqrt {-2 \, \sqrt {5} + 6}\right ) + \frac {1}{80} \, \sqrt {10} \sqrt {2} {\left (-2 \, \sqrt {5} + 6\right )}^{\frac {1}{4}} \log \left (-10 \, \sqrt {10} \sqrt {5} \sqrt {2} x {\left (-2 \, \sqrt {5} + 6\right )}^{\frac {1}{4}} + 100 \, x^{2} + 50 \, \sqrt {-2 \, \sqrt {5} + 6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(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(1/800*sqrt(10)*sqrt(10*sqrt(10)*sqr
t(5)*sqrt(2)*x*(2*sqrt(5) + 6)^(1/4) + 100*x^2 + 50*sqrt(2*sqrt(5) + 6))*(7*sqrt(5) - 15)*(2*sqrt(5) + 6)^(5/4
)*sqrt(sqrt(5) + 3) - 1/80*sqrt(10)*(7*sqrt(5)*x - 15*x)*(2*sqrt(5) + 6)^(5/4)*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(1/800*sqrt(10)*sqrt(-10*sqrt(10)*sqrt(5)*sqrt(2)*x*(2*sqrt(5) + 6)^(1/4) + 100*x^
2 + 50*sqrt(2*sqrt(5) + 6))*(7*sqrt(5) - 15)*(2*sqrt(5) + 6)^(5/4)*sqrt(sqrt(5) + 3) - 1/80*sqrt(10)*(7*sqrt(5
)*x - 15*x)*(2*sqrt(5) + 6)^(5/4)*sqrt(sqrt(5) + 3) - 1/8*(sqrt(5)*sqrt(2) - 3*sqrt(2))*sqrt(2*sqrt(5) + 6)*sq
rt(sqrt(5) + 3)) + 1/80*sqrt(10)*(sqrt(5) + 3)*sqrt(-sqrt(5) + 3)*(-2*sqrt(5) + 6)^(3/4)*arctan(1/800*sqrt(10)
*sqrt(10*sqrt(10)*sqrt(5)*sqrt(2)*x*(-2*sqrt(5) + 6)^(1/4) + 100*x^2 + 50*sqrt(-2*sqrt(5) + 6))*(7*sqrt(5) + 1
5)*sqrt(-sqrt(5) + 3)*(-2*sqrt(5) + 6)^(5/4) - 1/80*(sqrt(10)*(7*sqrt(5)*x + 15*x)*(-2*sqrt(5) + 6)^(5/4) + 10
*(sqrt(5)*sqrt(2) + 3*sqrt(2))*sqrt(-2*sqrt(5) + 6))*sqrt(-sqrt(5) + 3)) + 1/80*sqrt(10)*(sqrt(5) + 3)*sqrt(-s
qrt(5) + 3)*(-2*sqrt(5) + 6)^(3/4)*arctan(1/800*sqrt(10)*sqrt(-10*sqrt(10)*sqrt(5)*sqrt(2)*x*(-2*sqrt(5) + 6)^
(1/4) + 100*x^2 + 50*sqrt(-2*sqrt(5) + 6))*(7*sqrt(5) + 15)*sqrt(-sqrt(5) + 3)*(-2*sqrt(5) + 6)^(5/4) - 1/80*(
sqrt(10)*(7*sqrt(5)*x + 15*x)*(-2*sqrt(5) + 6)^(5/4) - 10*(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(10*sqrt(10)*sqrt(5)*sqrt(2)*x*(2*sqrt(5)
+ 6)^(1/4) + 100*x^2 + 50*sqrt(2*sqrt(5) + 6)) - 1/80*sqrt(10)*sqrt(2)*(2*sqrt(5) + 6)^(1/4)*log(-10*sqrt(10)
*sqrt(5)*sqrt(2)*x*(2*sqrt(5) + 6)^(1/4) + 100*x^2 + 50*sqrt(2*sqrt(5) + 6)) - 1/80*sqrt(10)*sqrt(2)*(-2*sqrt(
5) + 6)^(1/4)*log(10*sqrt(10)*sqrt(5)*sqrt(2)*x*(-2*sqrt(5) + 6)^(1/4) + 100*x^2 + 50*sqrt(-2*sqrt(5) + 6)) +
1/80*sqrt(10)*sqrt(2)*(-2*sqrt(5) + 6)^(1/4)*log(-10*sqrt(10)*sqrt(5)*sqrt(2)*x*(-2*sqrt(5) + 6)^(1/4) + 100*x
^2 + 50*sqrt(-2*sqrt(5) + 6))

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Sympy [A]
time = 0.95, size = 24, normalized size = 0.05 \begin {gather*} \operatorname {RootSum} {\left (40960000 t^{8} + 19200 t^{4} + 1, \left ( t \mapsto t \log {\left (- 51200 t^{5} - 12 t + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(40960000*_t**8 + 19200*_t**4 + 1, Lambda(_t, _t*log(-51200*_t**5 - 12*_t + x)))

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Giac [A]
time = 3.40, size = 239, normalized size = 0.53 \begin {gather*} \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {5 \, \sqrt {5} + 5} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {5 \, \sqrt {5} + 5} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {5 \, \sqrt {5} - 5} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {5 \, \sqrt {5} - 5} + \frac {1}{40} \, \sqrt {5 \, \sqrt {5} + 5} \log \left (625 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 625 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {5 \, \sqrt {5} + 5} \log \left (625 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 625 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {5 \, \sqrt {5} - 5} \log \left (4225 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 4225 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {5 \, \sqrt {5} - 5} \log \left (4225 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 4225 \, x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80*(pi + 4*arctan(x*sqrt(sqrt(5) - 1) + 1))*sqrt(5*sqrt(5) + 5) - 1/80*(pi + 4*arctan(-x*sqrt(sqrt(5) - 1) +
1))*sqrt(5*sqrt(5) + 5) - 1/80*(pi + 4*arctan(x*sqrt(sqrt(5) + 1) - 1))*sqrt(5*sqrt(5) - 5) + 1/80*(pi + 4*ar
ctan(-x*sqrt(sqrt(5) + 1) - 1))*sqrt(5*sqrt(5) - 5) + 1/40*sqrt(5*sqrt(5) + 5)*log(625*(x + sqrt(sqrt(5) + 1))
^2 + 625*x^2) - 1/40*sqrt(5*sqrt(5) + 5)*log(625*(x - sqrt(sqrt(5) + 1))^2 + 625*x^2) - 1/40*sqrt(5*sqrt(5) -
5)*log(4225*(x + sqrt(sqrt(5) - 1))^2 + 4225*x^2) + 1/40*sqrt(5*sqrt(5) - 5)*log(4225*(x - sqrt(sqrt(5) - 1))^
2 + 4225*x^2)

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Mupad [B]
time = 0.20, size = 454, normalized size = 1.01 \begin {gather*} \frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {3\,2^{3/4}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}}{2}-\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}}{2}\right )}-\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}}{2}-\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}}{2}\right )}\right )\,{\left (-\sqrt {5}-3\right )}^{1/4}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}\,3{}\mathrm {i}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}}{2}-\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}}{2}\right )}-\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}}{2}-\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}}{2}\right )}\right )\,{\left (-\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {3\,2^{3/4}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-3}}{2}+\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}}{2}\right )}+\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-3}}{2}+\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}}{2}\right )}\right )\,{\left (\sqrt {5}-3\right )}^{1/4}}{20}+\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}\,3{}\mathrm {i}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-3}}{2}+\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}}{2}\right )}+\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-3}}{2}+\frac {\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}}{2}\right )}\right )\,{\left (\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(3*x^4 + x^8 + 1),x)

[Out]

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

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